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The Standard Model Works
•Any discussion of the Standard Model has to start with its success
•This is unlikely to be an accident
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
Unitarity
• Consider 2 2 elastic scattering
• Partial wave decomposition of amplitude
• al are the spin l partial waves
2
264
1A
sd
d
0
)(cos)12(16l
ll aPlA
s=center of mass energy-squared
Unitarity
• Pl(cos) are Legendre polynomials:
)(cos)(coscos)12()12(8 1
1
*
00
llllll
PPdaalls
1
1
,
12
2)()(
lxPxdxP ll
ll
0
2)12(
16
llal
s
Sum of positive definite terms
More on Unitarity
0
2)12(
16)0(Im
1
llal
sA
s
2)Im( ll aa
2
1)Re( la
Optical theorem derived assuming only conservation of probability
Re(al)
Im(a
l)
• Optical theorem
• Unitarity requirement:
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: W+W-W+W-
A(WL+WL
- →WL+WL
-) =A(+ - → +-)+O(MW2/s)
Electroweak Equivalence theorem:
are Goldstone bosons which become the longitudinal components of massive W and Z gauge bosons
W+W-W+W-
• Consider Goldstone boson scattering: +-+
• Recall scalar potential
2222
222
22
2
28
222
zHv
MzHH
v
MH
MV HHH
2
22
2
22
2
2
2)(h
h
h
hh
Ms
i
v
Mi
Mt
i
v
Mi
v
MiiA
+-+-
• Two interesting limits:
– s, t >> MH2
– s, t << MH2
2
2
2)(v
MA H
2)(
v
uA
2
200 8 v
Ma H
200 32 v
sa
Use Unitarity to Bound Higgs
• High energy limit:
• Heavy Higgs limit
2
1)Re( la
2
200 8 v
Ma h
200 32 v
sa
MH < 800 GeV
Ec 1.7 TeV
New physics at the TeV scale
Can get more stringent bound from coupled channel analysis
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
)()()()1()1()(8
)( 525
2 pppu
k
kqv
giWWvvAt
W+(p+)
W-(p-)
cos,sin,0,12
cos,sin,0,12
1,0,0,12
1,0,0,12
WW
WW
sp
sp
sq
sp
sM
ppkt
qps
WW /41
)(
)(
2
22
2
W+W- pair production, 2
Interesting physics is in the longitudinal W sector:
Use Dirac Equation: pu(p)=0
s
MO
M
p W
W
2
)()1()(4
)( 52
2
pukqvM
giWWvvA
WLLt
s
MOsGWWvvA W
FLLt
2222
2sin2)(
Grows with energy
W+W- pair production, 3 SM has additional contribution from s-channel Z
exchange
For longitudinal W’s
)()()()()()()1()()(4
)(252
2
ppkpgkpgppg
M
kkgpuqv
Ms
giWWA
ZZs
)()1)()((4
)( 52
2
puppqvM
giWWA
WLLs
)()1()(4
)( 52
2
pukqvM
giWWvvA
WLLs
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)
No deviations from SM at LEP2
LEP EWWG, hep-ex/0312023
No evidence for Non-SM 3 gauge boson vertices
Contribution which grows like me
2s cancels between Higgs diagram and others
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
422
42HH
MV h
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
)(12121216 4222 gaugeggdt
dtt
2
2
logQ
tv
Mg tt
Does Spontaneous Symmetry Breaking Happen?
• SM requires spontaneous symmetry
• This requires
• For small
• Solve
)0()( VvV
42 1616 tgdt
d
2
2
2
4
log4
3)()(
v
gv t
Does Spontaneous Symmetry Breaking Happen?
() >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
2
2
2
22 log
2
3
v
vM H
GeVM H 130
What happens for large ?
• Consider HH→HH
(Q) blows up as Q (called Landau pole)
)(6
log89
1
6
2
Q
MQ
A
H
...log16
916
2
2
2
HM
QA
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)>0 gives upper limit on Mh
• Assume theory is valid to 1016 GeV– Gives upper limit of MH< 180 GeV
0)(
1
Q
2
2
222
log9
32
vQ
vM H
Bounds on SM Higgs Boson
• If SM valid up to Planck scale, only a small range of allowed Higgs Masses
(GeV)
MH (
GeV
)
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
Masses at one-loop
• First consider a fermion coupled to a massive complex Higgs scalar
• Assume symmetry breaking as in SM:
..)(22
chmiL RLFs
22
)( vM
vH FF
Masses at one-loop
• Calculate mass renormalization for
.....log32
32
2
2
2
F
FFF M
MM
To calculate with a cut-off, see my Trieste notes
Symmetry and the fermion mass
MF MF
– MF=0, then quantum corrections vanish
– When MF=0, Lagrangian is invariant under
LeiLL
ReiRR
– MF0 increases the symmetry of the theory
– 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
....log8
)( 222
2222
FFH
FHSH M
MMMM
Scalars • MH diverges quadratically
• Requires large cancellations (hierarchy problem)
• H does not obey decoupling theorem– Says that effects of heavy particles
decouple as M• MH0 doesn’t increase symmetry of theory
– Nothing protects Higgs mass from large corrections
2
22222
2
2
GeV200TeV 0.7
123624
tHZWF
H MMMMG
M
MH 200 GeV requires large cancellations
• Higgs mass grows with • No additional symmetry for MH=0, no
protection from large corrections
H H
Light Scalars are Unnatural
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
(...)12
02
HH MM
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