Perturbation Theory of Neutrino Oscillation with and without...

Perturbation Theory of Neutrino Oscillation with and without NSI �

Hisakazu Minakata Tokyo Metropolitan

University

Part I: Comments on Kamioka-Korea 2 detector setting �

•  Search for CP violation and ν mass hierarchy; discussion supplementary to Suzuki-san’s and Volpe-san’s talks �

September 20, 2009 Erice School 09: Neutrinos


Kamioka-Korea 2 detector setting

Why don’t you bring one of the 2 tanks to Korea? (@EPP2010)


Original idea: sensitive because dynamism in 2nd oscillation maximum


Two detector method is powerful


Spectral information solves intrinsic degeneracy

from 1000 page Ishitsuka file

SK momentum resolution ~30 MeV at 1 GeV

T2K T2KK

2 detector method powerful!

Ishitsuka-Kajita-HM-Nunokawa 05 �


Two-detector setting is powerful

•  With the same input parameter and Korean detector of 0.54 Mt the sign-Δm2 degeneracy is NOT completely resolved

T2KK Korea only


10 -2

10 -1

0 1 2 3 4 5 6

normal

10 -2

10 -1

0 1 2 3 4 5 6

inverted

s

in 2

2

13

10 -2

10 -1

0 1 2 3 4 5 6

normal

Kamioka 0.54 Mton

Kamioka 0.27 Mton + Korea 0.27 Mton

10 -2

10 -1

0 1 2 3 4 5 6

inverted

s

in 2

2

13

Total mass of the detectors = 0.54 Mton fid. mass 4 years neutrino beam + 4 years anti-neutrino beam

3 σ (thick) 2 σ (thin)

Mass hierarchy CP violation (sinδ≠0)

hep-ph/0504026

T2K�

T2KK�


Relative cross section error does matter

•  Identical 2 detector setting robust to larger systematic error •  It gives conservative lower bounds on sensitivity estimate

of mass hierarchy and CP

Barger et al. 07

T2K II

T2KK


T2KK can solve θ23 degeneracy in situ T2K-II + phase II reactor T2KK

δ=0 assumed si

n2 2θ 1

3

sin2 θ23

sin2

2θ 1

3

> 3σ 2~3σ

T2KK 2σ (rough)

T2KK has better sensitivity at sin2 2θ13 < 0.06~0.07 .

hep-ph/0601258

Conclusion: part I �

•  Kamioka-Korea 2 detector setting = reasonable thing to think about for sensitivities both CPV and mass hierarchy

•  Sensitivity to CPV comparable between in T2K and T2KK staged approach possible



Part II: This talk is originally meant to

be…

Perturbation Theory of Neutrino Oscillation with and without NSI �

•  This is meant to be a pedagogical talk for students

•  Nothing terribly exciting, but serves for better understanding of ν oscillations, a useful tool for understanding lepton flavor mixing �


General feature (standard interaction only) �


General feature (continued) �


Our goal �

Because Hamiltonian has this form, it is natural to define the tilde basis (next page) �

Tilde basis �


Akhmedov et al. 04�


Perturbation theory

of ν oscillation

Perturbation theory of ν oscillation �

•  No general perturbative scheme for ν oscillation many dimensionless parameters: Δm2

solar/Δm2atm, Δm2L/E, a/Δm2

•  Only known small parameter: Δm2solar/Δm2

atm ~ 0.03

•  Need to choose the right framework suitable for your problem (case by case)

•  I explain you only a particular example

September 20, 2009 Erice School 09: Neutrinos I work mostly for LBL experiments�

ε perturbation theory �

•  I take the assumption

•  I assume a/Δm2atm~O(1), Δm2

atm E/L~O(1) •  most natural perturbation theory of ν

oscillation Peµ consists only of order ε2 terms widely used formula (Cervera et al. 00)


Are you happy with such small θ13? �

Perturbation theory in tilde basis �


``Interaction representation’’ �

zeroth-order �1st-order �

We follow standard path �


Seµ and Peµ �


Cervera et al. 00�

Constant matter density approximation �

Matter hesitation �


Matter effect enters into Pαβ at second order in ε => no first order a-dependent term �

Highly nontrivial, in particular for Pee, but in fact very easy to prove !�

“matter hesitation” indicates that detecting the matter effect is not easy in LBL experiments �

order ~ ε2 �Kikuchi-HM-Uchinami JHEP09 �

Magic baseline= solar amplitude zero �


At magic baseline, = 0, the solar amplitude vanishes �

which occurs at L ~ 7200 km �

Then there is no δ dependence in Peµ �


Analytic treatment of degeneracy can be done with 2nd order formula

•  You can draw two ellipses from a point in P-Pbar space

•  Intrinsic degeneracy

•  doubled by the unknown sign of Δm2

•  4-fold degeneracy


ν oscillation with NSI

New physics at TeV scale? �

•  Waiting for LHC for new physics at TeV scale

•  <φ> = 250 GeV •  => E~TeV as a

new physics scale •  •  Extra dimensions •  Supersymmetry •  …. September 20, 2009 Erice School 09: Neutrinos


ν's Non-standard interactions (NSI) •  If there exists New Physics at TeV scale there

might be NSI of ν’s expressed by higher dimensional operators

•  But coefficient ε may be small (MNP = 1 TeV)

•  if dimension 6, ε~ (MW/MNP)2 = 0.01 •  if dimension 8, ε~ (MW/MNP)4 = 0.0001

Wolfenstein, Grossman, Berezhiani-Rossi, Davidson et al. … many people


Non Standard Interaction (NSI)

Grossmann, Ota-Sato-Yamashita, Huber et al. …

I this talk, I focus on effects of NSI in ν propagation in matter governed by evolution eq. => vector type interactions

•  It comes in into 3 places, production, propagation, & detection

ν oscillation with NSI �•  Given the structure

•  the system is highly nontrivial ! •  all εαβ may be related, but likely to be

present with similar magnitudes •  At the moment nobody knows how these

elements are related with each other •  No stringent constraints on particular εαβ

September 20, 2009 Erice School 09: Neutrinos include all of them �

A general theorem of phase reduction �


For Δm221=0, or at the magic baseline, number

of CP violating phases decreases by 1 �

natural because of effective 2 generations�

Kikuchi-HM-Uchinami JHEP09 �


Perturbation theory with

NSI

ε perturbation theory with NSI �•  We include NSI by assuming

•  and expand to ε2; NSI 2nd order formula •  If dimension 8, 1st order formula would be

enough but, 2nd order formula is simpler !


NSI 2nd order formula for Peµ �

SI formula -> NSI 2nd order formula �

c12s12 (Δm221 / a)

s13 e-iδ (Δm231/a)

c12s12 (Δm221 / a)

c12s12 (Δm221 / a) s13 e-iδ (Δm2

31/a)

s13 e-iδ (Δm231/a)

Cervera et al , hep-ph/0002108 September 20, 2009 Erice School 09: Neutrinos

SI formula -> NSI 2nd order formula �

September 20, 2009 Erice School 09: Neutrinos T. Kikuchi, H.M., S. Uchinami, arXiv:0809.3312 => JHEP �

Generalized atmospheric and solar variables �

“θ13” “θ12”

SI-NSI confusion ! �


Secret is in tilde hamiltonian !�

Invariance of tilde hamiltonian �


Seτ = Seµ (extended transf.) �

tilde-H is invariant under the extended transf. �

Undoing … �

How big is the contribution of εαβ? Bird-eye view

•  decoupling of εµτ and εµµ to P(νe -> νµ) to ε2

•  Determine εeµ and εeτ from P(νe -> νµ) •  εµτ and εµµ dependent term is common to all P(νµ -> ντ), P(νµ -> νµ) and P(ντ -> ντ)


If θ23 is maximal Matter hesitation Direct transition by NSI

Characteristics �

spectrum analysis required �

NSI perturbation theory; temporary summary �

•  Structure of ν oscillation with NSI is in fact very simple within ε perturbation theory

•  But in real life, it is complicated even with single εαβ because the system is enriched with 2 phases lepton KM phase + NSI phase requires hard work �


ν oscillation only with

single type of NSI εαβ is already a

complicated phenomenon;

(example with εeµ) �

September 20, 2009

0 1 2 3 4 5 6 7 8

P(!e

!!"#"$x10

-4]

0

1

2

3

4

5

6

7

8

P(!

e!!"#"$x

10

-4]

sin22"

%&"'"%(

#&

E = 20 GeV, |$e!)"'"(*((+

% : varied continuasly

&e!

: varied discontinuously

Erice School 09: Neutrinos

2 phase confusion with εeτ; synergy �


Gago-HM-Nunokawa-Uchinami-Zukanovich Funchal, arXiv:0904.3360 �

Degeneracy; εeτ�



Conclusion •  General relations between S matrix exist

because of the special structure of matter effect

(both with and without NSI) •  Qualitative understanding of neutrino

oscillation can be achieved by perturbative formulation

•  SI only: neat Peµ and Peτ 2nd-order formula, matter hesitation,

•  With NSI: generalized solar and atmospheric variables, εαβ dependence of Peµ, Peτ, Pµτ, etc.

diag(a, 0, 0)�

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