Niels Tuning (1) “Elementary Particles” Lecture 7 Niels Tuning Harry van der Graaf.
-
Upload
poppy-hoover -
Category
Documents
-
view
223 -
download
4
Transcript of Niels Tuning (1) “Elementary Particles” Lecture 7 Niels Tuning Harry van der Graaf.
Niels Tuning (1)
“Elementary Particles”Lecture 7
Niels Tuning
Harry van der Graaf
Thanks
• Ik ben schatplichtig aan:– Dr. Ivo van Vulpen (UvA)
– Prof. dr. ir. Bob van Eijk (UT)
– Prof. dr. Marcel Merk (VU)
Niels Tuning (2)
Homework
Exercises Lecture 6:
Niels Tuning (4)
• To be handed in next week
Plan
1) Intro: Standard Model & Relativity
2) Basis
1) Atom model, strong and weak force
2) Scattering theory
3) Hadrons
1) Isospin, strangeness
2) Quark model, GIM
4) Standard Model
1) QED
2) Parity, neutrinos, weak inteaction
3) QCD
6) e+e- and DIS
7) Higgs and CKM
Niels Tuning (5)
1900-1940
1945-1965
1965-1975
1975-2000
2000-2015
17 Feb
3 Mar
17 Mar
19 May
26 May
10 Feb
21 Apr
1) Higgs mechanism
2) Higgs discovery at ATLAS
3) CKM-mechanism
4) CP violations at LHCb
Outline for today:
Niels Tuning (6)
Summary Lects. 1-5
• Theory of relativity– Lorentz transformations (“boost”)
– Calculate energy in collissions
• 4-vector calculus
• High energies needed to make (new) particles
Lecture 1: Relativity
Niels Tuning (8)
• 4-vectors: – Space-time x
– Energie-momentum p
– 4-potential A
– Derivative ∂
– Covariant derivative D
– Gamma matrices
• Tensors– Metric g
– Electromagnetic tensor F
Lecture 1: 4-vector examples
Niels Tuning (9)
• Schrödinger equation– Time-dependence of wave function
• Klein-Gordon equation– Relativistic equation of motion of scalar particles
Dirac equation– Relativistically correct, and linear
– Equation of motion for spin-1/2 particles
– Described by 4-component spinors
– Prediction of anti-matter
Lecture 2: Quantum Mechanics & Scattering
Niels Tuning (10)
• Scattering Theory– (Relative) probability for certain process to happen
– Cross section
• Fermi’s Golden Rule
– Decay: “decay width” Γ
– Scattering: “cross section” σ
Lecture 2: Quantum Mechanics & Scattering
Niels Tuning (11)
ClassicClassic Scattering amplitude Scattering amplitude inQuantum Field Theory
a → b + c
a + b → c + d
• “Partice Zoo” not elegant
• Hadrons consist of quarks
Observed symmetries– Same mass of hadrons: isospin
– Slow decay of K, Λ: strangeness
– Fermi-Dirac statistics Δ++, Ω: color
• Combining/decaying particles with (iso)spin– Clebsch-Gordan coefficients
Lecture 3: Quarkmodel & Isospin
Niels Tuning (12)
• Arbitrary “gauge” Physics invariant
Introduce “gauge” fields in derivative
Interactions!
• QED
• Weak interactions
• QCD
Lecture 4: Gauge symmetry and Interactions
Niels Tuning (13)
2
exp8,1
aaa x
i
1 photon
3 weak bosons
8 gluons
EM coupling α
Strong coupling αS
Confinement
Asymptotic freedom
Lecture 5: Running couplings
Niels Tuning (14)
Feynman rules: Example
Niels Tuning (15)
• Process: e-μ- →μ- e-
Remember the 4-componentspinors in Dirac-space:
• Local SU(3) gauge transformation
• Introduce 8 Aμa gauge fields
• Non-“Abelian” theory,
• Self-interacting gluons– Gluons have (color) charge
• Different “running”
• Local U(1) gauge transformation
• Introduce 1 Aμ gauge field
• “Abelian” theory,
• No self-interacting photon– Photons do not have (electric) charge
• Different “running”
QED QCDQED QCD
Niels Tuning (16)
2
exp8,1
aaa x
i
QED and QCD
• e+e- scattering: QED at work: RR
– e+e-→μ+μ-
– e+e-→cc
– e+e-→qq g
– e+e-→Z
– e+e-→WW
• e+p scattering: QCD at work: FF22
– Quarkmodel: do quarks exist??
– Substructure
– Bjorken-x, sum rules
– Scaling
– ‘Parton density functions’ (pdf) and ‘structure functions’
– Scaling violations: more quarks at higher Q2 due to QCD
Lecture 5: e+e- scattering and DIS
Niels Tuning (17)
Standard Model
Niels Tuning (18)
Todo-list:
• No masses for W, Z !?– (LHC/ATLAS) Higgs mechanism, Yukawa couplings
• Interactions between the three families !?– (LHC/LHCb) CKM-mechanism, CP violation
FFmDiψ
4
1 L
Half-way there?!
Prof.dr. J. Ellis
Higgs mechanism
• Let’s give the photon a mass!
• Introduce a complex scalar field:
– with:
– and the Lagrangian is invariant under:
Higgs mechanism
Niels Tuning (21)
If 2 > 0:
• φ will acquire a vaccum expectation value v,
•“spontaneously” !
•System not any more
“spherical” symmetric
Scalar potential V(φ)
Niels Tuning (22)
:02 :02
Spontaneous Symmetry Breaking
If 2 > 0:
• φ will acquire a vaccum expectation value v
•Parameterize φ as:– h: Higgs boson
: Goldstone boson
– Both real scalar fields
Complex scalar field φ
Niels Tuning (23)
:02 vie
hv /
2
h
• Let’s give the photon a mass!
• Introduce a complex scalar field:
• with:
• and:
Then:
Higgs mechanism
Niels Tuning (24)
viehv /
2
int. ),(2
12
2
1
24
1 2222
hhhhAA
veevAFF L
Higgs mechanism
Niels Tuning (25)
int. ),(2
12
2
1
24
1 2222
hhhhAA
veevAFF L
Photon A with mass e2v2 Interactions Photon field Aμ
e2v2 22
Higgs h with mass 22
Higgs mechanism
Niels Tuning (26)
int. ),(2
12
2
1
24
1 2222
hhhhAA
veevAFF L
Photon A with mass e2v2 Interactions Photon field Aμ
e2v2 22
Higgs h with mass 22
• What about this field ?
• Unitary gauge:
Goldstone boson has been “eaten” by the photon mass
Higgs mechanism
(27)
int. ),(2
12
2
1
24
1 2222
hhhhAA
veevAFF L
Interactions Photon field AμPhoton A with mass e2v2
e2v2 22
Higgs h with mass 22
• Unitary gauge:
• Degrees of freedom– Before: massless photon: 2, complex scalar field φ: 2 Total: 4
– After: massive photon: 3, one real scalar field h: 1 Total: 4
Goldstone boson has been “eaten” by the photon mass
Higgs mechanism
(28)
int. ),(2
12
2
1
24
1 2222
hhhhAA
veevAFF L
Interactions Photon field AμPhoton A with mass e2v2
e2v2 22
Higgs h with mass 22
• Let’s give the W,Z a mass!
• Introduce a doublet of complex scalar fields:
Higgs mechanism in the Standard Model
Niels Tuning (29)
Spontaneous symmetry breaking
Niels Tuning (30)
Mass terms!
• How about the physical fields?
1) W1, W2 :
2) W3, B:
Let’s do a ‘trick’ and ‘rotate’ the W3 and B fields to get the Z and A fields
Rewriting in terms of physical gauge bosons
Niels Tuning (31)
1) W1, W2 :
2) W3, B:
Rewriting in terms of physical gauge bosons
Niels Tuning (32)
1) W1, W2 :
2) W3, B:
Rewriting in terms of physical gauge bosons
Niels Tuning (33)
1) W1, W2 :
2) W3, B:
Weak mixing angle (or Weinberg angle): W
Niels Tuning (34)
boson)-(Z sincos1
(photon) cossin1
33
22
33
22
ZBWBggWgg
ABWgBWggg
WW
WW
Rewriting in terms of physical gauge bosons
1) W1, W2 :
2) W3, B:
Rewriting in terms of physical gauge bosons
Niels Tuning (35)
Spontaneous symmetry breaking
Niels Tuning (36)
Mass terms!
• How about the physical fields?
(Keep the vacuum neutral)
Physical fields: Mass term Mass
Spontaneous symmetry breaking
Niels Tuning (37)
1) Introduce doublet of scalar fields:
2) With potential:
3) S.S.B.:
4) Mass terms for gauge fields:
Summary:
Niels Tuning (38)
• Photon couples to e:
• Prediction for ratio of masses:
• Veltman parameter:
• Higgs mass:
Boson masses?
Niels Tuning (39)
boson)-(Z sincos
(photon) cossin3
3
BWZ
BWA
WW
WW
FGv 2/1
• Add ad-hoc (!?) term to Lagrangian:
Fermion masses?
Niels Tuning (40)
Done
Prof.dr. J. Ellis
Let’s tackle the Yukawa couplings
Prof.dr. J. Ellis
First:Higgs discovery
ATLAS
LHCb
ALICECMS
Normal
How are discoveries made?
New ?
?
muon
muon
muon
muon
Higgs ZZ 4 leptonssmall number of beautiful eventsHiggs ZZ 4 leptonssmall number of beautiful events
120.000 Higgs bosons
with Higgs 68 events
‘other’ 52 events
Only 1 in 1000 Higgs bosons decays to 4 leptons
50% chance that ATLAS detector finds them
60 (Higgs 4 lepton) events
higgs
Z
Z
HZZ l+l-l+l-
l+
l-
l-
l-
peak !?
peak !?
Higgs 2 photons
Hγγ decay
higgs
photon
photon
Interpretation of excessInterpretation of excess
Probability of observing excess smaller than 1 in 1 milion
Claim discovery if:
Throwing 8 times 6 in a row
Discovery in slow-motion
dec/11
jul/11
dec/11
jul/11mar/12 dec/
11
jul/11mar/12
jul/12
dec/11
jul/11mar/12
aug/12
jul/12
Time-line higgs discovery
Discovery of Higgs particle on July 4, 2012
What is mass ?? Anno 1687
Mass is de ‘exchange rate’ between force and acceleration:
Does not describe what mass is ...
F = m x a
Newton
What is mass ?? Anno 1905
Mass is energy
Describes what mass is !
But not where it comes from …
E = m x c2
Einstein
What is mass ?? Anno 1964
Mass of elementary particles is due to
“friction” of ubiquitous ‘Higgs field’
Higgs
m: ψψH
What is mass ?? Anno 1964
Mass of elementary particles is due to
“friction” of ubiquitous ‘Higgs field’
m: ψψH
Next: Higgs’ properties as expected?Next: Higgs’ properties as expected?
Higgs boson mass (GeV)
mh=125 GeV
prediction measurement
Standard Model
Philosophy?
Higgs: Particle? Field?
Particle Particle FieldField
e+ e-
Photon (light particle)Photon (light particle) Electrical fieldElectrical field
Why is the Higgs particle so special?
Apparent mass
Particle
Higgs field
H production
higgs
proton
proton
As if the fish discovered the water he’s As if the fish discovered the water he’s in…in…
Particle Particle FieldField
What is mass?
Mass of elementary particles is due to
“friction” of ubiquitous ‘Higgs field’
Revolutionary – with spectaculair consequences : Revolutionary – with spectaculair consequences :
space is not empty, but filled with sort of ‘ether’space is not empty, but filled with sort of ‘ether’
Elementary particlein empty space:no rest-energy=no mass
Elementary particlein Higgs field:rest energy = interaction with Higgs field= mass!
Einstein: proton mass = binding energy
Another field: the Big BangAnother field: the Big Bang
One of Higgs’ properties match that of another field…
The inflaton that inflated the Universe between 10-33 and 10-32 seconds after the Big Bang
Another field: the Big BangAnother field: the Big Bang
Een ander veld: de Big BangEen ander veld: de Big Bang
Couplings across generations:
CKM
The Standard Model Lagrangian
Introduce the massless fermion fieldsRequire local gauge invariance existence of gauge bosons
Introduce Higgs potential with <φ> ≠ 0Spontaneous symmetry breaking
Ad hoc interactions between Higgs field & fermions
(3) (2) (1) (3) (1)SM C L Y C QG SU SU U SU U
W+, W-,Z0 bosons acquire a mass
Ivo van Vulpen (64)
Fields: Notation
Quarks:
Under SU2:Left handed doubletsRight handed singlets
5 51 1;
2 2L R
Fermions: with = QL, uR, dR, LL, lR, R
Ivo van Vulpen (65)
L i
Leptons:
Scalar field: Interaction representation: standard model
L i
SU(3)C SU(2)LHypercharge Y
Left-handedgeneration index
Interaction rep.
Fields: explicitly
Explicitly:
3
3
( )1 61 2
1 2
, , , , , ,(3,2,1 6) , ,
, , , , , ,
I I Ir r r
I Ig gI
Li
L L L
I I
I I Ib b b
I I Ib b b
I I Ir r
g
g g grI
IT
YT
u c t
d s b
u c t
d
c t
d sQ
bb
u
s
• The left handed quark doublet :
Ivo van Vulpen (66)
3
3
1 21 2
1 2(1,2, 1 2) , ,
II IeI
Li I IIL LL
TL Y
Te
• Similarly for the quark singlets:
2 3
1 3
(3,1, 2 3) , , , , , , , ,
(3,1, 1 3) , , , , , , , ,
IRi R R R
IR
I I Ir r r
I I Ir
I I Ir r r
I I Ir r
I I Ir r r
I I Ii Rr r r r Rr R r
Y
Y
t
d s b
u c tu
d d s b
cuu c t
d s b
• And similarly the (charged) singlets: (1,1, 1) , , 1I I I IRi R R Rl e Y
• The left handed leptons:
: Fermions + gauge bosons + interactions
Procedure: Introduce the fermion fields and demand that the theory is local gauge invariant under SU(3)CxSU(2)LxU(1)Y transformations.
Start with the Dirac Lagrangian:
Replace:
Fields:
Generators:
Ga : 8 gluons
Wb: weak bosons: W1, W2, W3
B: : hypercharge boson
La : Gell-Mann matrices: ½ a (3x3) SU(3)C
σb : Pauli Matrices: ½ b (2x2) SU(2)L
Y : Hypercharge: U(1)Y
:The Kinetic Part
For the remainder we only consider Electroweak: SU(2)L x U(1)YIvo van Vulpen (67)
: The Kinetic Part
1 1 2 2 3 3( , ) ,2
...2 2
IIWeak I
kinetic L LL
I I I I I I I IL L L L L L L L
uiu d i u d g W W W
d
g giu u id d u W d d W u
L
: ( ) ( )
, , , ,
kinetic
I I I I ILi Ri Ri Li Ri
i i D
with Q u d L l
L
Example: the term with QLiI becomes:
( )2 2
(
6
)I I Ikinetic Li Li Li
I ILi b La a b is
i i ig G gW g
Q iQ D Q
iQ B Q
L
Writing out only the weak part for the quarks:
dLI
gW+
uLI
L=JW
1
2
3
0 1
1 0
0
0
1 0
0 1
i
i
Ivo van Vulpen (68)
: The Higgs Potential
V()
Symmetry
Spontaneous Symmetry Breaking: The Higgs field adopts a non-zero vacuum expectation value
Procedure: Substitute:
And rewrite the Lagrangian (tedious):
V
1. . 2. The W+,W-,Z0 bosons acquire mass3. The Higgs boson H appears
: (3) (2) (1) (3) (1)SM C L Y C EMG SU SU U SU U
Ivo van Vulpen (69)
Broken symmetry
0v
= =
: The Yukawa Part
Since we have a Higgs field we can add (ad-hoc) interactions between Higgs field and the fermions in a gauge invariant way
. .LiYukawa ij Rj h cY L
. .I I I I I ILi Rj L
d u lij i Rj Liij j RjiY Y YQ d Q u L l h c
The result is:
are arbitrary complex matrices which operate in family space (3x3) flavour physics
doubletssinglet
0* *
2
0 1
1 0i
With:
(The CP conjugate of φ)
i, j : indices for the 3 generations!
Ivo van Vulpen (70)
Yijd Yij
u Yijl
~
: The Yukawa Part
0 0 0
0 0 0
0
11 12 13
21 22 13
31 32 0 33 0
, , ,
, , ,
, , ,
I I I I I IL L L L L L
I I I I I IL L L L L L
I
d d d
d
I I I I
d d
d IL L L L
dL L
d
u d u d u d
c s c s
Y Y Y
Y Y Y
Y
c
Y Y
s
t b t b t b
IR
IR
IR
d
s
b
Writing the first term explicitly:
0( , )I I
L LIR
dij jiY u dd
Ivo van Vulpen (71)
: The Fermion Masses
0
( , ) ... ...I I IYuk
d u lij ij jL Rj iL iu Yd YdY
L
Start with the Yukawa Lagrangian
After which the following mass term emerges:
, with
Ivo van Vulpen (72)
SSB
Spontaneous symmetry breaking
: The Fermion Masses
., , , ,, , .
I I
I I I I I I I I
L LI
I
I I
I
R
I
L
R
I
R
I
ed u
s u c t c e
b t
hs cd bMassd u lM M M
L
†f f fdiagonaL R l
f MV M V
Writing in an explicit form:
The matrices M can always be diagonalised by unitary matrices VLf and VR
f such that:
Then the real fermion mass eigenstates are given by:
dLI , uL
I , lLI are the weak interaction eigenstates
dL , uL , lL are the mass eigenstates (“physical particles”)
I ILi Lj Ri Rj
I ILi Lj Ri Rj
I ILi Lj R
d dL Rij ij
u uL Rij ij
l lL R Rjiiij j
d d d d
u u u
V V
V V
V V
u
l l l l
† †, ,
I
I I I I
LI
f f f fL R
f
R
L RV V
d
d s b sV
b
VM
Ivo van Vulpen (73)
SSB
: The Fermion Masses
,
., , .
, , ,L
d u
sL
R R
c
e
R
L
b t
Mass
m m
m m
h
m m
m
m
d s b
d u
s u c t c
b t
e
e
m
c
L
In terms of the mass eigenstates:
Mass u c t
d s b
e
uu cc tt
dd ss bb
m m m
m m m
m ee m m
L
In flavour space one can choose:
Weak basis: The gauge currents are diagonal in flavour space, but the flavour mass matrices are non-diagonal
Mass basis: The fermion masses are diagonal, but some gauge currents (charged weak interactions) are not diagonal in flavour space
What happened to the charged current interactions (in LKinetic) ? Ivo van Vulpen (74)
SSB
: The Charged Current
The charged current interaction for quarks in the interaction basis is:
The charged current interaction for quarks in the mass basis is:
, ,2
CKMLW
L
d
u c t V s
b
gW
L
†
2u
L L LW
dLi iu V
gV d W
L
The unitary matrix: †u dCKM L LV V V
is the Cabibbo Kobayashi Maskawa mixing matrix:
† 1CKM CKMV V
2I ILi LW i
gWu d
L
with:
Lepton sector: similarly †lMNS L LV V V
However, for massless neutrino’s: VL = arbitrary.
Choose it such that VMNS = 1 no mixing in the lepton sectorIvo van Vulpen (75)
Charged Currents
†
*
5 5 5 5
5 5
2 2
1 1 1 1
2 2 2 22 2
1 12 2
I I I ICC Li Li Li Li CC
ij ji
ij i
CC
i j j i
i j j ij
g gu W d d W u J W J W
g gu W d d W uV V
Vg g
u W d d W uV
L
The charged current term reads:
Ivo van Vulpen (76)
How do you measure those numbers?
• Magnitudes are typically determined from ratio of decay rates
• Example 1 – Measurement of Vud
– Compare decay rates of neutrondecay and muon decay
– Ratio proportional to Vud2
– |Vud| = 0.9735 ± 0.0008
– Vud of order 1
Ivo van Vulpen (77)
What do we know about the CKM matrix?
• Magnitudes of elements have been measured over time– Result of a large number of measurements and calculations
'
'
'
ud us ub
cd cs cb
td ts tb
d V V V d
s V V V s
b V V V b
0.9738 0.0002 0.227 0.001 0.00396 0.00009
0.227 0.001 0.9730 0.0002 0.0422 0.0005
0.0081 0.0005 0.0416 0.0005 0.99910 0.00004
ud us ub
cd cs cb
td ts tb
V V V
V V V
V V V
4 parameters•3 real•1 phase
Magnitude of elements shown only, no information of phase
Ivo van Vulpen (78)
Approximately diagonal form
• Values are strongly ranked: – Transition within generation favored
– Transition from 1st to 2nd generation suppressed by sin(c)
– Transition from 2nd to 3rd generation suppressed bu sin2(c)
– Transition from 1st to 3rd generation suppressed by sin3(c)
u
d
t
c
bsCKM magnitudes
3 2
2
3
=cos(c)=0.23
Why the ranking?We don’t know (yet)!
If you figure this out,you will win the nobelprize
Ivo van Vulpen (79)
LHCb experiment: study the B particle
1)1) Find differences between matter and anti-matterFind differences between matter and anti-matter
2)2) Find new particlesFind new particles
b
s
s
b
b s
μ
μ
23 sep 2010 19:49:24Run 79646 Event 143858637
The LHCb DetectorThe LHCb Detector
LHCb experiment: study the B particle
1)1) Find differences between matter and anti-matterFind differences between matter and anti-matter
CP violationCP violation
b
s
s
b
Final remarks: How about the leptons?
• We now know that neutrinos also have flavour oscillations– Neutrinos have mass
– Diagonalizing Ylij doesn’t come for free any longer
• thus there is the equivalent of a CKM matrix for them:– Pontecorvo-Maki-Nakagawa-Sakata matrix
Niels Tuning (83)
vs
Final remarks : How about the leptons?
• the equivalent of the CKM matrix– Pontecorvo-Maki-Nakagawa-Sakata matrix
• a completely different hierarchy!
Niels Tuning (84)
vs
Final remarks: How about the leptons?
• the equivalent of the CKM matrix– Pontecorvo-Maki-Nakagawa-Sakata matrix
• a completely different hierarchy!
Niels Tuning (85)
vs
ν1 ν2 ν3 d s b
See eg. PhD thesis R. de Adelhart Toorop
What’s going on??
• ??? Edward Witten, 17 Feb 2009…
• See “From F-Theory GUT’s to the LHC” by Heckman and Vafa (arXiv:0809.3452)
• Is 125 GeV coincidental?
Kabbalah!
Niels Tuning (87)
David d’Enterriahttp://arxiv.org/pdf/1208.1993v1.pdf
• More serious stuff!
Kabbalah?
Niels Tuning (88)
Shaposhnikov et alhttp://arxiv.org/pdf/1205.2893.pdf
Enough to wonder about…
• Couplings of Higgs to fermions, bosons?
• Why different masses?
• Relation between masses and W-couplings?
• Quark couplings and lepton couplings so different?
End
Niels Tuning (89)