Electroweak Precision Measurements and BSM Physics: (A) Triplet Models

(B) The 3- and 4-Site Models

S. Dawson (BNL)

February, 2009

S. Dawson and C. Jackson, arXiv:0810.5068; hep-ph/0703299

M. Chen, S. Dawson, and C. Jackson, arXiV:0809.4185

WARNING: THIS IS A THEORY TALK

Standard Model Renormalization

• 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 =MW

2/(MZ2c

2)=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

Muon Decay in the SM

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

• One loop radiative corrections included in parameter rSM

• Dominant contributions from 2-point functions

)1(

12 22

2 SM

WZ

W

r

MMM

G

e

e

Wr is a physical parameter

Part A: Triplet ModelModels with 1 at tree level are different

from the SM

• SM with Higgs Triplet• Left-Right Symmetric Models• Little Higgs Models• …..many more• These models need additional input parameter• Decoupling is not always obvious beyond tree

level

=MW2/(MZ

2c2)1

Higgs Triplet Model

• Add a real triplet

– vSM2=(246 GeV)2=v2+4v’2

– Real triplet doesn’t contribute to MZ

• At tree level, =1+4v’2/v21• PDG: v’ < 12 GeV

2

2222 '4

14 v

vvgMW

)(2

1 00

ihvH

0'v

Simplest extension of SM with 1

Motivated by Little Higgs modelsNeglects effects of scalar loops

Scalar Potential

aaHHHHHV 4

223424

1

222

221 24

0

0

0

0

hcs

sc

K

H

• 4 has dimensions of mass → doesn’t decouple

• Mass Eigenstates:

cs

sc

H

G

• 6 parameters in scalar sector: Take them to be:

MH0, MK0, MH+, v, ,

small since it is related to parameter

tan = 2 v’/v

Forbidden by T-parity

Decoupling at Tree Level

• Require no mixing between doublet-triplet sectors for decoupling

• v’→0 requires 4 →0 (custodial symmetry), or 3→ (invalidating perturbation theory)

• v’→0 implies MK0 MH+

3

4

v

32

222

22

2

1120 vvM

K

432

222

22 2

2

14 vvvM

H

Heavy Scalars → Small Mass Splittings

• Plots are restriction 2 < (4 )2

Forshaw, Vera, & White, hep-ph.0302256

MH+ (GeV)

MK

0 -

MH

+ (

Ge

V)

=0

Allowed region is between curves

Renormalization of Triplet Model

• At tree level, W mass related to input parameters:

• One loop radiative corrections included in parameter r

)1(2 2

2 rGs

MW

122

2

Z

W

Mc

M

For 1, 4 input parameters

Input 4 Measured Quantities(MZ, , G, sin eff)

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

eZaveiL ee )( 5

2

1,2

2

1 2 eeff

e asv

• Could equally well have used or MW as 4th parameter

• At tree level, SM and triplet model are identical in seff

scheme (SM inputs , G, sin eff here )

This scheme discussed by: Chen, Dawson, Krupovnickas, hep-ph/0604102; Blank and Hollik hep-ph/9703392

rGs

MeffW 1

2 2

2

Triplet Results

• Compare with SM in effective mixing angle scheme

• Input parameters: MZ, sineff, , G, MH0, MK0, MH+, =1/cos2 = (MW /MZ cos eff )2 predicts sin =.07 (v’=9 GeV)

– System is overconstrained (can’t let v’ run)

• Triplet model has extra contributions to r from K0, H+

• SM couplings are modified by factors of cos , cos

Quadratic dependence on Higgs mass

• Triplet model with MH0 << MK0 MH and small mixing

(...)sin(...)sin24

~2

22

2

0

H

HKSMtriplet

M

MM

srr

Toussaint, PRD18 (1978) 1626

Inputs different in triplet model and SM

Triplet model: MZ=91.1876 GeV is input

SM (in this scheme): MZ is calculated = 91.453 GeV

Perturbativity requires MK0~MH+ for large MH+

MW(SM)-MW(Triplet)

• For heavy H+, perturbativity requires MH+~MK0, and predictions of triplet model approach SM

• No large effects in perturbative regime

• SM not exactly recovered at large MH+ due to different MZ

inputs for 1-loop correctionsSimilar conclusions from Chivukula, Christensen, Simmons: arXiv:0712.0546

MH+ - MK=0, 10, 20 Gev

MH+ (GeV)

MW (

Me

V)

=.1,

MH0=120 GeV

Conclusions on Triplets

MW=80.399 0.025 GeV

Triplet model consistent with experimental data if MK ~MH+

Small mixing angles required

• Part B: Higgsless Models and Effective Lagrangians

• What can we learn from precision electroweak measurements?

The Usual Approach

• Build the model of the week• Assume new physics contributes primarily to gauge

boson 2-point functions• Calculate contributions of new particles to S, T, U• Extract limits on parameters of model

THIS

NOT THIS

STU Assumptions

• Assume dominant contribution of new physics is to 2-point functions

• Assume scale of new physics, >> MZ

– This means no new low energy particles

– Taylor expand in MZ/– Symmetry is symmetry of SM

• Assume reference values for MH, Mt

• Assume =1 at tree level– Otherwise you need 4-input parameters to renormalize

STU Definitions

Peskin & Takeuchi, PRD46 (1992) 381

Taylor expand 2-point functions:

)()0()(

)()0()(

)()0()(

)()0()(

222

222

222

222

qqq

qqq

qqq

qqq

ZZZZZZ

WWWWWW

ZZZ

Vanishes by EM gauge invariance

Fermion & scalar contributions vanish; gauge boson contributions non-zero

6 unknown functions to this order in MZ/

STU Definitions

22

2

2

2

22

2

22

222

222

222

22

)0()0()(2

)0()()0()(4

)0(2

)0()0(

)0(0)()()0()(4

ZW

Z

ZWZWW

Z

ZZWZZW

W

WWWWWW

Z

Z

W

W

Z

ZZ

W

WW

ZZZWW

WWZZZZZZZ

Z

WW

Ms

M

Msc

M

Mc

M

MsU

Mc

s

MMT

Msc

scMM

M

csS

Peskin & Takeuchi, PRD46 (1992) 381

6 unknowns:

3 fixed by SM renormalization, 3 free parameters

S is scaling of Z 2-point function from q2=0 to MZ2

T is isospin violation

U contributes mostly to MW

3 and 4 Site Higgsless ModelsPDG Fits

• Data are 1 constraints with MH=117 GeV

• Ovals are 90% CL contours

)09.0(09.002.0

)07.0(09.004.0

T

S

MH,ref=117 GeV (300 GeV)

S

T

What if There is No Higgs?

• Simplest possibility: No Higgs / No new light particles / No expanded gauge symmetry at EW scale– Electroweak chiral Lagrangian

• LSMnl doesn’t include Higgs, so it is non-renormalizable

• Assume global symmetry SU(2) x SU(2) →SU(2)V or U(1)– SU(2) x SU(2) →SU(2)V is symmetry of Goldstone Boson sector

of SM

• Li is an expansion in (Energy)2/2

inlSMeff LLL

Remember Han talk

No Higgs → Unitarity Violation

• Consider W+W-→W+W-

• Unitarity conservation requires• MH→

• If all resonances (Higgs, vector mesons…etc) much heavier than ~ few TeV

2

1)Re( la

200 32 v

sa

1.7 TeV

New physics at the TeV scale

4

2

2)(

v

sO

v

tsWWWWA

Electroweak Chiral Lagrangian

• Terms with 2 derivatives:

• Unitary gauge: =1– SM masses for W/Z gauge bosons

• This is SM without Higgs– SM W/Z/ interactions

DDTr

vL

4

2

2)/exp( vi

322

B

giW

igD ii

General Framework for studying BSM physics without a Higgs

E4 Terms in Chiral Lagrangian

• 3 operators contribute at tree level to gauge boson 2-point functions

22

88

11

211

4

12

14

1

TWTrgL

TWBTrggL

TVTrL

)(

2 3

DV

TT

Also contribute to gauge boson 3-point functions

Limits from LEP2/Tevatron

Gives tree level isospin violation

Apologies: my normalization is different from Han… ~ l(v/)2

E4 Terms continued

• Contribute to WW, WWZ vertices (but not to 2-point functions)

• Only contribute to quartic interactions

• Conserves CP, violates P

VVTTrTWTrgi

L

VVWTrigL

VVTTrBgi

L

,2

,

,2

99

33

22

2

1010

77

2

1

TVTrTVTrTrL

TVTrTVTrVVTrL

WVTrTVTrgL 1111

TVTrTVTrVVTrL

VVTrL

VVTrL

66

2

55

244

Appelquist & Longhitano

12 E4 Operators

• Assume custodial SU(2)V (=1at tree level)– L1’, L6, L7, L8, L9, L10 vanish

– 6 operators remain

– Assume P conservation, L11 vanishes

Simple format for BSM physics

Estimate coefficients in your favorite model

SU(2)SU(2)

Tree Level

• 2-point functions

• SM fit assumes a value for MH

– Contribution from heavy Higgs:

• Scale theory from to MZ, add back in contribution from MH(ref)– Approach assumes logarithms dominate

82

1

12

4

2

4

eU

T

eS

tree

tree

tree

Z

HH

Z

HH M

M

cT

M

MS ln

8

3,ln

6

12

82

1,

2

12

,

4

2ln8

3

4ln6

eU

McT

eM

S

tree

refHtree

refHtree

0074.0034.3 1 TeV

Extended Gauge Symmetries

• General model with gauged SU(2) x SU(2)N x U(1)

• After electroweak symmetry breaking, massless photon, plus tower of massive Wn

, Zn vector bosons

• Fermions:

• Calculate for general couplings, then apply to specific model

1

1

,,4

1

4

1 N

i

ai

aiG WWBBL

1

1

,,N

nnini WaW

1

1000

N

nnnZbbB

1

10,3

N

nninii ZbbW

0,

1

15

,

1

15

)(

.)1(22

00

0

ni

N

ni

VA

VViijV

nj

N

nij i

ijWf

Vggg

chWg

L

n

i

n

ij

n

3-Site Higgsless Model (aka BESS)

• Global SU(2) x SU(2) x SU(2) →SU(2)V symmetry

• Gauged SU(2)1 x SU(2)2 x U(1)

• Model looks like SM in limit

222

2

21

2

20

2

1

2

2 2

1

2

1

2

1

4

RTrg

VTrg

LTrg

DDTrf

Li

ii

RiiVD

ViiLD

2222

1111

1

1

1

2

1

0

g

gy

g

gx

222

220

4

4

W

W

cg

sg

1 2

L V R

g0 g1 g2

Scales

~10 TeV

MW’ ~ 1 TeV

MW

Strong Coupling

Weakly coupled non-linear model

Calculate log-enhanced contributions to S

3-Site Higgsless Model

• At tree level,

• Get around this by delocalizing fermions

• Ideal delocalization: pick x1 to make S vanish at tree level

2'

22

21

44

W

WWtree M

Ms

gS

LLf DixL 111 )(

2

2'1

2'

22

21

4

W

W

W

WWtree M

Mx

M

MsS

BJVxLxJL YLf 111

What happens at 1-loop?

Requires MW’> 3 TeV

Problem with unitarity

The Problem with Gauge Boson Loops

22222

222

222

1)(

)(

)(

VVV

GG

SS

Mq

qqg

Mq

iqD

Mq

igqD

Mq

igqD

i

i

=1 (Feynman), =0 (Landau), → (Unitary)

Independent of

+ …. Depends on

Pinch Technique

• STU extracted from 4-fermion interactions• Idea:

– Isolate gauge-dependent terms in vertex/box diagrams– Combine with 2-point diagrams– Result is gauge-independent– q4 and q6 terms cancel in unitary gauge

Degrassi, Kniehl, & Sirlin, PRD48 (1993) 3963; Papavassiliou & Sirlin, PRD50 (1994) 5951

Individually gauge dependent

=

Gauge independent

Pinch Technique

...

)())((

1)(...)...()(

2

)())((

1)()...)(...()(

2

)())((

1)(...)...()(

2

221

2212

12

221

2211112

221

22112

qkpkMkpupkpu

kdqkpkMk

puppkpkpukd

qkpkMkpukpkpu

kd

Wn

nW

n

nW

n

n

Extract terms from vertex/box corrections that look like 2-point functions

Associate this piece with propagator

Pinch Technique

• Calculate pinch 2-point functions with SM gauge sector plus W’,Z’

• Unitary gauge minimizes number of diagrams

• Neutral gauge boson 2-point functions:

Pinchloop

q

ABqVq

q

AA

2

042

2

01 2

Unknown Terms

• Unknown higher-dimensional operators

– Generalization of 1 term

• L2 gives poles at one loop: 1/→ Log (2/MW2)

– Poles absorbed in redefinition of arbitrary couplings, c1 and c2

• Approach only makes sense if logarithms dominate• Results have scheme dependence

)()(8 22

210 ccS

11012222114

VLTrggcBVTrggcL

Leading Chiral Logarithm

• To leading order in x=2MW/MW’

– Landau and also Feynman gauge (Chivukula, Simmons….)

• Unitary gauge, keeping subleading terms in x

2

'

21

2'

2

2

2'

1 ln4

3ln

24

41ln

12 WWW

Wloop Mx

x

MM

MS

02

2'

'2

2'

3 lnln SM

MAA

M

MASS

W

WSW

SW

W

WSWtreesite

Large Corrections in 3-Site Model at 1-loop

Ideal localization only fixes S problem at tree level

4-Site Model

• Gauge symmetry: SU(2)L x SU(2)V1 x SU(2)V2 x U(1)Y

• Why?– With f2f1 can “mimic” warped RS models

– Gauge couplings approximately SM strength if:

– Gauge sector: SM + 2 sets of heavy gauge bosons,

i, 0, i=1,2

1~1~

g

gy

g

gx

Accomando et al, arXiv:0807.5051; Chivukula, Simmons, arXiv: 0808.2017

f1 f2 f1

g g’g̃̃��g̃̃��

22

22 4

,4

cg

sg

L V1 V2 R

See talk by de Curtis

Compute Masses & Mixings

• Compute masses as expansion in x

• 3-site limit: f2→, z→0, 2 decouples

422

22

12

2

221

22

22

22

1

22

21

22

21

4

2~2

14

~

124

1

1

zxffg

M

xfgM

zxff

ffgM wW

22

21

1

2 ff

fz

412

1zzW

RgiVgiD

VgiVgiD

VgiigLD

33233

222122

11111

~

~~

~

3

1

2

4iii

i DDTrf

L

Delocalizing Fermions in 4-Site Model

• One-Site delocalization

• Contribution to S at tree level

• Pick x1 to minimize Stree

L

aaL

aaRRLLf PVTxgPLTxgBPYPYgL ,

11,

1~1

)(12

14

41

4

222

212

2

22

21

2

2

12

2

2

22

1

1

01

02

01

01

xOzM

Mxz

M

Ms

fg

Mx

M

M

M

MsS

W

WW

WWtree

LPDixL 111

Calculate S Using Pinch Technique

Just like 3-Site calculation, but more of it….

S at 1-Loop in 4-Site Model

)(1)()(18

ln14

3

24

331683

ln14

3

24

171741

ln12

12

14

34

212

2

22

21

42

2

2

221

642

2,

2

22

212

2

22

4

2

1

2

11

1

czccz

Mz

x

xzz

M

Mz

x

xzzz

M

Mz

M

Mxz

M

MsS

refHW

WWSite

Note scaling between different energy regimes

Dawson & Jackson, arXIv:0810.5068

One-Loop Results in 4-Site Model

Large fine tuning needed at 1-loop

2

1

M

Mz

The Moral of the Story is….

• Triplet models– Can’t use STU approach in triplet models– Triplet models can fit EW data if heavy scalars

roughly degenerate– Minimizing the potential requires small mixing in

neutral and charged sectors

• 2, 3, 4 Site Higgsless models– Ideal delocalization at tree level doesn’t solve problem

at 1-loop– Unknown coefficients make predictions problematic