Particle Physics: The Standard ModelThe Electroweak Theory Particle Physics: The Standard Model Dirk...
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The Electroweak Theory Particle Physics: The Standard Model Dirk Zerwas LAL [email protected] April 11 and April 18, 2013 Dirk Zerwas Particle Physics: The Standard Model
Transcript of Particle Physics: The Standard ModelThe Electroweak Theory Particle Physics: The Standard Model Dirk...
The Electroweak Theory
Particle Physics: The Standard Model
Dirk Zerwas
April 11 and April 18, 2013
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
charged leptons andphoton
quarks and gluon
neutrinos
W±, Z◦
H
(
uL
dL
) (
cL
sL
) (
tLbL
)
(
νeL
eL
) (
νµL
µL
) (
ντL
τL
)
uR cR tRdR sR bR
eR µR τR
γ
gW±, Z◦
H
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
charged leptons andphoton
quarks and gluon
neutrinos
W±, Z◦
H
(
uL
dL
) (
cL
sL
) (
tLbL
)
(
νeL
eL
) (
νµL
µL
) (
ντL
τL
)
uR cR tRdR sR bR
eR µR τR
γ
gW±, Z◦
H
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
charged leptons andphoton
quarks and gluon
neutrinos
W±, Z◦
H
(
uL
dL
) (
cL
sL
) (
tLbL
)
(
νeL
eL
) (
νµL
µL
) (
ντL
τL
)
uR cR tRdR sR bR
eR µR τR
γ
gW±, Z◦
H
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
charged leptons andphoton
quarks and gluon
neutrinos
W±, Z◦
H
(
uL
dL
) (
cL
sL
) (
tLbL
)
(
νeL
eL
) (
νµL
µL
) (
ντL
τL
)
uR cR tRdR sR bR
eR µR τR
γ
gW±, Z◦
H
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
History
1896 Henri Becquerel: β decay
1899 Ernest Rutherford: distinguishes α and β rays
1914 James Chadwick: the β decay has a continuousspectrum
1930 Wolfgang Pauli: postulates the neutrino (ballroom)
1933 Enrico Fermi: contact interaction
1953 Frederick Reines: νeL + p → n + e+
1956 Lee, Yang, Wu, Garwin et al: Parity violation
1961 Glashow, Salam, Weinberg, Higgs, EBKGH
1973 Lagarrigue, Faissner: neutral currents (Z◦ t-channel)
1984 Rubbia, van der Meer : W±, Z◦
2012 discovery of the Higgs boson
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
n → p + e− + νeL
Tfi ∼ G(pγµn)(eγµν)
∼ G(pγµn) 1q2−m2 (eγµν)
QED: m2 = 0 → 1q2
if m2 ≫ q2 neglect q2
constant in momentumspace → Dirac function inspace-time: contactinteraction
G ∼ 10−5GeV−2 ≪ αEM
Currents
ψψ scalar Sψγµψ vector Vψσµνψ tensor Tψγµγ5ψ axial vector Aψγ5ψ pseudo scalar PS
QED: VEW: V − AV − A violates parity(experiment)V − A quark/lepton level, nothadron level
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
n → p + e− + νeL
Tfi ∼ G(pγµn)(eγµν)
∼ G(pγµn) 1q2−m2 (eγµν)
QED: m2 = 0 → 1q2
if m2 ≫ q2 neglect q2
constant in momentumspace → Dirac function inspace-time: contactinteraction
G ∼ 10−5GeV−2 ≪ αEM
Currents
ψψ scalar Sψγµψ vector Vψσµνψ tensor Tψγµγ5ψ axial vector Aψγ5ψ pseudo scalar PS
QED: VEW: V − AV − A violates parity(experiment)V − A quark/lepton level, nothadron level
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Chirality
Chirality is the handed-ness ofthe particle:
ψ = PLψ + PRψ
= ψL + ψR
Definitions
Helicity: ~σ · ~pm = 0: Helicity = chirality
m = 0: ψ and γ5ψ solveDIRAC
Weyl basis
γ5 = iγ0γ1γ2γ3
γ25 = 1
0 = γ5γµ + γµγ5
γ5 =
(
−12 00 12
)
γ0 =
(
0 12
12 0
)
γ0† = γ0
γ5† = γ5
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Chirality
Chirality is the handed-ness ofthe particle:
ψ = PLψ + PRψ
= ψL + ψR
Definitions
Helicity: ~σ · ~pm = 0: Helicity = chirality
m = 0: ψ and γ5ψ solveDIRAC
Weyl basis
γ5 = iγ0γ1γ2γ3
γ25 = 1
0 = γ5γµ + γµγ5
γ5 =
(
−12 00 12
)
γ0 =
(
0 12
12 0
)
γ0† = γ0
γ5† = γ5
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Chirality
Chirality is the handed-ness ofthe particle:
ψ = PLψ + PRψ
= ψL + ψR
Definitions
Helicity: ~σ · ~pm = 0: Helicity = chirality
m = 0: ψ and γ5ψ solveDIRAC
Weyl basis
γ5 = iγ0γ1γ2γ3
γ25 = 1
0 = γ5γµ + γµγ5
γ5 =
(
−12 00 12
)
γ0 =
(
0 12
12 0
)
γ0† = γ0
γ5† = γ5
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Left and Right chirality
PL = 12(1 − γ5)
PR = 12(1 + γ5)
PL + PR = 1P2
L = 12(1 − γ5)
12(1 − γ5)
= 14(1 − γ5)(1 − γ5)
= 14(1 − γ5 − γ5 + γ2
5)
= 14(1 − γ5 − γ5 + 1)
= 12(1 − γ5)
PLPR = 12(1 − γ5)
12(1 + γ5)
= 14(1 − γ5)(1 + γ5)
= 14(1 − γ5 + γ5 − γ2
5)= 0
PLψ = PL
(
ψL
ψR
)
= ψL
ψPL = (PLγ0ψ)†
= ψ†R
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Left and Right chirality
PL = 12(1 − γ5)
PR = 12(1 + γ5)
PL + PR = 1P2
L = 12(1 − γ5)
12(1 − γ5)
= 14(1 − γ5)(1 − γ5)
= 14(1 − γ5 − γ5 + γ2
5)
= 14(1 − γ5 − γ5 + 1)
= 12(1 − γ5)
PLPR = 12(1 − γ5)
12(1 + γ5)
= 14(1 − γ5)(1 + γ5)
= 14(1 − γ5 + γ5 − γ2
5)= 0
PLψ = PL
(
ψL
ψR
)
= ψL
ψPL = (PLγ0ψ)†
= ψ†R
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
m = 0 helicity conserved → σ · ~p good QN
particle (p) → anti-particle −p: σ → −σ
ψR right-(left)handed (anti-)particle
ψL left-(right)handed (anti-)particle
EM current
jµ = −eψγµψ
= −eψ(PL + PR)γµ(PL + PR)ψ= −eψPLγ
µPLψ − eψPRγµPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψγµPRPLψ − eψγµPLPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψPRγµPLψ − eψPLγµPRψ
Perfect symmetry under parity: ~p → −~p
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
m = 0 helicity conserved → σ · ~p good QN
particle (p) → anti-particle −p: σ → −σ
ψR right-(left)handed (anti-)particle
ψL left-(right)handed (anti-)particle
EM current
jµ = −eψγµψ
= −eψ(PL + PR)γµ(PL + PR)ψ= −eψPLγ
µPLψ − eψPRγµPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψγµPRPLψ − eψγµPLPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψPRγµPLψ − eψPLγµPRψ
Perfect symmetry under parity: ~p → −~p
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
m = 0 helicity conserved → σ · ~p good QN
particle (p) → anti-particle −p: σ → −σ
ψR right-(left)handed (anti-)particle
ψL left-(right)handed (anti-)particle
EM current
jµ = −eψγµψ
= −eψ(PL + PR)γµ(PL + PR)ψ= −eψPLγ
µPLψ − eψPRγµPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψγµPRPLψ − eψγµPLPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψPRγµPLψ − eψPLγµPRψ
Perfect symmetry under parity: ~p → −~p
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
m = 0 helicity conserved → σ · ~p good QN
particle (p) → anti-particle −p: σ → −σ
ψR right-(left)handed (anti-)particle
ψL left-(right)handed (anti-)particle
EM current
jµ = −eψγµψ
= −eψ(PL + PR)γµ(PL + PR)ψ= −eψPLγ
µPLψ − eψPRγµPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψγµPRPLψ − eψγµPLPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψPRγµPLψ − eψPLγµPRψ
Perfect symmetry under parity: ~p → −~p
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
m = 0 helicity conserved → σ · ~p good QN
particle (p) → anti-particle −p: σ → −σ
ψR right-(left)handed (anti-)particle
ψL left-(right)handed (anti-)particle
EM current
jµ = −eψγµψ
= −eψ(PL + PR)γµ(PL + PR)ψ= −eψPLγ
µPLψ − eψPRγµPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψγµPRPLψ − eψγµPLPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψPRγµPLψ − eψPLγµPRψ
Perfect symmetry under parity: ~p → −~p
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
m = 0 helicity conserved → σ · ~p good QN
particle (p) → anti-particle −p: σ → −σ
ψR right-(left)handed (anti-)particle
ψL left-(right)handed (anti-)particle
EM current
jµ = −eψγµψ
= −eψ(PL + PR)γµ(PL + PR)ψ= −eψPLγ
µPLψ − eψPRγµPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψγµPRPLψ − eψγµPLPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψPRγµPLψ − eψPLγµPRψ
Perfect symmetry under parity: ~p → −~p
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
m = 0 helicity conserved → σ · ~p good QN
particle (p) → anti-particle −p: σ → −σ
ψR right-(left)handed (anti-)particle
ψL left-(right)handed (anti-)particle
EM current
jµ = −eψγµψ
= −eψ(PL + PR)γµ(PL + PR)ψ= −eψPLγ
µPLψ − eψPRγµPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψγµPRPLψ − eψγµPLPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψPRγµPLψ − eψPLγµPRψ
Perfect symmetry under parity: ~p → −~p
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
m = 0 helicity conserved → σ · ~p good QN
particle (p) → anti-particle −p: σ → −σ
ψR right-(left)handed (anti-)particle
ψL left-(right)handed (anti-)particle
EM current
jµ = −eψγµψ
= −eψ(PL + PR)γµ(PL + PR)ψ= −eψPLγ
µPLψ − eψPRγµPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψγµPRPLψ − eψγµPLPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψPRγµPLψ − eψPLγµPRψ
Perfect symmetry under parity: ~p → −~p
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
m = 0 helicity conserved → σ · ~p good QN
particle (p) → anti-particle −p: σ → −σ
ψR right-(left)handed (anti-)particle
ψL left-(right)handed (anti-)particle
EM current
jµ = −eψγµψ
= −eψ(PL + PR)γµ(PL + PR)ψ= −eψPLγ
µPLψ − eψPRγµPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψγµPRPLψ − eψγµPLPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψPRγµPLψ − eψPLγµPRψ
Perfect symmetry under parity: ~p → −~p
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
m = 0 helicity conserved → σ · ~p good QN
particle (p) → anti-particle −p: σ → −σ
ψR right-(left)handed (anti-)particle
ψL left-(right)handed (anti-)particle
EM current
jµ = −eψγµψ
= −eψ(PL + PR)γµ(PL + PR)ψ= −eψPLγ
µPLψ − eψPRγµPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψγµPRPLψ − eψγµPLPRψ − eψPRγµPLψ − eψPLγµPRψ
= −eψPRγµPLψ − eψPLγµPRψ
Perfect symmetry under parity: ~p → −~p
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
weak interaction: Left is not equal to Right
use vector bosons
ask for local gauge invariance
remember that U(1)EM is QED and was extremelysuccessful
unify electromagnetic and weak interactions
SU(2) × U(1)
SU(2): three generators (gauge bosons)
U(1): one generators (gauge boson)
SU(2) vector bosons must be massive (Fermi-contactinteraction)
massive vector bosons lead to a non-renomalizable theory
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
weak interaction: Left is not equal to Right
use vector bosons
ask for local gauge invariance
remember that U(1)EM is QED and was extremelysuccessful
unify electromagnetic and weak interactions
SU(2) × U(1)
SU(2): three generators (gauge bosons)
U(1): one generators (gauge boson)
SU(2) vector bosons must be massive (Fermi-contactinteraction)
massive vector bosons lead to a non-renomalizable theory
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
weak interaction: Left is not equal to Right
use vector bosons
ask for local gauge invariance
remember that U(1)EM is QED and was extremelysuccessful
unify electromagnetic and weak interactions
SU(2) × U(1)
SU(2): three generators (gauge bosons)
U(1): one generators (gauge boson)
SU(2) vector bosons must be massive (Fermi-contactinteraction)
massive vector bosons lead to a non-renomalizable theory
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
weak interaction: Left is not equal to Right
use vector bosons
ask for local gauge invariance
remember that U(1)EM is QED and was extremelysuccessful
unify electromagnetic and weak interactions
SU(2) × U(1)
SU(2): three generators (gauge bosons)
U(1): one generators (gauge boson)
SU(2) vector bosons must be massive (Fermi-contactinteraction)
massive vector bosons lead to a non-renomalizable theory
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
weak interaction: Left is not equal to Right
use vector bosons
ask for local gauge invariance
remember that U(1)EM is QED and was extremelysuccessful
unify electromagnetic and weak interactions
SU(2) × U(1)
SU(2): three generators (gauge bosons)
U(1): one generators (gauge boson)
SU(2) vector bosons must be massive (Fermi-contactinteraction)
massive vector bosons lead to a non-renomalizable theory
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
weak interaction: Left is not equal to Right
use vector bosons
ask for local gauge invariance
remember that U(1)EM is QED and was extremelysuccessful
unify electromagnetic and weak interactions
SU(2) × U(1)
SU(2): three generators (gauge bosons)
U(1): one generators (gauge boson)
SU(2) vector bosons must be massive (Fermi-contactinteraction)
massive vector bosons lead to a non-renomalizable theory
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
weak interaction: Left is not equal to Right
use vector bosons
ask for local gauge invariance
remember that U(1)EM is QED and was extremelysuccessful
unify electromagnetic and weak interactions
SU(2) × U(1)
SU(2): three generators (gauge bosons)
U(1): one generators (gauge boson)
SU(2) vector bosons must be massive (Fermi-contactinteraction)
massive vector bosons lead to a non-renomalizable theory
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
weak interaction: Left is not equal to Right
use vector bosons
ask for local gauge invariance
remember that U(1)EM is QED and was extremelysuccessful
unify electromagnetic and weak interactions
SU(2) × U(1)
SU(2): three generators (gauge bosons)
U(1): one generators (gauge boson)
SU(2) vector bosons must be massive (Fermi-contactinteraction)
massive vector bosons lead to a non-renomalizable theory
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
weak interaction: Left is not equal to Right
use vector bosons
ask for local gauge invariance
remember that U(1)EM is QED and was extremelysuccessful
unify electromagnetic and weak interactions
SU(2) × U(1)
SU(2): three generators (gauge bosons)
U(1): one generators (gauge boson)
SU(2) vector bosons must be massive (Fermi-contactinteraction)
massive vector bosons lead to a non-renomalizable theory
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
weak interaction: Left is not equal to Right
use vector bosons
ask for local gauge invariance
remember that U(1)EM is QED and was extremelysuccessful
unify electromagnetic and weak interactions
SU(2) × U(1)
SU(2): three generators (gauge bosons)
U(1): one generators (gauge boson)
SU(2) vector bosons must be massive (Fermi-contactinteraction)
massive vector bosons lead to a non-renomalizable theory
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
The free Lagrangian (L0) Remember QCD:
GaugeGroup SU(3)Gaugebosons 8Lorentz − Vectors Ga
µ(x)
Field − Tensor Gaµν = ∂µGa
ν(x) − ∂νGaµ(x) − gSfabcGb
µ(x)Gcν(x)
Structure [λa2 , λb
2 ] = ifabcλc2
SU(2)L:
GaugeGroup SU(2)Gaugebosons 3Lorentz − Vectors W a
µ (x)
Field − Tensor W aµν = ∂µW a
ν (x) − ∂νW aµ (x) − g2ǫabcW b
µ (x)W cν (x
Structure [Ta2 , Tb
2 ] = iǫabcTc2
(Ta)3×3
(
σa 00 0
)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
The free Lagrangian (L0) Remember QCD:
GaugeGroup SU(3)Gaugebosons 8Lorentz − Vectors Ga
µ(x)
Field − Tensor Gaµν = ∂µGa
ν(x) − ∂νGaµ(x) − gSfabcGb
µ(x)Gcν(x)
Structure [λa2 , λb
2 ] = ifabcλc2
SU(2)L:
GaugeGroup SU(2)Gaugebosons 3Lorentz − Vectors W a
µ (x)
Field − Tensor W aµν = ∂µW a
ν (x) − ∂νW aµ (x) − g2ǫabcW b
µ (x)W cν (x
Structure [Ta2 , Tb
2 ] = iǫabcTc2
(Ta)3×3
(
σa 00 0
)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
The free Lagrangian (L0) Remember QCD:
GaugeGroup SU(3)Gaugebosons 8Lorentz − Vectors Ga
µ(x)
Field − Tensor Gaµν = ∂µGa
ν(x) − ∂νGaµ(x) − gSfabcGb
µ(x)Gcν(x)
Structure [λa2 , λb
2 ] = ifabcλc2
SU(2)L:
GaugeGroup SU(2)Gaugebosons 3Lorentz − Vectors W a
µ (x)
Field − Tensor W aµν = ∂µW a
ν (x) − ∂νW aµ (x) − g2ǫabcW b
µ (x)W cν (x
Structure [Ta2 , Tb
2 ] = iǫabcTc2
(Ta)3×3
(
σa 00 0
)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
The free Lagrangian (L0) Remember QCD:
GaugeGroup SU(3)Gaugebosons 8Lorentz − Vectors Ga
µ(x)
Field − Tensor Gaµν = ∂µGa
ν(x) − ∂νGaµ(x) − gSfabcGb
µ(x)Gcν(x)
Structure [λa2 , λb
2 ] = ifabcλc2
SU(2)L:
GaugeGroup SU(2)Gaugebosons 3Lorentz − Vectors W a
µ (x)
Field − Tensor W aµν = ∂µW a
ν (x) − ∂νW aµ (x) − g2ǫabcW b
µ (x)W cν (x
Structure [Ta2 , Tb
2 ] = iǫabcTc2
(Ta)3×3
(
σa 00 0
)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
The free Lagrangian (L0) Remember QCD:
GaugeGroup SU(3)Gaugebosons 8Lorentz − Vectors Ga
µ(x)
Field − Tensor Gaµν = ∂µGa
ν(x) − ∂νGaµ(x) − gSfabcGb
µ(x)Gcν(x)
Structure [λa2 , λb
2 ] = ifabcλc2
SU(2)L:
GaugeGroup SU(2)Gaugebosons 3Lorentz − Vectors W a
µ (x)
Field − Tensor W aµν = ∂µW a
ν (x) − ∂νW aµ (x) − g2ǫabcW b
µ (x)W cν (x
Structure [Ta2 , Tb
2 ] = iǫabcTc2
(Ta)3×3
(
σa 00 0
)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
The free Lagrangian (L0) Remember QCD:
GaugeGroup SU(3)Gaugebosons 8Lorentz − Vectors Ga
µ(x)
Field − Tensor Gaµν = ∂µGa
ν(x) − ∂νGaµ(x) − gSfabcGb
µ(x)Gcν(x)
Structure [λa2 , λb
2 ] = ifabcλc2
SU(2)L:
GaugeGroup SU(2)Gaugebosons 3Lorentz − Vectors W a
µ (x)
Field − Tensor W aµν = ∂µW a
ν (x) − ∂νW aµ (x) − g2ǫabcW b
µ (x)W cν (x
Structure [Ta2 , Tb
2 ] = iǫabcTc2
(Ta)3×3
(
σa 00 0
)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
SU(2)L:
GaugeGroup SU(2)Gaugebosons 3Lorentz − Vectors W a
µ (x)
Field − Tensor W aµν = ∂µW a
ν (x) − ∂νW aµ (x) − gǫabcW b
µ (x)W cν (x)
Structure [Ta2 , Tb
2 ] = iǫabcTc2
U(1)Y Y weak hypercharge:
GaugeGroup U(1)Gaugeboson 1Lorentz − Vector Bµ(x)Field − Tensor Bµν = ∂µBν(x) − ∂νBµ(x)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
SU(2)L:
GaugeGroup SU(2)Gaugebosons 3Lorentz − Vectors W a
µ (x)
Field − Tensor W aµν = ∂µW a
ν (x) − ∂νW aµ (x) − gǫabcW b
µ (x)W cν (x)
Structure [Ta2 , Tb
2 ] = iǫabcTc2
U(1)Y Y weak hypercharge:
GaugeGroup U(1)Gaugeboson 1Lorentz − Vector Bµ(x)Field − Tensor Bµν = ∂µBν(x) − ∂νBµ(x)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
SU(2)L:
GaugeGroup SU(2)Gaugebosons 3Lorentz − Vectors W a
µ (x)
Field − Tensor W aµν = ∂µW a
ν (x) − ∂νW aµ (x) − gǫabcW b
µ (x)W cν (x)
Structure [Ta2 , Tb
2 ] = iǫabcTc2
U(1)Y Y weak hypercharge:
GaugeGroup U(1)Gaugeboson 1Lorentz − Vector Bµ(x)Field − Tensor Bµν = ∂µBν(x) − ∂νBµ(x)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
SU(2)L:
GaugeGroup SU(2)Gaugebosons 3Lorentz − Vectors W a
µ (x)
Field − Tensor W aµν = ∂µW a
ν (x) − ∂νW aµ (x) − gǫabcW b
µ (x)W cν (x)
Structure [Ta2 , Tb
2 ] = iǫabcTc2
U(1)Y Y weak hypercharge:
GaugeGroup U(1)Gaugeboson 1Lorentz − Vector Bµ(x)Field − Tensor Bµν = ∂µBν(x) − ∂νBµ(x)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
SU(2)L:
GaugeGroup SU(2)Gaugebosons 3Lorentz − Vectors W a
µ (x)
Field − Tensor W aµν = ∂µW a
ν (x) − ∂νW aµ (x) − gǫabcW b
µ (x)W cν (x)
Structure [Ta2 , Tb
2 ] = iǫabcTc2
U(1)Y Y weak hypercharge:
GaugeGroup U(1)Gaugeboson 1Lorentz − Vector Bµ(x)Field − Tensor Bµν = ∂µBν(x) − ∂νBµ(x)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Free Lagrangian Gauge Fields SU(2)L × U(1)Y
L0 = −14BµνBµν − 1
4W aµνW µνa
= −14BµνBµν − 1
2Tr(WµνW µν)
Wµν = W aµν
Ta2
Organize the Dirac Fields
eL(x) = PLe(x)eL(x) = (γ0PLe(x))†
ℓ(x) =
νeL(x)eL(x)eR(x)
No right-handed neutrinos
Define the weak Hypercharge
hypercharge left 6= right:
Y =
−12 0 0
0 −12 0
0 0 yR
SU(2) × U(1): yR to be chosenlater.....
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Free Lagrangian Gauge Fields SU(2)L × U(1)Y
L0 = −14BµνBµν − 1
4W aµνW µνa
= −14BµνBµν − 1
2Tr(WµνW µν)
Wµν = W aµν
Ta2
Organize the Dirac Fields
eL(x) = PLe(x)eL(x) = (γ0PLe(x))†
ℓ(x) =
νeL(x)eL(x)eR(x)
No right-handed neutrinos
Define the weak Hypercharge
hypercharge left 6= right:
Y =
−12 0 0
0 −12 0
0 0 yR
SU(2) × U(1): yR to be chosenlater.....
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Free Lagrangian Gauge Fields SU(2)L × U(1)Y
L0 = −14BµνBµν − 1
4W aµνW µνa
= −14BµνBµν − 1
2Tr(WµνW µν)
Wµν = W aµν
Ta2
Organize the Dirac Fields
eL(x) = PLe(x)eL(x) = (γ0PLe(x))†
ℓ(x) =
νeL(x)eL(x)eR(x)
No right-handed neutrinos
Define the weak Hypercharge
hypercharge left 6= right:
Y =
−12 0 0
0 −12 0
0 0 yR
SU(2) × U(1): yR to be chosenlater.....
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Free Lagrangian DiracFields
L0 = νeL(x)iγµ∂µνeL(x)
+ eL(x)iγµ∂µeL(x)
+ eR(x)iγµ∂µeR(x)
= ℓ(x)iγµ∂µℓ(x)
Minimal Substitution
∂µ → ∂µ + ig2W aµ
Ta2 + ig1BµY
Interaction Lagrangian
L′ = −ℓγµ(g2W aµ
Ta2 + g1BµY )ℓ
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Free Lagrangian DiracFields
L0 = νeL(x)iγµ∂µνeL(x)
+ eL(x)iγµ∂µeL(x)
+ eR(x)iγµ∂µeR(x)
= ℓ(x)iγµ∂µℓ(x)
Minimal Substitution
∂µ → ∂µ + ig2W aµ
Ta2 + ig1BµY
Interaction Lagrangian
L′ = −ℓγµ(g2W aµ
Ta2 + g1BµY )ℓ
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Free Lagrangian DiracFields
L0 = νeL(x)iγµ∂µνeL(x)
+ eL(x)iγµ∂µeL(x)
+ eR(x)iγµ∂µeR(x)
= ℓ(x)iγµ∂µℓ(x)
Minimal Substitution
∂µ → ∂µ + ig2W aµ
Ta2 + ig1BµY
Interaction Lagrangian
L′ = −ℓγµ(g2W aµ
Ta2 + g1BµY )ℓ
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
use form of Pauli matrices and W±µ = 1√
2(W 1
µ ∓ iW 2µ )
Investigate the Interaction
L′ = −ℓγµ(g2W aµ
Ta2 + g1BµY )ℓ
= −ℓγµ[g2(W 1µ
T12 + W 2
µT22 + W 3
µT32 ) + g1BµY ]ℓ
= −g2(νeL , eL)γµ(W 1µ
τ12 + W 2
µτ22 + W 3
µτ32 )
(
νeL
eL
)
+12g1νeLγ
µBµνeL + 12g1eLγµBµeL − yRg1eRγµBµeR
= − g2√2(W +
µ νeLγµeL + W−
µ eLγµνeL)
−12(g2W 3
µ − g1Bµ)νeLγµνeL
+12(g2W 3
µ + g1Bµ)eLγµeL − yRg1BµeRγµeR
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
use form of Pauli matrices and W±µ = 1√
2(W 1
µ ∓ iW 2µ )
Investigate the Interaction
L′ = −ℓγµ(g2W aµ
Ta2 + g1BµY )ℓ
= −ℓγµ[g2(W 1µ
T12 + W 2
µT22 + W 3
µT32 ) + g1BµY ]ℓ
= −g2(νeL , eL)γµ(W 1µ
τ12 + W 2
µτ22 + W 3
µτ32 )
(
νeL
eL
)
+12g1νeLγ
µBµνeL + 12g1eLγµBµeL − yRg1eRγµBµeR
= − g2√2(W +
µ νeLγµeL + W−
µ eLγµνeL)
−12(g2W 3
µ − g1Bµ)νeLγµνeL
+12(g2W 3
µ + g1Bµ)eLγµeL − yRg1BµeRγµeR
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
use form of Pauli matrices and W±µ = 1√
2(W 1
µ ∓ iW 2µ )
Investigate the Interaction
L′ = −ℓγµ(g2W aµ
Ta2 + g1BµY )ℓ
= −ℓγµ[g2(W 1µ
T12 + W 2
µT22 + W 3
µT32 ) + g1BµY ]ℓ
= −g2(νeL , eL)γµ(W 1µ
τ12 + W 2
µτ22 + W 3
µτ32 )
(
νeL
eL
)
+12g1νeLγ
µBµνeL + 12g1eLγµBµeL − yRg1eRγµBµeR
= − g2√2(W +
µ νeLγµeL + W−
µ eLγµνeL)
−12(g2W 3
µ − g1Bµ)νeLγµνeL
+12(g2W 3
µ + g1Bµ)eLγµeL − yRg1BµeRγµeR
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
use form of Pauli matrices and W±µ = 1√
2(W 1
µ ∓ iW 2µ )
Investigate the Interaction
L′ = −ℓγµ(g2W aµ
Ta2 + g1BµY )ℓ
= −ℓγµ[g2(W 1µ
T12 + W 2
µT22 + W 3
µT32 ) + g1BµY ]ℓ
= −g2(νeL , eL)γµ(W 1µ
τ12 + W 2
µτ22 + W 3
µτ32 )
(
νeL
eL
)
+12g1νeLγ
µBµνeL + 12g1eLγµBµeL − yRg1eRγµBµeR
= − g2√2(W +
µ νeLγµeL + W−
µ eLγµνeL)
−12(g2W 3
µ − g1Bµ)νeLγµνeL
+12(g2W 3
µ + g1Bµ)eLγµeL − yRg1BµeRγµeR
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
use form of Pauli matrices and W±µ = 1√
2(W 1
µ ∓ iW 2µ )
Investigate the Interaction
L′ = −ℓγµ(g2W aµ
Ta2 + g1BµY )ℓ
= −ℓγµ[g2(W 1µ
T12 + W 2
µT22 + W 3
µT32 ) + g1BµY ]ℓ
= −g2(νeL , eL)γµ(W 1µ
τ12 + W 2
µτ22 + W 3
µτ32 )
(
νeL
eL
)
+12g1νeLγ
µBµνeL + 12g1eLγµBµeL − yRg1eRγµBµeR
= − g2√2(W +
µ νeLγµeL + W−
µ eLγµνeL)
−12(g2W 3
µ − g1Bµ)νeLγµνeL
+12(g2W 3
µ + g1Bµ)eLγµeL − yRg1BµeRγµeR
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Identify the gauge bosons
charged bosons: W±µ = 1√
2(W 1
µ ∓ iW 2µ )
neutral boson: Zµ = 1√g2
1+g22
(g2W 3µ − g1Bµ)
neutral boson: Aµ = 1√g2
1+g22
(g1W 3µ + g2Bµ)
Aµ and Zµ are orthogonal
weak angle: sin θW = g1√g2
1+g22
, cos θW = g2√g2
1+g22
Bµ = 1√g2
1+g22
(g2Aµ − g1Zµ) = cos θW Aµ − sin θW Zµ
W 3µ = 1√
g21+g2
2
(g1Aµ + g2Zµ) = sin θW Aµ + cos θW Zµ
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Identify the gauge bosons
charged bosons: W±µ = 1√
2(W 1
µ ∓ iW 2µ )
neutral boson: Zµ = 1√g2
1+g22
(g2W 3µ − g1Bµ)
neutral boson: Aµ = 1√g2
1+g22
(g1W 3µ + g2Bµ)
Aµ and Zµ are orthogonal
weak angle: sin θW = g1√g2
1+g22
, cos θW = g2√g2
1+g22
Bµ = 1√g2
1+g22
(g2Aµ − g1Zµ) = cos θW Aµ − sin θW Zµ
W 3µ = 1√
g21+g2
2
(g1Aµ + g2Zµ) = sin θW Aµ + cos θW Zµ
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Identify the gauge bosons
charged bosons: W±µ = 1√
2(W 1
µ ∓ iW 2µ )
neutral boson: Zµ = 1√g2
1+g22
(g2W 3µ − g1Bµ)
neutral boson: Aµ = 1√g2
1+g22
(g1W 3µ + g2Bµ)
Aµ and Zµ are orthogonal
weak angle: sin θW = g1√g2
1+g22
, cos θW = g2√g2
1+g22
Bµ = 1√g2
1+g22
(g2Aµ − g1Zµ) = cos θW Aµ − sin θW Zµ
W 3µ = 1√
g21+g2
2
(g1Aµ + g2Zµ) = sin θW Aµ + cos θW Zµ
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Identify the gauge bosons
charged bosons: W±µ = 1√
2(W 1
µ ∓ iW 2µ )
neutral boson: Zµ = 1√g2
1+g22
(g2W 3µ − g1Bµ)
neutral boson: Aµ = 1√g2
1+g22
(g1W 3µ + g2Bµ)
Aµ and Zµ are orthogonal
weak angle: sin θW = g1√g2
1+g22
, cos θW = g2√g2
1+g22
Bµ = 1√g2
1+g22
(g2Aµ − g1Zµ) = cos θW Aµ − sin θW Zµ
W 3µ = 1√
g21+g2
2
(g1Aµ + g2Zµ) = sin θW Aµ + cos θW Zµ
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Identify the gauge bosons
charged bosons: W±µ = 1√
2(W 1
µ ∓ iW 2µ )
neutral boson: Zµ = 1√g2
1+g22
(g2W 3µ − g1Bµ)
neutral boson: Aµ = 1√g2
1+g22
(g1W 3µ + g2Bµ)
Aµ and Zµ are orthogonal
weak angle: sin θW = g1√g2
1+g22
, cos θW = g2√g2
1+g22
Bµ = 1√g2
1+g22
(g2Aµ − g1Zµ) = cos θW Aµ − sin θW Zµ
W 3µ = 1√
g21+g2
2
(g1Aµ + g2Zµ) = sin θW Aµ + cos θW Zµ
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Identify the gauge bosons
charged bosons: W±µ = 1√
2(W 1
µ ∓ iW 2µ )
neutral boson: Zµ = 1√g2
1+g22
(g2W 3µ − g1Bµ)
neutral boson: Aµ = 1√g2
1+g22
(g1W 3µ + g2Bµ)
Aµ and Zµ are orthogonal
weak angle: sin θW = g1√g2
1+g22
, cos θW = g2√g2
1+g22
Bµ = 1√g2
1+g22
(g2Aµ − g1Zµ) = cos θW Aµ − sin θW Zµ
W 3µ = 1√
g21+g2
2
(g1Aµ + g2Zµ) = sin θW Aµ + cos θW Zµ
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
−12(g2W 3
µ − g1Bµ)νeLγµνeL + 1
2(g2W 3µ + g1Bµ)eLγµeL
−yRg1BµeRγµeR
= −12 [g2
1√g2
1+g22
(g1Aµ + g2Zµ) − g11√
g21+g2
2
(g2Aµ − g1Zµ)]νeLγµνeL
+12 [g2(sin θW Aµ + cos θW Zµ) + g1(cos θW Aµ − sin θW Zµ)]eLγµeL
−yRg1(cos θW Aµ − sin θW Zµ)eRγµeR
= −√
g21 + g2
2Zµ[12νeLγ
µνeL − 12eLγµeL
− sin2 θW (−eLγµeL + yReRγµeR)]
− g1g2√g2
1+g22
Aµ(−eLγµeL + yReRγµeR)
deduce g1g2√g2
1+g22
= g1 cos θW = g2 sin θW = e
deduce yR = −1Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
−12(g2W 3
µ − g1Bµ)νeLγµνeL + 1
2(g2W 3µ + g1Bµ)eLγµeL
−yRg1BµeRγµeR
= −12 [g2
1√g2
1+g22
(g1Aµ + g2Zµ) − g11√
g21+g2
2
(g2Aµ − g1Zµ)]νeLγµνeL
+12 [g2(sin θW Aµ + cos θW Zµ) + g1(cos θW Aµ − sin θW Zµ)]eLγµeL
−yRg1(cos θW Aµ − sin θW Zµ)eRγµeR
= −√
g21 + g2
2Zµ[12νeLγ
µνeL − 12eLγµeL
− sin2 θW (−eLγµeL + yReRγµeR)]
− g1g2√g2
1+g22
Aµ(−eLγµeL + yReRγµeR)
deduce g1g2√g2
1+g22
= g1 cos θW = g2 sin θW = e
deduce yR = −1Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
−12(g2W 3
µ − g1Bµ)νeLγµνeL + 1
2(g2W 3µ + g1Bµ)eLγµeL
−yRg1BµeRγµeR
= −12 [g2
1√g2
1+g22
(g1Aµ + g2Zµ) − g11√
g21+g2
2
(g2Aµ − g1Zµ)]νeLγµνeL
+12 [g2(sin θW Aµ + cos θW Zµ) + g1(cos θW Aµ − sin θW Zµ)]eLγµeL
−yRg1(cos θW Aµ − sin θW Zµ)eRγµeR
= −√
g21 + g2
2Zµ[12νeLγ
µνeL − 12eLγµeL
− sin2 θW (−eLγµeL + yReRγµeR)]
− g1g2√g2
1+g22
Aµ(−eLγµeL + yReRγµeR)
deduce g1g2√g2
1+g22
= g1 cos θW = g2 sin θW = e
deduce yR = −1Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
−12(g2W 3
µ − g1Bµ)νeLγµνeL + 1
2(g2W 3µ + g1Bµ)eLγµeL
−yRg1BµeRγµeR
= −12 [g2
1√g2
1+g22
(g1Aµ + g2Zµ) − g11√
g21+g2
2
(g2Aµ − g1Zµ)]νeLγµνeL
+12 [g2(sin θW Aµ + cos θW Zµ) + g1(cos θW Aµ − sin θW Zµ)]eLγµeL
−yRg1(cos θW Aµ − sin θW Zµ)eRγµeR
= −√
g21 + g2
2Zµ[12νeLγ
µνeL − 12eLγµeL
− sin2 θW (−eLγµeL + yReRγµeR)]
− g1g2√g2
1+g22
Aµ(−eLγµeL + yReRγµeR)
deduce g1g2√g2
1+g22
= g1 cos θW = g2 sin θW = e
deduce yR = −1Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
−12(g2W 3
µ − g1Bµ)νeLγµνeL + 1
2(g2W 3µ + g1Bµ)eLγµeL
−yRg1BµeRγµeR
= −12 [g2
1√g2
1+g22
(g1Aµ + g2Zµ) − g11√
g21+g2
2
(g2Aµ − g1Zµ)]νeLγµνeL
+12 [g2(sin θW Aµ + cos θW Zµ) + g1(cos θW Aµ − sin θW Zµ)]eLγµeL
−yRg1(cos θW Aµ − sin θW Zµ)eRγµeR
= −√
g21 + g2
2Zµ[12νeLγ
µνeL − 12eLγµeL
− sin2 θW (−eLγµeL + yReRγµeR)]
− g1g2√g2
1+g22
Aµ(−eLγµeL + yReRγµeR)
deduce g1g2√g2
1+g22
= g1 cos θW = g2 sin θW = e
deduce yR = −1Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
L′ = − e√2 sin θW
(W +µ νeLγ
µeL + W−µ eLγµνeL)
− esin θW cos θW
Zµ[12νeLγ
µνeL − 12eLγµeL
− sin2 θW (−eLγµeL − eRγµeR)]
−eAµ(−eLγµeL − eRγµeR)
photon couples to charged particles only
charged gauge bosons ensure transition between chargedleptons and neutrinos
a neutral gauge boson is predicted
all gauge bosons are massless
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
L′ = − e√2 sin θW
(W +µ νeLγ
µeL + W−µ eLγµνeL)
− esin θW cos θW
Zµ[12νeLγ
µνeL − 12eLγµeL
− sin2 θW (−eLγµeL − eRγµeR)]
−eAµ(−eLγµeL − eRγµeR)
photon couples to charged particles only
charged gauge bosons ensure transition between chargedleptons and neutrinos
a neutral gauge boson is predicted
all gauge bosons are massless
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
L′ = − e√2 sin θW
(W +µ νeLγ
µeL + W−µ eLγµνeL)
− esin θW cos θW
Zµ[12νeLγ
µνeL − 12eLγµeL
− sin2 θW (−eLγµeL − eRγµeR)]
−eAµ(−eLγµeL − eRγµeR)
photon couples to charged particles only
charged gauge bosons ensure transition between chargedleptons and neutrinos
a neutral gauge boson is predicted
all gauge bosons are massless
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
L′ = − e√2 sin θW
(W +µ νeLγ
µeL + W−µ eLγµνeL)
− esin θW cos θW
Zµ[12νeLγ
µνeL − 12eLγµeL
− sin2 θW (−eLγµeL − eRγµeR)]
−eAµ(−eLγµeL − eRγµeR)
photon couples to charged particles only
charged gauge bosons ensure transition between chargedleptons and neutrinos
a neutral gauge boson is predicted
all gauge bosons are massless
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Introduce a complex scalardoublet:
φ(x) =
(
φ1(x)φ2(x)
)
Free Lagrangian
L0 = (∂µφ†)(∂µφ) − V (φ)V (φ) = κφ†φ + λ(φ†φ)2
Theory must be stable: λ > 0Minimum not at 0: κ = −µ2 < 0
The ground state is not unique:
φ = exp(iτa
2ϕa)
(
0√
µ2
2λ
)
Choose ϕ = 0 → SU(2)symmetry is broken
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Yukawa terms
LY = −yeeRφ†(
νeL
eL
)
h.c. −ye(νeL , eL)φeR
= −ye(eRφ†1νeL + eRφ
†2eL)
−ye(νeLφ1eR + eLφ2eR)
Deduce the hypercharge:
eL → eR + φ2
−12 = yH + −1
yH = 12
Minimal Subsitution
∂µφ → ∂µφ + ig2W aµ
τa2 φ
+ig1BµyHφ
∂µφ† → ∂µφ† − φ†ig2W aµ
τa2
−φ†ig1BµyH
Calculate the interaction terms
φ†(−ig2W aµ
τa2 − ig1BµyH)
(+ig2W aµ
τa2 + ig1BµyH)φ
φT = (0,
√
µ2
2λ) = (0, v√
2)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Yukawa terms
LY = −yeeRφ†(
νeL
eL
)
h.c. −ye(νeL , eL)φeR
= −ye(eRφ†1νeL + eRφ
†2eL)
−ye(νeLφ1eR + eLφ2eR)
Deduce the hypercharge:
eL → eR + φ2
−12 = yH + −1
yH = 12
Minimal Subsitution
∂µφ → ∂µφ + ig2W aµ
τa2 φ
+ig1BµyHφ
∂µφ† → ∂µφ† − φ†ig2W aµ
τa2
−φ†ig1BµyH
Calculate the interaction terms
φ†(−ig2W aµ
τa2 − ig1BµyH)
(+ig2W aµ
τa2 + ig1BµyH)φ
φT = (0,
√
µ2
2λ) = (0, v√
2)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Yukawa terms
LY = −yeeRφ†(
νeL
eL
)
h.c. −ye(νeL , eL)φeR
= −ye(eRφ†1νeL + eRφ
†2eL)
−ye(νeLφ1eR + eLφ2eR)
Deduce the hypercharge:
eL → eR + φ2
−12 = yH + −1
yH = 12
Minimal Subsitution
∂µφ → ∂µφ + ig2W aµ
τa2 φ
+ig1BµyHφ
∂µφ† → ∂µφ† − φ†ig2W aµ
τa2
−φ†ig1BµyH
Calculate the interaction terms
φ†(−ig2W aµ
τa2 − ig1BµyH)
(+ig2W aµ
τa2 + ig1BµyH)φ
φT = (0,
√
µ2
2λ) = (0, v√
2)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Yukawa terms
LY = −yeeRφ†(
νeL
eL
)
h.c. −ye(νeL , eL)φeR
= −ye(eRφ†1νeL + eRφ
†2eL)
−ye(νeLφ1eR + eLφ2eR)
Deduce the hypercharge:
eL → eR + φ2
−12 = yH + −1
yH = 12
Minimal Subsitution
∂µφ → ∂µφ + ig2W aµ
τa2 φ
+ig1BµyHφ
∂µφ† → ∂µφ† − φ†ig2W aµ
τa2
−φ†ig1BµyH
Calculate the interaction terms
φ†(−ig2W aµ
τa2 − ig1BµyH)
(+ig2W aµ
τa2 + ig1BµyH)φ
φT = (0,
√
µ2
2λ) = (0, v√
2)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
(0,
√
µ2
2λ)(−ig2W a
µτa2 − ig1BµyH)(+ig2W a
µτa2 + ig1BµyH)
(
0√
µ2
2λ
)
= (0,
√
µ2
2λ)(g2W a
µτa2 + g1BµyH)(g2W a
µτa2 + g1BµyH)
(
0√
µ2
2λ
)
= (0,
√
µ2
2λ)
2g2g1Aµ+(g22−g2
1)Zµ
2√
g21+g2
2
g2√2W +
µ
g2√2W−
µ −√
g21+g2
22 Zµ
2g2g1Aµ+(g22−g2
1)Zµ
2√
g21+g2
2
g2√2W +
µ
g2√2W−
µ −√
g21+g2
22 Zµ
(
0√
µ2
2λ
)
=g2
2v2
4 W−µ W µ+ +
(g21+g2
2)v2
8 ZµZ µ
The weak bosons have acquired a mass!Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
(0,
√
µ2
2λ)(−ig2W a
µτa2 − ig1BµyH)(+ig2W a
µτa2 + ig1BµyH)
(
0√
µ2
2λ
)
= (0,
√
µ2
2λ)(g2W a
µτa2 + g1BµyH)(g2W a
µτa2 + g1BµyH)
(
0√
µ2
2λ
)
= (0,
√
µ2
2λ)
2g2g1Aµ+(g22−g2
1)Zµ
2√
g21+g2
2
g2√2W +
µ
g2√2W−
µ −√
g21+g2
22 Zµ
2g2g1Aµ+(g22−g2
1)Zµ
2√
g21+g2
2
g2√2W +
µ
g2√2W−
µ −√
g21+g2
22 Zµ
(
0√
µ2
2λ
)
=g2
2v2
4 W−µ W µ+ +
(g21+g2
2)v2
8 ZµZ µ
The weak bosons have acquired a mass!Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
(0,
√
µ2
2λ)(−ig2W a
µτa2 − ig1BµyH)(+ig2W a
µτa2 + ig1BµyH)
(
0√
µ2
2λ
)
= (0,
√
µ2
2λ)(g2W a
µτa2 + g1BµyH)(g2W a
µτa2 + g1BµyH)
(
0√
µ2
2λ
)
= (0,
√
µ2
2λ)
2g2g1Aµ+(g22−g2
1)Zµ
2√
g21+g2
2
g2√2W +
µ
g2√2W−
µ −√
g21+g2
22 Zµ
2g2g1Aµ+(g22−g2
1)Zµ
2√
g21+g2
2
g2√2W +
µ
g2√2W−
µ −√
g21+g2
22 Zµ
(
0√
µ2
2λ
)
=g2
2v2
4 W−µ W µ+ +
(g21+g2
2)v2
8 ZµZ µ
The weak bosons have acquired a mass!Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
(0,
√
µ2
2λ)(−ig2W a
µτa2 − ig1BµyH)(+ig2W a
µτa2 + ig1BµyH)
(
0√
µ2
2λ
)
= (0,
√
µ2
2λ)(g2W a
µτa2 + g1BµyH)(g2W a
µτa2 + g1BµyH)
(
0√
µ2
2λ
)
= (0,
√
µ2
2λ)
2g2g1Aµ+(g22−g2
1)Zµ
2√
g21+g2
2
g2√2W +
µ
g2√2W−
µ −√
g21+g2
22 Zµ
2g2g1Aµ+(g22−g2
1)Zµ
2√
g21+g2
2
g2√2W +
µ
g2√2W−
µ −√
g21+g2
22 Zµ
(
0√
µ2
2λ
)
=g2
2v2
4 W−µ W µ+ +
(g21+g2
2)v2
8 ZµZ µ
The weak bosons have acquired a mass!Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
(0,
√
µ2
2λ)(−ig2W a
µτa2 − ig1BµyH)(+ig2W a
µτa2 + ig1BµyH)
(
0√
µ2
2λ
)
= (0,
√
µ2
2λ)(g2W a
µτa2 + g1BµyH)(g2W a
µτa2 + g1BµyH)
(
0√
µ2
2λ
)
= (0,
√
µ2
2λ)
2g2g1Aµ+(g22−g2
1)Zµ
2√
g21+g2
2
g2√2W +
µ
g2√2W−
µ −√
g21+g2
22 Zµ
2g2g1Aµ+(g22−g2
1)Zµ
2√
g21+g2
2
g2√2W +
µ
g2√2W−
µ −√
g21+g2
22 Zµ
(
0√
µ2
2λ
)
=g2
2v2
4 W−µ W µ+ +
(g21+g2
2)v2
8 ZµZ µ
The weak bosons have acquired a mass!Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Charged lepton masses
LY = −ye(eRφ†1νeL + eRφ
†2eL) − ye(νeLφ1eR + eLφ2eR)
= −ye(eRv√2
eL) − ye(eLv√2
eR)
= −yev√2(eReL + eLeR)
= −yev√2(ee)
Masses
me = yev√2
m2W± =
g22v2
4 = e2v2
4 sin2 θW
m2Z◦ =
(g21+g2
2)v2
4 = e2v2
4 sin2 θW cos2 θWL → EQM
m2W±
m2Z◦
= cos2 θW
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
Charged lepton masses
LY = −ye(eRφ†1νeL + eRφ
†2eL) − ye(νeLφ1eR + eLφ2eR)
= −ye(eRv√2
eL) − ye(eLv√2
eR)
= −yev√2(eReL + eLeR)
= −yev√2(ee)
Masses
me = yev√2
m2W± =
g22v2
4 = e2v2
4 sin2 θW
m2Z◦ =
(g21+g2
2)v2
4 = e2v2
4 sin2 θW cos2 θWL → EQM
m2W±
m2Z◦
= cos2 θW
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
QuantumNumbersweak IsospinSU(2)L offermions
weakhypercharge
Q = IW3 + Y
IW IW3 Y
12
12−1
2
16
(
uL
dL
) (
cL
sL
) (
tLbL
)
12
12−1
2− 1
2
(
νeL
eL
) (
νµL
µL
) (
ντL
τL
)
0 0 23 uR cR tR
0 0 − 13 dR sR bR
0 0 − 1 eR µR τR
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
QuantumNumbersweak IsospinSU(2)L offermions
weakhypercharge
Q = IW3 + Y
IW IW3 Y
12
12−1
2
16
(
uL
dL
) (
cL
sL
) (
tLbL
)
12
12−1
2− 1
2
(
νeL
eL
) (
νµL
µL
) (
ντL
τL
)
0 0 23 uR cR tR
0 0 − 13 dR sR bR
0 0 − 1 eR µR τR
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
QuantumNumbersweak IsospinSU(2)L offermions
weakhypercharge
Q = IW3 + Y
IW IW3 Y
12
12−1
2
16
(
uL
dL
) (
cL
sL
) (
tLbL
)
12
12−1
2− 1
2
(
νeL
eL
) (
νµL
µL
) (
ντL
τL
)
0 0 23 uR cR tR
0 0 − 13 dR sR bR
0 0 − 1 eR µR τR
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
eL = (γ0PLe)† = e†P†Lγ
0 = e†PLγ0 = e†γ0PR = ePR
The interactions
L′ = − e√2 sin θW
(W +µ νeLγ
µeL + W−µ eLγµνeL)
− esin θW cos θW
Zµ[12νeLγ
µνeL − 12eLγµeL
− sin2 θW (−eLγµeL − eRγµeR)]
−eAµ(−eLγµeL − eRγµeR)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
eL = (γ0PLe)† = e†P†Lγ
0 = e†PLγ0 = e†γ0PR = ePR
The interactions
L′ = − e√2 sin θW
(W +µ νeLγ
µeL + W−µ eLγµνeL)
− esin θW cos θW
Zµ[12νeLγ
µνeL − 12eLγµeL
− sin2 θW (−eLγµeL − eRγµeR)]
−eAµ(−eLγµeL − eRγµeR)
Electromagnetic Current
L′ = −eAµ(−ePRγµPLe − ePLγµPRe)
= −eAµeγµQe = −eAµeγµ(IW3 + Y )e
= −eAµjµEM
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
eL = (γ0PLe)† = e†P†Lγ
0 = e†PLγ0 = e†γ0PR = ePR
The interactions
L′ = − e√2 sin θW
(W +µ νeLγ
µeL + W−µ eLγµνeL)
− esin θW cos θW
Zµ[12νeLγ
µνeL − 12eLγµeL
− sin2 θW (−eLγµeL − eRγµeR)]
−eAµ(−eLγµeL − eRγµeR)
Charged Current
L′ = − e√2 sin θW
(W +µ νeLγ
µPLe + W−µ ePRγµνeL)
= − e√2 sin θW
(W +µ νeLγ
µPLe + W−µ eγµPLνeL)
= − e√2 sin θW
(W +µ jµCC + W−
µ jµCC†)
Dirk Zerwas Particle Physics: The Standard Model
The Electroweak Theory
FermiLeft and RightLagrangienHiggs
eL = (γ0PLe)† = e†P†Lγ
0 = e†PLγ0 = e†γ0PR = ePR
The interactions
L′ = − e√2 sin θW
(W +µ νeLγ
µeL + W−µ eLγµνeL)
− esin θW cos θW
Zµ[12νeLγ
µνeL − 12eLγµeL
− sin2 θW (−eLγµeL − eRγµeR)]
−eAµ(−eLγµeL − eRγµeR)
Neutral Current
L′ = − esin θW cos θW
Zµ[νeLγµIW
3 νeL + eLγµIW3 eL
− sin2 θW jµEM ]
= − esin θW cos θW
ZµjµNC
Dirk Zerwas Particle Physics: The Standard Model
