Ignasi Rosell Universidad CEU Cardenal Herrera IFIC, CSIC–Universitat de València The Oblique S Parameter in Higgsless Electroweak Models June 2012 In collaboration with: A. Pich (IFIC, Valencia, Spain) J.J. Sanz-Cillero (INFN, Bari, Italy) arXiv: [hep-ph] Related works: JHEP 07 (2008) 014 [arXiv: ]

2/15 OUTLINE 1) Motivation 2) Oblique Electroweak Observables 3) The Effective Lagrangian 4) The Calculation of S 5) High-energy Constraints 6) Phenomenology 7) Summary The Oblique S Parameter in Higgsless Electroweak Models, I. Rosell

1. Motivation The Higgs Hunting * Prelimiminary CMS, ATLAS, CDF and D0 Collaborations. 3/15 The Oblique S Parameter in Higgsless Electroweak Models, I. Rosell Higgsless Electroweak Models i) The Standard Model (SM) provides an extremely succesful description of the electroweak and strong interactions. ii) A key feature is the particular mechanism adopted to break the electroweak gauge symmetry to the electroweak subgroup, SU(2) L x U(1) Y  U(1) QED, so that the W and Z bosons become massive. iii) The LHC has already excluded a broad range of Higgs masses: M H in the range [117.5,118.5] GeV U [122.5,127.5] GeV*. iv) What if there is no Higgs? We should look for alternative mechanisms of mass generation. v) They should fulfilled the existing phenomenological tests. vi) Strongly-coupled models: usually they do not contain any fundamental Higgs, bringing instead resonances. Many possibilities in the market: Technicolour, Walking Technicolour, Conformal Technicolour, Extra Dimensions… Oblique Electroweak Observables** ** Peskin and Takeuchi ’92.

The Oblique S Parameter in Higgsless Electroweak Models, I. Rosell 4/15 Why am I talking about Higgsless Electroweaks Models at i) In the limit where the U(1) Y coupling g’ is neglected, the Lagrangian is invariant under global SU(2) L x SU(2) R transformations. The Electroweak Symmetry Breaking (EWSB) turns out to be SU(2) L x SU(2) R  SU(2) L+R (custodial symmetry). ii) Absolutely similar to the Chiral Symmetry Breaking (ChSB) occuring in QCD. So the same pion Lagrangian describes the Goldstone boson dynamics associated with the EWSB, being replaced f π by v=1/√2G F =246 GeV. Same procedure as in Chiral Perturbation Theory (ChPT)*. iii) In other words, the electroweak Goldstone dynamics is then parameterized through an Effective Lagrangian which contains the SM gauge symmetry realized nonlinearly. iv) We can introduce the resonance fields needed in strongly-coupled Higgsless modes in a similar way as in ChPT: Resonance Chiral Theory (RChT)**. v) Actually, the estimation of the S parameter in strongly-coupled EW models is equivalent to the determination of L 10 in ChPT***. *** Pich, IR, Sanz-Cillero ’08. * Weinberg ’79 * Gasser & Leutwyler ‘84 ‘85 * Bijnens et al. ‘99 ‘00 **Ecker et al. ’89 ** Cirigliano et al. ’06

5/15 The Oblique S Parameter in Higgsless Electroweak Models, I. Rosell 2. Oblique Electroweak Observables Universal oblique corrections, occurred via the EW boson self-energies, which are transverse in the Landau gauge, The vacuum polarization amplitudes are expected to contain the dominant contributions from physics beyond the SM. S parameterizes the new physics contribution to the difference between the Z self-energies at Q 2 =M Z 2 and Q 2 =0. In this work we follow the useful dispersive representation introduced by Peskin and Takeuchi*. The convergence of the integral needs a vanishing at short distances. Note that a reference value for the SM Higgs mass is required. * Peskin and Takeuchi ’92.

3. The Effective Lagrangian 6/15 The Oblique S Parameter in Higgsless Electroweak Models, I. Rosell Let us consider a low-energy effective theory containing the SM gauge bosons coupled to the electroweak Goldstones and the lightest vector and axial-vector resonances: We have seven resonance parameters: F V, G V, F A, κ, σ, M V and M A. The high-energy constraints are fundamental.

7/15 The Oblique S Parameter in Higgsless Electroweak Models, I. Rosell 4. The Calculation of S i) At leading-order (LO)* * Peskin and Takeuchi ’92.

8/15 The Oblique S Parameter in Higgsless Electroweak Models, I. Rosell ii) At next-to-leading order (NLO)** Dispersive relation Contributions with two-particle cuts with two Goldstones or one Goldstone plus one resonance. F R r and M R r are renormalized couplings which define the resonance poles at the one-loop level. * Barbieri et al.’08 * Cata and Kamenik ‘08 * Orgogozo and Rynchov ‘08

9/15 The Oblique S Parameter in Higgsless Electroweak Models, I. Rosell 5. High-energy Constraints We have seven resonance parameters: F V, G V, F A, κ, σ, M V and M A. The number of unknown couplings can be reduced by using short-distance information. In contrast with the QCD case, we ignore the underlying dynamical theory. i) Weinberg Sum Rules (WSR)* * Weinberg’67 * Bernard et al.’75. i.i) LO: 1 or 2 constraintsi.ii) Imaginary NLO i.iii) Real NLO: fixing of F V,A r or lower bounds** 3 or 4 constraints ** Pich et al.’08

10/15 The Oblique S Parameter in Higgsless Electroweak Models, I. Rosell ii) Additional short-distance constraints ii.i) W L W L  W L W L scattering* ii.ii) Vector Form Factor** ii.iii) Axial Form Factor*** * Bagger et al.’94 * Barbieri et al.’08 ** Ecker et al.’89 *** Pich et al.’08 3 additional constraints! We have up to 9 (7) constraints with 2 (1) WSR and 7 resonance parameters: we cannot consider all the constraints at the same time. We consider different scenarios: One or two WSR. The constraint coming from W L W L  W L W L is supposed to be too strong. One can consider the WSR at LO and at NLO or only at NLO.

11/15 The Oblique S Parameter in Higgsless Electroweak Models, I. Rosell 6. Phenomenology S = 0.04 ± 0.10 * (M H =0.120 TeV) * Gfitter * LEP EWWG * Zfitter i) LO results i.i) 1st and 2nd WSRs i.ii) Only 1st WSR At LO M V > 1.5 TeV at 3σ

12/15 The Oblique S Parameter in Higgsless Electroweak Models, I. Rosell ii) NLO results: 1st and 2nd WSRs Considering the 1st and the 2nd WSRs at LO and at NLO one has 6 constraints: the only free parameter is M V. We have found 8 solutions. Only 2 of them are approximately compatible with the VFF, AFF and scattering constraints (green). If alternatively we consider the 1st and the 2nd WSR only at NLO with the VFF and AFF constraints (6 constraints), a heavier result is found: M V > 2.4 TeV at 3σ. At NLO with the 1st and 2nd WSRs M V > 1.8 TeV at 3σ

13/15 The Oblique S Parameter in Higgsless Electroweak Models, I. Rosell iii) NLO results: only 1st WSR Considering the 1st WSR at NLO with the VFF and AFF constraints one has 5 constraints: the only free parameters are M V and M A. Without the 2nd WSR we can only derive lower bounds on S. Imposing that F v 2 – F A 2 > 0 we have found only 2 solutions. One of them (red) is clearly disfavoured: it violates sharply the 2nd WSR at LO and at NLO and implies a large NLO correction. At NLO with only the 1st WSR M V > 1.8 TeV at 3σ With only the 1st WSR it is possible to perform the analysis with only the ππ cut. In any case, the same result is found: M V > 1.8 TeV at 3σ.

7. Summary 3. Where? What if there is no Higgs? 2. Why? 1. What? 4. How? One-loop calculation of the oblique S parameter within Higgsless models of EWSB Dispersive representation of Peskin and Takeuchi’92. Effective approach a)EWSB: SU(2) L x SU(2) R  SU(2) L+R : similar to ChSB in QCD: ChPT. b)Strongly-coupled models of EWSB within Higgsless models are characterized by the presence of massive resonance states: RChT. c)Most Ingredients: general Lagrangian with at most two derivatives and short-distance information. The Standard Model (SM) provides an extremely succesful description of the electroweak and strong interactions. The most important missing piece is the mechanism to generate the W and Z masses. The standard answer is the Higgs. The LHC has already excluded a broad range of Higgs masses. We should look for alternative ways of mass generation.: strongly-coupled higgsless models. They should fulfilled the existing phenomenological tests. 14/15 The Oblique S Parameter in Higgsless Electroweak Models, I. Rosell

15/15 The Oblique S Parameter in Higgsless Electroweak Models, I. Rosell Improvements over previous NLO calculation: Dispersive calculation: no unphysical cut- offs. A more general Lagrangian. The short-distance information is one of our main ingredients. We have considered different possibilites: LO NLO with the 1st and 2nd WSR NLO with only the 1st WSR Similar results: At LO M V > 1.5 TeV at 3σ. At NLO M V > 1.8 TeV at 3σ. Therefore, the S parameter requires a high resonance mass scale, beyond the 1 TeV, at least if one considers the most reasonable strongly-coupled models of EWSB without Higgs.

Future work Oblique T parameter. Absence of a known dispersive representation. 16/15 The Oblique S Parameter in Higgsless Electroweak Models, I. Rosell