# Presentation: 725857

## Radiative Corrections to the Standard Model

- S. Dawson
- TASI06, Lecture 3

## Radiative Corrections

- Good References:
- Jim Wells, TASI05, hep-ph/0512342
- K. Matchev, TASI04, hep-ph/0402031
- M. Peskin and T. Takeuchi, Phys. Rev. D46, (1992) 381
- W. Hollik, TASI96, hep-ph/9602380

## Basics

- SM is SU(2) x U(1) theory
- Two gauge couplings: g and g’
- Higgs potential is V=-22+4
- Two free parameters
- Four free parameters in gauge-Higgs sector
- Conventionally chosen to be
-  =1/137.0359895(61)
-  GF =1.16637(1) x 10-5 GeV -2
-  MZ=91.1875  0.0021 GeV
-  MH
- Express everything else in terms of these parameters

## Example:  parameter

- At tree level,
- Many possible definitions of ; use

- At tree level GCC=GNC=GF

##   parameter (#2)

- Top quark contributes to W and Z 2-point functions

- qq piece of propagator connects to massless fermions, and doesn’t contribute here

## Top Corrections to  parameter (Example)

- 2-point function of Z
- Shift momentum, k’=k + px

- Nc=number of colors
- n=4-2 dimensions

## Top Corrections to  parameter (#2)

- Shift momenta, keep symmetric pieces
- Only need g pieces

## Top Corrections to  parameter (#3)

-  Only need answer at p2=0

-  Dimensional regularization gives:

## Top Corrections to  parameter (#4)

- 2-point function of W, Z

- Lt=1-4sW2/3
- Rt=-4sW2/3

## Top quark doesn’t decouple

- Longitudinal component of W couples to top mass-- eg, tbW coupling:
- Decoupling theorem doesn’t apply to particles which couple to mass

- For longitudinal W’s:

## Why doesn’t the top quark decouple?

- In QED, running of  at scale  not affected by heavy particles with M >> 
- Decoupling theorem:  diagrams with heavy virtual particles don’t contribute at scales  << M  if
*Couplings don’t grow with M *
*Gauge theory with heavy quark removed is still renormalizable*
- Spontaneously broken SU(2) x U(1) theory violates both conditions
*Longitudinal modes of gauge bosons grow with mass**	*
*Theory without top quark is not renormalizable*
- Effects from top quark grow with mt2

***Expect m******t ****** to have  large effect on precision observables***

## Modification of tree level relations

- r is a physical quantity which incorporates 1-loop corrections

## Corrections to GF

- GF defined in terms of muon lifetime
- Consider 4-point interaction,with QED corrections
- Gives precise value:
- Angular spectrum of decay electrons tests V-A properties

## GF at one loop

- Vertex and box corrections are small
- 	(but 1/ poles don’t cancel without them)

## Renormalizing Masses

- Propagator corrections form geometric series

## Running of , 1

- (0)=1/137.0359895(61)
- (p2)= (0)[1+]

## Running of , 2

- 

- 

- p

- k

- k+p

- Photon is massless:(0)=0

- Calculate in n=4-2 dimensions

## Running of , 3

- Contributions of heavy fermions decouple:

- Contributions of light fermions:

## Running of , 4

-  From leptons: lept(MZ)=0.0315

-  Light quarks require  at low p2
- Strong interactions not perturbative
- Optical theorem plus dispersion relation:

## Have we seen  run?

- Langacker, Fermilab Academic Series, 2005

## Final Ingredient is sW2

- sW is not an independent parameter

## Predict MW

- Use on-shell scheme:
- r incorporates the 1-loop radiative corrections and is a function of , s, Mh, mt,...

## Data prefer light Higgs

- Low Q2 data not included
- Doesn’t include atomic parity violation in cesium, parity violation in Moller scattering, & neutrino-nucleon scattering (NuTeV)
- Higgs fit not sensitive to low Q2 data
- Mh< 207 GeV
- 1-side 95% c.l. upper limit, including direct search limit

- Direct search limit from e+e-Zh

## Logarithmic sensitivity to Mh

- Data prefer a light Higgs

- 2006

## Understanding Higgs Limit

- MW(experiment)=80.404  0.030

## Light Higgs and Supersymmetry?

- Adding the extra  particles of a SUSY model changes the fit, but a light Higgs is still preferred

## How to define sin2W

- At tree level, equivalent expressions
- Can use these as definitions for renormalized sin2W
- They will all be different at the one-loop level

- On-shell

- Z-mass

- Effective sW(eff)

## sin2w depends on scale

- Moller scattering,
- e-e-e-e-
- -nucleon scattering
- Atomic parity violation in Cesium

## S,T,U formalism

- Suppose “new physics” contributes primarily to gauge boson 2 point functions
- cf r where vertex and box corrections are small
- Also assume “new physics” is at scale M>>MZ
- Two point functions for  , WW, ZZ, Z

## S,T,U, (#2)

- Taylor expand 2-point functions
- Keep first two terms
- Remember that QED Ward identity requires any amplitude involving EM current vanish at q2=0

## S,T,U (#3)

-  Use JZ=J3-sW2JQ

- Normalization of JZ differs by -1/2 from Lecture 2 here

## S,T,U (#4)

- To O(q2), there are 6 coefficients:
- Three combinations of parameters absorbed in , GF, MZ
- In general, 3 independent coefficients which can be extracted from data

## S,T,U, (#5)

- Advantages:  Easy to calculate
-                       Valid for many models
- 	           Experimentalists can give you model
-                                independent fits

- SM contributions in , GF, and MZ

## Limits on S & T

- A model with a heavy Higgs requires a source of large (positive) T
- Fit assumes Mh=150 GeV

## Higgs can be heavy in models with new physics

- Specific examples of heavy Higgs bosons in Little Higgs and Triplet Models
- Mh  450-500 GeV allowed with large isospin violation (T=) and higher dimension operators

- We don’t know what the model is which produces the operators which generate large T

## Compute S,T, U from Heavy Higgs

-  SM values of S, T, U are defined for a reference Mh0

## Using S,T, & U

- Assume new physics only contributes to gauge boson 2-point functions (Oblique corrections)
- Calculate observables in terms of SM contribution and S, T, and U

## Conclusion

- Radiative corrections necessary to fit LEP and Tevatron data
- Radiative corrections strongly limit new models