Electroweak chiral Lagrangian with a light Higgs and γγ→ZLZL, W+LW−L scattering at one loop

In these proceedings we provide a brief summary of the findings of a previous article where we have studied the photon-photon scattering into longitudinal weak bosons within the context of the electroweak chiral Lagrangian with a light Higgs, a low-energy effective field theory including a Higgs-like scalar singlet and where the electroweak would-be Goldstone bosons are non-linearly realized. We consider the relevant Lagrangian up to next-to-leading order in the chiral counting, which is explained in some detail here. We find that these amplitudes are ultraviolet finite and the relevant combinations of next-to-leading parameters (cγ and a1 − a2 + a3) do not get renormalized. We propose the joined analysis of γγ–scattering and other photon related observables (Γ(h → γγ), S –parameter and the γ∗ → W L W− L and γ∗ → hγ electromagnetic form-factors) in order to separate and determine each chiral parameter. Moreover, the correlations between observables provided by the NLO computations would lead to more stringent bounds on the new physics that is parametrized by means of this effective Lagrangian. We also show an explicit computation of the γγ–scattering up to next-to-leading order in the S O(5)/S O(4) minimally composite Higgs model.


γγ-scattering as a probe into new physics
Two years ago the Large Hadron Collider (LHC) discovered a new particle, most likely a scalar, with mass m h ≈ 125 GeV [1] and couplings so far compatible with what one would expect for the Standard Model (SM) Higgs boson. We are therefore in a scenario with small deviations from the SM and, apparently, a large mass gap (as no new particle has shown up below the TeV). Thus, the effective field theory (EFT) framework seems to be the most convenient one to confront current experimental data and to explore possible beyond Standard Model (BSM) effects in the electroweak (EW) sector.
In these proceedings we discuss some of the findings in a previous work [2]. Therein we studied the processes γγ → Z L Z L and γγ → W + L W − L in the context of a general EW low-energy effective field theory (EFT), which we will denote as Electroweak Chiral Lagrangian with a light Higgs (ECLh), with the EW would-be Goldstone bosons (WBGBs) denoted here by w a and non-linearly realized. In addition to be more general, this non-linear representation seems to be more appropriate in the case of strong interactions in the EW sector, as it is the case in Quantum Chromody-namics [3,4]. The three would-be Goldstone bosons w a from the spontaneous EW symmetry breaking are parametrized through a unitary matrix U that takes values in the S U(2) L × S U(2) R /S U(2) L+R coset. 1 The Higgs boson does not enter in the SM at treelevel in these γγ → V L V L (V = Z, W) processes (where in addition M(γγ → Z L Z L ) tree SM = 0 [6]). Nevertheless, one can search for new physics by studying the one-loop corrections [2], which are sensitive to deviations from the SM in the Higgs boson couplings. Our analysis [2] is performed in the Landau gauge and making use of the Equivalence Theorem (Eq.Th.) [7], 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 experimentally m h ∼ m W,Z we also neglect m h in our calculation. In summary, the applicability range in [2] is ECLh , (2) 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, where v = 246 GeV denotes the SM Higgs vacuum expectation value.
Although our derivation is general and does not assume any particular underlying BSM theory, it is obviously inspired by models where the Higgs is another (pseudo) Nambu-Goldstone boson (NGB). Indeed, in the final part of these proceedings we provide an explicit example for the S O(5)/S O(4) Minimally Composite Higgs Model (MCHM) [8].

ECLh up to next-to-leading order
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 † [9,10]. The basic building blocks employed to construct the relevant ECLh Lagrangian for our analysis are [2,9,10] 1 Two parametrizations of the coset were considered in Ref. [2]: exponential coordinates, U = exp{iτ a w a /v}; and spherical coordinates, U = 1 − w a w a /v 2 +iτ a w a /v. Both parametrizations are found to produce the same prediction for the γγ → w a w b amplitudes once the external particles are set on-shell. Other representations of U were recently studied in [5].
with well-defined transformation properties [2,10]. The Higgs field h is a singlet in the ECLh and enters in the Lagrangian operators via polynomials or their partial derivatives [11,12]. These building blocks are employed to construct ECLh operators with CP, Lorentz and S U(2) L × U(1) Y gauge invariance.
We consider the following scaling in powers of momentum p, (4) and the counting for the tensors above [2,13,14], Within the approximations of our analysis [2], the relevant ECLh operators for γγ → w a w b at leading order (LO) -O(p 2 )-and next-to-leading order (NLO) in the chiral counting -O(p 4 )-are [2, 10] where 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 the γγ-scattering [2]. The classification of the chiral order in the previous Lagrangian (6) provides a consistent perturbative expansion as we show now in more detail. First, we denote as O(p d ) any operator of the generic form with χ any bosonic field (h, w a , W a μ , B μ ), p refers to derivatives ∂ or light masses m h,W,Z acting appropriately on the fields, and f (d) k are the corresponding couplings of the operator ( f (2) Let us now consider an arbitrary diagram with L loops, I internal boson propagators and N d vertices from L d (with total number of vertices V = d N d ). Following Weinberg's dimensional arguments [3], it is not difficult to see that in dimensional regularization this amplitude will scale with p like [2,3,13] where we have used the topological identity I = L+V −1 in the last line. Finally, keeping track of the constant factors with powers of (16π 2 ) −1 (from loops) and v (coming with every field χ in (7)), and the coupling con- with N E the number of external boson legs, which shows up in the final expression after counting the total number of fields from all the L d vertices, and hence the total number of powers of v −1 : the diagram carries then the The various possible contributions to the amplitude of a given process can be then sorted in the form Observing Eq. (9) one can see that higher orders in the chiral expansion can be reached by either adding more loops L to the diagram or vertices of "chiral dimension" d ≥ 4. Notice that adding vertices from L 2 does not modify the scaling of the diagram with p, as far as the number of loops L remains the same. At LO, one needs to consider only the tree-level diagrams made out of L 2 vertices (L = 0, N 2 arbitrary, N d≥4 = 0); at NLO, one needs to compute the one-loop diagrams with L 2 vertices (L = 1, N 2 arbitrary, N d≥4 = 0) and tree-level diagrams with one vertex from L 4 and any number of vertices from L 2 (L = 0, N 2 arbitrary, N 4 = 1, N d≥6 = 0); the procedure is analogous for higher chiral orders. In our particular computation of M(γγ → w a w b ) up to NLO, the contributions we find are sorted out in the form [2] where e ∼ O(p/v) and a i stands for a general L 4 coupling. These three types of contributions can be better understood through the detailed analysis of the examples in Fig. 1, three of the many diagrams entering in γγ → w + w − up to NLO [2]: • a) The tree-level amplitude in Fig. 1  • b) The one-loop amplitude in Fig. 1 with each γw + w − vertex scaling like e p, each hw + w − vertex like p 2 /v and each internal propagator like p −2 . This amplitude actually comes together with logarithms of the energy and ultraviolet (UV) divergences.
• c) The tree-level amplitude in Fig. 1.c with one vertex from L 4 and vertices from L 2 scales like [2] M c ∼ with the γγh vertex from L 4 scaling like 2 c γ e 2 p 2 /v, the hw + w − vertex like p 2 /v and the intermediate Higgs propagator like p −2 . In general, the cancelation of the UV divergences in the one-loop NLO diagrams will require the renormalization of the L 4 couplings, e.g., c r γ = c γ + δc γ , a r i = a i + δa i .
The M(γ(k 1 , 1 )γ(k 2 , 2 ) → w a (p 1 )w b (p 2 )) amplitudes, with w a w b = zz, w + w − , have the Lorentz decomposition [2,15,16] , u), (12) written in terms of the two independent Lorentz structures T (1) μν ∼ O(p 2 ) and T (2) μν ∼ O(p 4 ) involving the external momenta, 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 initial photons. At LO and NLO we find for the neutral channel [2], and for the charged one [2] A 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 (6) yield the tree-level contribution to γγ → w + w − proportional to (a 1 − a 2 + a 3 ). Independent diagrams are in general UV divergent and have complicated logarithmic and Lorentz structure. 3 However, in dimensional regularization, when all the different contributions (10 and 39 loop diagrams for the neutral and charged channels, respectively) are put together the final one-loop amplitude turns out to be UV finite and free of logs in the limits considered in our analysis [2], both in γγ → zz and γγ → w + w − . Therefore the combinations of NLO couplings c γ and 3 For instance, the diagram shown in Fig. 1.c corresponds to the diagram 14 in App. B.2 in Ref. [2], given by the complicate structure +2 ( 1 2 ) t + ( 1 2 ) u) + B 0 (t, 0, 0)((( 1 Δ) + ( 1 k 2 ))(( 2 Δ) − ( 2 k 1 )) − ( 1 2 ) t)) + ( 1 Δ) (( 2 Δ) + 3 ( 2 k 1 ))(−s) + ( 1 k 2 ) (( 2 k 1 ) (23t + 11u)  Table 2. Running of the relevant ECLh parameters and their combinations appearing in the six selected observables [2]. The third column provides the corresponding running for the Higgsless EW Chiral Lagrangian [18]. The table has been completed with the running of a 4 and a 5 from WW-scattering analyses [17].

Relevant combinations Observables of parameters
(a 1 − a 2 + a 3 ) which enter here do not need to be renormalized: a r 1 − a r 2 + a r 3 = a 1 − a 2 + a 3 (like in the Higgsless case [15,16]) and c r γ = c γ are renormalization group invariant [2]. All the UV divergences and renormalizations occur at O(p 4 ) and the L 2 couplings (like a, for instance) do not get renormalized within the approximations considered in this work [2].
Our γγ-scattering amplitudes depend on three combinations of parameters (a, c r γ and a r 1 − a r 2 + a r 3 ). This tells us that in order to extract each coupling separately one needs to study other observables. However, other related photon processes are ruled by the same parameters. In Ref. [2] we provide a list of four additional observables, computed with the ECLh under the same assumptions of this work and depending on different combinations of a, c r γ , a r 1 and (a r 2 − a r 3 ): the h → γγ partial width, the oblique S -parameter and the electromagnetic form-factors for γ * → w + w − and γ * → hγ. In table 1 one can see the combinations of couplings that rule each quantity. This gives six observables and four relevant combinations. Thus, the ECLh allows us to extract the couplings from four observables and make a definite prediction for the other two. Notice that a global fit with the non-linear EFT must incorporate both NLO loops and NLO tree-level contributions (both are of the same order in the chiral counting), otherwise one may eventually run into inconsistent determinations.
These six observables provide in addition a consistent set of renormalization conditions (a 1 and a 2 −a 3 do need to be renormalized). The corresponding running for the O(p 4 ) couplings C r = c r γ , a r i are summarized in Table 2, where the constants Γ C therein are given by For the sake of completeness, we have also included in the last two lines of Table 2 the running of a r 4 and a r 5 determined in WW-scattering analyses [17]. A remarkable feature of the one-loop photon-photon amplitudes is that individual diagrams carry the usual chiral suppression O p 2 /(16π 2 v 2 ) with respect to the LO. However, the full one-loop amplitude shows a stronger suppression O (1 − a 2 )p 2 /(16π 2 v 2 ) , where experimentally a is found to be close to 1 within O(10%) uncertainties [1].
We would like to finish this section with the preliminary phenomenological analysis for γγ → W + L W − L shown in Fig. 2. The fact that the Equivalence Theorem works with an error lower that 2% in the SM for M γγ = √ s > 0.5 TeV reassures us about the validity of our analysis. The SM cross section behaves at high energies like 1/s for γγ → W + L W − L . On the other hand, the O(p 4 ) NLO terms in the amplitude (15) add a contribution to the cross section that grows with s and turns more and more important at higher and higher energies. We observe the impact of possible new physics by varying the couplings within typical ranges for the chiral couplings [4,16]: a r 1 − a r 2 + a r 3 = 2 × 10 −3 , 4 × 10 −3 , 6 × 10 −3 (respectively from bottom to top in Fig. 2), and the other couplings set to their SM values, a = 1 and c r γ = 0. The deviation from the SM is negligible at very low energies. Nonetheless, it grows with M γγ and for a 1 − a 2 + a 3 = 2 × 10 −3 (4 × 10 −3 ; 6 × 10 −3 ) the cross section exceeds the SM one by 20% for M γγ > 2.6 TeV (1.8 TeV; 1.5 TeV). The signal keeps turning more and more intense beyond these values of M γγ . A more detailed study will be provided in a forthcoming work. In order to study this subprocess in colliders (LHC or future e + e − accelerators) we will have to convolute this γγ cross sections with the corresponding photon luminosity functions. Although preliminary studies show that one can get a measurable amount of events for integrated luminosities of the order of 1 ab −1 , the key-point will be the discrimination and separation of SM background through convenient cuts [19,20,21] and the minimization of theoretical uncertainties. For instance, the nonzero h, W and Z masses produce corrections suppressed by m h,W,Z /M γγ , which may turn important if one studies this reaction below the TeV. This also means going beyond the Equivalence Theorem and computing the full one-loop γγ → V L V L amplitude. It can be also interesting to analyze within this framework the reverted subprocess VV → γγ via vector boson fusion at LHC.

γγ-scattering in MCHM
In this section we show an explicit example of how our EFT description describes the small momentum regime of any underlying theory with the same symmetries and low-energy particle content.
In the context of the so called S O(5)/S O(4) MCHM [8] it is assumed that some global symmetry breaking takes place at some scale 4π f > 4πv so that with the G-fundamental representation vector Φ parametrizing the NGBs in the way with s = sin θ, c = cos θ and θ being the vacuum misalignment angle, with sin θ = v/ f [8]. The S 4 metric is given by The γγ-scattering was considered in the framework of general S O(N + 1)/S O(N) gauged NLσM [22] for low-energy QCD and the one-loop computation only involves the bubble and triangle diagrams (Fig. 3). The one-loop result at NLO is simply A(γγ → zz) NLO−loop = A(γγ → hh) NLO−loop = − 1 4π 2 f 2 , and B NLO−loop = 0 in all cases. We find this in agreement with our ECLh result in Eqs. (14) and (15) by means of the relation (1 − a 2 ) = v 2 / f 2 between f , v and the hW + W − coupling a in the S O(5)/S O(4) MCHM [8].
We want to remark that the L MCHM 2 is often written in exponential coordinates rather than the S 4 parametrization used in this computation [8], leading to a lowenergy Lagrangian with exactly the same structure as L 2 in Eq. (6) but with precise predictions for the ECLh couplings. One can then use this Lagrangian in terms of exponential coordinates and compute the γγ-scattering in the way done in this work (in that same parametrization), with all its complication and tricky diagrammatic cancelations. The final outcome agrees with (19), as expected. The lesson one draws is that, even though all the coset parametrizations yield the same outcome for a given on-shell amplitude, computations can be simpler for some choices of the NGB coordinates (we already saw this in our ECLh calculation in the previous section, where some vertices are absent in spherical coordinates and one has fewer diagrams to compute [2]). Likewise, in the exponential parametrization individual loop diagrams are suppressed with respect to the LO by O p 2 /(16π 2 v 2 ) and only after summing up all of them one finds that the full one-loop amplitude is actually suppressed by O p 2 /(16π 2 f 2 ) . On the other hand, in the S 4 coordinates each single diagram shown in Fig. 3 already carries the final suppression p 2 /(16π 2 f 2 ) with respect to the LO.