# Presentation: 702654

## Sally Dawson, BNLIntroduction 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

## Science Timeline

- Tevatron

- LHC

- LHC Upgrade

- ILC

- 2006

- 2007

- 2012

## 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

- 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:

## Abelian Higgs Model, 3

- Case 1: 2 >  0
- QED with MA=0 and m=
- Unique minimum at =0

- By convention,  > 0

## Abelian Higgs Model, 4

- Case 2: 2 < 0
- Minimum energy state at:

- 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 –22 > 0
- Massless scalar field  *(Goldstone Boson)*

-  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

## Higgs  Mechanism summarized

***Spontaneous breaking of a gauge theory***
***by 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 Invariance

- What 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

## More on R gauges

- 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)

- For SU(2): a=a/2

## Non-Abelian Higgs Mechanism, 2

- In exact analogy to the Abelian case
*********a************0 ************0***       Massive vector boson + Goldstone boson
*********a************0******=0       *** Massless vector boson + massive scalar field

## 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:

## 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: 

- Electroweak

- QCD

## 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?

- Invariant under  - 

## 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:

- Justify later: Y=1

## More on SM Higgs Mechanism

- With massive gauge bosons:
-    W  = (W1    W2) /2
-    Z 0 = (g W3 - g’B )/ (g2+g’2)
- Orthogonal combination to Z is massless photon
-    A 0 = (g’ W3+gB )/ (g2+g’2)
- Weak mixing angle defined
- Z = - sin WB + cosWW3
- A =  cos WB +  sinWW3

- MW=gv/2
- MZ=(g2+g’2)v/2

- MW=MZ cos W

## 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

## 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

- 

- 

- 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

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

- In terms of mass eigenstates
- Re-arrange couplings

## 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

- 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

## After spontaneous symmetry breaking....

- Couplings to leptons fixed:

- Le=I3-2Qemsin2W
- Re=-2Qemsin2W

## Higgs VEV Conserves Electric Charge

## Muon decay

- Consider  e e
- Fermi Theory:

-  

- e

- 

- e

-  EW Theory:

**For ********k********<< M****W****, 2********2G********=g****2****/2M****W****2**

**For ********k********>> M****W****,  ********1/E****2**

- 

- 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

## 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=91.18760.0021 GeV
- Plus masses

## Decay widths trivial

- Calculate decay widths from:
- Z decays measured precisely at LEP

- 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:

- Qem=(I3+Y)/2

-  i=1,2,3  for color

## Fermions come in generations

- 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

- 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

| 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*

## 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

- W+(p+)

- W-(p-)

## W+W- pair production, 2

- Interesting physics is in the longitudinal W sector:
- Use Dirac Equation:  pu(p)=0

- Grows with energy

## W+W- pair production, 3

- SM has additional contribution from s-channel Z exchange
- For longitudinal W’s

- 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

- p-

- p+

- pZ

## No deviations from SM at LEP2

- LEP EWWG, hep-ex/0312023

- No evidence for Non-SM 3 gauge boson vertices

- Contribution which grows like me2s 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:

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

- 

## Fermion Masses, 2

- Mu from c=i2* (not allowed in SUSY)
- For 3 generations, , =1,2,3 (flavor indices)

## 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

-    Charged current:

- 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***

- v=246 GeV

## Review of Higgs Boson Feynman Rules

- Higgs couples to heavy particles
***No tree level coupling to photons (************) or gluons (g)***
- Mh2=2v2large 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=Ph2=Mh2
-  s-2 s EZ+MZ2= Mh2
- LEP2 limit, ***M******h ******> 114.1 GeV***

## Conclusion

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