Ignasi Rosell Universidad CEU Cardenal Herrera IFIC, CSIC–Universitat de València

1. Ignasi RosellUniversidad CEU Cardenal HerreraIFIC, CSIC–Universitat de València ICHEP2012, 4 - 11 July 2012 One-LoopCalculation of the Oblique S Parameter in HiggslessElectroweakModels In collaboration with: A. Pich (IFIC, Valencia, Spain) J.J. Sanz-Cillero (INFN, Bari, Italy) arXiv:1206.3454 [hep-ph] Related works: JHEP 07 (2008) 014 [arXiv:0803.1567]

2. OUTLINE • Motivation • Oblique Electroweak Observables • TheEffectiveLagrangian • TheCalculation of S • High-energyConstraints • Phenomenology • Summary The Oblique S Parameter in HiggslessElectroweakModels, I. Rosell

3. 1. Motivation i) The Standard Model(SM) providesanextremelysuccesfuldescription of theelectroweakand stronginteractions. ii) A keyfeatureisthe particular mechanismadoptedto break theelectroweak gauge symmetrytotheelectroweaksubgroup,SU(2)L x U(1)Y  U(1)QED, so thattheW and Z bosonsbecomemassive. TheHiggsHunting iii) TheLHC has justdiscovered a new particlearound125 GeV*. iv) Whatifthis new particleisnot a Higgsboson? Or a notstandardone? Or a scalarresonance? Weshould look foralternativemechanisms of massgeneration. HiggslessElectroweak Models vi) Strongly-coupledmodels: usuallythey do containresonances. Manypossibilities in themarket:Technicolour, WalkingTechnicolour, ConformalTechnicolour, Extra Dimensions… Oblique Electroweak Observables** v) Theyshouldfulfilledtheexistingphenomenologicaltests. * Prelimiminary CMS and ATLAS Collaborations. The Oblique S Parameter in HiggslessElectroweakModels, I. Rosell ** Peskin and Takeuchi’92.

4. SimilaritiestoChiralSimmetryBreaking in QCD i) In thelimitwherethe U(1)Ycoupling g’ isneglected, theLagrangianisinvariantunder global SU(2)L x SU(2)Rtransformations. TheElectroweakSymmetryBreaking(EWSB) turnsoutto be SU(2)L x SU(2)R SU(2)L+R(custodial symmetry). ii) Absolutely similar totheChiralSymmetryBreaking(ChSB) occuring in QCD. So thesame pion Lagrangian describes theGoldstonebosondynamicsassociatedwiththe EWSB, beingreplacedfπbyv=1/√(2GF)=246 GeV.Sameprocedure as in ChiralPerturbationTheory (ChPT)*. iii) We can introduce theresonancefieldsneeded in strongly-coupledHiggslessmodelsin a similar way as in ChPT: ResonanceChiralTheory(RChT)**. • Note theimplications of a naïverescalingfrom QCD toEW: iv) Actually, theestimation of the S parameter in strongly-coupled EW modelsisequivalenttothedetermination of L10 in ChPT***. * Weinberg ’79 * Gasser & Leutwyler ‘84 ‘85 * Bijnens et al. ‘99 ‘00 **Eckeret al. ’89 ** Cirigliano et al. ’06 The Oblique S Parameter in HiggslessElectroweakModels, I. Rosell *** Pich, IR, Sanz-Cillero ’08.

5. 2. Oblique Electroweak Observables • Universal obliquecorrectionsviatheEW bosonself-energies(transverse in theLandau gauge) • S parameter:new physics in thedifferencebetweenthe Z self-energies at Q2=MZ2 and Q2=0. • WefollowtheusefuldispersiverepresentationintroducedbyPeskin and Takeuchi*. • Theconvergence of the integral needs a vanishingat short distances. • S parameterisdefinedfor a referencevaluefortheSM Higgsmass. The Oblique S Parameter in HiggslessElectroweakModels, I. Rosell * Peskin and Takeuchi’92.

6. 3. TheEffectiveLagrangian Letusconsider a low-energyeffectivetheorycontainingtheSM gauge bosonscoupledtotheelectroweakGoldstonesand thelightestvector and axial-vector resonances: Wehavesevenresonanceparameters: FV, GV, FA, κ, σ, MV and MA. Thehigh-energyconstraintsare fundamental. The Oblique S Parameter in HiggslessElectroweakModels, I. Rosell

7. 4. TheCalculation of S i) At leading-order (LO)* The Oblique S Parameter in HiggslessElectroweakModels, I. Rosell * Peskin and Takeuchi’92.

8. ii) At next-to-leadingorder (NLO)** • Once-subtracteddispersiverelation • Contributionsfromππ, Vπ and Aπ cuts, sincehighercuts are supposedto be suppressed. • FRr and MRrare renormalizedcouplingswhich define theresonancepoles at theone-looplevel. * Barbieri et al.’08 * Cata and Kamenik ‘08 * Orgogozo and Rynchov ‘08 The Oblique S Parameter in HiggslessElectroweakModels, I. Rosell

9. 5. High-energyConstraints • Wehavesevenresonanceparameters: FV, GV, FA, κ, σ, MV and MA. • Thenumber of unknowncouplings can be reducedbyusingshort-distanceinformation. • In contrastwiththeQCD case, we ignore theunderlyingdynamicaltheory. i) Weinberg Sum Rules (WSR)* i.i) LO i.ii) Imaginary NLO i.iii) Real NLO: fixing of FV,Ar orlowerbounds** 1or 2 constraints 3 or 4 constraints ConstraintsonFVr and FAr • * Weinberg’67 • * Bernard et al.’75. The Oblique S Parameter in HiggslessElectroweakModels, I. Rosell ** Pichet al.’08

10. ii) Additional short-distanceconstraints ii.i) WLWL WLWLscattering* 3 additional constraints! ii.ii) Vector Form Factor** ii.iii) Axial Form Factor*** • Wehave up to9 (7) constraintswith2 (1) WSR and 7 resonanceparameters: wecannotconsideralltheconstraints at thesame time, someapproximately. • As a check of consistencyweconsiderdifferentcombination of constraints. • * Baggeret al.’94 • * Barbieri et al.’08 The Oblique S Parameter in HiggslessElectroweakModels, I. Rosell • ** Ecker et al.’89 • *** Pich et al.’08

11. 6. Phenomenology S = 0.04 ± 0.10 * (MH=0.120 TeV) i) LO results i.i) 1st and 2nd WSRs i.ii) Only1st WSR At LO MV > 1.5 TeV at 3σ The Oblique S Parameter in HiggslessElectroweakModels, I. Rosell • * Gfitter • * LEP EWWG • * Zfitter

12. ii) NLO results: 1st and 2nd WSRs • 1st and 2nd WSRs at LO and at NLO: • 6 constraints • MV theonly free parameter • 8 solutions. • Only2approximately compatible withVFF, AFF and scatteringconstraints(green). • If, alternatively, weconsiderthe1st and the2nd WSR only at NLO withthe VFF and AFF constraints (6 constraints), a heavierresultisfound: MV > 2.4 TeV at 3σ. At NLO withthe 1st and 2nd WSRs MV > 1.8 TeV at 3σ The Oblique S Parameter in HiggslessElectroweakModels, I. Rosell

13. iii) NLO results: only 1st WSR • 1st WSR at NLO + VFF and AFF constraints: • 5 constraints • MV and MA theonly free parameters are. • Withoutthe 2nd WSR we can only derive lowerboundson S. • Imposingthat Fv2 – FA2 > 0 wehavefoundonly2solutions. • One of them (red) isclearlydisfavoured: • Sharplyviolation of the 2nd WSR at LO and at NLO • Large NLO correction • Big splittingbetween MV and MA. At NLO withonlythe 1st WSR MV > 1.8 TeV at 3σ • Withoutthe 2nd WSR itispossibletheanalysiswithonlythe ππ cut. Thesameresultisfound: MV > 1.8 TeV at 3σ. The Oblique S Parameter in HiggslessElectroweakModels, I. Rosell

14. 7. Summary 1. What? One-loopcalculation of theoblique S parameterwithinHiggslessmodels of EWSB • Weshould look foralternativewaysof massgeneration: strongly-coupledhiggslessmodels. • Theyshouldfulfilledtheexistingphenomenologicaltests. Whatifthis new particlearound125 GeVisnot a Higgsbosson? 2. Why? • EWSB: SU(2)L x SU(2)R SU(2)L+R: similar toChSB in QCD:ChPT. • Strongly-coupledHiggslessmodels: similar toresonancesin QCD: RChT. • General Lagrangianwith at mosttwoderivatives and short-distanceinformation. Effective approach 3. Where? 4. How? Dispersiverepresentation of Peskin and Takeuchi’92. The Oblique S Parameter in HiggslessElectroweakModels, I. Rosell

15. Improvementsoverprevious NLO calculation: • Dispersivecalculation: no unphysicalcut-offs. • A more general Lagrangian. • Short-distanceinformation as a crucial ingredient. • Wehaveconsidereddifferentpossibilites: • LO • NLOwiththe1st and 2nd WSR • NLOwithonlythe1st WSR • Similar results: • At LO MV > 1.5 TeVat 3σ. • At NLO MV > 1.8 TeV at 3σ. • In thesereasonablestronglycoupledmodelsthe S parameterrequires a highresonancemassscale, beyondthe 1 TeV. The Oblique S Parameter in HiggslessElectroweakModels, I. Rosell

16. 8*. Futurework • Consideration of thisnew scalarwith a massaround125 GeV in ourcalculation: • Higgsboson? Whichone? • Ascalarresonance of strongly-coupledmodels? A new Sπ or Hπ cut!, butonly at NLO. • Oblique T parameter Absenceof a knowndispersiverepresentation. The Oblique S Parameter in HiggslessElectroweakModels, I. Rosell