The Top Quark and Precision Measurements

S. Dawson

BNL

April, 2005

M.-C. Chen, S. Dawson, and T. Krupovnikas, in preparation

M.-C. Chen and S. Dawson, hep-ph/0311032

Standard Model Case is Well Known

• EW sector of SM is SU(2) x U(1) gauge theory– 3 inputs needed: g, g’, v, plus fermion/Higgs masses– Trade g, g’, v for precisely measured G, MZ,

– SM has =MW2/(MZ

2c2)=1 at tree level• s is derived quantity

– Models with =1 at tree level include• MSSM• Models with singlet or doublet Higgs bosons• Models with extra fermion families

2

22

2 ZMGsc

2/122

22

2 24

'4

Gv

cg

sg

We have a model….And it works to the 1% level

EW Measurements test consistency of SM

Consistency of precision measurements at multi-loop level used to constrain models with new physics

2005

Models with 1 at tree level are different

• SM with Higgs Triplet

• Left-Right Symmetric Models

• Little Higgs Models

• …..many more

• These models need additional input parameter

• Decoupling is not so obvious beyond tree level

NEWSM LLL

As the scale of the new physics becomes large, the SM is not always recovered, violating our intuition

i

ii

NEW Oc

L2

Lore: Effects of LNEW become very small as

Muon Decay in the SM

• At tree level, muon decay related to input parameters:

• One loop radiative corrections included in parameter rZ

• Where:

22222 22 WZ MsMcsG

2

2

2

22

2

2

s

s

c

sc

M

M

G

Gr

Z

Z

)1(2 222

rMcs

GZ

e

e

W

122

2

Z

W

Mc

M

If 1, there would be 4 input parameters

Calculate top quark contribution to rZ

(mt2 dependence only)

• Muon decay constant:

• Vertex and box corrections, V-B small neglect

• Vacuum polarization, /, has no quadratic top mass dependence

• Z-boson 2-point function:

BVW

WW

MG

G

2

)0(

2

11)1(

4

32)0( 2

2

2

2

2

tt

cWW mm

QgN

1)1(

4

32 2

2

2

2

22

2

2

2

tZ

tc

Z

Z

m

Q

M

m

c

gN

M

M

Calculate top quark contribution to rZ

(continued)

• Need s2/s2

• From SM relation using on-mass shell definition for s2

2

22 1

Z

W

M

Ms

2

2

2

2

2

2

2

2

2

2

2

2

2

2

64 W

tc

W

W

Z

Z

M

mNg

s

c

M

M

M

M

s

c

s

s

MW and MZ are physical masses

s2/s

2 not independent parameter

Includes all known corrections

2

2

2

282t

cSMt m

s

cNGr

Predict MW in terms of input parameters and mt

2005

What’s different with a Higgs Triplet?

• SM: SU(2) x U(1)– Parameters, g, g’, v

• Add a real triplet, (+,0,-), 0=v

– Parameters in gauge sector: g, g’, v, v

– vSM2=(246 GeV)2=v2+4v

2

– Real triplet doesn’t contribute to MZ

• At tree level, =1+4v2/v21

• Return to muon decay:

2

2222 4

14 v

vvgMW

Blank & Hollik, hep-ph/9703392

2

2

2

22

2

2

s

s

c

sc

M

M

G

Gr

Z

Ztriplet

)1(2 222

rMcs

GZ

Need Four Input Parameters With Higgs Triplet

• Use effective leptonic mixing angle at Z resonance as 4th parameter

• Variation of s:

241 sa

v

e

e

eZaveiL ee )( 5

2

22

2

222

22

2

2

2

2

log3

4

2

1

3

2

)()(

2)(

)(

t

e

ZzeeA

e

ZzeeVe

eeA

e

ee

Z

ZZ

m

Qs

s

a

M

v

M

cs

vm

a

av

M

M

s

c

s

s

2

1,2

2

1 2 ee asv

This is definition of s:

Proportional to meneglect

Contrast with SM where s2 is proportional to mt

2

* Could equally well have used as 4th parameter

SM with triplet, cont.

2

2

2

22

2

2

s

s

c

sc

M

M

G

Gr

Z

Ztriplett

2

2

2

2

2

2

2

2

s

s

c

s

M

M

M

M

Z

Z

W

W

• Putting it all together:

• Finally,

mt2 dependence cancels

mt2 dependence cancels

rttriplet depends logarithmically on mt

2

If there is no symmetry which forces v=0, then no matter how small v is, you still need 4 input parameters

v 0 then 1

Triplet mass, M gv Two possible limits:

• g fixed, then light scalar in spectrum

• M fixed, then g and theory is non-renormalizable

SU(2)L x SU(2)R x U(1)B-L Model

• Minimal model

• Physical Higgs bosons: 4 H0, 2A0, 2H

• Count parameters:

(g, g’, , ’, vR) (e , MW1, MW2, MZ1, MZ2)

'0

00,

2

1,

2

1

Czakon, Zralek, Gluza, hep-ph/9906356

EWSB

0

00)2,0,1(

LL v

0

00)2,1,0(

RR v

SU(2)R x U(1)B-L U(1)Y

Assume vL=0 (could be used to generate neutrino masses)

Assume gL=gR=g

Renormalization of s in LR Model

2

22

2222222

22222

'21

2

1

)'(22

12

1

21

12

12

g

gvgMM

ggvgMM

vgMM

RZZ

RZZ

RWW

22222222

2

1

2

112 RRWW vgvgMM

2cos',

eg

s

eg

• Expand equations to incorporate one-loop corrections:

22222

222222

22222

22222222

2222

222222

))()((

)2()2(

2

1

))()((

))(())((

2

1

)()(

)()(2

1212

212121

1212

12121212

1212

1212

WWZZ

ZZZZZZ

WWZZ

WWZZZZWW

WWZZ

WWZZ

MMMM

MMMMMM

MMMM

MMMMMMMM

MMMM

MMMMcs

etc

• Gauge boson masses after symmetry breaking:

+2=2+’2

• Solve for s2 using

Renormalization of s in LR Model, cont.

22222

222222

22222

22222222

2222

222222

))()((

)2()2(

2

1

))()((

))(())((

2

1

)()(

)()(2

1212

212121

1212

12121212

1212

1212

WWZZ

ZZZZZZ

WWZZ

WWZZZZWW

WWZZ

WWZZ

MMMM

MMMMMM

MMMM

MMMMMMMM

MMMM

MMMMcs

• Scale set by: 222

22

22222

121212 2cos

1

2cos2 WWRWWZZ MMvg

MMMM

• At leading order in MW12/MW2

2 v2/vR2:

)()(

24

)()(2

)()(

)()(2

22

2222

2

2

2

22

222

2

2

2222

2222

2

2

2

2

12

1

12

1

1212

1212

WW

WtcF

WW

W

WWZZ

WWZZ

MM

MmNsc

s

cG

MM

Msc

s

c

MMMM

MMMM

s

c

s

s

Very different from SM!

• As MW22, s2/s2 0

• The SM is not recovered!

Thoughts on Decoupling

Limit MW22, s20

SM is not recovered

4 input parameters in Left-Right model: 3 input parameters in SM

No continuous limit from Left-Right model to SM

Even if vR is very small, still need 4 input parameters

No continuous limit which takes a theory with =1 at tree level to 1 at tree level

Results on Top Mass Dependence

Scale fixed to go through data pointAbsolute scale arbitrary

Plots include only mt dependence

Final example: Littlest Higgs Model

• EW precision constraints in SM require Mh light

• To stabilize Mh introduce new states to cancel quadratic dependence on higher scales– Classic model of this type is MSSM

• Littlest Higgs model: non-linear model based on SU(5)/SO(5)– Global SU(5) Global SO(5) with – Gauged [SU(2) x U(1)]1 x [SU(2) x U(1)]2SU(2) x U(1)SM

is complex Higgs triplet

22

22

1

x

x

I

I

fie /2

2/

2/2/

2/*

Th

hh

h

Littlest Higgs Model, continued

• Model has complex triplet (1) at tree level

– Requires 4 input parameters

• Quadratic divergences cancelled at one-loop by new states• W, Z, B WH, ZH, BH

• t T

• H

• Cancellation between states with same spin statistics– Naturalness requires f ~ few TeV

• Just like in SM with triplet, dependence of r on charge 2/3 quark, T, is logarithmic!

T T T

T tb

2

2

(...)1f

v

Littlest Higgs Model, continued

• One loop contributions numerically important– Tree level corrections (higher order terms in chiral perturbation

theory) v2/f2

– One loop radiative corrections 1/162

– Large cancellations between tree level and one-loop corrections

– Low cutoff with f 2 TeV is still allowed for some parameters.

– Contributions grow quadratically with scalar masses

Quadratic contributions cancel between these

Quadratic contribution remains from mixed diagrams

Fine Tuned set of parameters in LH Model

Parameters chosen for large cancellations

Models with triplets have Quadratic dependence on Higgs mass

• Mh0 is lightest neutral Higgs

• In SM:

• Quadratic dependence on Mh0 in LR Model:

• Quadratic dependence also found in little Higgs model

22

2

2

222

22 0

22

114)21(

224

1h

ZW

WFLRh M

M

c

M

scMG

sr

Czakon, Zralek J. Gluza, hep-ph/9906356

2

2

2

2

log192

11

W

hSMh M

Mgr

M.-C. Chen and S. Dawson, hep-ph/0311032

Conclusion

• Models with 1 at tree level require 4 input parameters in gauge sector for consistent renormalization– Cannot write models as one-loop SM contribution plus tree level new

physics contribution in general

• Models with extended gauge symmetries can have very different behaviour of EW quantities from SM beyond tree level– Obvious implications for moose models, little Higgs models, LR

models, etc