Sally Dawson, BNLIntroduction to the Standard Model

TASI, 2006• Introduction to the Standard Model

– Review of the SU(2) x U(1) Electroweak theory– Experimental status of the EW theory– Constraints from Precision Measurements

• Theoretical problems with the Standard Model• Beyond the SM

– Why are we sure there is physics BSM?– What will the LHC and Tevatron tell us?

Lecture 1

• Introduction to the Standard Model– Just the SU(2) x U(1) part of it….

• Some good references:– Chris Quigg, Gauge Theories of the Strong, Weak,

and Electromagnetic Interactions– Michael Peskin, An Introduction to Quantum Field

Theory– Sally Dawson, Trieste lectures, hep-ph/9901280

Tevatron LHC LHC Upgrade ILC

2006 2007 2012

Science Timeline

Large Hadron Collider (LHC)

• proton-proton collider at CERN (2007)

• 14 TeV energy– 7 mph slower than the

speed of light– cf. 2TeV @ Fermilab ( 307 mph slower than

the speed of light)

• Typical energy of quarks and gluons 1-2 TeV

LHC is Big….

• ATLAS is 100 meters underground, as deep as Big Ben is tall

• The accelerator circumscribes 58 square kilometers, as large as the island of Manhattan

LHC Will Require Detectors of Unprecedented Scale

• CMS is 12,000 tons (2 x’s ATLAS)

• ATLAS has 8 times the volume of CMS

What we know

• The photon and gluon appear to be massless

• The W and Z gauge bosons are heavy

– MW=80.404 0.030 GeV

– MZ =92.1875 0.0021 GeV

• There are 6 quarks

– Mt=172.5 2.3 GeV

– Mt >> all the other quark masses

• There appear to be 3 distinct neutrinos with small but non-zero masses

• The pattern of fermions appears to replicate itself 3 times

– Why not more?

Abelian Higgs Model

• Why are the W and Z boson masses non-zero?

• U(1) gauge theory with single spin-1gauge field, A

• U(1) local gauge invariance:

• Mass term for A would look like:

• Mass term violates local gauge invariance

• We understand why MA = 0

AAF

FFL

4

1

)()()( xxAxA

AAmFFL 2

2

1

4

1

Gauge invariance is guiding principle

Abelian Higgs Model, 2

• Add complex scalar field, , with charge –e:

• Where

• L is invariant under local U(1) transformations:

)(4

1 2

VDFFL

2222)(

V

ieAD

)()(

)()()()( xex

xxAxAxie

AAF

Abelian Higgs Model, 3

• Case 1: 2 > 0

– QED with MA=0 and m=

– Unique minimum at =0

)(4

1 2

VDFFL

2222)(

V

ieAD

By convention, > 0

Abelian Higgs Model, 4

• Case 2: 2 < 0

• Minimum energy state at:

2222)( V

22

2 v

Vacuum breaks U(1) symmetry

Aside: What fixes sign (2)?

Abelian Higgs Model, 5

• Rewrite

• L becomes:

• Theory now has:

– Photon of mass MA=ev

– Scalar field h with mass-squared –22 > 0

– Massless scalar field (Goldstone Boson)

hve vi

2

1

)nsinteractio,(2

1

22

1

24

1 2222

h

hhhAAve

evAFFL

and h are the 2 degrees of freedom of the complex Higgs field

Abelian Higgs Model, 6• What about mixed -A propagator?

– Remove by gauge transformation

• field disappears

– We say that it has been eaten to give the photon mass field called Goldstone boson

– This is Abelian Higgs Mechanism

– This gauge (unitary) contains only physical particles

ev

AA1'

)(2

1

24

1 22

hVhhAAve

FFL

Higgs Mechanism summarized

Spontaneous breaking of a gauge theoryby a non-zero VEV of a scalar field results in the disappearance of a Goldstone boson and its transformation into the longitudinal component of a massive gauge boson

Gauge InvarianceWhat about gauge invariance? Choice above called

unitarity gauge

– No field

– Bad high energy behavior of A propagator

– R gauges more convenient:

– LGF= (1/2)(A+ev)2

222

)(AA M

kkg

Mk

ik

222

222222

22

1

22

111

2

1

ve

hhhAvegAL

More on R gauges

)1(2222

AA Mk

kkg

Mk

i

22hMk

i

22AMk

i

Mass of Goldstone boson depends on =1: Feynman gauge with massive =0: Landau gauge: Unitarity gauge

Higgs, h

Goldstone Boson, , or Faddeev-Popov ghost

Gauge Boson, A

Aside on gauge boson counting

• Massless photon has 2 transverse degrees of freedom– p=(k,0,0,|k|) =(0,1, i,0)/2

• Massive gauge boson has 3 degrees of freedom (2 transverse, 1 longitudinal) L=(|k|,0,0,k0)/MV

• Count: Abelian Higgs Model– We start with: Massless gauge boson (2 dof), complex

scalar (2 dof)– We end with: Massive gauge boson (3 dof), physical

scalar (1 dof)

Non-Abelian Higgs Mechanism

• Vector fields Aa(x) and scalar fields i(x) of SU(N) group

• L is invariant under the non-Abelian symmetry:

a are group generators, a=1…N2-1 for SU(N)

22 )()(

),())((

V

VDDL

jijaa

i i )1(

N

.

.

.1

For SU(2): a=a/2

aa AigD

Non-Abelian Higgs Mechanism, 2

• In exact analogy to the Abelian case

a0 0 Massive vector boson + Goldstone boson a0=0 Massless vector boson + massive scalar field

...)()(...

...)()(...))((

002

2

0

bai

bi

a

bai

bi

a

AAg

AAgDD

Non-Abelian Higgs Mechanism, 3

• Consider SU(2) example

• Suppose gets a VEV:

• Gauge boson mass term

• Using the property of group generators, a,b=ab/2

• Mass term for gauge bosons:

a

a

AigD2

v

0

2

1

baba AA

vvgD

0,0

2

1 22

aamass AA

vgL

8

22

Standard Model Synopsis

• Group: SU(3) x SU(2) x U(1)

• Gauge bosons:– SU(3): G

i, i=1…8– SU(2): W

i, i=1,2,3– U(1): B

• Gauge couplings: gs, g, g• SU(2) Higgs doublet:

ElectroweakQCD

SM Higgs Mechanism

• Standard Model includes complex Higgs SU(2) doublet

• With SU(2) x U(1) invariant scalar potential

• If 2 < 0, then spontaneous symmetry breaking

• Minimum of potential at:

– Choice of minimum breaks gauge symmetry

– Why is 2 < 0?

043

21

2

1

i

i

22 )( V Invariant under -

v

0

2

1

More on SM Higgs Mechanism

• Couple to SU(2) x U(1) gauge bosons (Wi, i=1,2,3; B)

• Gauge boson mass terms from:

BYg

iWg

iD

VDDL

ii

S

22

)()()('

...)()()(8

...

...0

))((,08

1...)(

232222122

BggWWgWgv

vBggWBggWvDD bbaa

Justify later: Y=1

More on SM Higgs Mechanism• With massive gauge bosons:

W = (W

1 W2) /2

Z 0 = (g W3 - g’B )/ (g2+g’2)

• Orthogonal combination to Z is massless photon

A 0 = (g’ W3+gB )/ (g2+g’2)

• Weak mixing angle defined

• Z = - sin WB + cosWW3

• A = cos WB + sinWW3

MW=gv/2MZ=(g2+g’2)v/2

MW=MZ cos W

2222sincos

gg

g

gg

gWW

More on SM Higgs Mechanism

• Generate mass for W,Z using Higgs mechanism– Higgs VEV breaks SU(2) x U(1)U(1)em

– Single Higgs doublet is minimal case

• Just like Abelian Higgs model

– Goldstone Bosons

• Before spontaneous symmetry breaking:

– Massless Wi, B, Complex • After spontaneous symmetry breaking:

– Massive W,Z; massless ; physical Higgs boson h

vi ii

e

2

1' zi ,

Fermi Model

• Current-current interaction of 4 fermions

• Consider just leptonic current

• Only left-handed fermions feel charged current weak interactions (maximal P violation)

• This induces muon decay

JJGL FFERMI 22

hceJ elept

2

1

2

1 55

sG2

e

e

GF=1.16639 x 10-5 GeV-2

This structure known since Fermi

Fermion Multiplet Structure

L couples to W (cf Fermi theory)

– Put in SU(2) doublets with weak isospin I3=1

R doesn’t couple to W

– Put in SU(2) doublets with weak isospin I3=0

• Fix weak hypercharge to get correct coupling to photon

Fermions in U(1) Theory

• Lagrangian is invariant under global U(1) symmetry

• Gauge the symmetry by requiring a local symmetry, (x)

• Local symmetry requires minimal substitution

)( ff miL

eiQ fe

)(xeAiQD f

Coupling Fermions to SU(2) x U(1) Gauge Fields

• In terms of mass eigenstates

• Re-arrange couplings

BY

giWigD aa

22

23IY

Qem

YAgg

ggiYggZ

ggiWW

giD

3

22

232

22 22

1

22

22 gg

gge

ieQAIZg

iWWg

iD WW

23 sin2

cos222

W

eg

sin

Now include Leptons

• Simplest case, include an SU(2) doublet of left-handed leptons

• Only right-handed electron, eR=(1+5)e/2

– No right-handed neutrino

• Define weak hypercharge, Y, such that Qem=(I3+Y)/2

– YeL=-1

– YeR=-2

eeL

L

L

)1(2

1

)1(2

1

5

5

I3=1

To make charge come out right

*Standard Model has massless neutrinos—discovery of non-zero neutrino mass evidence for physics beyond the SM

Leptons, 2• By construction Isospin, I3, commutes with weak

hypercharge [I3,Y]=0

• Couple gauge fields to leptons

• Write in terms of charged and neutral currents

emZleptons JeAJZJWJWgkineticL )(

LiiL

RRleptons

Wg

iYBg

ii

eYBg

iieL

22

2

RWRLWLLLZ

LL

emem

eeeeJ

eJ

eeQJ

22 sin2sin21cos2

12

1

2

1

2

1 55

RL

After spontaneous symmetry breaking….

• Couplings to leptons fixed:

)1()1(cos4

:

)1(4

:

)1(22

:

:

55

5

5

eeW

e

LRg

ieZe

giZ

gieW

ieQieee

Le=I3-2Qemsin2W

Re=-2Qemsin2W

Higgs VEV Conserves Electric Charge

23IY

Qem

v

0

2

1

02

0

00

01

vQem

10

01Y

10

013I

Muon decay

• Consider e e

• Fermi Theory:

e

e

• EW Theory:

eF uuuugGie

2

1

2

122 55

eW

uuuugMk

ige

2

1

2

11

255

22

2

For k<< MW, 22G=g2/2MW2

For k>> MW, 1/E2

e

e

W

Higgs Parameters

• G measured precisely

• Higgs potential has 2 free parameters, 2,

• Trade 2, for v2, Mh2

– Large Mh strong Higgs self-coupling

– A priori, Higgs mass can be anything

22 )( V

42

23

22

2

822h

v

Mh

v

Mh

MV hhh

22

22

2

2

vM

v

h

22

2

2

1

82 vM

gG

W

F 212 )246()2( GeVGv F

Parameters of SU(2) x U(1) Sector

• g, g,,v, Mh Trade for:

=1/137.03599911(46) from (g-2)e and quantum Hall effect

– GF=1.16637(1) x 10-5 GeV-2 from muon lifetime

– MZ=91.18760.0021 GeV

– Plus masses

Decay widths trivial

• Calculate decay widths from:

– Z decays measured precisely at LEP

81

2

1)(

2A

MffV

V

222

2

212)(

212)(

eeZF

ZF

LRMG

eeZ

MGZ

f

f

Z

Now Add Quarks to Standard Model

• Include color triplet quark doublet

– Right handed quarks are SU(2) singlets, uR=(1+5)u, dR=(1+5)d

• With weak hypercharge

– YuR=4/3, YdR=-2/3, YQL=1/3

• Couplings of charged current to W and Z’s take the form:

iL

iLL d

uQ

Qem=(I3+Y)/2

uWddWu

gLWqq )1()1(

2255

qZRLq

gL qq

WZqq )1()1(

cos4 55

Wemq

Wemq

QR

QIL

2

23

sin2

sin2

i=1,2,3 for color

Fermions come in generations

R

L

RR

L

ee

dud

u,, ,

R

L

RR

L

scs

c

,, ,

R

L

RR

L

btb

t

,, ,

Except for masses, the generations look identical

Need Complete Generations

• Complete generations needed for anomaly

cancellation

• Triangle diverges; grows with energy

• Anomaly independent of mass; depends only on gauge properties

3

1

)2( k

kdT

n

n

RLi

cbai

for

tttTrT

,1

,,

Sensible theories can’t have anomalies

Anomalies, 2

• Standard Model Particles:

• V=B, A=Wa (SU(2) gauge bosons)

• Nc=3 is number of colors; Ng is number of generations

• Anomaly cancellation requires complete generation

0113

11

3

21 2

222

3

ggcgcem NNNNNQT

Particle SU(3) SU(2)L U(1)Y

(uR,dR) 3 2 1/3

uR 3 1 -1/3

dR 3 1 1/3

(L,eL) 1 2 -1

eR 1 1 2

Gauge boson self-interactions

• Yang-Mills for gauge fields:

• In terms of physical fields:

• Triple and quartic gauge boson couplings fundamental prediction of model Related to each other

2

2

2

)()(2

1

)(4

1

)(4

1

ZWZWs

cieAWAWieWW

WWWWs

cieZZ

WWWWieAAL

W

W

W

W

YM

BBWWL aa

YM 4

1

4

1 ,

BBB

WWgWWW cbabcaaa

Consider W+W- pair production

Example: W+W-

t-channel amplitude:

In center-of-mass frame:

(p)

(q)

e(k) k=p-p+=p--q

)()()()1()1()(8

)( 525

2 pppu

k

kqv

giWWvvAt

W+(p+)

W-(p-)

cos,sin,0,12

cos,sin,0,12

1,0,0,12

1,0,0,12

WW

WW

sp

sp

sq

sp

sM

ppkt

qps

WW /41

)(

)(

2

22

2

W+W- pair production, 2

Interesting physics is in the longitudinal W sector:

Use Dirac Equation: pu(p)=0

s

MO

M

p W

W

2

)()1()(4

)( 52

2

pukqvM

giWWvvA

WLLt

s

MOsGWWvvA W

FLLt

2222

2sin2)(

Grows with energy

W+W- pair production, 3 SM has additional contribution from s-channel Z exchange

For longitudinal W’s

)()()()()()()1()()(4

)(252

2

ppkpgkpgppg

M

kkgpuqv

Ms

giWWA

ZZs

)()1)()((4

)( 52

2

puppqvM

giWWA

WLLs

)()1()(4

)( 52

2

pukqvM

giWWvvA

WLLs

Contributions which grow with energy cancel between t- and s- channel diagrams

Depends on special form of 3-gauge boson couplings

Z(k)

W+(p+)

W-(p-)

(p)

(q)

Feynman Rules for Gauge Boson Vertices

gppgppgpppppV ZZZ )()()(),,(

),,( ZWWV pppVig

WWWZ

WW

eg

eg

cot

p-

p+

pZ

ggggggigWWVV 2

WWWZZ

WZWW

WW

eg

eg

eg

22

2

2

cot

cot

ggggggig 22

No deviations from SM at LEP2

LEP EWWG, hep-ex/0312023

No evidence for Non-SM 3 gauge boson vertices

Contribution which grows like me

2s cancels between Higgs diagram and others

What about fermion masses?

• Fermion mass term:

• Left-handed fermions are SU(2) doublets

• Scalar couplings to fermions:

• Effective Higgs-fermion coupling

• Mass term for down quark:

RRLLmmL

..chdQL RLdd

L

L d

uQ

..0

),(2

1chd

hvduL RLLdd

v

M dd

2

Forbidden by SU(2)xU(1) gauge invariance

Fermion Masses, 2

• Mu from c=i2* (not allowed in SUSY)

• For 3 generations, , =1,2,3 (flavor indices)

0

*

hcuQL RLu *v

M uu

2

,

..2

)(chdduu

hvL RLdRLuY

Fermion masses, 3• Unitary matrices diagonalize mass matrices

– Yukawa couplings are diagonal in mass basis– Neutral currents remain flavor diagonal– Not necessarily true in models with extended

Higgs sectors

mRdR

mRuR

mLdL

mLuL

dVduVu

dUduUu

• Charged current:m

LdumLLL dVUuduJ

)(

2

1

2

1

CKM matrix

Review of Higgs Couplings

• Higgs couples to fermion mass– Largest coupling is to heaviest fermion

– Top-Higgs coupling plays special role?– No Higgs coupling to neutrinos

• Higgs couples to gauge boson masses

• Only free parameter is Higgs mass!• Everything is calculable….testable theory

hffffv

mfhf

v

mL LRRL

ff

....cos

hZZgM

hWWgMLW

ZW

v=246 GeV

WWW

F eM

Gg

22

222

sin

4

sin8

2

Review of Higgs Boson Feynman Rules

• Higgs couples to heavy particles

• No tree level coupling to photons () or gluons (g)

• Mh2=2v2large Mh is strong

coupling regime– Mh is parameter which

separates perturbative/non-perturbative regimes

Higgs Searches at LEP2• LEP2 searched for e+e-Zh• Rate turns on rapidly after threshold, peaks just above threshold,

3/s• Measure recoil mass of Higgs; result independent of Higgs decay

pattern– Pe-=s/2(1,0,0,1)– Pe+=s/2(1,0,0,-1)– PZ=(EZ, pZ)

• Momentum conservation:– (Pe-+Pe+-PZ)2=Ph

2=Mh2

– s-2 s EZ+MZ2= Mh

2

• LEP2 limit, Mh > 114.1 GeV

Conclusion

• Standard Model consistent theory– Well tested (see lectures 2&3)– Theoretical issues (see lectures 4&5)