# Presentation: 802053

## What do we know about the Standard Model?

- Sally Dawson
- Lecture 4
- TASI, 2006

## The Standard Model Works

- Any discussion of the Standard Model has to start with its success
- This is unlikely to be an accident!

## 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

## Example 1: W+W-W+W-

- Recall scalar potential (Include Goldstone Bosons in Unitarity gauge)
- Consider Goldstone boson scattering:
-      +-+

##  +-+-

- Two interesting limits:
- s, t  >> Mh2
- s, t  << Mh2

## 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

## Electroweak Equivalence Theorem

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

## Plausibility argument for Electroweak Equivalence Theorem

- Compute (hWL+WL-) for Mh>>MW

- (hWW)  Mh
- for Mh 1.4 TeV

## Landau Pole

- Mh is a free parameter in the Standard Model
- Can we derive limits on the basis of 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.

## 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

## Bounds on SM Higgs Boson

- If SM valid up to Planck scale, only a small range of allowed Higgs Masses

## 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

## 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 

## 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!
- This implies quadratic sensitivity to high mass scales

## Scalars (#2)

- Mh diverges quadratically!
- Requires large cancellations (hierarchy problem)
- Can do this in Quantum Field Theory
- h does not obey decoupling theorem
- Says that effects of heavy particles decouple as M
- 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

## 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

## We expect something 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 requirement of theoretical consistency