# Presentation: 454795

## What Do We Know About EWSB?

- Sally Dawson (BNL)
- Lecture 2
- XIII Mexican School of Particles and Fields, 2008

## 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

## Basics, #2

- Chose parameters in gauge/Higgs sector
-  =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

## Inadequacy of Tree Level Calculations

- Mixing angle is predicted quantity
- On-shell definition cos2W=MW2/MZ2
-  Predict MW
- Plug in numbers:
- MW predicted =80.939 GeV
- MW(exp) =80.399  0.025 GeV
- Need to calculate beyond tree level

## Quantum Corrections

- Relate tree level to one-loop corrected masses
- Majority of corrections at one-loop are from 2-point functions

- Note sign conventions for 2-point functions

## Example of Quantum Corrections

- Example:

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

## Top Quark Corrections to  Parameter

- 2-point functions of W, Z

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

- (Neglecting log(mt) pieces and using on-shell definition of sinW)

## Heavy Higgs Contribution to 

- In on-shell scheme, Higgs contributes logarithmically to quantum corrections & top quark contributes quadratically

## Modification of tree level relations

- Contributions to r from top quark and Higgs loops

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

- Extreme sensitivity of precision measurements to mt

## Understanding Higgs Limit

- Theory:  Input MZ, GF, 
- 	→ Predict MW

- Consistency between direct and indirect measurements of MW and mt a strong test of theory!

## Precision Measurements Limit MH

- LEP EWWG (July, 2008):
- mt=172.4  1.2 GeV
- MH=84+34-26 GeV
- MH < 154 GeV (one-sided 95% cl)
- MH < 185 GeV (Precision measurements plus direct search limit)

***Best fit in region excluded from direct searches***

## Caveats

- Low Q2 data not included in fit
- Doesn’t include atomic parity violation in cesium, parity violation in Moller scattering, & neutrino-nucleon scattering (NuTeV)
- Higgs fit not hugely sensitive to low Q2 data
- MH< 185 GeV
- Higgs limit moves around with mt

*Higgs limit assumes SM!*

## EW Measurements test SM

- We have a model....
- And it works to the 1% level

- Consistency  of precision measurements at multi-loop level used to constrain models with new physics
- If a new model predicts some deviation from the SM, it has to be small

- This fit ASSUMES SM

## Limits on MH Assume SM

- 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 

- MH (GeV)

- T

## 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)

- 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, (#4)

- 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
- S/T/U approach subtracts off SM contributions

## Theoretical Limits on MH

- Unitarity
- If unitarity is violated, interactions grow with energy (cf longitudinal W’s)
- Perturbativity of couplings
- If perturbativity is violated, loop corrections may be larger than tree
- Limits tell us where minimal SM is valid
- Unitarity and perturbativity provide strong limits on beyond the SM physics
- These are model builders tools
- Renormalization of Higgs mass
- What about naturalness?

## Unitarity

- Consider 2  2 elastic scattering
- Partial wave decomposition of amplitude
- al are the spin *l* partial waves

## Unitarity

- Pl(cos) are Legendre polynomials:

-  Sum of positive definite terms

## More on Unitarity

- Optical theorem
- Unitarity requirement:

- Optical theorem derived assuming only conservation of probability

## More on Unitarity

- Idea:  Use unitarity to limit parameters of theory

- Cross sections which grow with energy always violate unitarity at some energy scale

-  Remember WL(p) with L~p/MW

## Aside on WW Scattering

- This is a statement about scattering amplitudes, NOT individual Feynman diagrams

-  , z are Goldstone bosons which are eaten by the Higgs mechanism to give the W & Z bosons their longitudinal components

## W+W-W+W-

- Recall scalar potential
- +-+

- p

- q

- p’

- q’

##  +-+-

- Two interesting limits:
- s, t  >> MH2
- s, t  << MH2

- Remember: This is a physical process

## Use Unitarity to Bound Higgs

- High energy limit:
- Heavy Higgs limit

- MH < 800  GeV

- Ec 1.7 TeV
-  ***New physics at the TeV scale***

- Can get more stringent bound from coupled channel analysis

## Another Sort of Limit: Landau Pole

- MH is a free parameter in the Standard Model
- Can we derive limits from consistency?
- Consider a scalar potential:
- This is potential at electroweak scale
- Parameters evolve with energy in a calculable way

## Consider HHHH

- Real scattering, s+t+u=4MH2
- Consider momentum space-like and off-shell: s=t=u=Q2<0
- Tree level: iA0=-6i

##  HHHH, #2

- One loop:
- A=A0+As+At+Au

## HHHH, #3

- Sum the geometric series to define running coupling
- (Q) blows up as Q (called Landau pole)

## HHHH, #4 

- This is independent of starting point
- BUT.... Without 4 interactions, theory is non-interacting
- Require quartic coupling be finite

## HHHH, #5

- Use =MH2/(2v2) and approximate log(Q/MH)  log(Q/v)
- Requirement for 1/(Q)>0 gives upper limit on MH
- Assume theory is valid to 1016 GeV
- Gives upper limit on MH< 180 GeV
- Can add fermions, gauge bosons, etc.

- We expect Higgs at electroweak scale

## High Energy Behavior of 

- Renormalization group scaling
*Large **** (Heavy Higgs):* self coupling causes   to grow with scale
*Small  ****  (Light Higgs):* coupling to top quark causes   to become negative

## Does Spontaneous Symmetry Breaking Happen? 

- SM requires spontaneous symmetry
- This requires
- For small 
- Solve

## Does Spontaneous Symmetry Breaking Happen? (#2)

- () >0 gives lower bound on MH
- If Standard Model valid to 1016 GeV
- For any given scale, , there is a theoretically consistent range for MH

## Light Higgs Theoretically Attractive

- Extrapolate Higgs potential to high scale 
-                                V=  (+ - v2)2

- Standard Model is only consistent to GUT scale for small range of Higgs masses

- Forbidden

- Forbidden

- Allowed

- Heavy Higgs implies new physics at some low scale

## Problems with the Higgs Mechanism

- We often say that the SM cannot be the entire story because of the quadratic divergences of the Higgs Boson mass
- Whether this is a problem or not is somewhat a matter of taste

## Masses at One-Loop

- First consider a fermion coupled to a massive complex Higgs scalar
- Assume symmetry breaking as in SM:

## Masses at One-Loop, #2

- Calculate mass renormalization for 

- H

## Renormalized Fermion Mass

- Do integral in Euclidean space

## Renormalized Fermion Mass, #2

- Renormalization of fermion mass:

## Symmetry and the Fermion Mass

- mF  mF
- mF=0, then quantum corrections vanish
- When mF=0, Lagrangian is invariant under
- LeiLL
- ReiRR
- mF0 increases the symmetry of the threoy
- Yukawa coupling (proportional to mass) breaks symmetry and so corrections  mF

## Scalars are very different

- MH diverges quadratically!
***Quadratic sensitivity to high mass scales***

- H

## Scalars (#2)

- MH diverges quadratically!
- Requires large cancellations (hierarchy problem)
- Can do this in Quantum Field Theory
- H does not obey decoupling theorem
- Effects of heavy particle (H) does not decouple as MH
- MH0 doesn’t increase symmetry of theory
- Nothing protects Higgs mass from large corrections

## Light Scalars are Unnatural

- Higgs mass grows with scale of new physics, 
- No additional symmetry for MH=0, no protection from large corrections

- H

- H

## What’s the problem?

- Compute MH in dimensional regularization and absorb infinities into definition of MH
- Perfectly valid approach
- Except we know there is a high scale (associated with gravity)

## Try to cancel quadratic divergences by adding new particles

- SUSY models add scalars with same quantum numbers as fermions, but different spin
- Little Higgs models cancel quadratic divergences with new particles with same spin

- New particles assumed to be at TeV scale for cancellation of quadratic divergences

## We expect something new at the TeV scale

- If it’s a SM Higgs then we have to think hard about what the quadratic divergences are telling us
- SM Higgs mass is highly restricted by requirements of agreement with precision electroweak data and theoretical consistency