# Presentation: 716438

## 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 =MW2/(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

- 

- 

- e

- e

- W

- r is a physical parameter

## Part A:  Triplet ModelModels 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/(MZ2c2)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/v21
- PDG: v’ < 12 GeV

- Simplest extension of SM with 1

*Motivated by Little Higgs models*

- Neglects effects of scalar loops

## Scalar Potential

-  4 has dimensions of mass → doesn’t decouple
-  Mass Eigenstates:

-  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+

## Heavy Scalars → Small Mass Splittings 

-  Plots are restriction 2 < (4 )2

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

- MH+ (GeV)

- MK0 - MH+ (GeV)

- =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

- For 1,  4 input parameters

##  Input 4 Measured Quantities(MZ, , G, sin eff)

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

-    Could equally well have used  or MW as 4th parameter
-    At tree level, SM and triplet model are identical in seff scheme (SM inputs , G, sin eff here )

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

## Triplet Results

- Compare with SM in effective  mixing angle scheme
- Input parameters:  MZ, sineff, , 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

*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)

- 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 corrections

*Similar conclusions from Chivukula, Christensen, Simmons: arXiv:0712.0546*

- MH+ - MK=0, 10, 20 Gev

- MH+ (GeV)

- MW (MeV)

- =.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:

- Vanishes by EM gauge invariance

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

- 6 unknown functions to this order in MZ/

## STU Definitions

- 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 ModelsPDG Fits

- Data are 1 constraints with MH=117 GeV
- Ovals are 90% CL contours

- 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

- 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

-  1.7 TeV
-  New physics at the TeV scale

## 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

- 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

- 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

- Appelquist & Longhitano

## 12 E4 Operators

- 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

## 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

## 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

- 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

- What happens at 1-loop?

- Requires MW’> 3 TeV
- Problem with unitarity

## The Problem with Gauge Boson Loops

- =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

- 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:

## 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

## Leading Chiral Logarithm

- To leading order in x=2MW/MW’
- Landau and also Feynman gauge (Chivukula, Simmons....)
- Unitary gauge, keeping subleading terms in x

## 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 f2f1 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

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

- f1

- f2

- f1

- g

- g’

- g̃̃

- g̃̃

- 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

## Delocalizing Fermions in 4-Site Model

- One-Site delocalization
- Contribution to S at tree level
- Pick x1 to minimize Stree

## Calculate S Using Pinch Technique

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

## S at 1-Loop in 4-Site Model

- 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

## 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