What do we know about the Standard Model?

Sally Dawson

Lecture 2

SLAC Summer Institute

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

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

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