Electroweak chiral Lagrangians and the Higgs properties at the one-loop level

In these proceedings we explore the use of (non-linear) electroweak chiral Lagrangians for the description of possible beyond the Standard Model strong dynamics in the electroweak sector. Experimentally one observes an approximate electroweak symmetry breaking pattern $SU(2)_L\times SU(2)_R/SU(2)_{L+R}$. Quantum Chromodynamics shows a similar chiral structure and, in spite of the differences (in the electroweak theory $SU(2)_L\times U(1)_Y$ is gauged), it has served for years as a guide for this type of studies. Examples of one-loop computations in the low-energy effective theory and the theory including the first vector and axial-vector resonances are provided, yielding, respectively, predictions for $\gamma\gamma\to Z_LZ_L,W^+_LW^-_L$ and the oblique parameters $S$ and $T$.


Introduction: strong dynamics and chiral Lagrangians
A non-linear realization of the EW would-be Goldstone bosons (WBGBs) is considered to build the EW low-energy effective field theory (EFT), which is described by an EW chiral Lagrangian with a light Higgs (ECLh). It includes the Standard Model (SM) content: the EW Goldstones w a , the EW gauge bosons W a μ and B μ and a singlet Higgs h (the fermion sector is not discussed here). In particular, in Sec. 2 we explain the chiral counting in the ECLh [5,6] and provide an example of a next-to-leading order (NLO) computation: we calculate γγ → W + L W − L , Z L Z L within this framework up to the one-loop level [5] at energies below new possible composite resonances, √ s Λ ECLh ∼min{M R , 4πv} (with v = ( √ 2G F ) −1/2 and 4πv 3 TeV). Analogous works on WW-scattering can be found in Refs. [7]. However, in the case of having heavy composite resonance, the EFT stops being valid when the energy becomes of the order of their masses (expected to be of the order of M R ∼ 4πv ∼ 3 TeV). One has to introduce these new degrees of freedom in our EW Lagrangian following a procedure analogous to that in QCD [8]. Likewise, under reasonable ultraviolet (UV) completion hypotheses like, e.g., the Weinberg sum-rules (WSRs) fulfilled by certain types of theories [9][10][11][12][13], one can make predictions on low-energy observables. In Sec. 3 we write down the relevant S U(2) L × S U(2) R invariant Lagrangian including the SM content and a multiplet of V and A resonances and extract one-loop limits on the resonance masses and the Higgs coupling g hWW [14] from the experimental values of oblique parameters S and T [15]. Alternative one-loop analyses can be found in Refs. [10,16]. a e-mail: juanj.sanz@uam.es I would like to thank the organizers for their work and the lively discussion during the workshop; also for their patience. This  The Higgs boson does not enter in the SM at tree-level in these processes (where one also has M(γγ → ZZ) tree SM = 0). Nevertheless, one can search for new physics by studying the one-loop corrections [5], which are sensitive to deviations from the SM in the Higgs boson couplings. Our analysis [5] is performed in the Landau gauge and making use of the Equivalence Theorem (Eq.Th.) [17], valid in the energy regime m 2 W , m 2 Z s. The EW gauge boson masses m W,Z are then neglected in our computation. Furthermore, since m h ∼ m W,Z 4πv 3 TeV we also neglect m h in our calculation. In summary, the applicability range in [5] is with the upper limit given by the EFT cut-off Λ ECLh , expected to be of the order of 4πv 3 TeV or the mass of possible heavy BSM particles. The WBGBs are described by a matrix field U that takes values in the S U(2) L ×S U(2) R /S U(2) L+R coset, and transforms as U → LUR † [2,3]. The relevant ECLh with the basic building blocks is with well-defined trasnformation properties [3,5,14]. Two particular parametrizations of the unitary matrix U (exponential and spherical) were considered in [5], both leading to the same predictions for the physical (on-shell) observables. 1 We consider the counting [5,6]. We require the ECLh Lagrangian to be CP invariant, Lorentz invariant and S U(2) L × U(1) Y gauge invariant. Here we focus ourselves on the relevant terms for γγ → w a w b at leading order (LO) -O(p 2 )-and NLO in the chiral counting -O(p 4 )- [3,5]: where X stands for the trace of the 2×2 matrix X, one has the photon field strength A μν = ∂ μ A ν −∂ ν A μ and the dots stand for operators not relevant within our approximations for γγ-scattering [5].
written in terms of the two independent Lorentz structures T (1,2) μν ∼ O(p 2 ) involving the external momenta, which can be found in [5]. The Mandelstam variables are defined as s = (p 1 + p 2 ) 2 , t = (k 1 − p 1 ) 2 and u = (k 1 − p 2 ) 2 and the i 's are the polarization vectors of the external photons.
In dimensional regularization, our NLO computation of the M(γγ → w a w b ) amplitudes can be systematically sorted out in the form [5] 1 Other representations have been recently studied in Ref. [18].

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where e ∼ O(p/v) and A and B are given up to NLO by [5] A(γγ → zz) LO The term with c r γ comes from the Higgs tree-level exchange in the s-channel, the term proportional to (a 2 − 1) comes from the one-loop diagrams with L 2 vertices, and the Higgsless operators in Eq. (5) yield the tree-level contribution to γγ → w + w − proportional to (a 1 − a 2 + a 3 ). Independent diagrams are in general UV divergent. However, in dimensional regularization, the final one-loop amplitude turns out to be UV finite and one has a r 1 − a r 2 + a r 3 = a 1 − a 2 + a 3 , c r γ = c γ [5], as in the Higgsless case [19].
In order to pin down each of the relevant combinations of ECLh couplings in Eq. (7) (a, c r γ and a r 1 − a r 2 + a r 3 ) one must combine our γγ-scattering analysis with other observables that depend on this same set of parameters. It is not difficult to find that other processes involving photons depend on these parameters. In Ref. [5] we computed 4 more observables of this kind: the h → γγ decay width (depending on a and c γ ), the oblique S -parameter (depending on a and a 1 ), and the γ * → w + w − (depending on a and a 2 − a 3 ) and γ * γ → h (depending on c γ ) electromagnetic form-factors. The one-loop contribution in these six relevant amplitudes is found to be UV-divergent in some cases. These divergences are absorbed by means of the generic O(p 4 ) renormalizations a r i (μ) = a i + δa i . As expected, the renormalization in the six observables gives a fully consistent set of renormalization conditions and fixes the running of the renormalized couplings in the way given in Table 1.

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QCD@Work 2014  [4]. For the sake of completeness, we have added the running of the ECLh parameters a r 4 and a r 5 , which has been recently determined in the one-loop analysis of WW-scattering within the framework of chiral Lagrangians [7]. One can see that in the SM limit (a = b = 1) these L 4 coefficients do not run, in agreement with the fact that these higher order operators are absent in the SM.

Impact of spin-1 composite resonances on the oblique parameters
One can extend the range of validity and predictability of the ECLh by adding possible new states to the theory. Thus, the lightest V and A resonances are added to the EW Lagrangian in Ref. [14] in order to describe the oblique parameters S and T [9]. The relevant EW chiral invariant Lagrangian is given by the kinetic and Yang-Mills terms and the interactions [14] 2, 3 In order to compute S and T up to the one-loop level we use the dispersive representations [9,14], with ρ S (t) the spectral function of the W 3 B correlator [9,21] and ρ T (t) the spectral function of the difference of the neutral and charged Goldstone self-energies [14]. The calculation of T above has been simplified by means of the Ward-Takahashi relation T = Z (w + ) /Z (w 0 ) − 1 [20]. Only the lightest two-particle cuts have been considered in ρ S (t) and ρ T (t), respectively, {ww, wh} and {Bw, Bh}. Since ρ S (t) SM t→∞ −→ 0, the convergence of the Peskin-Takeuchi sum-rule requires ρ S (t) t→∞ −→ 0. Furthermore, assuming that weak isospin and parity are good symmetries of the BSM strong dynamics, the W 3 B correlator is proportional to the difference of the vector and axial-vector two-point Green's functions [9]. In asymptotically-free gauge theories this difference vanishes at s → ∞ as 1/s 3 [12], implying the (tree-level) LO WSRs [13], 2 Here we follow the notation f μν ± = u †Ŵμν u ± uB μν u † from Ref. [14,21], where there is a global sign difference with [5] in the definitions ofŴ μ andB μ . The spin-1 resonances are described in the antisymmetric tensor formalism [8].
However, although the 1st WSR is expected to be true in gauge theories with non-trivial ultraviolet fixed points [10,11], the 2nd WSR is questionable in some of these models. Thus, two alternative scenarios are studied in Ref. [14]: one assuming the two WSRs and another assuming just the 1st WSR.
At tree-level one has the the LO determinations [9,14,21] S LO = 4π In the first case, the two WSRs imply M V < M A and determine F V and F A in terms of the resonance masses [8,9,14,21]. In the second case, it is not possible to extract a definite prediction with just the 1st WSR but one can still derive the inequality above if one assumes a similar mass hierarchy M V < M A . On the other hand, this inequality flips direction if M A < M V or turns into an equality in the degenerate case M V = M A [14]. At NLO the computed W 3 B correlator is given by the ww and hw cuts, whose contributions to the ρ S (t) spectral function would have an unphysical grow at high energies unless F V G V = v 2 and F A λ hA 1 = av [8,14,21]. Thus, we obtain the NLO prediction [14] S = 4πv 2 In the two-WSRs scenario, in order to enforce the 2nd WSR at NLO one needs the additional constraint a = M 2 V /M 2 A (hence restricted to the range 0 ≤ a ≤ 1). Again, the inequality in the last line flips direction or turns into an equality when, respectively, At LO, ρ T (t) is zero and one has T LO = 0. At NLO, where we enforce the ρ S (t) constraints F V G V = v 2 and F A λ hA 1 = av, we find that ρ T (t) t→∞ −→ 0 and obtain the NLO prediction In Fig. 1, we show the compatibility between the experimental determinations for S and T [15] and our NLO determinations in both scenarios. The numerical results in Table 2 show that the precision 00053-p.5 electroweak data requires resonance masses over the TeV and the hWW coupling to be close to the SM one (a SM = 1), in agreement with present LHC bounds [22]. To conclude, we emphasize that, remarkably, just by considering the experimental m h (the only LHC input) and the EW precision observables (LEP input), the allowed region concentrates around a 1 for reasonable values of the splitting M V /M A ∼ O(1) (see Fig. 1 and Table 2).