The Importance of the TeV Scale

Sally Dawson

Lecture 3

FNAL LHC Workshop, 2006

The Standard Model Works

Any discussion of the Standard Model has to start with its success

This is unlikely to be an accident!

Issues with the Standard Model Unitarity Landau pole Triviality Dependence of Higgs mass on high scale

physics

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

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

Optical theorem

Unitarity requirement:

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

More on Unitarity

Idea: Use unitarity to limit parameters of theory

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

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

Feynman Rules for Gauge Boson Vertices

gppgppgpppppV ZZZ )()()(),,(

),,( ZWWV pppVig WWWZ

WW

eg

eg

cot

p-

p+

pZ

ggggggigWWVV 2

WWWZZ

WZWW

WW

eg

eg

eg

22

2

2

cot

cot

ggggggig 22

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

Example: W+W-W+W-

Recall scalar potential (Include Goldstone Bosons)

Consider Goldstone boson scattering:

+-+

2222

2

222

22

28

222

zhv

M

zhhv

Mh

MV

h

hh

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 >> Mh

2

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

Electroweak Equivalence Theorem

2

2

1111 )......()()()......(

E

MO

AiiVVVVA

W

NNNNN

LLNLL

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

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

422

42hh

MV h

Consider hhhh

Real scattering, s+t+u=4Mh2

Consider momentum space-like and off-shell: s=t=u=Q2<0

Tree level: iA0=-6i

hhhh, #2

One loop:

A=A0+As+At+Au

)1()()4(8

9

)()2(2

1)6(

2222

2

222222

xxQM

Mqpk

i

Mk

ikdiiA

h

hh

n

s

...)1()()4(

16

916 222

2

xxQMA h

hhhh, #3

Sum the geometric series to define running coupling

(Q) blows up as Q (called Landau pole)

)(6

log89

1

6

2

Q

MQ

A

h

...log16

916

2

2

2

hM

QA

hhhh, #4

This is independent of starting point BUT…. Without 4 interactions, theory is non-

interacting Require quartic coupling be finite

0)(

1

Q

hhhh, #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.

2

2

222

log9

32

vQ

vM 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

Q

Qlog(...)

)(

1

)(

1

)(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? (#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

2

2

2

22 log

2

3

v

vM h

GeVM h 130

Bounds on SM Higgs Boson

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

More Problems 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:

..)(22

chmiL RLFs

22

)( vm

vh FF

Masses at one-loop, #2

Calculate mass renormalization for

.....log32

32

2

2

2

F

FFF m

mm

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 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

22

2

1

22

222

2222

11)

22(

log8

)(

Om

mI

mm

mmmmM

F

ssF

FFs

FsSh

1

0

1 )1(1log)( xaxdxaI

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

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 scale of new physics, • 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