Prospects for Slepton Searchesin Future Experiments

（将来実験におけるスレプトン探索索の展望）

Presented by Takahiro Yoshinaga

博士論論文審査会

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo K Hamaguchi T Kitahara and TY JHEP 11 013 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

45 Slides

emsp　博士論論文の概要 2

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　博士論論文の内容 3

1  Introduction2  Foundation (Review)3  Supersymmetric Standard Model (Review)4  Prospects for Slepton Searches (Main)

5  Conclusion

41 Foundation Model amp Mass Bound42 SelectronSmuon43 Stau

emsp　本日の内容 4

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　本日の内容 5

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

ミューオンの異異常磁気モーメント (Muon g-‐‑‒2)

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

emsp　Muon g-‐‑‒2 6

l ミューオンの磁気モーメント

l  g-‐‑‒factor

H = m middotB m =Ccedil

e2mmicro

aringsg

-‐‑‒  g = 2 Tree Level-‐‑‒  g ne 2 Radiative Correction

amicro g 2

2

emsp　Muon g-‐‑‒2 7

SM Prediction

Contributions Value (10-‐‑‒10)

QED (O(α5)) 116584718951 (00080)

EW (NLO) 1536 (01)

Hadronic(LO)

[HLMNT] 69491 (427)

[DHMZ] 6923 (42)

Hadronic (HO) -‐‑‒984 (007)

Hadronic(LbL)

[RdRV] 105 (26)

[NJN] 116 (39)

Total SM [HLMNT] 116591828 (49)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

had

QED EW

had

Experiment

[E821 Muon g-‐‑‒2実験のHome pageより]

116592089 (63) times 10-‐‑‒10aexp

micro =

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Muon g-‐‑‒2 8

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

aμ times 1010 -‐‑‒ 11659000

SM

Exp

[Hagiwara Liao Martin Nomura Teubner ʼrsquo11]

emsp　Muon g-‐‑‒2 9

Error [20] [21] Futurea

SMmicro 49 50 35

a

HLOmicro 42 43 26

a

HLbLmicro 26 26 25

(aEXPmicro a

SMmicro ) 80 80 40

Figure 9 Estimated uncertainties amicro in units of 1011 according to Refs [20 21] and (lastcolumn) prospects for improved precision in the e+e hadronic cross-section measurementsThe final row projects the uncertainty on the dicrarrerence with the Standard Model amicro Thefigure give the comparison between a

SMmicro and a

EXPmicro DHMZ is Ref [20] HLMNT is Ref [21]

ldquoSMXXrdquo is the same central value with a reduced error as expected by the improvementon the hadronic cross section measurement (see text) ldquoBNL-E821 04 averdquo is the currentexperimental value of amicro ldquoNew (g-2) exprdquo is the same central value with a fourfold improvedprecision as planned by the future (g-2) experiments at Fermilab and J-PARC

References

[1] J Schwinger Phys Rev 73 (1948) 416 and Phys Rev 76 (1949) 790 The formerpaper contains a misprint in the expression for ae that is corrected in the longer paper

[2] T Aoyama M Hayakawa T Kinoshita and M Nio Phys Rev Lett 109 (2012)111808

[3] J P Miller E de Rafael B L Roberts and D Stockinger Ann Rev Nucl Part Sci62 (2012) 237

[4] D Stockinger in Advanced Series on Directions in High Energy Physics - Vol 20 LeptonDipole Moments eds B L Roberts and W J Marciano World Scientific (2010) p393

[5] D Hanneke S Fogwell and G Gabrielse Phys Rev Lett 100 (2008) 120801

[6] G W Bennett et al (The g 2 Collab) Phys Rev D73 (2006) 072003

[7] M Davier in Advanced Series on Directions in High Energy Physics - Vol 20 Lep-

ton Dipole Moments eds B L Roberts and W J Marciano World Scientific (2010)chapter 8

[8] R Bouchendira P Clade S Guellati-Khelifa F Nez and F Biraben Phys Rev Lett106 (2011) 080801

16

Fermilab amp JPARCNext 3-‐‑‒5 years

e+e-‐‑‒rarrhadrons実験

中心値が変わらなければ近い将来gt5σの感度度が期待

[Snowmass white paperより]

SMXX1815plusmn35

emsp　Muon g-‐‑‒2 10

中心値が変わらなければ近い将来gt5σの感度度が期待

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

「今」Muon g-‐‑‒2の不不一致の原因を調べることは非常に重要

博士論論文での立立場

emsp　Muon g-‐‑‒2 11

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+NP

Exp

Muon g-‐‑‒2の不不一致は新しい物理理(NP)の寄与が原因

新しい物理理

-‐‑‒  博士論論文では超対称模型を仮定-‐‑‒  model-‐‑‒independentにSUSYの寄与を確かめる

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

NP

emsp　本日の内容 12

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Supersymmetry 13

「BosonとFermionの間の対称性」

MSSM＝Minimal Supersymmetric Standard ModelSM粒粒子のパートナー(superparticle)を予言

emsp　Supersymmetry 14

SUSY contribution to the muon g-‐‑‒2

aSUSYmicro crarr2 tan

4

m2micro

m2SUSY

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

Chargino-‐‑‒sneutrino Neutralino-‐‑‒smuonCcedil 150 1010

100GeVmSUSY

2

tan10

aring

mSUSY 典型的なSUSY粒粒子の質量量 tanβ Higgsの真空期待値の比

mSUSY = O(100)GeV tanβ = O(10)のとき

超対称模型の寄与によりMuon g-‐‑‒2の不不一致を説明可能[Lopez Nanopoulos Wang ʼrsquo93][Chattopadhyay Nath ʼrsquo95][Moroi ʼrsquo95]

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Supersymmetry 15

Representative SUSY contributions

Contributions O(100)GeV particles Diagram

Chagino-‐‑‒sneutrino

Wino Higgsinos (Bino) smuons

Neutralino-‐‑‒smuon Bino smuons

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  Muon g-‐‑‒2を説明するminimalな解の一つl  特徴的な性質を持つ

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　博士論論文の概要 2

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　博士論論文の内容 3

1  Introduction2  Foundation (Review)3  Supersymmetric Standard Model (Review)4  Prospects for Slepton Searches (Main)

5  Conclusion

41 Foundation Model amp Mass Bound42 SelectronSmuon43 Stau

emsp　本日の内容 4

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　本日の内容 5

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

ミューオンの異異常磁気モーメント (Muon g-‐‑‒2)

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

emsp　Muon g-‐‑‒2 6

l ミューオンの磁気モーメント

l  g-‐‑‒factor

H = m middotB m =Ccedil

e2mmicro

aringsg

-‐‑‒  g = 2 Tree Level-‐‑‒  g ne 2 Radiative Correction

amicro g 2

2

emsp　Muon g-‐‑‒2 7

SM Prediction

Contributions Value (10-‐‑‒10)

QED (O(α5)) 116584718951 (00080)

EW (NLO) 1536 (01)

Hadronic(LO)

[HLMNT] 69491 (427)

[DHMZ] 6923 (42)

Hadronic (HO) -‐‑‒984 (007)

Hadronic(LbL)

[RdRV] 105 (26)

[NJN] 116 (39)

Total SM [HLMNT] 116591828 (49)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

had

QED EW

had

Experiment

[E821 Muon g-‐‑‒2実験のHome pageより]

116592089 (63) times 10-‐‑‒10aexp

micro =

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Muon g-‐‑‒2 8

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

aμ times 1010 -‐‑‒ 11659000

SM

Exp

[Hagiwara Liao Martin Nomura Teubner ʼrsquo11]

emsp　Muon g-‐‑‒2 9

Error [20] [21] Futurea

SMmicro 49 50 35

a

HLOmicro 42 43 26

a

HLbLmicro 26 26 25

(aEXPmicro a

SMmicro ) 80 80 40

Figure 9 Estimated uncertainties amicro in units of 1011 according to Refs [20 21] and (lastcolumn) prospects for improved precision in the e+e hadronic cross-section measurementsThe final row projects the uncertainty on the dicrarrerence with the Standard Model amicro Thefigure give the comparison between a

SMmicro and a

EXPmicro DHMZ is Ref [20] HLMNT is Ref [21]

ldquoSMXXrdquo is the same central value with a reduced error as expected by the improvementon the hadronic cross section measurement (see text) ldquoBNL-E821 04 averdquo is the currentexperimental value of amicro ldquoNew (g-2) exprdquo is the same central value with a fourfold improvedprecision as planned by the future (g-2) experiments at Fermilab and J-PARC

References

[1] J Schwinger Phys Rev 73 (1948) 416 and Phys Rev 76 (1949) 790 The formerpaper contains a misprint in the expression for ae that is corrected in the longer paper

[2] T Aoyama M Hayakawa T Kinoshita and M Nio Phys Rev Lett 109 (2012)111808

[3] J P Miller E de Rafael B L Roberts and D Stockinger Ann Rev Nucl Part Sci62 (2012) 237

[4] D Stockinger in Advanced Series on Directions in High Energy Physics - Vol 20 LeptonDipole Moments eds B L Roberts and W J Marciano World Scientific (2010) p393

[5] D Hanneke S Fogwell and G Gabrielse Phys Rev Lett 100 (2008) 120801

[6] G W Bennett et al (The g 2 Collab) Phys Rev D73 (2006) 072003

[7] M Davier in Advanced Series on Directions in High Energy Physics - Vol 20 Lep-

ton Dipole Moments eds B L Roberts and W J Marciano World Scientific (2010)chapter 8

[8] R Bouchendira P Clade S Guellati-Khelifa F Nez and F Biraben Phys Rev Lett106 (2011) 080801

16

Fermilab amp JPARCNext 3-‐‑‒5 years

e+e-‐‑‒rarrhadrons実験

中心値が変わらなければ近い将来gt5σの感度度が期待

[Snowmass white paperより]

SMXX1815plusmn35

emsp　Muon g-‐‑‒2 10

中心値が変わらなければ近い将来gt5σの感度度が期待

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

「今」Muon g-‐‑‒2の不不一致の原因を調べることは非常に重要

博士論論文での立立場

emsp　Muon g-‐‑‒2 11

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+NP

Exp

Muon g-‐‑‒2の不不一致は新しい物理理(NP)の寄与が原因

新しい物理理

-‐‑‒  博士論論文では超対称模型を仮定-‐‑‒  model-‐‑‒independentにSUSYの寄与を確かめる

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

NP

emsp　本日の内容 12

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Supersymmetry 13

「BosonとFermionの間の対称性」

MSSM＝Minimal Supersymmetric Standard ModelSM粒粒子のパートナー(superparticle)を予言

emsp　Supersymmetry 14

SUSY contribution to the muon g-‐‑‒2

aSUSYmicro crarr2 tan

4

m2micro

m2SUSY

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

Chargino-‐‑‒sneutrino Neutralino-‐‑‒smuonCcedil 150 1010

100GeVmSUSY

2

tan10

aring

mSUSY 典型的なSUSY粒粒子の質量量 tanβ Higgsの真空期待値の比

mSUSY = O(100)GeV tanβ = O(10)のとき

超対称模型の寄与によりMuon g-‐‑‒2の不不一致を説明可能[Lopez Nanopoulos Wang ʼrsquo93][Chattopadhyay Nath ʼrsquo95][Moroi ʼrsquo95]

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Supersymmetry 15

Representative SUSY contributions

Contributions O(100)GeV particles Diagram

Chagino-‐‑‒sneutrino

Wino Higgsinos (Bino) smuons

Neutralino-‐‑‒smuon Bino smuons

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  Muon g-‐‑‒2を説明するminimalな解の一つl  特徴的な性質を持つ

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　博士論論文の内容 3

1  Introduction2  Foundation (Review)3  Supersymmetric Standard Model (Review)4  Prospects for Slepton Searches (Main)

5  Conclusion

41 Foundation Model amp Mass Bound42 SelectronSmuon43 Stau

emsp　本日の内容 4

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　本日の内容 5

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

ミューオンの異異常磁気モーメント (Muon g-‐‑‒2)

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

emsp　Muon g-‐‑‒2 6

l ミューオンの磁気モーメント

l  g-‐‑‒factor

H = m middotB m =Ccedil

e2mmicro

aringsg

-‐‑‒  g = 2 Tree Level-‐‑‒  g ne 2 Radiative Correction

amicro g 2

2

emsp　Muon g-‐‑‒2 7

SM Prediction

Contributions Value (10-‐‑‒10)

QED (O(α5)) 116584718951 (00080)

EW (NLO) 1536 (01)

Hadronic(LO)

[HLMNT] 69491 (427)

[DHMZ] 6923 (42)

Hadronic (HO) -‐‑‒984 (007)

Hadronic(LbL)

[RdRV] 105 (26)

[NJN] 116 (39)

Total SM [HLMNT] 116591828 (49)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

had

QED EW

had

Experiment

[E821 Muon g-‐‑‒2実験のHome pageより]

116592089 (63) times 10-‐‑‒10aexp

micro =

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Muon g-‐‑‒2 8

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

aμ times 1010 -‐‑‒ 11659000

SM

Exp

[Hagiwara Liao Martin Nomura Teubner ʼrsquo11]

emsp　Muon g-‐‑‒2 9

Error [20] [21] Futurea

SMmicro 49 50 35

a

HLOmicro 42 43 26

a

HLbLmicro 26 26 25

(aEXPmicro a

SMmicro ) 80 80 40

Figure 9 Estimated uncertainties amicro in units of 1011 according to Refs [20 21] and (lastcolumn) prospects for improved precision in the e+e hadronic cross-section measurementsThe final row projects the uncertainty on the dicrarrerence with the Standard Model amicro Thefigure give the comparison between a

SMmicro and a

EXPmicro DHMZ is Ref [20] HLMNT is Ref [21]

ldquoSMXXrdquo is the same central value with a reduced error as expected by the improvementon the hadronic cross section measurement (see text) ldquoBNL-E821 04 averdquo is the currentexperimental value of amicro ldquoNew (g-2) exprdquo is the same central value with a fourfold improvedprecision as planned by the future (g-2) experiments at Fermilab and J-PARC

References

[1] J Schwinger Phys Rev 73 (1948) 416 and Phys Rev 76 (1949) 790 The formerpaper contains a misprint in the expression for ae that is corrected in the longer paper

[2] T Aoyama M Hayakawa T Kinoshita and M Nio Phys Rev Lett 109 (2012)111808

[3] J P Miller E de Rafael B L Roberts and D Stockinger Ann Rev Nucl Part Sci62 (2012) 237

[4] D Stockinger in Advanced Series on Directions in High Energy Physics - Vol 20 LeptonDipole Moments eds B L Roberts and W J Marciano World Scientific (2010) p393

[5] D Hanneke S Fogwell and G Gabrielse Phys Rev Lett 100 (2008) 120801

[6] G W Bennett et al (The g 2 Collab) Phys Rev D73 (2006) 072003

[7] M Davier in Advanced Series on Directions in High Energy Physics - Vol 20 Lep-

ton Dipole Moments eds B L Roberts and W J Marciano World Scientific (2010)chapter 8

[8] R Bouchendira P Clade S Guellati-Khelifa F Nez and F Biraben Phys Rev Lett106 (2011) 080801

16

Fermilab amp JPARCNext 3-‐‑‒5 years

e+e-‐‑‒rarrhadrons実験

中心値が変わらなければ近い将来gt5σの感度度が期待

[Snowmass white paperより]

SMXX1815plusmn35

emsp　Muon g-‐‑‒2 10

中心値が変わらなければ近い将来gt5σの感度度が期待

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

「今」Muon g-‐‑‒2の不不一致の原因を調べることは非常に重要

博士論論文での立立場

emsp　Muon g-‐‑‒2 11

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+NP

Exp

Muon g-‐‑‒2の不不一致は新しい物理理(NP)の寄与が原因

新しい物理理

-‐‑‒  博士論論文では超対称模型を仮定-‐‑‒  model-‐‑‒independentにSUSYの寄与を確かめる

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

NP

emsp　本日の内容 12

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Supersymmetry 13

「BosonとFermionの間の対称性」

MSSM＝Minimal Supersymmetric Standard ModelSM粒粒子のパートナー(superparticle)を予言

emsp　Supersymmetry 14

SUSY contribution to the muon g-‐‑‒2

aSUSYmicro crarr2 tan

4

m2micro

m2SUSY

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

Chargino-‐‑‒sneutrino Neutralino-‐‑‒smuonCcedil 150 1010

100GeVmSUSY

2

tan10

aring

mSUSY 典型的なSUSY粒粒子の質量量 tanβ Higgsの真空期待値の比

mSUSY = O(100)GeV tanβ = O(10)のとき

超対称模型の寄与によりMuon g-‐‑‒2の不不一致を説明可能[Lopez Nanopoulos Wang ʼrsquo93][Chattopadhyay Nath ʼrsquo95][Moroi ʼrsquo95]

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Supersymmetry 15

Representative SUSY contributions

Contributions O(100)GeV particles Diagram

Chagino-‐‑‒sneutrino

Wino Higgsinos (Bino) smuons

Neutralino-‐‑‒smuon Bino smuons

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  Muon g-‐‑‒2を説明するminimalな解の一つl  特徴的な性質を持つ

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　本日の内容 4

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　本日の内容 5

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

ミューオンの異異常磁気モーメント (Muon g-‐‑‒2)

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

emsp　Muon g-‐‑‒2 6

l ミューオンの磁気モーメント

l  g-‐‑‒factor

H = m middotB m =Ccedil

e2mmicro

aringsg

-‐‑‒  g = 2 Tree Level-‐‑‒  g ne 2 Radiative Correction

amicro g 2

2

emsp　Muon g-‐‑‒2 7

SM Prediction

Contributions Value (10-‐‑‒10)

QED (O(α5)) 116584718951 (00080)

EW (NLO) 1536 (01)

Hadronic(LO)

[HLMNT] 69491 (427)

[DHMZ] 6923 (42)

Hadronic (HO) -‐‑‒984 (007)

Hadronic(LbL)

[RdRV] 105 (26)

[NJN] 116 (39)

Total SM [HLMNT] 116591828 (49)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

had

QED EW

had

Experiment

[E821 Muon g-‐‑‒2実験のHome pageより]

116592089 (63) times 10-‐‑‒10aexp

micro =

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Muon g-‐‑‒2 8

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

aμ times 1010 -‐‑‒ 11659000

SM

Exp

[Hagiwara Liao Martin Nomura Teubner ʼrsquo11]

emsp　Muon g-‐‑‒2 9

Error [20] [21] Futurea

SMmicro 49 50 35

a

HLOmicro 42 43 26

a

HLbLmicro 26 26 25

(aEXPmicro a

SMmicro ) 80 80 40

Figure 9 Estimated uncertainties amicro in units of 1011 according to Refs [20 21] and (lastcolumn) prospects for improved precision in the e+e hadronic cross-section measurementsThe final row projects the uncertainty on the dicrarrerence with the Standard Model amicro Thefigure give the comparison between a

SMmicro and a

EXPmicro DHMZ is Ref [20] HLMNT is Ref [21]

ldquoSMXXrdquo is the same central value with a reduced error as expected by the improvementon the hadronic cross section measurement (see text) ldquoBNL-E821 04 averdquo is the currentexperimental value of amicro ldquoNew (g-2) exprdquo is the same central value with a fourfold improvedprecision as planned by the future (g-2) experiments at Fermilab and J-PARC

References

[1] J Schwinger Phys Rev 73 (1948) 416 and Phys Rev 76 (1949) 790 The formerpaper contains a misprint in the expression for ae that is corrected in the longer paper

[2] T Aoyama M Hayakawa T Kinoshita and M Nio Phys Rev Lett 109 (2012)111808

[3] J P Miller E de Rafael B L Roberts and D Stockinger Ann Rev Nucl Part Sci62 (2012) 237

[4] D Stockinger in Advanced Series on Directions in High Energy Physics - Vol 20 LeptonDipole Moments eds B L Roberts and W J Marciano World Scientific (2010) p393

[5] D Hanneke S Fogwell and G Gabrielse Phys Rev Lett 100 (2008) 120801

[6] G W Bennett et al (The g 2 Collab) Phys Rev D73 (2006) 072003

[7] M Davier in Advanced Series on Directions in High Energy Physics - Vol 20 Lep-

ton Dipole Moments eds B L Roberts and W J Marciano World Scientific (2010)chapter 8

[8] R Bouchendira P Clade S Guellati-Khelifa F Nez and F Biraben Phys Rev Lett106 (2011) 080801

16

Fermilab amp JPARCNext 3-‐‑‒5 years

e+e-‐‑‒rarrhadrons実験

中心値が変わらなければ近い将来gt5σの感度度が期待

[Snowmass white paperより]

SMXX1815plusmn35

emsp　Muon g-‐‑‒2 10

中心値が変わらなければ近い将来gt5σの感度度が期待

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

「今」Muon g-‐‑‒2の不不一致の原因を調べることは非常に重要

博士論論文での立立場

emsp　Muon g-‐‑‒2 11

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+NP

Exp

Muon g-‐‑‒2の不不一致は新しい物理理(NP)の寄与が原因

新しい物理理

-‐‑‒  博士論論文では超対称模型を仮定-‐‑‒  model-‐‑‒independentにSUSYの寄与を確かめる

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

NP

emsp　本日の内容 12

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Supersymmetry 13

「BosonとFermionの間の対称性」

MSSM＝Minimal Supersymmetric Standard ModelSM粒粒子のパートナー(superparticle)を予言

emsp　Supersymmetry 14

SUSY contribution to the muon g-‐‑‒2

aSUSYmicro crarr2 tan

4

m2micro

m2SUSY

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

Chargino-‐‑‒sneutrino Neutralino-‐‑‒smuonCcedil 150 1010

100GeVmSUSY

2

tan10

aring

mSUSY 典型的なSUSY粒粒子の質量量 tanβ Higgsの真空期待値の比

mSUSY = O(100)GeV tanβ = O(10)のとき

超対称模型の寄与によりMuon g-‐‑‒2の不不一致を説明可能[Lopez Nanopoulos Wang ʼrsquo93][Chattopadhyay Nath ʼrsquo95][Moroi ʼrsquo95]

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Supersymmetry 15

Representative SUSY contributions

Contributions O(100)GeV particles Diagram

Chagino-‐‑‒sneutrino

Wino Higgsinos (Bino) smuons

Neutralino-‐‑‒smuon Bino smuons

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  Muon g-‐‑‒2を説明するminimalな解の一つl  特徴的な性質を持つ

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　本日の内容 5

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

ミューオンの異異常磁気モーメント (Muon g-‐‑‒2)

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

emsp　Muon g-‐‑‒2 6

l ミューオンの磁気モーメント

l  g-‐‑‒factor

H = m middotB m =Ccedil

e2mmicro

aringsg

-‐‑‒  g = 2 Tree Level-‐‑‒  g ne 2 Radiative Correction

amicro g 2

2

emsp　Muon g-‐‑‒2 7

SM Prediction

Contributions Value (10-‐‑‒10)

QED (O(α5)) 116584718951 (00080)

EW (NLO) 1536 (01)

Hadronic(LO)

[HLMNT] 69491 (427)

[DHMZ] 6923 (42)

Hadronic (HO) -‐‑‒984 (007)

Hadronic(LbL)

[RdRV] 105 (26)

[NJN] 116 (39)

Total SM [HLMNT] 116591828 (49)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

had

QED EW

had

Experiment

[E821 Muon g-‐‑‒2実験のHome pageより]

116592089 (63) times 10-‐‑‒10aexp

micro =

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Muon g-‐‑‒2 8

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

aμ times 1010 -‐‑‒ 11659000

SM

Exp

[Hagiwara Liao Martin Nomura Teubner ʼrsquo11]

emsp　Muon g-‐‑‒2 9

Error [20] [21] Futurea

SMmicro 49 50 35

a

HLOmicro 42 43 26

a

HLbLmicro 26 26 25

(aEXPmicro a

SMmicro ) 80 80 40

Figure 9 Estimated uncertainties amicro in units of 1011 according to Refs [20 21] and (lastcolumn) prospects for improved precision in the e+e hadronic cross-section measurementsThe final row projects the uncertainty on the dicrarrerence with the Standard Model amicro Thefigure give the comparison between a

SMmicro and a

EXPmicro DHMZ is Ref [20] HLMNT is Ref [21]

ldquoSMXXrdquo is the same central value with a reduced error as expected by the improvementon the hadronic cross section measurement (see text) ldquoBNL-E821 04 averdquo is the currentexperimental value of amicro ldquoNew (g-2) exprdquo is the same central value with a fourfold improvedprecision as planned by the future (g-2) experiments at Fermilab and J-PARC

References

[1] J Schwinger Phys Rev 73 (1948) 416 and Phys Rev 76 (1949) 790 The formerpaper contains a misprint in the expression for ae that is corrected in the longer paper

[2] T Aoyama M Hayakawa T Kinoshita and M Nio Phys Rev Lett 109 (2012)111808

[3] J P Miller E de Rafael B L Roberts and D Stockinger Ann Rev Nucl Part Sci62 (2012) 237

[4] D Stockinger in Advanced Series on Directions in High Energy Physics - Vol 20 LeptonDipole Moments eds B L Roberts and W J Marciano World Scientific (2010) p393

[5] D Hanneke S Fogwell and G Gabrielse Phys Rev Lett 100 (2008) 120801

[6] G W Bennett et al (The g 2 Collab) Phys Rev D73 (2006) 072003

[7] M Davier in Advanced Series on Directions in High Energy Physics - Vol 20 Lep-

ton Dipole Moments eds B L Roberts and W J Marciano World Scientific (2010)chapter 8

[8] R Bouchendira P Clade S Guellati-Khelifa F Nez and F Biraben Phys Rev Lett106 (2011) 080801

16

Fermilab amp JPARCNext 3-‐‑‒5 years

e+e-‐‑‒rarrhadrons実験

中心値が変わらなければ近い将来gt5σの感度度が期待

[Snowmass white paperより]

SMXX1815plusmn35

emsp　Muon g-‐‑‒2 10

中心値が変わらなければ近い将来gt5σの感度度が期待

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

「今」Muon g-‐‑‒2の不不一致の原因を調べることは非常に重要

博士論論文での立立場

emsp　Muon g-‐‑‒2 11

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+NP

Exp

Muon g-‐‑‒2の不不一致は新しい物理理(NP)の寄与が原因

新しい物理理

-‐‑‒  博士論論文では超対称模型を仮定-‐‑‒  model-‐‑‒independentにSUSYの寄与を確かめる

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

NP

emsp　本日の内容 12

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Supersymmetry 13

「BosonとFermionの間の対称性」

MSSM＝Minimal Supersymmetric Standard ModelSM粒粒子のパートナー(superparticle)を予言

emsp　Supersymmetry 14

SUSY contribution to the muon g-‐‑‒2

aSUSYmicro crarr2 tan

4

m2micro

m2SUSY

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

Chargino-‐‑‒sneutrino Neutralino-‐‑‒smuonCcedil 150 1010

100GeVmSUSY

2

tan10

aring

mSUSY 典型的なSUSY粒粒子の質量量 tanβ Higgsの真空期待値の比

mSUSY = O(100)GeV tanβ = O(10)のとき

超対称模型の寄与によりMuon g-‐‑‒2の不不一致を説明可能[Lopez Nanopoulos Wang ʼrsquo93][Chattopadhyay Nath ʼrsquo95][Moroi ʼrsquo95]

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Supersymmetry 15

Representative SUSY contributions

Contributions O(100)GeV particles Diagram

Chagino-‐‑‒sneutrino

Wino Higgsinos (Bino) smuons

Neutralino-‐‑‒smuon Bino smuons

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  Muon g-‐‑‒2を説明するminimalな解の一つl  特徴的な性質を持つ

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

ミューオンの異異常磁気モーメント (Muon g-‐‑‒2)

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

emsp　Muon g-‐‑‒2 6

l ミューオンの磁気モーメント

l  g-‐‑‒factor

H = m middotB m =Ccedil

e2mmicro

aringsg

-‐‑‒  g = 2 Tree Level-‐‑‒  g ne 2 Radiative Correction

amicro g 2

2

emsp　Muon g-‐‑‒2 7

SM Prediction

Contributions Value (10-‐‑‒10)

QED (O(α5)) 116584718951 (00080)

EW (NLO) 1536 (01)

Hadronic(LO)

[HLMNT] 69491 (427)

[DHMZ] 6923 (42)

Hadronic (HO) -‐‑‒984 (007)

Hadronic(LbL)

[RdRV] 105 (26)

[NJN] 116 (39)

Total SM [HLMNT] 116591828 (49)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

had

QED EW

had

Experiment

[E821 Muon g-‐‑‒2実験のHome pageより]

116592089 (63) times 10-‐‑‒10aexp

micro =

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Muon g-‐‑‒2 8

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

aμ times 1010 -‐‑‒ 11659000

SM

Exp

[Hagiwara Liao Martin Nomura Teubner ʼrsquo11]

emsp　Muon g-‐‑‒2 9

Error [20] [21] Futurea

SMmicro 49 50 35

a

HLOmicro 42 43 26

a

HLbLmicro 26 26 25

(aEXPmicro a

SMmicro ) 80 80 40

Figure 9 Estimated uncertainties amicro in units of 1011 according to Refs [20 21] and (lastcolumn) prospects for improved precision in the e+e hadronic cross-section measurementsThe final row projects the uncertainty on the dicrarrerence with the Standard Model amicro Thefigure give the comparison between a

SMmicro and a

EXPmicro DHMZ is Ref [20] HLMNT is Ref [21]

ldquoSMXXrdquo is the same central value with a reduced error as expected by the improvementon the hadronic cross section measurement (see text) ldquoBNL-E821 04 averdquo is the currentexperimental value of amicro ldquoNew (g-2) exprdquo is the same central value with a fourfold improvedprecision as planned by the future (g-2) experiments at Fermilab and J-PARC

References

[1] J Schwinger Phys Rev 73 (1948) 416 and Phys Rev 76 (1949) 790 The formerpaper contains a misprint in the expression for ae that is corrected in the longer paper

[2] T Aoyama M Hayakawa T Kinoshita and M Nio Phys Rev Lett 109 (2012)111808

[3] J P Miller E de Rafael B L Roberts and D Stockinger Ann Rev Nucl Part Sci62 (2012) 237

[4] D Stockinger in Advanced Series on Directions in High Energy Physics - Vol 20 LeptonDipole Moments eds B L Roberts and W J Marciano World Scientific (2010) p393

[5] D Hanneke S Fogwell and G Gabrielse Phys Rev Lett 100 (2008) 120801

[6] G W Bennett et al (The g 2 Collab) Phys Rev D73 (2006) 072003

[7] M Davier in Advanced Series on Directions in High Energy Physics - Vol 20 Lep-

ton Dipole Moments eds B L Roberts and W J Marciano World Scientific (2010)chapter 8

[8] R Bouchendira P Clade S Guellati-Khelifa F Nez and F Biraben Phys Rev Lett106 (2011) 080801

16

Fermilab amp JPARCNext 3-‐‑‒5 years

e+e-‐‑‒rarrhadrons実験

中心値が変わらなければ近い将来gt5σの感度度が期待

[Snowmass white paperより]

SMXX1815plusmn35

emsp　Muon g-‐‑‒2 10

中心値が変わらなければ近い将来gt5σの感度度が期待

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

「今」Muon g-‐‑‒2の不不一致の原因を調べることは非常に重要

博士論論文での立立場

emsp　Muon g-‐‑‒2 11

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+NP

Exp

Muon g-‐‑‒2の不不一致は新しい物理理(NP)の寄与が原因

新しい物理理

-‐‑‒  博士論論文では超対称模型を仮定-‐‑‒  model-‐‑‒independentにSUSYの寄与を確かめる

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

NP

emsp　本日の内容 12

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Supersymmetry 13

「BosonとFermionの間の対称性」

MSSM＝Minimal Supersymmetric Standard ModelSM粒粒子のパートナー(superparticle)を予言

emsp　Supersymmetry 14

SUSY contribution to the muon g-‐‑‒2

aSUSYmicro crarr2 tan

4

m2micro

m2SUSY

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

Chargino-‐‑‒sneutrino Neutralino-‐‑‒smuonCcedil 150 1010

100GeVmSUSY

2

tan10

aring

mSUSY 典型的なSUSY粒粒子の質量量 tanβ Higgsの真空期待値の比

mSUSY = O(100)GeV tanβ = O(10)のとき

超対称模型の寄与によりMuon g-‐‑‒2の不不一致を説明可能[Lopez Nanopoulos Wang ʼrsquo93][Chattopadhyay Nath ʼrsquo95][Moroi ʼrsquo95]

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Supersymmetry 15

Representative SUSY contributions

Contributions O(100)GeV particles Diagram

Chagino-‐‑‒sneutrino

Wino Higgsinos (Bino) smuons

Neutralino-‐‑‒smuon Bino smuons

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  Muon g-‐‑‒2を説明するminimalな解の一つl  特徴的な性質を持つ

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Muon g-‐‑‒2 7

SM Prediction

Contributions Value (10-‐‑‒10)

QED (O(α5)) 116584718951 (00080)

EW (NLO) 1536 (01)

Hadronic(LO)

[HLMNT] 69491 (427)

[DHMZ] 6923 (42)

Hadronic (HO) -‐‑‒984 (007)

Hadronic(LbL)

[RdRV] 105 (26)

[NJN] 116 (39)

Total SM [HLMNT] 116591828 (49)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

had

QED EW

had

Experiment

[E821 Muon g-‐‑‒2実験のHome pageより]

116592089 (63) times 10-‐‑‒10aexp

micro =

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Muon g-‐‑‒2 8

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

aμ times 1010 -‐‑‒ 11659000

SM

Exp

[Hagiwara Liao Martin Nomura Teubner ʼrsquo11]

emsp　Muon g-‐‑‒2 9

Error [20] [21] Futurea

SMmicro 49 50 35

a

HLOmicro 42 43 26

a

HLbLmicro 26 26 25

(aEXPmicro a

SMmicro ) 80 80 40

Figure 9 Estimated uncertainties amicro in units of 1011 according to Refs [20 21] and (lastcolumn) prospects for improved precision in the e+e hadronic cross-section measurementsThe final row projects the uncertainty on the dicrarrerence with the Standard Model amicro Thefigure give the comparison between a

SMmicro and a

EXPmicro DHMZ is Ref [20] HLMNT is Ref [21]

ldquoSMXXrdquo is the same central value with a reduced error as expected by the improvementon the hadronic cross section measurement (see text) ldquoBNL-E821 04 averdquo is the currentexperimental value of amicro ldquoNew (g-2) exprdquo is the same central value with a fourfold improvedprecision as planned by the future (g-2) experiments at Fermilab and J-PARC

References

[1] J Schwinger Phys Rev 73 (1948) 416 and Phys Rev 76 (1949) 790 The formerpaper contains a misprint in the expression for ae that is corrected in the longer paper

[2] T Aoyama M Hayakawa T Kinoshita and M Nio Phys Rev Lett 109 (2012)111808

[3] J P Miller E de Rafael B L Roberts and D Stockinger Ann Rev Nucl Part Sci62 (2012) 237

[4] D Stockinger in Advanced Series on Directions in High Energy Physics - Vol 20 LeptonDipole Moments eds B L Roberts and W J Marciano World Scientific (2010) p393

[5] D Hanneke S Fogwell and G Gabrielse Phys Rev Lett 100 (2008) 120801

[6] G W Bennett et al (The g 2 Collab) Phys Rev D73 (2006) 072003

[7] M Davier in Advanced Series on Directions in High Energy Physics - Vol 20 Lep-

ton Dipole Moments eds B L Roberts and W J Marciano World Scientific (2010)chapter 8

[8] R Bouchendira P Clade S Guellati-Khelifa F Nez and F Biraben Phys Rev Lett106 (2011) 080801

16

Fermilab amp JPARCNext 3-‐‑‒5 years

e+e-‐‑‒rarrhadrons実験

中心値が変わらなければ近い将来gt5σの感度度が期待

[Snowmass white paperより]

SMXX1815plusmn35

emsp　Muon g-‐‑‒2 10

中心値が変わらなければ近い将来gt5σの感度度が期待

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

「今」Muon g-‐‑‒2の不不一致の原因を調べることは非常に重要

博士論論文での立立場

emsp　Muon g-‐‑‒2 11

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+NP

Exp

Muon g-‐‑‒2の不不一致は新しい物理理(NP)の寄与が原因

新しい物理理

-‐‑‒  博士論論文では超対称模型を仮定-‐‑‒  model-‐‑‒independentにSUSYの寄与を確かめる

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

NP

emsp　本日の内容 12

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Supersymmetry 13

「BosonとFermionの間の対称性」

MSSM＝Minimal Supersymmetric Standard ModelSM粒粒子のパートナー(superparticle)を予言

emsp　Supersymmetry 14

SUSY contribution to the muon g-‐‑‒2

aSUSYmicro crarr2 tan

4

m2micro

m2SUSY

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

Chargino-‐‑‒sneutrino Neutralino-‐‑‒smuonCcedil 150 1010

100GeVmSUSY

2

tan10

aring

mSUSY 典型的なSUSY粒粒子の質量量 tanβ Higgsの真空期待値の比

mSUSY = O(100)GeV tanβ = O(10)のとき

超対称模型の寄与によりMuon g-‐‑‒2の不不一致を説明可能[Lopez Nanopoulos Wang ʼrsquo93][Chattopadhyay Nath ʼrsquo95][Moroi ʼrsquo95]

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Supersymmetry 15

Representative SUSY contributions

Contributions O(100)GeV particles Diagram

Chagino-‐‑‒sneutrino

Wino Higgsinos (Bino) smuons

Neutralino-‐‑‒smuon Bino smuons

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  Muon g-‐‑‒2を説明するminimalな解の一つl  特徴的な性質を持つ

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Muon g-‐‑‒2 8

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

aμ times 1010 -‐‑‒ 11659000

SM

Exp

[Hagiwara Liao Martin Nomura Teubner ʼrsquo11]

emsp　Muon g-‐‑‒2 9

Error [20] [21] Futurea

SMmicro 49 50 35

a

HLOmicro 42 43 26

a

HLbLmicro 26 26 25

(aEXPmicro a

SMmicro ) 80 80 40

Figure 9 Estimated uncertainties amicro in units of 1011 according to Refs [20 21] and (lastcolumn) prospects for improved precision in the e+e hadronic cross-section measurementsThe final row projects the uncertainty on the dicrarrerence with the Standard Model amicro Thefigure give the comparison between a

SMmicro and a

EXPmicro DHMZ is Ref [20] HLMNT is Ref [21]

ldquoSMXXrdquo is the same central value with a reduced error as expected by the improvementon the hadronic cross section measurement (see text) ldquoBNL-E821 04 averdquo is the currentexperimental value of amicro ldquoNew (g-2) exprdquo is the same central value with a fourfold improvedprecision as planned by the future (g-2) experiments at Fermilab and J-PARC

References

[1] J Schwinger Phys Rev 73 (1948) 416 and Phys Rev 76 (1949) 790 The formerpaper contains a misprint in the expression for ae that is corrected in the longer paper

[2] T Aoyama M Hayakawa T Kinoshita and M Nio Phys Rev Lett 109 (2012)111808

[3] J P Miller E de Rafael B L Roberts and D Stockinger Ann Rev Nucl Part Sci62 (2012) 237

[4] D Stockinger in Advanced Series on Directions in High Energy Physics - Vol 20 LeptonDipole Moments eds B L Roberts and W J Marciano World Scientific (2010) p393

[5] D Hanneke S Fogwell and G Gabrielse Phys Rev Lett 100 (2008) 120801

[6] G W Bennett et al (The g 2 Collab) Phys Rev D73 (2006) 072003

[7] M Davier in Advanced Series on Directions in High Energy Physics - Vol 20 Lep-

ton Dipole Moments eds B L Roberts and W J Marciano World Scientific (2010)chapter 8

[8] R Bouchendira P Clade S Guellati-Khelifa F Nez and F Biraben Phys Rev Lett106 (2011) 080801

16

Fermilab amp JPARCNext 3-‐‑‒5 years

e+e-‐‑‒rarrhadrons実験

中心値が変わらなければ近い将来gt5σの感度度が期待

[Snowmass white paperより]

SMXX1815plusmn35

emsp　Muon g-‐‑‒2 10

中心値が変わらなければ近い将来gt5σの感度度が期待

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

「今」Muon g-‐‑‒2の不不一致の原因を調べることは非常に重要

博士論論文での立立場

emsp　Muon g-‐‑‒2 11

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+NP

Exp

Muon g-‐‑‒2の不不一致は新しい物理理(NP)の寄与が原因

新しい物理理

-‐‑‒  博士論論文では超対称模型を仮定-‐‑‒  model-‐‑‒independentにSUSYの寄与を確かめる

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

NP

emsp　本日の内容 12

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Supersymmetry 13

「BosonとFermionの間の対称性」

MSSM＝Minimal Supersymmetric Standard ModelSM粒粒子のパートナー(superparticle)を予言

emsp　Supersymmetry 14

SUSY contribution to the muon g-‐‑‒2

aSUSYmicro crarr2 tan

4

m2micro

m2SUSY

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

Chargino-‐‑‒sneutrino Neutralino-‐‑‒smuonCcedil 150 1010

100GeVmSUSY

2

tan10

aring

mSUSY 典型的なSUSY粒粒子の質量量 tanβ Higgsの真空期待値の比

mSUSY = O(100)GeV tanβ = O(10)のとき

超対称模型の寄与によりMuon g-‐‑‒2の不不一致を説明可能[Lopez Nanopoulos Wang ʼrsquo93][Chattopadhyay Nath ʼrsquo95][Moroi ʼrsquo95]

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Supersymmetry 15

Representative SUSY contributions

Contributions O(100)GeV particles Diagram

Chagino-‐‑‒sneutrino

Wino Higgsinos (Bino) smuons

Neutralino-‐‑‒smuon Bino smuons

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  Muon g-‐‑‒2を説明するminimalな解の一つl  特徴的な性質を持つ

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Muon g-‐‑‒2 9

Error [20] [21] Futurea

SMmicro 49 50 35

a

HLOmicro 42 43 26

a

HLbLmicro 26 26 25

(aEXPmicro a

SMmicro ) 80 80 40

Figure 9 Estimated uncertainties amicro in units of 1011 according to Refs [20 21] and (lastcolumn) prospects for improved precision in the e+e hadronic cross-section measurementsThe final row projects the uncertainty on the dicrarrerence with the Standard Model amicro Thefigure give the comparison between a

SMmicro and a

EXPmicro DHMZ is Ref [20] HLMNT is Ref [21]

ldquoSMXXrdquo is the same central value with a reduced error as expected by the improvementon the hadronic cross section measurement (see text) ldquoBNL-E821 04 averdquo is the currentexperimental value of amicro ldquoNew (g-2) exprdquo is the same central value with a fourfold improvedprecision as planned by the future (g-2) experiments at Fermilab and J-PARC

References

[1] J Schwinger Phys Rev 73 (1948) 416 and Phys Rev 76 (1949) 790 The formerpaper contains a misprint in the expression for ae that is corrected in the longer paper

[2] T Aoyama M Hayakawa T Kinoshita and M Nio Phys Rev Lett 109 (2012)111808

[3] J P Miller E de Rafael B L Roberts and D Stockinger Ann Rev Nucl Part Sci62 (2012) 237

[4] D Stockinger in Advanced Series on Directions in High Energy Physics - Vol 20 LeptonDipole Moments eds B L Roberts and W J Marciano World Scientific (2010) p393

[5] D Hanneke S Fogwell and G Gabrielse Phys Rev Lett 100 (2008) 120801

[6] G W Bennett et al (The g 2 Collab) Phys Rev D73 (2006) 072003

[7] M Davier in Advanced Series on Directions in High Energy Physics - Vol 20 Lep-

ton Dipole Moments eds B L Roberts and W J Marciano World Scientific (2010)chapter 8

[8] R Bouchendira P Clade S Guellati-Khelifa F Nez and F Biraben Phys Rev Lett106 (2011) 080801

16

Fermilab amp JPARCNext 3-‐‑‒5 years

e+e-‐‑‒rarrhadrons実験

中心値が変わらなければ近い将来gt5σの感度度が期待

[Snowmass white paperより]

SMXX1815plusmn35

emsp　Muon g-‐‑‒2 10

中心値が変わらなければ近い将来gt5σの感度度が期待

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

「今」Muon g-‐‑‒2の不不一致の原因を調べることは非常に重要

博士論論文での立立場

emsp　Muon g-‐‑‒2 11

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+NP

Exp

Muon g-‐‑‒2の不不一致は新しい物理理(NP)の寄与が原因

新しい物理理

-‐‑‒  博士論論文では超対称模型を仮定-‐‑‒  model-‐‑‒independentにSUSYの寄与を確かめる

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

NP

emsp　本日の内容 12

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Supersymmetry 13

「BosonとFermionの間の対称性」

MSSM＝Minimal Supersymmetric Standard ModelSM粒粒子のパートナー(superparticle)を予言

emsp　Supersymmetry 14

SUSY contribution to the muon g-‐‑‒2

aSUSYmicro crarr2 tan

4

m2micro

m2SUSY

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

Chargino-‐‑‒sneutrino Neutralino-‐‑‒smuonCcedil 150 1010

100GeVmSUSY

2

tan10

aring

mSUSY 典型的なSUSY粒粒子の質量量 tanβ Higgsの真空期待値の比

mSUSY = O(100)GeV tanβ = O(10)のとき

超対称模型の寄与によりMuon g-‐‑‒2の不不一致を説明可能[Lopez Nanopoulos Wang ʼrsquo93][Chattopadhyay Nath ʼrsquo95][Moroi ʼrsquo95]

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Supersymmetry 15

Representative SUSY contributions

Contributions O(100)GeV particles Diagram

Chagino-‐‑‒sneutrino

Wino Higgsinos (Bino) smuons

Neutralino-‐‑‒smuon Bino smuons

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  Muon g-‐‑‒2を説明するminimalな解の一つl  特徴的な性質を持つ

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Muon g-‐‑‒2 10

中心値が変わらなければ近い将来gt5σの感度度が期待

gt3σの不不一致が観測261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

「今」Muon g-‐‑‒2の不不一致の原因を調べることは非常に重要

博士論論文での立立場

emsp　Muon g-‐‑‒2 11

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+NP

Exp

Muon g-‐‑‒2の不不一致は新しい物理理(NP)の寄与が原因

新しい物理理

-‐‑‒  博士論論文では超対称模型を仮定-‐‑‒  model-‐‑‒independentにSUSYの寄与を確かめる

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

NP

emsp　本日の内容 12

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Supersymmetry 13

「BosonとFermionの間の対称性」

MSSM＝Minimal Supersymmetric Standard ModelSM粒粒子のパートナー(superparticle)を予言

emsp　Supersymmetry 14

SUSY contribution to the muon g-‐‑‒2

aSUSYmicro crarr2 tan

4

m2micro

m2SUSY

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

Chargino-‐‑‒sneutrino Neutralino-‐‑‒smuonCcedil 150 1010

100GeVmSUSY

2

tan10

aring

mSUSY 典型的なSUSY粒粒子の質量量 tanβ Higgsの真空期待値の比

mSUSY = O(100)GeV tanβ = O(10)のとき

超対称模型の寄与によりMuon g-‐‑‒2の不不一致を説明可能[Lopez Nanopoulos Wang ʼrsquo93][Chattopadhyay Nath ʼrsquo95][Moroi ʼrsquo95]

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Supersymmetry 15

Representative SUSY contributions

Contributions O(100)GeV particles Diagram

Chagino-‐‑‒sneutrino

Wino Higgsinos (Bino) smuons

Neutralino-‐‑‒smuon Bino smuons

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  Muon g-‐‑‒2を説明するminimalな解の一つl  特徴的な性質を持つ

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Muon g-‐‑‒2 11

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+NP

Exp

Muon g-‐‑‒2の不不一致は新しい物理理(NP)の寄与が原因

新しい物理理

-‐‑‒  博士論論文では超対称模型を仮定-‐‑‒  model-‐‑‒independentにSUSYの寄与を確かめる

CHAPTER 2 FOUNDATION 23

micro micro

γ

(A)

micro micro

γ

(B)

micro micro

γ

(C)

Figure 21 (A) The Feynman diagram which contributes to the anomalous magnetic momentof the muon (B) The hadronic vacuum-polarization contributions to the muon gminus2 (C) Thehadronic light-by-light contributions to the muon g minus 2

The amplitude can be interpreted as the Born approximation to the scattering of theelectron from a potential (For detail see textbook [19]) The interaction Hamiltonian whichcorresponds to such potential is given by

Hint = minuslangminusrarrmicro rang middotminusrarrB (217)

where

langminusrarrmicro rang= 2

F1(0) + F2(0)times eQℓ

2mℓξprimedaggerminusrarrσ2ξ (218)

The factor ξprimedagger(minusrarrσ 2)ξ can be interpreted as the spin of the leptonsminusrarrS Comparing Eq (218)

with Eqs (23) and (24) the coefficient 2

F1(0) + F2(0)

becomes

2

F1(0) + F2(0)= 2(1+ aℓ) = gℓ (219)

It is just the g-value

222 The Standard Model prediction of the muon g minus 2

The SM prediction of the muon g minus 2 has been precisely evaluated through many efforts ofseveral groups of theorists Fig 21 (A) shows the Feynman diagram which contribute to themuon g minus 2 The theoretical uncertainty reaches to about 04 ppm In this section we brieflyreview the SM prediction of the muon g minus 2

QED Contribution

The quantum electromagnetic dynamics (QED) contributions are the dominant contributionsto the muon g minus 2 (99993) and come from the diagrams with leptons and photons Theyhave been calculated analytically up to ( (α3) and recently the full five-loop (( (α5)) contri-bution has been calculated [2021] The present QED value is reported as

amicro(QED) = (116584718951plusmn 00080)times 10minus10 (220)

NP

emsp　本日の内容 12

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Supersymmetry 13

「BosonとFermionの間の対称性」

MSSM＝Minimal Supersymmetric Standard ModelSM粒粒子のパートナー(superparticle)を予言

emsp　Supersymmetry 14

SUSY contribution to the muon g-‐‑‒2

aSUSYmicro crarr2 tan

4

m2micro

m2SUSY

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

Chargino-‐‑‒sneutrino Neutralino-‐‑‒smuonCcedil 150 1010

100GeVmSUSY

2

tan10

aring

mSUSY 典型的なSUSY粒粒子の質量量 tanβ Higgsの真空期待値の比

mSUSY = O(100)GeV tanβ = O(10)のとき

超対称模型の寄与によりMuon g-‐‑‒2の不不一致を説明可能[Lopez Nanopoulos Wang ʼrsquo93][Chattopadhyay Nath ʼrsquo95][Moroi ʼrsquo95]

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Supersymmetry 15

Representative SUSY contributions

Contributions O(100)GeV particles Diagram

Chagino-‐‑‒sneutrino

Wino Higgsinos (Bino) smuons

Neutralino-‐‑‒smuon Bino smuons

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  Muon g-‐‑‒2を説明するminimalな解の一つl  特徴的な性質を持つ

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　本日の内容 12

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Supersymmetry 13

「BosonとFermionの間の対称性」

MSSM＝Minimal Supersymmetric Standard ModelSM粒粒子のパートナー(superparticle)を予言

emsp　Supersymmetry 14

SUSY contribution to the muon g-‐‑‒2

aSUSYmicro crarr2 tan

4

m2micro

m2SUSY

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

Chargino-‐‑‒sneutrino Neutralino-‐‑‒smuonCcedil 150 1010

100GeVmSUSY

2

tan10

aring

mSUSY 典型的なSUSY粒粒子の質量量 tanβ Higgsの真空期待値の比

mSUSY = O(100)GeV tanβ = O(10)のとき

超対称模型の寄与によりMuon g-‐‑‒2の不不一致を説明可能[Lopez Nanopoulos Wang ʼrsquo93][Chattopadhyay Nath ʼrsquo95][Moroi ʼrsquo95]

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Supersymmetry 15

Representative SUSY contributions

Contributions O(100)GeV particles Diagram

Chagino-‐‑‒sneutrino

Wino Higgsinos (Bino) smuons

Neutralino-‐‑‒smuon Bino smuons

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  Muon g-‐‑‒2を説明するminimalな解の一つl  特徴的な性質を持つ

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Supersymmetry 13

「BosonとFermionの間の対称性」

MSSM＝Minimal Supersymmetric Standard ModelSM粒粒子のパートナー(superparticle)を予言

emsp　Supersymmetry 14

SUSY contribution to the muon g-‐‑‒2

aSUSYmicro crarr2 tan

4

m2micro

m2SUSY

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

Chargino-‐‑‒sneutrino Neutralino-‐‑‒smuonCcedil 150 1010

100GeVmSUSY

2

tan10

aring

mSUSY 典型的なSUSY粒粒子の質量量 tanβ Higgsの真空期待値の比

mSUSY = O(100)GeV tanβ = O(10)のとき

超対称模型の寄与によりMuon g-‐‑‒2の不不一致を説明可能[Lopez Nanopoulos Wang ʼrsquo93][Chattopadhyay Nath ʼrsquo95][Moroi ʼrsquo95]

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Supersymmetry 15

Representative SUSY contributions

Contributions O(100)GeV particles Diagram

Chagino-‐‑‒sneutrino

Wino Higgsinos (Bino) smuons

Neutralino-‐‑‒smuon Bino smuons

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  Muon g-‐‑‒2を説明するminimalな解の一つl  特徴的な性質を持つ

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Supersymmetry 14

SUSY contribution to the muon g-‐‑‒2

aSUSYmicro crarr2 tan

4

m2micro

m2SUSY

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

Chargino-‐‑‒sneutrino Neutralino-‐‑‒smuonCcedil 150 1010

100GeVmSUSY

2

tan10

aring

mSUSY 典型的なSUSY粒粒子の質量量 tanβ Higgsの真空期待値の比

mSUSY = O(100)GeV tanβ = O(10)のとき

超対称模型の寄与によりMuon g-‐‑‒2の不不一致を説明可能[Lopez Nanopoulos Wang ʼrsquo93][Chattopadhyay Nath ʼrsquo95][Moroi ʼrsquo95]

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

emsp　Supersymmetry 15

Representative SUSY contributions

Contributions O(100)GeV particles Diagram

Chagino-‐‑‒sneutrino

Wino Higgsinos (Bino) smuons

Neutralino-‐‑‒smuon Bino smuons

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  Muon g-‐‑‒2を説明するminimalな解の一つl  特徴的な性質を持つ

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Supersymmetry 15

Representative SUSY contributions

Contributions O(100)GeV particles Diagram

Chagino-‐‑‒sneutrino

Wino Higgsinos (Bino) smuons

Neutralino-‐‑‒smuon Bino smuons

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  Muon g-‐‑‒2を説明するminimalな解の一つl  特徴的な性質を持つ

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Supersymmetry 16

博士論論文での立立場

21Dec 2 2013 SUSY Model-building and Phenomenology Teppei KITAHARA -The Univ of Tokyo

Status of the muon g-2

SM Value

experimentDHMZ (11)

The latest result of the muon g-2

[HagiwaraLiaoMartinNomuraTeubnerJ Phys G 38 (2011)085033] [DavierHoeckerMalaescuZhangEur Phys J C 71(2011)1515]

33 σ36 σ

(possibly a signal of new physics)muon g-2 anomaly3

SM+SUSY

Exp

Muon g-‐‑‒2の不不一致は の寄与が原因

知りたいこと

-‐‑‒ どのような性質を持つか-‐‑‒ 実験で検証できるか

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　本日の内容 17

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Minimal SUSY model 18

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Minimal SUSY model 19

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

[GeV]g~m600 700 800 900 1000 1100 1200 1300 1400 1500 1600

[GeV

]10 χsim

m

200

400

600

800

1000

forbi

dden

10χsimt t

rarrg~

= 8 TeVs) g~) gtgt m(q~ m(10χsimt trarrg~ production g~g~ ICHEP 2014

ATLASPreliminary

ExpectedObservedExpectedObservedObservedExpected

10 jetsge0-lepton 7 -

3 b-jetsge0-1 lepton

3 b-jetsge2SS3 leptons 0 -

arXiv 13081841

arXiv 14070600

arXiv 14042500

]-1 = 203 fbint

[L

]-1 = 201 fbint

[L

]-1 = 203 fbint

[L

not includedtheorySUSYσ95 CL limits

Colored superparticles gt ~sim1TeV

[httpstwikicernchtwikibinviewAtlasPublicSupersymmetryPublicResultsSummary_plots_status_July_2014]

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Minimal SUSY model 20

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

Scalar top gtgt 1TeV

[Hahn Heinemeyer Hollik Rzehak Weiglein ʼrsquo13]

3

the two-loop level obtained with the FD approach in theOS scheme while (∆M2

h)RGE are the leading and sub-

leading logarithmic contributions (either up to a certainloop order or summed to all orders) obtained in the RGEapproach as evaluated via Eq (2) In all terms of Eq (4)the top-quark mass is parametrised in terms of mt therelation between XMS

t and XOSt is given by

XMSt = XOS

t [1 + 2L (αsπ minus (3αt)(16π))] (5)

up to non-logarithmic terms and there are no logarithmiccontributions in the relation between MMS

S and MOSS

Since the higher-order corrections beyond 2-loop orderhave been derived under the assumption MA ≫ MZ toa good approximation these corrections can be incorpo-rated as a shift in the prediction for the φ2φ2 self-energy(where ∆M2

h enters with a coefficient 1 sin2β) In thisway the new higher-order contributions enter not onlythe prediction for Mh but also all other Higgs sectorobservables that are evaluated in FeynHiggs The latestversion of the code FeynHiggs2100 which is availableat feynhiggsde contains those improved predictions aswell as a refined estimate of the theoretical uncertaintiesfrom unknown higher-order corrections Taking into ac-count the leading and subleading logarithmic contribu-tions in higher orders reduces the uncertainty of the re-maining unknown higher-order corrections Accordinglythe estimate of the uncertainties arising from correctionsbeyond two-loop order in the topstop sector is adjustedsuch that the impact of replacing the running top-quarkmass by the pole mass (see Ref [7]) is evaluated only forthe non-logarithmic corrections rather than for the fulltwo-loop contributions implemented in FeynHiggs Fur-ther refinements of the RGE resummed result are pos-sible in particular extending the result to the case ofa large splitting between the left- and right-handed softSUSY-breaking terms in the scalar top sector [25] andto the region of small values of MA (close to MZ) aswell as including the corresponding contributions fromthe (s)bottom sector We leave those refinements for fu-ture work

III NUMERICAL ANALYSIS

In this section we briefly analyze the phenomenologi-cal implications of the improved Mh prediction for largestop mass scales as evaluated with FeynHiggs2100The upper plot of Fig 1 shows Mh as a function ofMS for Xt = 0 and XtMS = 2 (which correspondsto the minimum and the maximum value of Mh as afunction of XtMS respectively here and in the fol-lowing Xt denotes XOS

t ) The other parameters areMA = M2 = micro = 1000 GeV mg = 1600 GeV (M2 is theSU(2) gaugino mass term micro the Higgsino mass parameterand mg the gluino mass) and tanβ = 10 The plot showsfor the two values of XtMS the fixed-order FD resultcontaining corrections up to the two-loop level (labelledas ldquoFH295rdquo which refers to the previous version of the

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

FH2953-loop4-loop5-loop6-loop7-loopLL+NLL

FeynHiggs 2100

Xt = 0

XtMS = 2

5000 10000 15000 20000MS [GeV]

115

120

125

130

135

140

145

150

155

M h [GeV

]

3-loop O(αt αs2)

3-loop fullLL+NLLH3m

FeynHiggs 2100

A0 = 0 tanβ = 10

FIG 1 Upper plot Mh as a function ofMS for Xt = 0 (solid)and XtMS = 2 (dashed) The full result (ldquoLL+NLLrdquo) iscompared with results containing the logarithmic contribu-tions up to the 3-loop 7-loop level and with the fixed-orderFD result (ldquoFH295rdquo) Lower plot comparison of FeynHiggs(red) with H3m (blue) In green we show the FeynHiggs 3-loopresult at O(αtα

2s) (full) as dashed (solid) line

code FeynHiggs) as well as the latter result supplementedwith the analytic solution of the RGEs up to the 3-loop 7-loop level (labelled as ldquo3-looprdquo ldquo7-looprdquo) Thecurve labelled as ldquoLL+NLLrdquo represents our full resultwhere the FD contribution is supplemented by the lead-ing and next-to-leading logarithms summed to all ordersOne can see that the impact of the higher-order logarith-mic contributions is relatively small for MS = O(1 TeV)while large differences between the fixed-order result andthe improved results occur for large values of MS The 3-loop logarithmic contribution is found to have the largestimpact in this context but forMS

gtsim 2500(6000) GeV forXtMS = 2(0) also contributions beyond 3-loop are im-portant A convergence of the higher-order logarithmiccontributions towards the full resummed result is visibleAt MS = 20 TeV the difference between the 7-loop resultand the full resummed result is around 900(200) MeV forXtMS = 2(0) The corresponding deviations stay below100 MeV for MS

ltsim 10 TeV The plot furthermore showsthat for MS asymp 10 TeV (and the value of tanβ = 10chosen here) a predicted value of Mh of about 126 GeVis obtained even for the case of vanishing mixing in thescalar top sector (Xt = 0) Since the predicted value ofMh grows further with increasing MS it becomes appar-

Mh Higgs massMs Scalar top mass

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Minimal SUSY model 21

Mass spectrum

eB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

l  軽い(O(100)GeV)粒粒子 Bino sleptonsl  他の超対称粒粒子 Decoupled

現在のLHCの制限および126GeV Higgs massと無矛盾

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Minimal SUSY model 22

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

Neutralino-‐‑‒smuon contribution

右巻きと左巻きsleptonの混合に比例例

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Minimal SUSY model 23

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Minimal SUSY model 24

右巻きと左巻きsmuonの混合に比例例

-‐‑‒ Large μtimestanβ enhanced-‐‑‒ Large smuon mass suppressed

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

Muon g-‐‑‒2

1σ2σ

加速器実験の探索索可能領領域を超えてしまう [Endo Hamaguchi Kitahara TY ʻlsquo13]

Neutralino-‐‑‒smuon contribution

emsp　スカラーポテンシャルの安定性条件emsp　により上限が存在する

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Minimal SUSY model 25

まとめとこれから議論論すること

SmuonがO(1)TeVでも極端に大きな混合を取ればMuon g-‐‑‒2の不不一致を説明できてしまう

emsp　emsp　がMuon g-‐‑‒2の不不一致の原因

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroR

m2e

LR

極端に大きな混合はスカラーポテンシャルの安定性条件により制限

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　本日の内容 26

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Slepton Mass Bound 27

真空の安定性条件-‐‑‒  SleptonとHiggsの3点結合

v 真空期待値 h SM Higgs 混合はμgt0のときnegative

EW vacuumの寿命 gt 宇宙年年齢とすると混合の大きさに上限が付く

EW vacuum Charge-‐‑‒breaking minima

大きすぎる混合

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Slepton Mass Bound 28

真空の安定性条件-‐‑‒  数値公式

Slepton mass bound

[Kitahara TY ʻlsquo13][Endo Hamaguchi Kitahara TY ʻlsquo13]

Calculated by CosmoTransitions η~sim1 (leptonのフレーバーに弱く依存)

Muon g-‐‑‒2の不不一致を説明するという条件と合わせるとsmuonのmassに上限

(すべて縮退)のとき

Stau-‐‑‒Higgsポテンシャルで強く制限

Vacuum bound

〜～400GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Slepton Mass Bound 29

Stau mass dependence

Massの制限はstauとsmuonのmassの比に依存

Vacuum bound

〜～400GeV 〜～600GeV

mememicro = 1 mememicro = 2 mememicro 1

stauによって制限(3点結合はYukawaに比例例)

memicro AElig 500GeV

smuonによって制限(stauがdecoupleしたとき)

memicro AElig 2TeV

[Endo Hamaguchi Kitahara TY ʻlsquo13]

stauが重くなると制限が弱くなる

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Slepton Mass Bound 30

まとめとこれから議論論すること

Universal Non-‐‑‒Universal

Spectrum

Vacuum Severe (stau) Mild (smuon)

Mass Bound Tight (lt500GeV) Loose (lt2TeV)

Collider LHC ILC

FlavorCP GIM Sensitive

まとめ

これから

mee = memicro = me mee = memicro me

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　本日の内容 31

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Universal Case 32

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuon mass lt 500GeVに制限される

g-‐‑‒21σ

2σ

Excluded by long-‐‑‒lived stau search

pp ee `+` + EmissT

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Universal Case 33

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

(Dilepton search)が有効

[ATLAS Collaboration JHEP 05 (2014) 035]

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

pp ee `+` + EmissT

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Universal Case 34

-‐‑‒  Mass

-‐‑‒  混合

-‐‑‒  LHC

mee = memicro = me (縮退)

真空の安定性条件を満たす中で最大の値を各点で選ぶ

LHC Status

ee

e01e0

1

` `

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

Excluded

[Endo Hamaguchi Kitahara TY ʻlsquo13]

pp ee `+` + EmissT

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Universal Case 35

-‐‑‒  Cross section (LO)

Future Prospects

Smuonの質量量がNeutralinoの質量量より十分大きい領領域は14TeV LHCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

(emicro1emicro1) =

8gtgtltgtgt

1fb (radics = 8TeV のとき)memicro = 330GeV

memicro = 450GeV1fb (radics = 14TeV のとき)harr 現在のLHCの制限

harr (Naiumlveな)将来の感度度

LHC

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Universal Case 36

Future Prospects

Smuonの質量量とNeutralinoの質量量が近い領領域は1TeV ILCで探索索可能

[Endo Hamaguchi Kitahara TY ʻlsquo13]

-‐‑‒  ILCkinematicalに許される質量量領領域ならば探索索可能

Closing the loopholes

At the ILC a systematic search for the NLSP is possible without leaving loopholes covering even the casesthat may be very difficult to test at the LHC

In the case of a very small mass difference between the LSP and the NLSP - less than a few GeV - theclean environment at the ILC nevertheless allows for a good detection efficiency If

radics is much larger than

the threshold for the NLSP-pair production the NLSPs themselves will be highly boosted in the detectorframe and most of the spectrum of the decay products will be easily detected In this case the preciseknowledge of the initial state at the ILC is of paramount importance to recognize the signal by the slightdiscrepancy in energy momentum and acolinearity between signal and background from pair productionof the NLSPrsquos SM partner In the case the threshold is not much below

radics the background to fight is

γγ rarr f f where the γrsquos are virtual ones radiated off the beam-electrons The beam-electrons themselvesare deflected so little that they leave the detector undetected through the outgoing beam-pipes Under theclean conditions at the ILC this background can be kept under control by demanding that there is a visibleISR photon accompanying the soft NLSP decay products If such an ISR is present in a γγ event thebeam-remnant will also be detected and the event can be rejected

If the LSP is unstable due to R-parity violation the ILC reach would be better or equal to the R-conserving case both for long-lived and short-lived LSPrsquos and whether the LSP is charged or neutral

Also in the case of an NLSP which is a mass-state mixed between the hyper-charge states the procedureis viable One will have one more parameter - the mixing angle However as the couplings to the Z of bothstates are known from the SUSY principle so is the coupling with any mixed state There will then beone mixing-angle that represents a possible ldquoworst caserdquo which allows to determine the reach whatever themixing is - namely the reach in this ldquoworst caserdquo

Finally the case of ldquoseveralrdquo NLSPsndash ie a group of near-degenerate sparticlesndash can be disentangled dueto the possibility to precisely choose the beam energy at the ILC This will make it possible to study theldquorealrdquo NLSP below the threshold of its nearby partner

0

50

100

150

200

250

0 50 100 150 200 250

Exclusion

Discovery

Exclud

able

at 95

CL

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

0

50

100

150

200

250

240 242 244 246 248 250

Exclusion

Discovery

NLSP microR

MNLSP [GeV]

MLS

P [G

eV]

Figure 3 Discovery reach for a microR NLSP after collecting 500 fbminus1 atradics = 500 GeV Left full scale Right

zoom to last few GeV before the kinematic limit

The strategy

At an e+eminus-collider the following typical features of NLSP production and decay can be exploited missingenergy and momentum high acolinearity expected particle or jet flavor identification as well as invariantdi-jetdi-lepton mass conditions optionally using constrained kinematic fitting A very powerful feature due

9

[Baer etal ʻlsquo13]

ex) radics = 500GeV Δm=O(10-‐‑‒100)MeV

Massの差はO(10)GeVILCで探索索可能

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Universal Case 37

Muon g-‐‑‒2を1σで説明できる領領域 (1σ Region)のほとんどは現在のLHCのデータですでに排除

残りの質量量領領域も将来のLHC ILC実験で探索索可能な見見通し

Stau-‐‑‒Higgsポテンシャルの安定性条件からslepton mass lt 500GeVに制限される

まとめ

ILCにおけるStau探索索に関してHiggsとDiphotonの結合の情報も有効[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　本日の内容 38

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Non-‐‑‒Universal Case 39

Smuon mass bound

500 1000 1500 2000

500

1500

2500

3500

memicro lt me01

Smuon-‐‑‒Higgsポテンシャルの安定性条件から smuon mass lt 2TeVに制限される

-‐‑‒  Mass

-‐‑‒  混合

mee = memicro me (stauはdecouple)memicroL

= memicroR tan = 40

真空の安定性条件を満たす中で最大の値を各点で選ぶ

Smuon mass lt2TeVは加速器実験の探索索可能領領域を超えている

[Endo Hamaguchi Kitahara TY ʻlsquo13]

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Non-‐‑‒Universal Case 40

Fermionを対角にする基底でSlepton質量量行行列列に非対角項が出現

Sleptonの質量量が縮退していない場合GIM機構が働かないためLFVCPVを誘発

LFVCPV

emsp　emsp　emsp　emsp　emsp　emsp　emsp　emsp　 (対角) である模型でもFermionを対角にする基底とは一般に揃わない

m2e

i j= diag

m2ee m2emicro m2e

LFVCPVはSlepton質量量行行列列の非対角項に敏感

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Non-‐‑‒Universal Case 41[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

l  Stauは12世代のSleptonより重いl  SelectronとSmuonは縮退1-‐‑‒2世代間の混合の寄与はsuppress

mee = memicro me

仮定

Stauが12世代のSleptonより十分重い場合μrarreγとMuon g-‐‑‒2の比はBino slepton massと独立立

Flavor mixingの定義

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Non-‐‑‒Universal Case 42[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) μrarreγ崩壊

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Br(micro e)

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

Stauが十分重い(R≫1)ときL

13R

23

AElig 3 106

L

13R

23

AElig 1078

(Current)

(Future)

VubVcb 104

cf) CKM-‐‑‒like

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Non-‐‑‒Universal Case 43

Sleptonのmassが縮退していない場合一般に大きなLFVCPVを誘発

Smuon-‐‑‒Higgsポテンシャルの安定性条件からsmuon mass lt 2TeVに制限される

まとめ

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　本日の内容 44

IntroMuon g-‐‑‒2

Supersymmetric Standard Model

MainldquoMinimalrdquo SUSY model

Slepton Mass Bound

Conc Summary

mee = memicro = me mee = memicro me

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　まとめ 45

「Muon g-‐‑‒2を説明する超対称模型の現象論論」

Muon g-‐‑‒2

gt3σ deveB eL eR

eg eq fW middot middot middot

≫1TeV

O(100)GeV

ldquoMinimalrdquo SUSY model

LHCILC

LFVEDM

m ee= m emicro

= m e

mee = memicro

me① massに制限

② 相補的に探索索可能

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　補足パート 46

Backup

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Muon g-‐‑‒2 47

測り方

磁場中でのMuonの歳差運動の周波数から求める

①  一様磁場中に偏極した(反)muonを入れる②  サイクロトロン運動(磁場)amp歳差運動(スピン)③  g=2 歳差運動はサイクロトロン運動に追従

gne2 歳差運動の方がわずかに速くなる(異異常歳差運動)④  崩壊先の陽電子の数の時間変動の測定から歳差運動の振動数を求める

⑤  歳差運動の振動数からg-‐‑‒2を読み取る

a e

mmicroamicroB

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Muon g-‐‑‒2 48

BNLとJPARCの違い

l  BNLMagic momentum

l  JPARC電場ゼロ

電場速度度

Lorentz因子

異異常歳差運動の振動数

γ=294 (p = 3094GeVc) のとき第二項が無視できる

Fermilab (FNAL) はBNLと同じsetup

極冷冷muonを用いることで収束電場を不不要にするEDMの観測も目標

a = e

mmicro

24amicroB

amicro 1

2 1

E

c

2

B35

EDM(無視)

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Muon g-‐‑‒2 49

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

条件SM EW contributionと同程度度

261 (80) times 10-‐‑‒10amicro aexp

micro ath

micro =

aSM EWmicro crarr2

4

m2micro

m2W= 1536 (01) times 10-‐‑‒10

αNP mNP ~sim O(EWボソンの結合 質量量)そのような粒粒子が存在したら既に発見見されているはずhellip

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Muon g-‐‑‒2 50

(Naive) NP contribution

aNPmicro

crarrNP

4

mmicrom2

NP

Muon g-2

g = 2 at tree level (Dirac equation) g ne 2 by radiative corrections

Magnetic moment

g-factor

gNP gNP

mNP

2種類の可能性

l  αNP~sim O(10-‐‑‒6) (weak) mNP~sim O(100)MeV (small)

l  αNP~sim O(01-‐‑‒1) (strong) mNP~sim O(100-‐‑‒1000)GeV (heavy)

eg Hidden Photon

eg Supersymmetry

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Supersymmetry 51

Higgs mass

l  Tree level l  Large top-‐‑‒stop correction

mtreeh mZ

m2

h m2

Z cos

2

2 +3

42

Y 2

t sin

2

ntildem2

t log

M2

S

m2

t+

X 2

t

M2

S

Ccedil1

X 2

t

12M2

S

aring+ middot middot middotocirc

Ms =p

met1

met2

Xt = At micro cot

-‐‑‒  Heavy stop Ms ≫ 1TeV Xt = 0-‐‑‒  Maximal mixing Ms ~sim 1TeV Xt = radic6Ms

Scalar top gt 1TeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Supersymmetry 52

CMSSM (mSUGRA)

m0

A0m12

sgn(micro) tan

Scalar mass

Scalar coupling

m0

Masses of Hud

Gaugino mass

Other parameters

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Supersymmetry 53

Muon g-‐‑‒2 vs CMSSM

CMSSMの枠組では現在のLHCの制限とHiggs massとMuon g-‐‑‒2を同時に説明することができない

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʻlsquo11]

[GeV]0m0 1000 2000 3000 4000 5000 6000

[GeV

]1

2m

300

400

500

600

700

800

900

1000

(2400 GeV)

q ~

(1600 GeV)

q ~

(1000 GeV)g~

(1400 GeV)g~

h (122 GeV)

h (124 GeV)

h (126 GeV)

ExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObservedExpectedObserved

gt 0micro 0 = -2m0

) = 30 AβMSUGRACMSSM tan( Status ICHEP 2014

ATLAS Preliminary = 8 TeVs -1 L dt = 201 - 207 fbint

τsim

LSP not includedtheory

SUSYσ95 CL limits

0-lepton 2-6 jets

0-lepton 7-10 jets

0-1 lepton 3 b-jets

1-lepton + jets + MET

1-2 taus + 0-1 lept + jets + MET

3 b-jetsge2SS3 leptons 0 -

arXiv 14057875

arXiv 13081841

arXiv 14070600

ATLAS-CONF-2013-062

arXiv 14070603

arXiv 14042500

mh ~sim126GeV

g-‐‑‒2

tanβ=20

m0[GeV]

m12[GeV]

Higgs massは (brarrsγの制限を満たす範囲で) A-‐‑‒termを調節して説明

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Supersymmetry 54

Muon g-‐‑‒2 vs Higgs mass and LHC (model)

l Messenger sectorを拡張

l  Extra vector-‐‑‒like matterを加える

l 新しいgauge対称性を加える

[Endo Hamaguchi Iwamoto Yokozaki ʼrsquo1112][Endo Hamaguchi Ishikawa Iwamoto Yokozaki ʼrsquo12][Moroi Sato Yanagida ʼrsquo12]

[Evans Ibe Yanagida ʼrsquo11][Evans Ibe Shirai Yanagida ʼrsquo11][Ibe Matsumoto Yanagida Yokozaki ʼrsquo12][Bhattacharyya Bhattacherjee Yanagida Yokozaki ʼrsquo13]

[Endo Hamaguchi Iwamoto Nakayama Yokozaki ʼrsquo11]

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Supersymmetry 55

Electron g-‐‑‒2

115965218073(28)times10-‐‑‒13aexp

e =

athe = 115965218187(06)(04)(02)(76)(01)times10-‐‑‒13

ae = aexp

e ath

e = -‐‑‒114 (81) times 10-‐‑‒13

l  実験値と理理論論値はConsistent (14σ)

l  新しい物理理の寄与

ae = amicro

3 109

07 1013 (NaiumlveにYukawaでスケール)

-‐‑‒  SUSYでSleptonが縮退している場合はMuon g-‐‑‒2に比べて感度度が弱い-‐‑‒  Non-‐‑‒universalの場合はElectron Muon g-‐‑‒2の両方がenhanceし得る(符号逆LFVCPVを誘発するのでちと厳しいがhellip)

ae 2 1012ex) のときmee = 100GeV tan = 30

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Minimal SUSY model 56

Muon g-‐‑‒2 (Higher order corr 1 ~sim10)

l  QEDのLeading Log correction l  Bino couplingに重い粒粒子がdecoupleした効果

1+2loop =

Ccedil1 4crarr

ln

msoft

mmicro

aring1+

1

4

2crarrYb+

9

4

crarr2

ln

Msoft

msoft

egL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

amicro = (1+2loop)a1loop

micro

Leading Log Bino coupling

重い粒粒子のスケール

軽い粒粒子のスケール

b =41

6

( of the generations of light sleptons)

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Minimal SUSY model 57

Muon g-‐‑‒2 (Higher order corr 2 ~sim10)

l  Yukawaの補正 (Non-‐‑‒Holomorphic Yukawa)

-‐‑‒  MSSMのYukawaとSMのYukawaのmatchingの際に出現

m` =

`L

`R

+hHdi hHui

`L

`R

SUSYSUSY

Y MSSM

` =m`

v cos

1

1+`

`

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Minimal SUSY model 58

Bino coupling

SUSY limitではU(1)Y ゲージ結合と等しい(重い粒粒子の) SUSY breakingの効果でズレが発生

Coupling

EnergyMsoft

msoft

egBino

= gY

gY

egBino

egBino

重い粒粒子decouple

くりこみ群のrunが変化

Lint = 1p

2egL eB`Le

L +p

2egR eB`R eR + hc

-‐‑‒  Interaction

-‐‑‒  Bino couplingegL = gY +egL gY

1+

1

4

crarrYb+

9

4

crarr2

ln

Msoft

msoft

egR = gY +egR gY

1+crarrY

4b ln

Msoft

msoft

b =41

6

( of the generations of light sleptons)

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Slepton Mass Bound 59

スカラーポテンシャルl  Up-‐‑‒type HiggsとSlepton

l  Slepton-‐‑‒Higgsの3点相互作用

l  Sleptonの4点相互作用

V = (m2Hu+micro2)|Hu|2 +m2

eL|eL |2 +m2e

R|eR|2 (Y`microHueLeR + hc) + Y 2

` |eLeR|2

+g2

8(|eL |2 + |Hu|2)2 +

g2Y

8(|eL |2 2|eR|2 |Hu|2)2 +

g2 + g2Y

8H |Hu|4

mixing Flavor dep

-‐‑‒  Sleptonの混合に比例例-‐‑‒  極端に大きいとEW vacuumが不不安定になる

-‐‑‒  LeptonのYukawaに比例例 (＝tanβが大きい極限でにtanβに比例例)-‐‑‒  tanβを大きくすると真空の安定性条件をゆるくする方向に働く-‐‑‒  Stauの場合は若若干依存SelectronSmuonは依存性無視できる

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Slepton Mass Bound 60

真空崩壊l  Decay rate

(Volume) = A middot eB

~sim(100GeV)4

Bounce解から評価

-‐‑‒  偽の真空の基底状態のエネルギーの虚部から出現

-‐‑‒  境界条件

-‐‑‒  境界条件のもとでの運動方程式の解Bounce解

= 2ImE0

E0

= lim

T1

1

Tln

CcedilZ[D]exp(SE[])

aring

Φ=Up-‐‑‒type Higgs Sleptons SE はユークリッド作用

limT1

x plusmn T

2

= f

B = SE[] SE[ f ]

偽の真空での場の配位

[Coleman ʻlsquo77][Callan Coleman ʻlsquo77]

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Slepton Mass Bound 61

真空崩壊l  Bounce解 (Path deformation method)

-‐‑‒  運動方程式 (O(4) symmetric zero temp α = 3)

-‐‑‒  最初にPathにあたりをつける

-‐‑‒  Pathを変形させて運動方程式の解になるものを探す

guess =

(x)

d

d x

= 1 (仮定)

Figure 4 Path deformation in two dimensions The normal force exerted on the startingpath (blue straight line) pushes it in the direction of the true tunneling solution (greencurved line)

speed Equation 7 does not ecrarrect the motion of the particle but insteaddetermines the normal force N that the track must exert on the particle to

keep it from falling ocrarr N = d

2~

dx

2

dx

d

2

rV (~) For the right path N = 0

Given a starting guess one can deform the path to the correct solution bycontinually pushing it in the direction of N (see figure 4)

I implement this general method in the CosmoTransitions package usingB-splines in the module pathDeformationpy Each path is written as a lin-ear combination of spline basis functions plus a linear component connectingits ends ~(y) =

Pi

i

~

i

(y)+(~0 ~

F

)y+ ~

F

where y parametrizes the path

(0 y 1 and generallyd~dy

6= 1) and ~

i

(0) = ~

i

(1) = 0 This fixes the

pathrsquos endpoints at ~

0 ~

(=0) and ~

F

Generally only a small numberof basis functions are needed to accurately model the path ( 10 per fielddirection) unless it contains sharp bends or many dicrarrerent curves

Before any deformation the algorithm first calculates the bubble profilealong the starting path using the overshootundershoot method Then itdeforms the path in a series of steps without recalculating either the one-

dimensional profile ord~d

At each step it calculates the normal force for

a relatively large number of points ( 100) along the path rescales thenormal force by |~

T

~

F

||rV |max

and moves the points in that directiontimes some small stepsize If the one-dimensional solution is thick-walled

8

[CosmoTransitions ʻlsquo11]

d2d2 +

crarr

d

d=rV (

)

d

2x

d2 +crarr

d x

d=

x

V ( )

d

2d2

d x

d

2=rV (

)

(Path方向)

(垂直方向)

V

true

false

cf) OvershootUndershoot method

Multi-‐‑‒dimensionだと一意に求められない

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Slepton Mass Bound 62

Fitting Formula

200 300 400 500 600

200

300

400

500

600

l  Stauの場合

micro tan tan = 70

30

20TeV

40

50

60

70

90

80

[Kitahara TY ʻlsquo13][Endo Kitahara TY ʻlsquo14]

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Slepton Mass Bound 63

Fitting Formula l  tanβ依存性

20 40 60 80 100080

085

090

095

100

105

110

115

120

-‐‑‒  スカラーの4点はYukawaに依存 ( )-‐‑‒  tanβが大きい極限で(tanβ)2に比例例-‐‑‒  tanβを大きくするとVacuumの制限が若若干改善

[Kitahara TY ʻlsquo13]

改善悪化

ここで1に規格化

12世代のsleptonの場合はYukawaが小さいためtanβ依存性は無視できる

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Universal Case 64

ILC (smuon)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

Smuonにmassの差がある場合はILCが有効偏極ビームを用いることで断面積がenhance

ps = 1TeV

ps = 1TeV

Polarization

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Universal Case 65

ILC (selectron)

[Endo Hamaguchi Kitahara TY ʻlsquo13]

t-‐‑‒channel Bino exchangeのため断面積がenhanceBino couplingの測定にも有効

ps = 1TeV -‐‑‒  Diagram (t-‐‑‒channel)

-‐‑‒  Bino coupling

eB

ee

eee+

e

uuml  重い粒粒子の効果でO()変化uuml  高精度度のCouplingの測定から重いスケールを探れるかも

[Nojiri Fujii Tsukamoto ʻlsquo96][Nojiri Pierce Yamada ʼrsquo97]

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Non-‐‑‒Universal Case 66[Endo Hamaguchi Kitahara TY ʻlsquo13]

Non-‐‑‒Universality

Smuonが重いほどMuon g-‐‑‒2の説明のために大きなNon-‐‑‒Universalityが要求される

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Non-‐‑‒Universal Case 67[Endo Hamaguchi Kitahara TY ʻlsquo13]

Flavor mixingの定義l  SleptonがFlavor diagonalである基底でのYukawal  Fermion massを対角にする基底でのYukawa

l  Mixing matrix世代間の混合をparameterize

Mixing matrix 基底間のズレ

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Non-‐‑‒Universal Case 68[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  Effective operator (双極子型)

l  Wilson係数

l  Flavor mixing

Leff em`i

2`imicro

AL

i j PL + ARi j PR

` j F

micro + hc

Higher-‐‑‒order correction

Loop function

ALi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUR

ii b

hM`i

ba

hUdagger

L

ia j

Fab

ARi j = (1+

2loop)crarrY

8

M1

micro tan

m` j

X

ab=123

hUL

iia

hMdagger

`

iab

hUdagger

R

ib j

Fab

hUL

i1a

hMdagger`

iab

hUdagger

R

ib2

Fab = mmicro

1+micro(L)12

AumlF12 F22

auml

+m

1+(L)13(R)23

AumlF12 F13 F32 + F33

auml

SelectronとSmuonが縮退していればキャンセル

Sleptonが全て縮退していればキャンセル

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Non-‐‑‒Universal Case 69[Endo Hamaguchi Kitahara TY ʻlsquo13]

ex) Lepton FCNCsl  μrarreγ崩壊

l  電子の電気双極子モーメント

l  Muon g-‐‑‒2との相関

46 Dissertation Takahiro Yoshinaga

microL microR

Wplusmn

νmicro

γ

(a)

microL microR

microL microR

B

γ

(b)

microL microR

microL

W 0 H0d

γ

(c)

microL microR

microL

B H0d

γ

(d)

microL microR

microR

H0d

B

γ

(e)

Figure 34 The diagrams of the SUSY contributions to the muon g minus 2 in gauge eigenstatesThe diagram (a) comes from the charginondashmuon sneutrino diagram the diagrams (b)ndash(e) arefrom the neutralinondashsmuon diagram

∆a χ0

micro = minusmmicro

48π2

6sum

X=1

4sum

A=1

1m2ℓX

$mmicro4

ampampampN L(e)2AX

ampampamp2+ampampampNR(e)

2AX

ampampamp2

F N1

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

+mχ0

A

2real

N L(e)lowast2AX NR(e)

2AX

-F N

2

⎛⎝

m2χ0

A

m2ℓX

⎞⎠

(333)

where F C12 and F N

12 are the loop functions

F C1 (x) =

2(1minus x)4

2+ 2x minus 6x2+ x3+ 6x log x

- (334)

F C2 (x) =

32(1minus x)3

minus3+ 4x minus x2minus 2 log x

- (335)

F N1 (x) =

2(1minus x)4

1minus 6x + 3x2+ 2x3minus 6x2 log x

- (336)

F N2 (x) =

3(1minus x)3

1minus x2+ 2x log x

- (337)

333 Formulae in gauge eigenstates

Even though the formulae in mass eigenstates (332) and (333) are useful for numerical cal-culations they are not suitable for understanding their dependence of the MSSM parametersThe parameter dependences are hidden by the electroweak symmetry breaking which causes

fW eH emicroL emicroReeR

eR

de

e=

me

2Imicirc

AL11 AR

11

oacute

B(micro e) 483crarr

G2F

Auml|AL

12|2 + |AR12|2auml

Br(micro e)aNeutralino

micro

2

1

(micro all)crarrmmicro16

13

23

2Ccedil

mmmicro

aring2

+ (higher order)

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

Stauが十分重いときSUSY粒粒子の質量量と独立立

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Non-‐‑‒Universal Case 70[Endo Hamaguchi Kitahara TY ʻlsquo13]

SleptonがLHC ILCで発見見されなくてもレプトンフレーバー実験から高感度度で探索索可能

＝Non-‐‑‒Universality

現在のBound

将来の感度度

-‐‑‒  Mass

-‐‑‒  Formula

mee = memicro me

Stauが十分重い(R≫1)とき(Current)

(Future)

ex) 電子の電気双極子モーメント

deeaNeutralino

micro

1

2mmicroIm[(R)13

(L)13

]mmmicro+ (higher order)

ImL

13R

13

AElig 6 108

ImL

13R

13

AElig 6 10910

dee

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Extra-‐‑‒Slides 71

Future Prospectsfor Stau

博士論論文 Sec43

Based on T Kitahara and TY JHEP 05 035 (2013) M Endo T Kitahara and TY JHEP 04 139 (2014)

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Stau 72

Higgs Oblique Correction l  Loop-‐‑‒induced Higgs coupling

l  新しい物理理にsensitive

-‐‑‒  Higgs to digluon diphoton Zγ-‐‑‒  間接的に新しい物理理を制限

-‐‑‒  SMtree levelで出現しない-‐‑‒  新しい物理理の影響を受けやすい

NPh

V

V制限

予言

実験 新しい物理理

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Stau 73

Higgs coupling to diphoton κγl  Current accuracy (LHC)

l  Joint analysis

gh

gh(SM)= 1+ 15

-‐‑‒  Br(hrarrγγ)Br(hrarrZZ)はHL-‐‑‒LHCで精密測定 (36と仮定)-‐‑‒  Br(hrarrZZ)は初期ステージでのILCで精密測定 (sub level)-‐‑‒  両者の結果を合わせると単体での測定よりもκγを精密に決定可能

1 2

[Peskin ʻlsquo13]

5

4

3

2

1

γCMS-1 CMS-2

ILC

ILC +LHC BRratio

CMS 250 500 500up 1000up1000

6

7

-‐‑‒  HL-‐‑‒LHC ILC単体だとO(1-‐‑‒10)-‐‑‒  Joint analysis

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Stau 74

Higgs coupling to diphoton κγl  Loop-‐‑‒induced Higgs couplingl  新しい物理理にsensitivel  HL-‐‑‒LHCと初期のILCから1-‐‑‒2の決定精度度が期待

将来excessが見見えた場合に分かる新しい物理理の性質を考察しておくことは重要

l  博士論論文では超対称模型を仮定l  Muon g-‐‑‒2を考慮すると軽いStauが予想l  初期のILCまでで数のexcessが観測されたと仮定

新しい物理理

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

l  Stau κγのみに大きな寄与l  3つのParameterでcontroll (mass mixing angle)

l  Stauが軽い大きな混合を持つ場合にenhance (O(10))

l  極端に大きな混合は真空の安定性条件により制限

emsp　Stau 75

Stau contribution to κγ

54 Dissertation Takahiro Yoshinaga

τR

τL

h

γ

γ

Figure 36 Feynman diagram of the stau contribution to the Higgs coupling to di-photon

where OLR are the unitary matrices which diagonalize the chargino mass matrix as seen inSec 331 δm2

fLLRRand δm2

fLRare defined as

δm2fLLRR=

2v(m2

f + Df LR) δm2fLR=

1v

m2fLR

(368)

where Df LR are D-terms Df = m2Z cos2β

T 3

f minusQ f sin2 θW

and m2

fLRis the left-right mixing

of the sfermion mass matrices as

m2fLR=

12(m2

f1minusm2

f2) sin2θf (369)

Here θf is mixing angles of the sfermions defined as

Uf =

$cosθf sinθfminus sinθf cosθf

(370)

The mass matrix of the sfermion is diagonalize by Uf as Uf 2f Udagger

f = diag(m2f1

m2f2) Here

we neglect effects of flavorCP violations These contributions are expected to be suppressedsince they are strongly constrained by lepton flavor experiments

The contributions of the charged Higgs bosons and the charginos becomes relevant whenthese masses are order of 100 GeV For charged Higgs the contributions are negligible sincewe assume that the CP-odd Higgs mass mA is decoupled This assumption is reasonable sincenone of them is discovered On the other hand mass regions at several hundred GeV forcharginos are still remained As we will discuss in Sec 43 the chargino contributions to κγmight become (1) In this dissertation we take them as theoretical uncertainty Detailedestimation will be discussed in Sec 43

Then let us consider the contributions of sfermions Light sfermions with large left-rightmixing ie the stop sbottom and stau can provide large contributions to the Higgs couplingto di-photon [95] We assume that the colored superparticles are heavy enough to evade cur-rent LHC bound In our assumption stau may have a large contribution to the Higgs coupling

m2eLR=

12

m2e1m2e2

sin 2e

m2e1

m2e2

e

UeMeUdagger

e = diag(m2

e1

me2

)

Ue =

cose sine sine cose

cf) Stauの質量量行行列列

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Stau 76

Stau contribution to κγ

[Endo Kitahara TY ʻlsquo13]

真空の安定性条件からκγの大きさは制限10-‐‑‒15が取りうるexcessの最大値

Large mixing angle真空の安定性条件で制限

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

emsp　Stau 77

Stau mass region

[Endo Kitahara TY ʻlsquo13]

δκγgt4ならばemsp　emsp　emsp　250GeVに制限radics =500GeV ILCまでで発見見可能

me1lt

Higgs coupling to diphotonsin 2e = 1

真空の安定性条件を満たすなかで最大の値を選択tan = 20

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

l  仮定

l  Sample point

l  Reconstruction (ILC at radics = 500GeV)

emsp　Stau 78

Stau searches (Reconstruction)

-‐‑‒  Stau12 mixing angle全てがILC (radics = 500GeV)で観測-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

88 Dissertation Takahiro Yoshinaga

Table 43 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 100 GeV 230 GeV 092 90 GeV 36

43 Stau

So far we studied the prospects for the selectionsmuon searches In this section we discussthe stau searches If the staus have masses of (100)GeV and large left-right mixing theHiggs coupling to the di-photon κγ is deviated from the SM prediction Such a large left-rightmixing is constrained by the vacuum meta-stability condition of the stau-Higgs potential asmentioned in Sec 413 Then we discussed that once the deviation from the SM prediction ofκγ were observed the mass region for the staus were determined by the condition in Sec 414

Once the stau is discovered at ILC its properties including the mass are determined Par-ticularly it is important to measure the mixing angle of the stau θτ When sin2θτ is sizablethe angle can be measured at ILC [124 160ndash162] As observed in Fig 48 it is likely to besizable to enhance κγ In particular if sin2θτ is large enough to be measurable the heavieststau is likely to be light Thus it may be possible to discover the heaviest stau and measureits mass at ILC Then the stau contribution to κγ can be reconstructed by using the measuredmasses and mixing angle This provides a direct test whether the stau contribution is the ori-gin of the deviation of κγ On the other hand the heaviest stau is not always discovered at theearly stage of ILC even if the stau mixing angle is measured If θτ as well as mτ1

is measuredmτ2

may be estimated in order to explain the excess of κγ In this section we will study thereconstruction of the stau contribution to κγ The mass of the heaviest stau and theoreticaluncertainties will also be discussed

431 Reconstruction

If both of τ1 and τ2 are measured the stau contribution to κγ can be reconstructed Thecontribution is determined by the parameters in Eq (422) In this subsection we assume thesituation that both of τ1 and τ2 are observed by ILC Then we discuss how and how accuratelythe model parameters are measured at ILC and consequently the stau contribution to κγ isreconstructed

Let us first specify a model point to quantitatively study the accuracies In Tab 43 thestau masses the stau mixing angle and the Bino (-like neutralize) mass are shown Thepoint is not so far away from the SPS1arsquo benchmark point [163] where ILC measurementshave been studied (see eg Ref [160]) The stau mixing angle is chosen to enhance theHiggs coupling as δκγ = 36 The staus masses are within the kinematical reach of ILCat

s = 500GeV The point is consistent with the vacuum meta-stability condition and thecurrent bounds from LHC and LEP The most tight bound on the stau mass has been obtained

[Endo Kitahara TY ʻlsquo13]

me1 01 GeV me2

6 GeV sin 2esin 2e

2

05

excessがStau起源かcheck可能

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV

l  仮定

l  Sample point (stau2はradics = 500GeVでは未発見見）

l  Prediction for stau2 (ILC at radics = 1TeV)

emsp　Stau 79

Stau searches (Prediction)

-‐‑‒  Stau1 mixing angleがILC (radics = 500GeV)で観測stau2は未発見見-‐‑‒  κγは数のexcessが観測測定精度度は2と仮定

[Endo Kitahara TY ʻlsquo13]

1

Table 1 Model parameters at our sample point In addition tanβ = 5 and Aτ = 0 are setthough the results are almost independent of them

Parameters mτ1mτ2

sin2θτ mχ01

δκγValues 150 GeV 400 GeV 091 140 GeV 56

me1 01 GeV

sin2esin 2e

25 2

radics = 1TeV ILCで発見見が期待ビームエネルギーを調節するためのヒントを与える

me2= 400plusmn 53 GeV