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Introduction to 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

(Structure Of The Standard Model, hep-ph/0304186)

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

The Weak Interactions

Radioactivity (Becquerel, 1896)

decay appeared to violate energy(Meitner, Hahn; 1911)

Neutrino hypothesis (Pauli, 1930)

e (Reines, Cowan; 1953)

(Lederman, Schwartz, Steinberger;1962)

(DONUT, 2000) ( , 1975)

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Fermi theory (1933)

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

pe

e

n

J J

e e

e e

J J e

e

e e

e e

W

pe

e

n

g g W +

e e

e e

g g

Typeset by FoilTEX 1

H GFJJ

J pn+ee [np, ee]

J np+ee [pn, ee ( ee)]

GF ' 1.17105 GeV2 [Fermi constant]

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Fermi theory modified to include , decay

strangeness (Cabibbo)

quark model

heavy quarks (CKM)

mass and mixing

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

Fermi theory correctly describes (at tree level) Nuclear/neutron decay (npee)

, decays (ee; , , )

, K decays (++, 0e+e; K++, 0e+e, +0)

hyperon decays (p; n; +e+e)

heavy quark decays (cse+e; bc, c)

scattering (ee; np| {z }elastic

; NX| {z }deepinelastic

)

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

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

pe

e

n

J J

e e

e e

J J e

e

e e

e e

W

pe

e

n

g g W +

e e

e e

g g

Typeset by FoilTEX 1

(eeee)G2F s

, s E2CM

pure S-wave unitarity: < 16s

fails for ECM2

GF 500 GeV

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

Fermi theory: divergent integralsd4k6 k +mek2 m2e

6 kk2

pe

e

n

J J

e e

e e

J J e

e

e e

e e

W

pe

e

n

g g W +

e e

e e

g g

Typeset by FoilTEX 1

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

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

pe

e

n

J J

e e

e e

J J e

e

e e

e e

W

pe

e

n

g g W +

e e

e e

g g

Typeset by FoilTEX 1

pe

e

n

J J

e e

e e

J J e

e

e e

e e

W

pe

e

n

g g W +

e e

e e

g g

Typeset by FoilTEX 1

GF2

g2

8M2Wfor MW Q

no longer pure S-wave

eeee better behaved

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

e

W W +

e e+

g g W 0

W W +

e e+

g

g

Z

d s

d sK0

K0

Typeset by FoilTEX 2

but, e+eW+W violatesunitarity for

s > 500 GeV

k/MW for longitudinalpolarization (non-renormalizable)

introduce W 0 to cancel

fixes W 0W+W and e+eW 0

vertices

requires[J, J

] J0

(like SU(2)U(1))

not realistic

e

W W +

e e+

g g W 0

W W +

e e+

g

g

Z

d s

d sK0

K0

Typeset by FoilTEX 2

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Glashow model (1961) (W, Z, , 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)

c discovered (1974)

e

W W +

e e+

g g W 0

W W +

e e+

g

g

Z

d s

d sK0

K0

Typeset by FoilTEX 2

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Weak neutral current(1973)

QCD (1970s)

W,Z (1983)

Precision tests (1989-2000)

CKM unitarity ( 1995-)

t quark (1995)

mass (1998-2002)

Measurement Fit |Omeas!Ofit|/"meas0 1 2 3

0 1 2 3

#$had(mZ)#$(5) 0.02758 0.00035 0.02767

mZ [GeV]mZ [GeV] 91.1875 0.0021 91.1874%Z [GeV]%Z [GeV] 2.4952 0.0023 2.4959"had [nb]"

0 41.540 0.037 41.478RlRl 20.767 0.025 20.742AfbA

0,l 0.01714 0.00095 0.01643Al(P&)Al(P&) 0.1465 0.0032 0.1480RbRb 0.21629 0.00066 0.21579RcRc 0.1721 0.0030 0.1723AfbA

0,b 0.0992 0.0016 0.1038AfbA

0,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 0.1480sin2'effsin

2'lept(Qfb) 0.2324 0.0012 0.2314mW [GeV]mW [GeV] 80.410 0.032 80.377%W [GeV]%W [GeV] 2.123 0.067 2.092mt [GeV]mt [GeV] 172.7 2.9 173.3

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

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

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

QED

Free electron equation,(i

xm

) = 0,

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

xm

) = 0, where ei

Not invariant under local (gauge) transf.,

ei(x), x (~x, t)

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Introduce vector field A ( ~A, ):(i

x+eA m

) = 0,

(e > 0 is gauge coupling) is invariant under

ei(x), AA 1

e

x

Quantization of A massless gaugeboson

Gauge invariance , long rangeforce, prescribed (up to e) amplitudefor emission/absorption

e p

e p

e e

Typeset by FoilTEX 1

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Non-Abelian

n non-interacting fermions of same mass m:(i

xm

)a = 0, a = 1 n,

invariant under (global) SU(n) group, 1...n

exp(i Ni=1

iLi)

1...n

.Li are nn generator matrices (N = n21); i are real parameters

[Li, Lj] = icijkLk

(cijk are structure constants)

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Gauge (local) transformation: ii(x)(i

xabg

Ni=1

AiLiab mab

)b = 0

Invariant under

1...n

U~A ~L ~A ~L U ~A ~LU

1 +i

g(U)U1

U ei~~L

(1)

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Gauge invariance implies:

N (apparently) massless gaugebosons Ai

Specified interactions (up to gaugecoupling g, group, representations),including self interactions

Ai

b

a

igLiab

Typeset by FoilTEX 1

g g2

Typeset by FoilTEX 1

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

Chiral Projections: L(R) 12(1 5)(Chirality = helicity up to O(m/E))

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

The Standard Model

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

)L

(ud

)L

(ud

)L

(ee

)L

uR uR uR eR(?)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, eL eL (W); phases (W 0)

U(1): phases (B)

Heavy families (c, s, , ), (t, b, , )

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Quantum Chromodynamics (QCD)

LSU(3) = 1

4F iF

i +r

qri 6D qr

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

F i = Gi G

i gsfijk G

j G

k

is field strength tensor for the gluon fields Gi, i = 1, , 8.gs = QCD gauge coupling constant. No gluon masses.

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

[i, j] = 2ifijkk

where i are the Gell-Mann matrices.

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

i = i 00 0

, i = 1, 2, 3

4 =

0@ 0 0 10 0 01 0 0

1A 5 =0@ 0 0 i0 0 0i 0 0

1A6 =

0@ 0 0 00 0 10 1 0

1A 7 =0@ 0 0 00 0 i

0 i 0

1A8 = 1

3

0@ 1 0 00 1 00 0 2

1A

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

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Quark interactions given by qri 6D qrqr = rth quark flavor; , = 1, 2, 3 are color indices

Gauge covariant derivative

D = (D) = + igs Gi L

i,

for triplet representation matrices Li = i/2.

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Quark color interactions:

Diagonal in flavor

Off diagonal in color

Purely vector (parity conserving)

Gi

u

u

igs2 i

Typeset by FoilTEX 1

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

L = g2s

322FiF

i, F i 12F i

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

QCD now very well established

Short distance behavior (asymptotic freedom)

Confinement, light hadron spectrum (lattice)

gs = O(1) (s(MZ) = g2s/4 0.12) Strength + gluon self-interactions confinement Yukawa model dipole-dipole

Approximate global SU(3)LSU(3)R symmetry and breaking(,K, are pseudo-goldstone bosons)

Unique field theory of strong interactions

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Quasi-Chiral Exotics

(J. Kang, PL, B. Nelson, in progress)

Exotic fermions (anomaly-cancellation)

Examples in 27-plet of E6 DL + DR (SU(2) singlets, chiral wrt U(1))

(E0

E

)

L

+

(E0

E

)

R

(SU(2) doublets, chiral wrt U(1))

Pair produce D + D by QCD processes (smaller rate for exotic leptons)

Lightest may decay by mixing; by diquark or leptoquark coupling;or be quasi-stable

22nd Henry Primakoff Lecture Paul Langacker (3/1/2006)

Quantum Chromodynamics (QCD)

Modern theory of the strong interactions

NYS APS (October 15, 2004) Paul Langacker (Penn)

9. Quantum chromodynamics 7

0.1 0.12 0.14

Average

Hadronic Jets

Polarized DIS

Deep Inelastic Scattering (DIS)

! decays

Z width

Fragmentation

Spectroscopy (Lattice)

ep event shapes

Photo-production

" decay

e+e

- rates

#s(M

Z)

Figure 9.1: Summary of the value of s(MZ) from various processes. The valuesshown indicate the process and the measured value of s extrapolated to = MZ .The error shown is the total error including theoretical uncertainties. The averagequoted in this report which comes from these measurements is also shown. See textfor discussion of errors.

theoretical estimates. If the nonperturbative terms are omitted from the fit, the extractedvalue of s(m ) decreases by 0.02.

For s(m ) = 0.35 the perturbative series for R is R 3.058(1+0.112+0.064+0.036).The size (estimated error) of the nonperturbative term is 20% (7%) of the size of theorder 3s term. The perturbation series is not very well convergent; if the order 3s termis omitted, the extracted value of s(m ) increases by 0.05. The order 4s term has beenestimated [47] and attempts made to resum the entire series [48,49]. These estimates canbe used to obtain an estimate of the errors due to these unknown terms [50,51]. Anotherapproach to estimating this 4s term gives a contribution that is slightly larger than the3s term [52].

R can be extracted from the semi-leptonic branching ratio from the relationR = 1/(B( e) 1.97256); where B( e) is measured directly or extractedfrom the lifetime, the muon mass, and the muon lifetime assuming universality of lepton

December 20, 2005 11:23

18 9. Quantum chromodynamics

0

0.1

0.2

0.3

1 10 102

GeV!

s(

)

Figure 9.2: Summary of the values of s() at the values of where they aremeasured. The lines show the central values and the 1 limits of our average.The figure clearly shows the decrease in s() with increasing . The data are,in increasing order of , width, decays, deep inelastic scattering, e+e eventshapes at 22 GeV from the JADE data, shapes at TRISTAN at 58 GeV, Z width,and e+e event shapes at 135 and 189 GeV.

The value of s at any scale corresponding to our average can be obtainedfrom http://www-theory.lbl.gov/ianh/alpha/alpha.html which uses Eq. (9.5) tointerpolate.

References:1. R.K. Ellis et al., QCD and Collider Physics (Cambridge 1996).2. For reviews see, for example, A.S. Kronfeld and P.B. Mackenzie, Ann. Rev. Nucl.

and Part. Sci. 43, 793 (1993);H. Wittig, Int. J. Mod. Phys. A12, 4477 (1997).

3. For example see, P. Gambino, International Conference on Lepton PhotonInteractions, Fermilab, USA, (2003); J. Butterworth International Conference onLepton Photon Interactions, Upsala, Sweden, (2005).

December 20, 2005 11:23

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

The Electroweak Sector

LSU(2)U(1) = Lgauge + L + Lf + LYukawa

Gauge part

Lgauge = 1

4F iF

i 1

4BB

Field strength tensors

B = B BF i = W

i W

i gijkW

jW

k , i = 1 3

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

Three and four-point self-interactions for the Wi

B and W3 will mix to form , Z

g g2

Typeset by FoilTEX 1

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

U(1): j exp(igyj)j, yj = qj t3j = weak hypercharge

Scalar part

L = (D)D V ()

where =+

0

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

Gauge covariant derivative:

D =

( + ig

i

2W i +

ig

2B

)

where i are the Pauli matrices

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

Typeset by FoilTEX 1

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Higgs potential

V () = +2+ ()2

Allowed by renormalizability and gaugeinvariance

Spontaneous symmetry breaking for 2 < 0

Vacuum stability: > 0.

Quartic self-interactions+

0

0

Typeset by FoilTEX 1

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Fermion part

LF =Fm=1

(q0mLi 6Dq

0mL + l

0mLi 6Dl

0mL

+ u0mRi 6Du0mR + d

0mRi 6Dd

0mR + e

0mRi 6De

0mR+

0mRi 6D

0mR

)L-doublets

q0mL =(u0md0m

)L

l0mL =(0me0m

)L

R-singlets

u0mR, d0mR, e

0mR,

0mR(?)

(F 3 families; m = 1 F = family index;0 = weak eigenstates (definite SU(2) rep.), mixtures of mass eigenstates (flavors);quark color indices = r, g, b suppressed (e.g., u0mL). )

Can add gauge singlet 0mR for Dirac neutrino mass term

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

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 (quark-leptoncancellations)

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Gauge covariant derivatives

Dq0mL =

( +

ig

2 iW i + i

g

6B

)q0mL

Dl0mL =

( +

ig

2 iW i i

g

2B

)l0mL

Du0mR =

( + i

2

3gB

)u0mR

Dd0mR =

( i

g

3B

)d0mR

De0mR = ( ig

B) e0mR

Read off W and Bcouplings to fermions W i

ig2i

15

2

Bigy

15

2

Typeset by FoilTEX 1

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Yukawa couplings (couple L to R)

LYukawa =F

m,n=1

[umnq

0mLu

0mR +

dmnq

0mLd

0nR

+ emnl0mne

0nR (+

mnl

0mL

0mR)

]+ H.C.

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

d, e terms require doublet =(+

0

)with Y = 1/2

u (and ) terms require doublet

=(

0

)with Y = 1/2

nR

mL

mn

Typeset by FoilTEX 1

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

In SU(2) the 2 and 2 are similar i 2 =(

0

)transforms as a 2 with Y = 12 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)LSU(2)RU(1) and SU(5))

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Spontaneous Symmetry Breaking

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||0 = constant

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

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Minimize V (v) to find v and quantize = v

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

=(+

0

)=

(12(1 i2)

12(3 i4

),

where i = i .

V () =1

22

(4i=1

2i

)+

1

4

(4i=1

2i

)2

is O(4) invariant.

w.l.o.g. choose 0|i|0 = 0, i = 1, 2, 4 and 0|3|0 =

V ()V (v) =1

222 +

1

44

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

For 2 < 0, minimum at

V () = (2 + 2) = 0

=(2/

)1/2SSB for 2 = 0 also; must

consider loop corrections

12

(0

) v the generators L1, L2, and L3 Y

spontaneously broken, L1v 6= 0, etc (Li = i2 , Y = 12I)

Qv = (L3 + Y )v =(

1 00 0

)v = 0 U(1)Q unbroken

SU(2)U(1)YU(1)Q

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)

Quantize around classical vacuum

Kibble transformation: introduce new variables i for rollingmodes

=1

2eiPiLi

(0

+H

) H = H is the Higgs scalar No potential for i massless Goldstone bosons for global

symmetry

Disappear from spectrum for gauge theory (eaten) Display particle content in unitary gauge

= eiPiLi =

1

2

(0

+H

)+ corresponding transformation on gauge fields

PiTP 2007 (July 16, 2007) Paul Langacker (IAS)