the standard model - university of...

The Standard Model

• Origins of the Electroweak Theory

• Gauge Theories

• The Standard Model Lagrangian

• Spontaneous Symmetry Breaking

• The Gauge Interactions

• Problems With the Standard Model

References: 2008 TASI lectures: arXiv:0901.0241 [hep-ph] and The Standard Model and Beyond, CRC Press

PreSUSY 2011, Chicago (August, 2011) Paul Langacker (IAS)

http://arXiv.org/abs/0901.0241

http://www.crcpress.com/product/isbn/9781420079067

The Weak Interactions

• Radioactivity (Becquerel, 1896)

• β decay appeared to violate energy(Chadwick, 1914)

• Neutrino hypothesis (Pauli, 1930)

– νe (Reines, Cowan; 1956)

– νµ (Lederman, Schwartz, Steinberger; 1962)

– ντ (DONUT, 2000) (τ , 1975)


• Fermi theory of β decay (n→ pe−ν) (1934)

– Loosely like QED, but zero range(non-renormalizable) and non-diagonal(charged current)

pe−

νe

n

J†µ Jµ

e− νe

νe e−

Jµ J†µ e− νe

νe e−

νe e−

→W −

pe−

νe

n

g g →W +

e− νe

νe e−

g g

– Typeset by FoilTEX – 1

H ∼ GFJ†µJµ

J†µ ∼ pγµn+νeγµe− [n→ p, e−→ νe]

Jµ ∼ nγµp+eγµνe [p→ n, νe→ e− ( × → e−νe)]

GF ' 1.17×10−5 GeV−2 [Fermi constant]


• Fermi theory modified to include

– parity violation (V −A) (Lee, Yang; Wu; Feynman-Gell-Mann)

– µ, τ decay

– strangeness (Cabibbo)

– quark model

– heavy quarks and CP violation (CKM)

– ν mass and mixing

• Fermi theory correctly describes (at tree level)

– Nuclear/neutron β decay/inverse (n→ pe−νe; e−p→ νen)

– µ, τ decays (µ− → e−νeνµ; τ− → µ−νµντ , ντπ−, · · · )– π, K decays (π+ → µ+νµ, π

0e+νe; K+ → µ+νµ, π0e+νe, π

+π0)

– hyperon decays (Λ→ pπ−; Σ− → nπ−; Σ+ → Λe+νe)

– heavy quark decays (c→ se+νe; b→ cµ−νµ, cπ−)

– ν scattering (νµe− → µ−νe; νµn→ µ

−p︸︷︷︸

“elastic′′; νµN → µ

−X︸︷︷︸

deep−inelastic

)


• Fermi theory violates unitarity at high energy (non-renormalizable)

pe−

νe

n

J†µ Jµ

e− νe

νe e−

Jµ J†µ e− νe

νe e−

νe e−

→W −

pe−

νe

n

g g →W +

e− νe

νe e−

g g


– σ(νee−→ e−νe)→

G2F s

π(s ≡ E2

CM)

– pure S-wave unitarity: σ < 16πs

– fails for ECM2≥√

πGF∼ 500 GeV

– Born not unitary; often restored by H.O.T.

– Fermi theory: divergent integrals∫d4k

( 6kk2

)( 6kk2

)p

e−

νe

n

J†µ Jµ

e− νe

νe e−

Jµ J†µ e− νe

νe e−

νe e−

→W −

pe−

νe

n

g g →W +

e− νe

νe e−

g g



• Intermediate vector boson theory (Yukawa, 1935; Schwinger, 1957)

pe−

νe

n

J†µ Jµ

e− νe

νe e−

Jµ J†µ e− νe

νe e−

νe e−

→W −

pe−

νe

n

g g →W +

e− νe

νe e−

g g


pe−

νe

n

J†µ Jµ

e− νe

νe e−

Jµ J†µ e− νe

νe e−

νe e−

→W −

pe−

νe

n

g g →W +

e− νe

νe e−

g g


GF√2∼

g2

8M2W

for MW � Q

– no longer pure S-wave ⇒

– νee−→ νee

− better behaved


νe

W − W +

e− e+

g g W 0

W − W +

e− e+

g

g

Z

d s

d sK0

K0


– but, e+e− → W+W− violatesunitarity for

√s & 500 GeV

– εµ ∼ kµ/MW for longitudinalpolarization (non-renormalizable)

– introduce W 0 to cancel

– fixes W 0W+W− and e+e−W 0

vertices

– requires[J, J†

]∼ J0

(like SU(2))

– not realistic

νe

W − W +

e− e+

g g W 0

W − W +

e− e+

g

g

Z

d s

d sK0

K0



• Glashow model (1961) (W±, Z, γ, but no mechanism for MW,Z)

• Weinberg-Salam (1967): Higgs mechanism → MW,Z

• Renormalizable (1971) (’t Hooft, · · · )

• Flavor changing neutral currents (FCNC)

– very large K0 ↔ K0 mixing

– GIM mechanism (c quark)(1970) (mc ∼ 1.5 GeV (1974))

– c discovered (1974)

J/ψ = cc (BNL, SLAC)

νµd(s)→ µ−c,c→ sµ+νµ (dimuons)

Z

s d

d sK0

K0

K0

K0

u u

W

W

d s

s d



• Weak neutral current(1973)

• QCD (1970’s)

• W,Z (1983)

• Precision tests (1989-)

• Precision K, B, Dphysics (∼ 2000-)

• CKM unitarity (∼ 1995-)

• t quark (1995)

• ν mass (1998-)

Measurement Fit |Omeas−Ofit|/σmeas

0 1 2 3

0 1 2 3

∆αhad(mZ)∆α(5) 0.02758 ± 0.00035 0.02768mZ [GeV]mZ [GeV] 91.1875 ± 0.0021 91.1875ΓZ [GeV]ΓZ [GeV] 2.4952 ± 0.0023 2.4957σhad [nb]σ0 41.540 ± 0.037 41.477RlRl 20.767 ± 0.025 20.744AfbA0,l 0.01714 ± 0.00095 0.01645Al(Pτ)Al(Pτ) 0.1465 ± 0.0032 0.1481RbRb 0.21629 ± 0.00066 0.21586RcRc 0.1721 ± 0.0030 0.1722AfbA0,b 0.0992 ± 0.0016 0.1038AfbA0,c 0.0707 ± 0.0035 0.0742AbAb 0.923 ± 0.020 0.935AcAc 0.670 ± 0.027 0.668Al(SLD)Al(SLD) 0.1513 ± 0.0021 0.1481sin2θeffsin2θlept(Qfb) 0.2324 ± 0.0012 0.2314mW [GeV]mW [GeV] 80.398 ± 0.025 80.374ΓW [GeV]ΓW [GeV] 2.140 ± 0.060 2.091mt [GeV]mt [GeV] 170.9 ± 1.8 171.3


Gauge Theories

Standard Model is remarkably successful gauge theory of themicroscopic interactions

• Gauge symmetry⇒ (apparently) massless spin-1 (vector, gauge) bosons

• Interactions ⇔ group, representations, gauge coupling

• Like QED (U(1)), but gauge self interactions for non-abelian

• Application to strong (short range) ⇒ confinement

• Application to weak (short range)⇒ spontaneous symmetry breaking(Higgs or dynamical)

• Unique renormalizable field theory for spin-1


QED

• Free electron equation,(iγµ

∂

∂xµ−m

)ψ = 0,

is invariant under U(1) (phase) transformations,(iγµ

∂

∂xµ−m

)ψ′ = 0, where ψ′ ≡ e−iβψ

• Not invariant under local (gauge) transf.,

ψ → ψ′ ≡ e−iβ(x)ψ, x ≡ (~x, t)


• Introduce vector field Aµ ≡ ( ~A, φ):(iγµ

∂

∂xµ+eγµAµ −m

)ψ = 0,

(e > 0 is gauge coupling) is invariant under

ψ → e−iβ(x)ψ, Aµ→ Aµ −1

e

∂β

∂xµ

• Quantization of Aµ⇒ massless gaugeboson

• Gauge invariance ⇒ γ, long rangeforce, prescribed (up to e) amplitude foremission/absorption

γ

e− p

e− p

e e



Non-Abelian

• n non-interacting fermions of same mass m:(iγµ

∂

∂xµ−m

)ψa = 0, a = 1 · · ·n,

invariant under (global) SU(n) group, ψ1...ψn

→ exp(i

N∑i=1

βiLi)

ψ1...ψn

.Li are n×n generator matrices (N = n2−1); βi are real parameters

[Li, Lj] = icijkLk

(cijk are structure constants)


• Gauge (local) transformation: βi→ βi(x)⇒(iγµ

∂

∂xµδab−g

N∑i=1

γµAiµLiab −mδab

)ψb = 0

• Invariant under

Φ ≡

ψ1...ψn

→ Φ′ ≡ UΦ

~Aµ · ~L → ~A′µ · ~L ≡ U ~Aµ · ~LU−1 +i

g(∂µU)U−1

U ≡ ei~β·~L

(1)


• Gauge invariance implies:

– N (apparently) massless gauge bosons Aiµ

– Specified interactions (up to gauge coupling

g, group, representations), including selfinteractions

Aiµ

ψb

ψa

−igLiabγ

µ


g g2


• Generalize to other groups, representations, chiral (L 6= R)

– Chiral Projections: ψL(R) ≡ 12(1∓ γ5)ψ

(Chirality = helicity up to O(m/E))


The Standard Model

• Gauge group SU(3)× SU(2)× U(1); gauge couplings gs, g, g′(ud

)L

(ud

)L

(ud

)L

(νee−

)L

uR uR uR νeR(?)

dR dR dR e−R( L = left-handed, R = right-handed)

• SU(3): u ↔ u ↔ u, d ↔ d ↔ d (8 gluons)

• SU(2): uL↔ dL, νeL↔ e−L (W±); phases (W 0)

• U(1): phases (B)

• Heavy families (c, s, νµ, µ−), (t, b, ντ , τ

−)


Quantum Chromodynamics (QCD)

LSU(3) = −1

4F iµνF

iµν +∑r

qαr i 6Dβα qrβ

F 2 term leads to three and four-point gluon self-interactions.

F iµν = ∂µGiν − ∂νG

iµ − gsfijk G

jµ G

kν

is field strength tensor for the gluon fields Giµ, i = 1, · · · , 8.

gs = QCD gauge coupling constant. No gluon masses.

Structure constants fijk (i, j, k = 1, · · · , 8), defined by

[λi, λj] = 2ifijkλk

where λi are the Gell-Mann matrices.


λi =

(τ i 0

0 0

), i = 1, 2, 3

λ4 =

0 0 1

0 0 0

1 0 0

λ5 =

0 0 −i0 0 0

i 0 0

λ6 =

0 0 0

0 0 1

0 1 0

λ7 =

0 0 0

0 0 −i0 i 0

λ8 = 1√

3

1 0 0

0 1 0

0 0 −2

The SU(3) (Gell-Mann) matrices.


Quark interactions given by qαr i 6Dβα qrβ

qr = rth quark flavor; α, β = 1, 2, 3 are color indices

Gauge covariant derivative

Dµβα = (Dµ)αβ = ∂µδαβ + igs G

iµ Liαβ,

for triplet representation matrices Li = λi/2.


Quark color interactions:

Diagonal in flavor

Off diagonal in color

Purely vector (parity conserving)

Giµ

uβ

uα

−igs2 λi

αβγµ


Bare quark mass allowed by QCD, but forbidden by chiral symmetryof LSU(2)×U(1) (generated by spontaneous symmetry breaking)

Additional ghost and gauge-fixing terms

Can add (unwanted) CP-violating term

Lθ =θg2s

32π2FiµνF

iµν, F iµν ≡ 12εµναβF iαβ


QCD now very well established

• Short distance behavior (asymptotic freedom)

• Confinement, light hadron spectrum (lattice)

– gs = O(1) (αs(MZ) = g2s/4π ∼ 0.12)

– Strength + gluon self-interactions⇒ confinement

– Yukawa model ⇒ dipole-dipole

• Approximate global SU(3)L × SU(3)R symmetry and breaking(π,K, η are pseudo-goldstone bosons)

• Unique field theory of strong interactions


0.11 0.12 0.13

α (Μ )s Z

Quarkonia (lattice)

DIS F2 (N3LO)

τ-decays (N3LO)

DIS jets (NLO)

e+e– jets & shps (NNLO)

electroweak fits (N3LO)

e+e– jets & shapes (NNLO)

Υ decays (NLO)

QCD α (Μ ) = 0.1184 ± 0.0007s Z

0.1

0.2

0.3

0.4

0.5

αs (Q)

1 10 100Q [GeV]

Heavy Quarkoniae+e– Annihilation

Deep Inelastic Scattering

July 2009


The Electroweak Sector

LSU(2)×U(1) = Lgauge + Lφ + Lf + LYukawa

Gauge part

Lgauge = −1

4F iµνF

µνi −1

4BµνB

µν

Field strength tensors

Bµν = ∂µBν − ∂νBµF iµν = ∂µW

iν − ∂νW

iµ − gεijkW

jµW

kν , i = 1 · · · 3

g(g′) is SU(2) (U(1)) gauge coupling; εijk is totally antisymmetric symbol

Three and four-pointself-interactions for the Wi

B and W3 will mix to form γ, Z

g g2



U(1): Φj → exp(ig′yjβ)Φj, yj = qj − t3j = weak hypercharge

Scalar part

Lφ = (Dµφ)†Dµφ− V (φ)

where φ =

(φ+

φ0

)is the (complex) Higgs doublet with yφ = 1/2.

Gauge covariant derivative:

Dµφ =

(∂µ + ig

τ i

2W iµ +

ig′

2Bµ

)φ

where τ i are the Pauli matrices

Three and four-point interactionsbetween gauge and scalar fields g g2



Higgs potential

V (φ) = +µ2φ†φ+ λ(φ†φ)2

φ†φ = φ−︸︷︷︸φ+†

φ+ + φ0†φ0

Allowed by renormalizability and gaugeinvariance

Spontaneous symmetry breaking for µ2 < 0

Vacuum stability: λ > 0.

Quartic self-interactionsφ+

φ0

φ−

φ0†

λ



Fermion part

LF =

F∑m=1

(q0mLi 6Dq

0mL + l0mLi 6Dl

0mL

+ u0mRi 6Du

0mR + d0

mRi 6Dd0mR + e0

mRi 6De0mR+ν0

mRi 6Dν0mR

)L-doublets

q0mL =

(u0m

d0m

)L

l0mL =

(ν0m

e−0m

)L

R-singlets

u0mR, d

0mR, e

−0mR, ν

0mR(?)

(F ≥ 3 families; m = 1 · · ·F = family index;0 = weak eigenstates (definite SU(2) rep.), mixtures of mass eigenstates (flavors);

quark color indices α = r, g, b suppressed (e.g., u0mαL). )

Can add gauge singlet ν0mR for Dirac neutrino mass term


Different (chiral) L and R representations lead to parity and chargeconjugation violation (maximal for SU(2))

Fermion mass terms forbidden by chiral symmetry

Triangle anomalies absent for chosen hypercharges and 3 colors(includes quark-lepton cancellations)

Jiµ

Jjν

Jkρ

i

k

j



Gauge covariant derivatives

Dµq0mL =

(∂µ +

ig

2τ iW i

µ + ig′

6Bµ

)q0mL

Dµl0mL =

(∂µ +

ig

2τ iW i

µ − ig′

2Bµ

)l0mL

Dµu0mR =

(∂µ + i

2

3g′Bµ

)u0mR

Dµd0mR =

(∂µ − i

g′

3Bµ

)d0mR

Dµe0mR = (∂µ − ig′Bµ) e0

mR

Read off W and Bcouplings to fermions W i

µ

−ig2τ iγµ

“1−γ5

2

”

Bµ−ig′yγµ

“1∓γ5

2

”



Yukawa couplings (couple L to R)

− LYukawa =

F∑m,n=1

[Γumnq

0mLφu

0mR + Γdmnq

0mLφd

0nR

+ Γemnl0mnφe

0nR (+Γνmnl

0mLφν

0mR)

]+ h.c.

Γmn are completely arbitrary Yukawa matrices, which determinefermion masses and mixings

d, e terms require doublet φ =(φ+

φ0

)with Yφ = 1/2

u (and ν) terms require doublet

Φ =

(Φ0

Φ−

)with YΦ = −1/2

φ

ψnR

ψmL

Γmn



In SU(2) the 2 and 2∗ are similar ⇒ φ ≡ iτ 2φ† =

(φ0†

−φ−

)transforms as a 2 with Yφ = −1

2⇒ only one doublet needed.

Does not generalize to SU(3), most extra U(1)′, supersymmetry,SO(10) etc ⇒ need two doublets.(Does generalize to SU(2)L × SU(2)R × U(1) and SU(5))


Spontaneous Symmetry Breaking (Higgs mechanism)

Gauge invariance implies massless gauge bosons and fermions

Weak interactions short ranged ⇒ spontaneous symmetry breakingfor mass; also for fermions

Color confinement for QCD ⇒ gluons remain massless

Allow classical (ground state) expectation value for Higgs field

v = 〈0|φ|0〉 = constant, φ =

(φ+

φ0

)

∂µv 6= 0 increases energy, but important for monopoles, strings,domain walls, phase transitions (e.g., EWPT, baryogenesis)


Minimize V (v) to find v and quantize φ′ = φ− v

SU(2)× U(1): introduce Hermitian basis

φ =

(φ+

φ0

)=

(1√2(φ1 + iφ2)

1√2(φ3 + iφ4)

)

where φi = φ†i .

V (φ) =1

2µ2

(4∑i=1

φ2i

)+

1

4λ

(4∑i=1

φ2i

)2

is O(4) invariant.

w.l.o.g. choose 〈0|φi|0〉 = 0, i = 1, 2, 4 and 〈0|φ3|0〉 = ν

V (φ)→ V (v) =1

2µ2ν2 +

1

4λν4


For µ2 < 0, minimum at

V ′(ν) = ν(µ2 + λν2) = 0

⇒ ν =(−µ2/λ

)1/2

SSB for µ2 = 0 also; mustconsider loop corrections

φ → 1√2

(0ν

)≡ v ⇒ the generators L1, L2, and L3 − Y

spontaneously broken, L1v 6= 0, etc (Li = τi

2 , Y = 12I)

Qv = (L3 + Y )v =

(1 00 0

)v = 0 ⇒ U(1)Q unbroken ⇒

SU(2)× U(1)Y → U(1)Q


Quantize around classical vacuum

• Kibble transformation: introduce new variables ξi for rollingmodes

φ =1√

2ei∑ξiLi

(0

ν +H

)• H = H† is the Higgs scalar

• No potential for ξi ⇒ massless Goldstone bosons for globalsymmetry

• Disappear from spectrum for gauge theory (“eaten”)

• Display particle content in unitary gauge

φ→ φ′ = e−i∑ξiLiφ =

1√

2

(0

ν +H

)+ corresponding transformation on gauge fields


Rewrite Lagrangian in New Vacuum

Physical Higgs scalar (oscillations around minimum): MH =√

2λν

Higgs covariant kinetic energy terms:

(Dµφ)†Dµφ =1

2(0 ν)

[g

2τ iW i

µ +g′

2Bµ

]2(0ν

)+H terms

→ M2WW

+µW−µ +M2Z

2ZµZµ

+ H kinetic energy and gauge interaction terms

W

W

ν

ν

g2

eR

eL

νhe


MW =gν

2

me =heν√

2W

W

ν

ν

g2

eR

eL

νhe



Mass eigenstate bosons: W, Z, and A (photon)

W± =1√

2(W 1 ∓ iW 2)

Z = − sin θWB + cos θWW3

A = cos θWB + sin θWW3

Weak angle: tan θW ≡ g′/g

Masses:

MW =gν

2, MZ =

√g2 + g′2

ν

2=

MW

cos θW, MA = 0

(Goldstone scalars “eaten”→ longitudinal components of W±, Z )


Will show: Fermi constant GF/√

2 ∼ g2/8M2W

(GF = 1.166364(5)× 10−5 GeV−2 from muon lifetime)

Electroweak scale:

ν = 2MW/g ' (√

2GF )−1/2 ' 246 GeV

Will show: g = e/ sin θW (α = e2/4π ∼ 1/137.036) ⇒

MW = MZ cos θW =gν

2∼

(πα/√

2GF )1/2

sin θW

Weak neutral current: sin2 θW ∼ 0.23 ⇒ MW ∼ 78 GeV, andMZ ∼ 89 GeV (increased by ∼ 2 GeV by loop corrections)

Discovered at CERN: UA1 and UA2, 1983


The Higgs Scalar H

Gauge interactions: ZZH,ZZH2,W+W−H,W+W−H2

φ→1√

2

(0

ν +H

)

Lφ = (Dµφ)†Dµφ− V (φ)

=1

2(∂µH)

2+M2

WWµ+W−µ

(1 +

H

ν

)2

+1

2M2ZZ

µZµ

(1 +

H

ν

)2

− V (φ)

(quartic and induced cubic interactions)


Higgs potential:

V (φ) = +µ2φ†φ+ λ(φ†φ)2

→ −µ4

4λ− µ2H2 + λνH3 +

λ

4H4

Fourth term: Quartic self-interaction

Third: Induced cubic self-interaction

Second: (Tree level) H mass-squared, MH =√−2µ2 =

√2λν

λ

ν

λ



First term in V : vacuum energy

〈0|V |0〉 = −µ4/4λ

No effect on microscopic interactions, but gives negativecontribution to cosmological constant

|ΛSSB| = 8πGN |〈0|V |0〉| ∼ 1050|Λobs|

Require fine-tuned cancellation

Λcosm = Λbare + ΛSSB

Also, QCD contribution from SSBof global chiral symmetry


Fermion Masses and Mixings

• Yukawa (Higgs-fermion) interaction (+ analogous u, e, ν terms)

−LY uk =

F∑m,n=1

Γdmnq0mLφd

0nR + h.c.

=

F∑m,n=1

Γdmn[u0mLφ

+d0nR + d 0

mLφ0d0nR

]+ h.c.

φ+

d0nR

u0mL

−iΓdmn φ0

d0nR

d0mL

−iΓdmn

φ−

u0mL

d0nR

−iΓd∗mn φ0†

d0mL

d0nR

−iΓd∗mn



• For φ→ 1√2

(0

ν +H

)(unitary gauge)

−LY uk ⇒F∑

m,n=1

Mdmnd

0mLd

0nR

(1 +

H

ν

)+ h.c.

=∑i

midiLdiR

(1 +

H

ν

)+ h.c. =

∑i

mididi

(1 +

H

ν

)with Md ≡ Γd ν/

√2

• di = diL + diR are mass eigenstates of mass mi

• For F = 3

Ad†LMdAdR =

md 0 00 ms 00 0 mb

, dsb

L,R

= Ad†L,R︸︷︷︸unitary

d01

d02

d03

L,R

• Higgs (H) couplings to fermions are diagonal in flavor and ∝ mass


Typical estimates: mu = 1.5− 3 MeV, md = 3− 7 MeV, ms =70− 120 MeV, mc ∼ 1.3 GeV, mb ∼ 4.2 GeV, mt = 172.9±1.1 GeV

Implications for global SU(3)L × SU(3)R of QCD

These are current quark masses. Mi = mi + Mdyn, Mdyn ∼ΛMS ∼ 300 MeV from chiral condensate 〈0|qq|0〉 6= 0

mt is pole mass; others, running masses at m or at 2 GeV2


Yukawa couplings of Higgs to fermions

LYukawa =∑i

ψi

(−mi −

gmi

2MW

H

)ψi

Coupling gmi/2MW is flavor diagonal and small except t quark

H → bb dominates for MH . 2MW (H →W+W−, ZZ dominate

when allowed because of larger gauge coupling)

Flavor diagonal because only one doublet couples to fermions ⇒fermion mass and Yukawa matrices proportional

Often flavor changing Higgs couplings in extended models withtwo doublets coupling to same kind of fermion (not MSSM)

Stringent limits, e.g., tree-level Higgs contribution to KL−KS

mixing (loop in standard model) ⇒hds/MH < 10−6 GeV−1


The Weak Charged Current

Fermi Theory incorporated in SM and made renormalizable

W -fermion interaction

L = −g

2√

2

(JµWW

−µ + Jµ†WW

+µ

)

Charge-raising current (ignoring ν masses)

Jµ†W =

F∑m=1

[ν0mγ

µ(1− γ5)e0m + u0

mγµ(1− γ5)d0

m

]= (νeνµντ)γ

µ(1− γ5)

e−

µ−

τ−

+ (u c t)γµ(1− γ5)V

dsb

.PreSUSY 2011, Chicago (August, 2011) Paul Langacker (IAS)

Ignore ν masses for now

Pure V − A ⇒ maximal P and C violation; CP conserved exceptfor phases in V

V = Au†L AdL is F × F unitary Cabibbo-Kobayashi-Maskawa (CKM)

matrix from mismatch between weak and Yukawa interactions

Cabibbo matrix for F = 2

V =

(cos θc sin θc− sin θc cos θc

)

sin θc ' 0.22 ≡ Cabibbo angle

Good zeroth-order description since third family almost decouples

General unitary 2× 2: 1 angle and 3 (unobservable) qL phases


CKM matrix for F = 3 involves 3 angles and 1 CP -violating phase(after removing unobservable qL phases) (new interations involving qR

could make observable)

V =

Vud Vus VubVcd Vcs VcbVtd Vtd Vtd

Extensive studies, especially in B decays, to test unitarity of V as

probe of new physics and test origin of CP violation

Need additional source of CP breaking for baryogenesis


Effective zero- range 4-fermi interaction (Fermi theory)

For |Q| � MW ,neglect Q2 in Wpropagator

−Lcceff =

(g

2√

2

)2

JµW

( −gµνQ2 −M2

W

)J†νW ∼

g2

8M2W

JµWJ†Wµ

Fermi constant: GF√2' g2

8M2W

= 12ν2

Muon lifetime: τ−1 =G2Fm

5µ

192π3 ⇒ GF = 1.17× 10−5 GeV−2

Weak scale: ν =√

2〈0|φ0|0〉 ' 246 GeV

Excellent description of β, K, hyperon, heavy quark, µ, and τdecays, νµe→ µ−νe, νµn→ µ−p, νµN → µ−X


Full theory probed:

e±p→(−)ν e X at high energy (HERA)

Electroweak radiative corrections (loop level)(Very important. Only calculable in full theory.)

MKS −MKL, kaon CP violation, B ↔ B mixing (loop level)

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1 1.5 2

sin 2!

sol. w/ cos 2! < 0(excl. at CL > 0.95)

excluded at CL > 0.95

"

"

#

#

$md

$ms & $md

%K

%K

|Vub/Vcb|

sin 2!

sol. w/ cos 2! < 0(excl. at CL > 0.95)

excluded at CL > 0.95

#

!"

&

'

excluded area has CL > 0.95

C K Mf i t t e r

EPS 2005

(CKMFITTER group:

http://ckmfitter.in2p3.fr/)


Quantum Electrodynamics (QED)

Incorporated into standard model

Lagrangian:

L = −gg′√g2 + g′2

JµQ(cos θWBµ + sin θWW3µ)

Photon field:

Aµ = cos θWBµ + sin θWW3µ

Positron electric charge: e = g sin θW , where tan θW ≡ g′/g


Electromagnetic current:

JµQ =

F∑m=1

[2

3u0mγ

µu0m −

1

3d0mγ

µd0m − e

0mγ

µe0m

]

=

F∑m=1

[2

3umγ

µum −1

3dmγ

µdm − emγµem]

Electric charge: Q = T 3 + Y , where Y = weak hypercharge(coefficient of ig′Bµ in covariant derivatives)

Flavor diagonal: Same form in weak and mass bases because fieldswhich mix have same charge

Purely vector (parity conserving): L and R fields have same charge(qi = t3i + yi is the same for L and R fields, even though t3i and yi are not)


Experiment Value of α−1 Precision ∆e

ae = (ge − 2)/2 137.035 999 683 (94) [6.9× 10−10] –

h/m (Rb, Cs) 137.035 999 35 (69) [5.0× 10−9] 0.33± 0.69

Quantum Hall 137.036 003 0 (25) [1.8× 10−8] −3.3± 2.5

h/m (neutron) 137.036 007 7 (28) [2.1× 10−8] −8.0± 2.8

γp,3He

(J. J.) 137.035 987 5 (43) [3.1× 10−8] 12.2± 4.3

µ+e− hyperfine 137.036 001 7 (80) [5.8× 10−8] −2.0± 8.0

Spectacularly successful:

Most precise: e anomalous magnetic moment → α

Many low energy tests to few ×10−8

mγ < 6× 10−17 eV, qγ < 5× 10−30|e|Running α(Q2) observed

Muon g − 2 sensitive to new physics. Anomaly?

Muonic Lamb shift. Anomaly? Proton radius?


The Weak Neutral Current

Prediction of SU(2)× U(1)

L = −√g2 + g′2

2JµZ

(− sin θWBµ + cos θWW

3µ

)= −

g

2 cos θWJµZZµ

Neutral current process and effective 4-fermi interaction for|Q| �MZ


Neutral current:

JµZ =∑m

[u0mLγ

µu0mL − d

0mLγ

µd0mL + ν0

mLγµν0mL − e

0mLγ

µe0mL

]−2 sin2 θWJ

µQ

=∑m

[umLγ

µumL − dmLγµdmL + νmLγµνmL − emLγµemL

]−2 sin2 θWJ

µQ

Flavor diagonal: Same form in weak and mass bases because fieldswhich mix have same charge

GIM mechanism: c quark predicted so that sL could be in doubletto avoid unwanted flavor changing neutral currents (FCNC) attree and loop level

Parity and charge conjugation violated but not maximally: first termis pure V −A, second is V


Effective 4-fermi interaction for |Q2| �M2Z:

−LNCeff =GF√

2JµZJZµ

Coefficient same as WCC because

GF√2

=g2

8M2W

=g2 + g′2

8M2Z


10

10 2

10 3

10 4

10 5

0 20 40 60 80 100 120 140 160 180 200 220

Centre-of-mass energy (GeV)

Cro

ss-s

ecti

on (

pb)

CESRDORIS

PEP

PETRATRISTAN

KEKBPEP-II

SLC

LEP I LEP II

Z

W+W-

e+e−→hadrons

0.0001 0.001 0.01 0.1 1 10 100 1000

µ [GeV]

0.228

0.23

0.232

0.234

0.236

0.238

0.24

0.242

0.244

0.246

0.248

0.25

sin2 θ W

(µ)

QW(APV)QW(e)

ν-DIS

LEP 1

SLC

Tevatron

e-DIS

MOLLER

Qweak

0

10

20

30

160 180 200

!s (GeV)

"W

W (

pb

)

YFSWW/RacoonWW

no ZWW vertex (Gentle)

only #e exchange (Gentle)

LEPPRELIMINARY

17/02/2005


The Z, the W , and the Weak Neutral Current

• Primary prediction and test of electroweak unification

• WNC discovered 1973 (Gargamelle at CERN, HPW at FNAL)

• 70’s, 80’s: weak neutral current experiments (few %)

– Pure weak: νN , νe scattering

– Weak-elm interference in eD, e+e−, atomic parity violation

– Model independent analyses (νe, νq, eq)

– SU(2)× U(1) group/representations; t and ντ exist; mt limit;hint for SUSY unification; limits on TeV scale physics

• W , Z discovered directly 1983 (UA1, UA2)


• 90’s: Z pole (LEP, SLD), 0.1%; lineshape, modes, asymmetries

• LEP 2: MW , Higgs search , gauge self-interactions

• Tevatron: mt, MW , Higgs search

• 4th generation weak neutral current experiments (atomic parity

(Boulder); νe; νN (NuTeV); polarized Møller asymmetry (SLAC))


• SM correct and unique to zerothapprox. (gauge principle, group,representations)

• SM correct at loop level (renormgauge theory; mt, αs, MH)

• TeV physics severely constrained(unification vs compositeness)

• Consistent with light elementaryHiggs

• Precise gauge couplings (SUSYgauge unification)

Measurement Fit |Omeas−Ofit|/σmeas

0 1 2 3

0 1 2 3

∆αhad(mZ)∆α(5) 0.02758 ± 0.00035 0.02768mZ [GeV]mZ [GeV] 91.1875 ± 0.0021 91.1874ΓZ [GeV]ΓZ [GeV] 2.4952 ± 0.0023 2.4959σhad [nb]σ0 41.540 ± 0.037 41.478RlRl 20.767 ± 0.025 20.742AfbA0,l 0.01714 ± 0.00095 0.01645Al(Pτ)Al(Pτ) 0.1465 ± 0.0032 0.1481RbRb 0.21629 ± 0.00066 0.21579RcRc 0.1721 ± 0.0030 0.1723AfbA0,b 0.0992 ± 0.0016 0.1038AfbA0,c 0.0707 ± 0.0035 0.0742AbAb 0.923 ± 0.020 0.935AcAc 0.670 ± 0.027 0.668Al(SLD)Al(SLD) 0.1513 ± 0.0021 0.1481sin2θeffsin2θlept(Qfb) 0.2324 ± 0.0012 0.2314mW [GeV]mW [GeV] 80.399 ± 0.023 80.379ΓW [GeV]ΓW [GeV] 2.098 ± 0.048 2.092mt [GeV]mt [GeV] 173.1 ± 1.3 173.2

August 2009


Problems with the Standard Model

Lagrangian after symmetry breaking:

L = Lgauge + LHiggs +∑i

ψi

(i 6∂ −mi −

miH

ν

)ψi

−g

2√

2

(JµWW

−µ + Jµ†WW

+µ

)− eJµQAµ −

g

2 cos θWJµZZµ

Standard model: SU(2) × U(1) (extended to include ν masses) +QCD + general relativity

Mathematically consistent, renormalizable theory

Correct to 10−16 cm


However, too much arbitrariness and fine-tuning: O(27) parameters(+ 2 for Majorana ν) and electric charges

• Gauge Problem

– complicated gauge group with 3 couplings

– charge quantization (|qe| = |qp|) unexplained

– Possible solutions: strings; grand unification; magneticmonopoles (partial); anomaly constraints (partial)

• Fermion problem

– Fermion masses, mixings, families unexplained

– Neutrino masses, nature? Probe of Planck/GUT scale?

– CP violation inadequate to explain baryon asymmetry

– Possible solutions: strings; brane worlds; family symmetries;compositeness; radiative hierarchies. New sources of CPviolation.


• Higgs/hierarchy problem

– Expect M2H = O(M2

W )

– higher order corrections:δM2

H/M2W ∼ 1034

H

H Hλ

W

H Hg2

W

W

H Hg g

f

f

H Hh h


Possible solutions: supersymmetry; dynamical symmetry breaking;large extra dimensions; Little Higgs; anthropically motivated fine-tuning (split supersymmetry) (landscape)

• Strong CP problem

– Can add θ32π2g

2sF F to QCD (breaks, P, T, CP )

– dN ⇒ θ < 10−9, but δθ|weak ∼ 10−3

– Possible solutions: spontaneously broken global U(1) (Peccei-Quinn) ⇒ axion; unbroken global U(1) (massless u quark);spontaneously broken CP + other symmetries


• Graviton problem

– gravity not unified

– quantum gravity not renormalizable

– cosmological constant: ΛSSB = 8πGN〈V 〉 > 1050Λobs

(10124 for GUTs, strings)

Possible solutions:

– supergravity and Kaluza Klein unify

– strings yield finite gravity

– Λ? Anthropically motivated fine-tuning (landscape)?


• Necessary new ingredients

– Mechanism for small neutrino masses

. Planck/GUT scale? Small Dirac (intermediate scale)?

– Mechanism for baryon asymmetry?

. Electroweak transition (Z′ or extended Higgs?)

. Heavy Majorana neutrino decay (seesaw)?

. Decay of coherent field? CPT violation?

– What is the dark energy?

. Cosmological Constant? Quintessence?

. Related to inflation? Time variation of couplings?

– What is the dark matter?

. Lightest supersymmetric particle? Axion?

– Suppression of flavor changing neutral currents? Proton decay?Electric dipole moments?

. Automatic in standard model, but not in extensions


New Physics

• A new layer at the TeV scale

– Compositeness, Little Higgs, twin Higgs, Higgless, dynamicalsymmetry breaking, strong dynamics

– Precision electroweak constraints, FCNC, UV completions?

• Large and/or warped extra dimensions; possible low fundamentalscale

• Unification at the Planck scale, MP = G−1/2N ∼ 1019 GeV

– Supersymmetry (between fermions and bosons), grand unification,strings?

– Top-down remnants: Z′, W ′, extended Higgs, exotic fermions,· · ·


Conclusions

• The standard model is spectacularly successful, but is incomplete

• Promising theoretical ideas at Planck and TeV scale

• Eagerly anticipate guidance from LHC


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