What do we know about the Standard Model

1
What do we know about the Standard Model?

• Sally Dawson
• Lecture 2
• SLAC Summer Institute

2
The Standard Model Works

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

3
Theoretical Limits on Higgs Sector

• Unitarity
• Really we mean perturbative unitarity
• Violation of perturbative unitarity leads to
  consideration of strongly interacting models of
  EWSB such as technicolor, Higgless
• Consistency of Standard Model
• Triviality (What happens to couplings at high
  energy?)
• Does spontaneous symmetry breaking actually
  happen?
• Naturalness
• Renormalization of Higgs mass is different than
  renormalization of fermion mass
• One motivation for supersymmetric models

4
Unitarity

• Consider 2 ? 2 elastic scattering
• Partial wave decomposition of amplitude
• al are the spin l partial waves

scenter of mass energy-squared
5
Unitarity

• Pl(cos?) are Legendre polynomials

Sum of positive definite terms
6
More on Unitarity

• Optical theorem
• Unitarity requirement

Optical theorem derived assuming only
conservation of probability
Im(al)
Re(al)
7
More on Unitarity

• Idea Use unitarity to limit parameters of
  theory

Cross sections which grow with energy always
violate unitarity at some energy scale
8
Example WW-?WW-
Electroweak Equivalence theorem
A(WLWL - ?WLWL-) A(?? - ? ??-)O(MW2/s)
? are Goldstone bosons which become the
longitudinal components of massive W and Z gauge
bosons
9
WW-?WW-

• Consider Goldstone boson scattering ??-???
• Recall scalar potential

10
??-???-

• Two interesting limits
• s, t gtgt MH2
• s, t ltlt MH2

11
Use Unitarity to Bound Higgs

• High energy limit
• Heavy Higgs limit

MH lt 800 GeV
Ec ?1.7 TeV ? New physics at the TeV scale
Can get more stringent bound from coupled channel
analysis
12
Consider WW- pair production

• Example ???WW-
• t-channel amplitude
• In center-of-mass frame

?(p)
W?(p)
kp-pp--q
e(k)
?(q)
W?-(p-)
13
WW- pair production, 2

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

Grows with energy
14
WW- pair production, 3

• SM has additional contribution from s-channel Z
  exchange
• For longitudinal Ws

W?(p)
?(p)
Z(k)
W?-(p-)
?(q)
Contributions which grow with energy cancel
between t- and s- channel diagrams
Depends on special form of 3-gauge boson couplings
15
No deviations from SM at LEP2
No evidence for Non-SM 3 gauge boson vertices
Contribution which grows like me2s cancels
between Higgs diagram and others
LEP EWWG, hep-ex/0312023
16
Limits on Scalar Potential

• 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

17
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

18
Does Spontaneous Symmetry Breaking Happen?

• SM requires spontaneous symmetry
• This requires
• For small ?
• Solve

19
Does Spontaneous Symmetry Breaking Happen?

• ?(?) gt0 gives lower bound on MH
• If Standard Model valid to 1016 GeV
• For any given scale, ?, there is a theoretically
  consistent range for MH

20
What happens for large ??

• Consider HH?HH
• ?(Q) blows up as Q?? (called Landau pole)

21
Landau Pole

• ?(Q) blows up as Q??, independent of starting
  point
• BUT. Without ?H4 interactions, theory is
  non-interacting
• Require quartic coupling be finite
• Requirement for 1/?(Q)gt0 gives upper limit on Mh
• Assume theory is valid to 1016 GeV
• Gives upper limit of MHlt 180 GeV

22
Bounds on SM Higgs Boson

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

MH (GeV)
? (GeV)
23
Naturalness

• We often say that the SM cannot be the entire
  story because of the quadratic divergences of the
  Higgs Boson mass
• Renormalization of scalar and fermion masses are
  fundamentally different

24
Masses at one-loop

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

25
Masses at one-loop

• Calculate mass renormalization for ?

To calculate with a cut-off, see my Trieste notes
26
Symmetry and the fermion mass

• ?MF ? MF
• MF0, then quantum corrections vanish
• When MF0, Lagrangian is invariant under
• ?L?ei?L?L
• ?R?ei?R?R
• MF?0 increases the symmetry of the theory
• Yukawa coupling (proportional to mass) breaks
  symmetry and so corrections ? MF

27
Scalars are very different

• MH diverges quadratically!
• This implies quadratic sensitivity to high mass
  scales

28
Scalars

• MH diverges quadratically
• Requires large cancellations (hierarchy problem)
• H does not obey decoupling theorem
• Says that effects of heavy particles decouple as
  M??
• MH?0 doesnt increase symmetry of theory
• Nothing protects Higgs mass from large corrections

29
Light Scalars are Unnatural

• Higgs mass grows with ?
• No additional symmetry for MH0, no protection
  from large corrections

H
H
MH ? 200 GeV requires large cancellations
30
Whats 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

31
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

32
We expect something at the TeV scale

• If its 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
• Expect that Tevatron or LHC will observe SM Higgs
  (or definitively exclude it)