A few words about Resonances in the
Electroweak Effective Lagrangian111Talk given at XIth Quark Confinement and Hadron Spectrum, 8-12th September (2014), Saint Petersburg (Russia). Preprint numbers: IFIC-15-04, IFT-UAM-CSIC-15-009, FTUAM-15-3, FTUV-15-2801.
Contrary to a widely spread believe, we have demonstrated that strongly coupled electroweak models including both a light Higgs-like boson and massive spin- resonances are not in conflict with experimental constraints on the oblique and parameters. We use an effective Lagrangian implementing the chiral symmetry breaking that contains the Standard Model gauge bosons coupled to the electroweak Goldstones, one Higgs-like scalar state with mass GeV and the lightest vector and axial-vector resonance multiplets and . We have considered the one-loop calculation of and in order to study the viability of these strongly-coupled scenarios, being short-distance constraints and dispersive relations the main ingredients of the calculation. Once we have constrained the resonance parameters, we do a first approach to the determination of the low energy constants of the electroweak effective theory at low energies (without resonances). We show this determination in the case of the purely Higgsless bosonic Lagrangian.
:12.39.Fe, 12.60.Fr, 12.60.Nz, 12.60.Rc
,altaddress=Departamento de Ciencias Físicas, Matemáticas y de la Computación, Universidad CEU Cardenal Herrera,
c/ Sant Bartomeu 55, 46115 Alfara del Patriarca, València, Spain
address=Departament de Física Teòrica, IFIC, Universitat de València – CSIC,
Apt. Correus 22085, 46071 València, Spain
address=Departament de Física Teòrica, IFIC, Universitat de València – CSIC,
Apt. Correus 22085, 46071 València, Spain
address=Departamento de Física Teórica and Instituto Física Teórica, IFT-UAM/CSIC,
Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
A Higgs-like boson around GeV was discovered at the LHC. Although its properties are being measured yet, it complies with the expected behaviour and therefore it is a very compelling candidate to be the Standard Model (SM) Higgs. An obvious question to address is to which extent alternative scenarios of Electroweak Symmetry Breaking (EWSB) can be already discarded or strongly constrained. In particular, what are the implications for strongly-coupled models where the electroweak symmetry is broken dynamically?
The existing phenomenological tests have confirmed the pattern of symmetry breaking, giving rise to three Goldstone bosons which, in the unitary gauge, become the longitudinal polarizations of the gauge bosons. When the coupling is neglected, the electroweak Goldstone dynamics is described at low energies by the same Lagrangian as the QCD pions, replacing the pion decay constant by the EWSB scale GeV [1, 2]. In most strongly-coupled scenarios the symmetry is nonlinearly realized and one expects the appearance of massive resonances generated by the non-perturbative interaction.
The dynamics of Goldstones and massive resonance states can be analyzed in a generic way by using an effective Lagrangian, based on symmetry considerations. The theoretical framework is completely analogous to the Resonance Chiral Theory description of QCD at GeV energies .
Using these techniques, we have investigated in Ref. , and as an update of Ref. , the oblique and parameters , characterizing the new physics contributions in the electroweak boson self-energies, within strongly-coupled models that incorporate a light Higgs-like boson. Adopting a dispersive approach and imposing a proper high-energy behaviour, it has been shown there that it is possible to calculate and at the next-to-leading order, i.e. at one-loop. Note that these results do not depend on unnecessary ultraviolet cut-offs. We concluded that there is room for these models, but they are stringently constrained. The vector and axial-vector states should be heavy enough (with masses above the TeV scale), the mass splitting between them is highly preferred to be small and the Higgs-like scalar should have a coupling close to the Standard Model one. Previous one-loop analyses can be found in Ref. .
As a continuation , and as a first approach to the determination of the low energy constants of the electroweak effective theory at low energies (without resonances), we do this estimation in the case of the purely Higgsless bosonic Lagrangian (withouth resonances), i.e. the Longhitano’s Lagrangian .
2 Constraining the Resonance Theory from Phenomenology
We have considered a low-energy effective theory containing the SM gauge bosons coupled to the electroweak Goldstones, one light scalar state with mass GeV and the lightest vector and axial-vector resonance multiplets and . We have only assumed the SM pattern of EWSB, i.e. the theory is symmetric under and becomes spontaneously broken to the diagonal subgroup . is taken to be singlet under , while and are triplets. The underlying theory is also assumed to preserve parity in this analysis. To build the Lagrangian we have only considered operators with the lowest number of derivatives, as higher-derivative terms are either proportional to the equations of motion or tend to violate the expected short-distance behaviour . In order to determine the oblique and parameters one only needs the interactions 
plus the standard gauge boson and resonance kinetic terms. We have followed the notation of Ref. . The first term in (2) gives the Goldstone Lagrangian, present in the SM, plus the scalar-Goldstone interactions. For one recovers the vertex of the SM.
The oblique parameter receives tree-level contributions from vector and axial-vector exchanges , while is identically zero at lowest-order (LO):
To compute next-to-leading order (NLO) contributions we have used the dispersive representation of introduced by Peskin and Takeuchi , whose convergence requires a vanishing spectral function at short distances:
The calculation of is simplified by noticing that, up to corrections of , , being and the wave-function renormalization constants of the charged and neutral Goldstone bosons computed in the Landau gauge . A further simplification occurs by setting to zero, which does not break the custodial symmetry, so only the -boson exchange produces an effect in . This approximation captures the lowest order contribution to in its expansion in powers of and .
Requiring the spectral function to vanish at high energies channel by channel leads to a good convergence of the Goldstone self-energies, at least for the cuts we have considered. Then, their difference obeys an unsubtracted dispersion relation, which enables us to compute through the dispersive integral ,
with the spectral function of the difference of the neutral and charged Goldstone self-energies.
It is quite interesting to remark the main assumptions we have done in our approach:
Only operators with at most two derivatives have been kept in the action. Considering the equations of motion, field redefinitions and the high-energy behavior of form factors, it is possible to justify the absence of higher derivative operators . This procedure works very well in the QCD case .
Only the lightest vector and axial-vector resonance multiplets have been considered. QCD phenomenology supports this “single-resonance” approximation, owing to the kinematical suppression of heavier resonance contributions .
Only contributions to the dispersive relations of (2) and (3) coming from the lightest two-particle channels without heavy resonances have been considered, i.e. two Goldstones or one Goldstone plus one Higgs-like scalar resonance for and the boson plus one Goldstone or one Higgs-like scalar resonance for . Note that contributions from higher cuts are kinematically suppressed: the or weights in the sum rules (2) and (3) enhance the contribution from the lightest thresholds and suppress channels with heavy states . and contributions were shown to be suppressed in a previous Higgsless analysis . Again, it is known that this procedure gives a very good approximation to the corresponding integrals in the QCD case .
Unlike what happens in QCD, the underlying theory is not known. Therefore, although we have worked at lowest order in and , the perturbative chiral counting in powers of momenta is not well defined. We only know that loops are suppressed ( counting in the loop expansion) and that it works in QCD in the framework of the expansion, with the number of colours.
Figure 1 shows the computed one-loop absorptive contributions to and .
2.1 High-energy constraints
Fixing GeV, one has seven undetermined parameters: , , , , , and . This number can be reduced using short-distance information :
Vector form factor. The two-Goldstone matrix element of the vector current defines the vector form factor. Imposing that it vanishes at , one finds :
Weinberg Sum Rules (WSRs). At leading-order the first and the second Weinberg sum rules  imply, respectively,
Note that a small splitting between the vector and axial-vector resonances would imply , that is, close to the SM value.
As a conclusion, we have seven resonance parameters and up to five constraints. Taking into account that the second WSR is questionable in some scenarios, we have also studied the consequence of discarding the second WSR.
We have taken the SM reference point at GeV, so the global fit gives the results and , with a correlation coefficient of .
LO. Considering the first and the second WSRs, becomes 
The resonance masses need to be heavy enough to comply with the experimental bound, this is, much higher than the Higgs mass. From this point of view it is interesting to note that the Higss mass GeV is light in comparison with those resonances and the EW scale TeV. One finds a big gap between the lightest two particle cuts and the next ones (including vector and axial-vector resonances) . As it has been explained previously, one expects therefore the NLO corrections to S to be widely dominated by the and cuts.
where terms of have been neglected.
In Figure 2 (left) we show the compatibility between the “experimental” values and these determinations . The Higgs-like scalar should have a coupling very close to the SM one. At 68% (95%) CL, one gets (), in nice agreement with LHC evidence, but more restrictive. Moreover, the vector and axial-vector states should be very heavy (and quite degenerate); one finds TeV ( TeV) at 68% (95%) CL.
where has been assumed and again terms of have been neglected.
Figure 2 (right) gives the allowed 68% CL region in the space of parameters and , varying between and . Note, however, that values of very different from the SM can only be obtained with a large splitting of the vector and axial-vector masses. In general, there is no solution for . Requiring , leads to at 68% CL, while the allowed vector mass stays above TeV.
To sum up, the principal conclusions of this analysis have been the following ones :
Strongly-coupled electroweak models with massive resonance states are still allowed by the current experimental data. In any case, these models are stringently constrained.
The Higgs-like boson with mass GeV must have a coupling close to the SM one (). In those scenarios, such as asymptotically-free theories, where the second WSR is satisfied, the and constraints force to be in the range at 95% CL, as shown in Figure 2 (left). From Figure 2 (right) it follows that larger departures from the SM value can be accommodated when the second WSR does not apply, but one needs to introduce a correspondingly large mass splitting between the vector and axial-vector states.
The vector and axial-vector states should be heavy enough (above the TeV scale), see Figure 2.
The mass splitting between the vector and axial-vector resonance fields is very small when the second WSR is valid (consider (7) and the restrictions on ).
3 Constraining the Electroweak Effective Theory from the Resonance Theory
Once we have constrained the Resonance Theory by using short-distance constraints and the phenomenology, we want to use the Resonance Theory to determine the Low Energy Constants (LECs) of the electroweak effective theory at low energies (without resonances) . As we have pointed out above, and as a first approach to this issue, we do this estimation in the case of the purely Higgsless bosonic case (without fermions). This exercise is similar to the estimation of the LECs of Chiral Perturbation Theory by using Resonance Chiral Theory .
where we use the same notation as before and assume that parity is a good symmetry of the bosonic sector.
The use of short-distance constraints is again very important in order to reduce the number of resonance parameters. In this way, (4) and (6) allow to determine , and in terms of , and , so (13) becomes:
Acknowledgements.We wish to thank the organizers of the conference for the pleasant conference. This work has been supported in part by the Spanish Government and the European Commission [FPA2010-17747, FPA2011-23778, FPA2013-44773-P, SEV-2012-0249 (Severo Ochoa Program), CSD2007-00042 (Consolider Project CPAN)], the Generalitat Valenciana [PrometeoII/2013/007] and the Comunidad de Madrid [HEPHACOS S2009/ESP-1473].
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