# Coupling *WW*, *ZZ* unitarized amplitudes to \(\gamma \gamma \) in the TeV region

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## Abstract

We define and calculate helicity partial-wave amplitudes for processes linking the Electroweak Symmetry Breaking Sector (EWSBS) to \(\gamma \gamma \), employing (to NLO) the Higgs-EFT (HEFT) extension of the Standard Model and the Equivalence Theorem, while neglecting all particle masses. The resulting amplitudes can be useful in the energy regime (\(500~\mathrm{{GeV}}{-}3~\mathrm{{TeV}}\)). We also deal with their unitarization so that resonances of the EWSBS can simultaneously be described in the \(\gamma \gamma \) initial or final states. Our resulting amplitudes satisfy unitarity, perturbatively in \(\alpha \), but for all *s* values. In this way we improve on the HEFT that fails as interactions become stronger with growing *s* and we provide a natural framework for the decay of dynamically generated resonances into *WW*, *ZZ* and \(\gamma \gamma \) pairs.

## Keywords

Partial Wave Chiral Expansion Riemann Sheet Minimal Composite Higgs Model Elastic Amplitude## 1 Introduction

### 1.1 The electroweak symmetry-breaking sector

Electroweak symmetry breaking happens at a scale of \(v=246\) GeV for reasons still unsettled, and the LHC is trying to discern whether the Higgs-like scalar boson found there [1, 2, 3, 4] couples to other known particles as dictated by the Standard Model. If the LHC finds new particles or perhaps broad resonances in the TeV region under exploration, it is natural (by the energy scale involved) to guess that they play a role in breaking electroweak symmetry.

Meanwhile, it makes sense to formulate theory in terms of the particles already known to be active in that Electroweak Symmetry breaking sector, namely the new Higgs-like boson *h* and the longitudinal components of the *W* and *Z* electroweak bosons. The resulting, most general, effective field theory that does not assume *h* to be part of an electroweak doublet, has come to be known as Higgs Effective Field Theory (HEFT) [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], and has been built upon the old Higgsless Electroweak Chiral Lagrangian [16, 17, 18, 19, 20, 21]. An effort to constrain the parameters of this HEFT from low-energy observables is under way employing LHC data [22]. We adopt the parity-conserving HEFT as our starting point.

*WW*/

*hh*threshold observables, but rather at the possible resonance region above 500 GeV, we adopt the Equivalence Theorem [23, 24, 25, 26, 27, 28, 29, 30]. This is valid for \(s\gg M_h^2,M_W^2,M_Z^2\simeq (100~\mathrm{{GeV}})^2\), and it allows one to identify the longitudinal gauge bosons with the pseudo-Goldstone bosons of symmetry breaking \(\omega ^a\) (\(a=1\dots 3\)) in their scattering amplitude

*T*. For example one has

In principle, HEFT is a usable theory through \(E\sim 4\pi v =3\) TeV, but if new strong interactions and resonances appear in that interval, its applicability region quickly contracts. Even at low energies, truncated HEFT may run into convergence difficulties. These are not alleviated much by increasing the order of the chiral expansion, and on the contrary the number of chiral parameters swiftly increases reducing predictive power. We therefore follow a different strategy to extend the low-energy regime by using dispersion relations (DR) compatible with analyticity and unitarity. This approach is extremely useful in the original hadron ChPT [34, 35] and we have long advocated it for the SBS of the strongly interaction sector of the SM [36].

Dispersion relations are identities that do not include all dynamical information, but they become much more predictive when the low-energy scattering amplitudes are known (for example from the HEFT). Even so, some model dependence remains in the treatment of amplitude cuts other than that extending for physical *s* (normally, this applies to the left cut, LC). To constrain it, we employ two different methods, the Inverse Amplitude Method (IAM) and the N/D method. We have detailed both of them in this context in [33] where a complete discussion and references may be found, as well as further unitarization strategies. As will also be exposed later (see Fig. 8), both methods are qualitatively equivalent, but to describe the same resonance they need parameters of the underlying NLO HEFT that are different from each other by 25%.

In the end, the resulting unitarized HEFT (UHEFT) provides an analytical and unitary description of higher-energy dynamics which is essentially unique up to the first resonances. These appear as poles in the second Riemann sheet thanks to the adequate analytical behavior of the amplitudes.

### 1.2 Coupling to \(\gamma \gamma \)

The \(\gamma \gamma \) channel is by itself not part of the EWSBS, but because photons propagate to the detectors (being reconstructed, e.g. at the electromagnetic calorimeters) they are direct messengers from the collision in the final state. Studies of new particles decaying into two photons have been pursued since the dawn of particle physics [37].

Conversely, photons also provide interesting production mechanisms from the initial state. With slight virtuality they accompany high-energy beam particles: the photon can be thought of as a parton of the proton [38] or the electron in *pp* and \(e^-e^+\) colliders, respectively. Thus, high-energy colliders can, in a sense, be thought of as photon colliders. The small electromagnetic \(\alpha \) lowers the photon flux, but in exchange the initial state is very clean and directly couples to the EWSBS (since the \(W^\pm \) are charged particles). For example, the CMS collaboration [39] is already setting bounds to anomalous quartic gauge couplings from an analysis of precisely \(\gamma \gamma \rightarrow W^+ W^-\). Moreover, thanks to Compton backscattering, photon colliders driven by lepton beams are perhaps also a future option [40, 41].

Thus, their coupling to the EWSBS is of much interest. Within the context of the HEFT, the perturbative Feynman amplitudes at the one-loop level have already been reported in [42].

In this work we extend the amplitudes to the resonance region. Because unitarity is most easily expressed in terms of partial waves, and because the partial-wave series converges quickly in the “low-energy” domain where HEFT is valid, we have projected the EWSBS as well as the \(\gamma \gamma \) over good angular momentum *J*. In the case of the Goldstone or the Higgs bosons, \(L=J\), but when photons are involved, their intrinsic spin is also at play. We have employed the helicity basis to carry out the computations.

While custodial isospin is presumably conserved by the EWSBS (as suggested by LEP), the electromagnetic coupling to the \(\gamma \gamma \) state violates its conservation. Still, we can label the partial-wave amplitudes from the initial \(\omega \omega \)-state isospin in photon–photon production (or the final \(\omega \omega \) at a photon collider).

The helicity basis and the corresponding amplitudes are constructed below in Sect. 3. Their partial-wave projections in turn appear in Sect. 4. We show their single- and coupled-channel unitarization in Sect. 5 and some selected numerical examples thereof in Sect. 6; finally, we add a few remarks in Sect. 7.

## 2 The chiral Lagrangian and its parameterizations

*h*. This particle content is valid for the energy range \(M_h,M_W,M_Z\simeq (100~\mathrm{{GeV}})^2 \ll s\ll 4\pi \nu \simeq 3~\mathrm{{TeV}}\) and exhausts the known Electroweak Symmetry Breaking sector. Resonances of these particles’ scattering are possible in this interval and we will describe them employing scattering theory based on the effective Lagrangian instead of introducing them as new fields. The starting point to expose the Lagrangian for the \(\omega \) and

*h*bosons, whose elements are immediately discussed, may be taken as

^{1}The extension that includes \(\gamma \gamma \) states can be found in Ref. [42], the covariant derivative being

*U*, and \(\tau _i\) are Pauli matrices. Finally, we note the definition of charge basis, \(\omega ^\pm = \frac{\omega ^1\mp i\omega ^2}{\sqrt{2}}\), \(\omega ^0 = \omega ^3\). Thus we are using a \(SU(2)_L \times U(1)_Y\) gauged non-linear sigma model corresponding to the coset \(SU(2)_L \times SU(2)_R/ SU(2)_{L+R}\) coupled to the

*h*singlet, where \(SU(2)_{L+R}\) is called the custodial group.

Different values of the parameters *a* and *b* in Eq. (3) make the Lagrangian density in Eq. (2) represent the low-energy limit of different theoretical models (and the NLO parameters specified shortly will also depend on the underlying theory). For instance, \(a^2=b=0\) corresponds to the old, Higgsless Electroweak Chiral Lagrangian [16, 17, 18, 19, 20, 21] (which had no explicit Higgs field and thus seems ruled out), \(a^2=1-(v/f)^2\), \(b=1-2(v/f)^2\) is the low-energy limit of a *SO*(5) / *SO*(4) Minimal Composite Higgs Model [43, 44, 45], \(a^2=b=(v/f)^2\) can be obtained from dilaton-type models [46, 47], and finally \(a^2=b=1\) represents the SM with a light Higgs (this is the current experimental situation).

There is no strong direct limit over the *b* parameter, because of the difficulty of measuring a 2-Higgs state. However, an indirect limit arises because of the coupling between the *hh* decay and the elastic \(\omega \omega \) scattering, as we showed in earlier work [48]. The current direct claimed limits over the *a* parameter, at a confidence level of \(2\sigma \) (\({\approx }95\%\)) are, from CMS [49], \(a\in (0.87, 1.14)\); and from ATLAS [50], \(a\in (0.96,1.34)\). Actual experimental analysis may be tracked from [51], which also details the LHC constraints over a number of SM extensions.

### 2.1 WBGBs scattering and coupling to \(\gamma \gamma \)

The one-loop computation for \(\omega \omega \rightarrow \omega \omega \), \(\omega \omega \rightarrow hh\) and \(hh\rightarrow hh\) processes was reported in [31, 33]. Because of the equivalence theorem regime, \(e^2,g^2,g^{'2}\ll s/v^2\), the electric charge coupling the photon can be introduced as a perturbation. Thus, the strong physics of the \(\omega \omega \) (longitudinal \(W_L\) modes) and *hh* sector dominates over the transverse modes (\(W_T\), \(\gamma \)) and provides the driving force to saturate unitarity. One can then work to leading non-vanishing order when incorporating the transverse modes. The minimum set of counterterms needed to renormalize those scattering amplitudes to one loop is that corresponding to the \(a_4\), \(a_5\), *g*,^{2} *d* and *e* parameters (see Refs. [31, 33]).

Current (\(2\sigma \)) bounds on those NLO parameters are \(c_\gamma \in (\frac{-1}{16\pi ^2},\frac{0.5}{16\pi ^2})\) [12]; \(a_1<10^{-3}\), \(a_2\in (-0.26,0.26)\) and \(a_3\in (-0.1,0.04)\) [53]. It is practical to quote these bounds for the \(a_i\) in terms of the only combination that will be needed in this work which is (conservatively adding them) \((a_1-a_2+a_3)\in (-0.36,0.3)\). We will employ these limits when we illustrate the parameter dependence later on in Sect. 6.

## 3 Matrix elements for \(\gamma \gamma \) to \(\omega \omega \) and *hh* scattering at NLO

*Z*is a neutral particle,

Lorentz structures \(\varepsilon _1^\mu \cdot \varepsilon _2^\nu T_{\mu \nu }^{(1)}\) and \(\varepsilon _1^\mu \cdot \varepsilon _2^\nu T_{\mu \nu }^{(2)}\) (Eqs. (18) and (19)). All are invariant under \(\theta \rightarrow \pi -\theta \), that is, *t*–*u* exchange, a consequence of Bose symmetry that guarantees \(\mathscr {M}(\gamma \gamma \rightarrow \omega ^+\omega ^-)_\mathrm{LO, NLO}=\mathscr {M}(\gamma \gamma \rightarrow \omega ^-\omega ^+)_\mathrm{LO, NLO}\)

\((\lambda _1\lambda _2)\) | \((++)\) | \((+-)\) | \((-+)\) | (- -) |
---|---|---|---|---|

\([\varepsilon _1^\mu (\lambda _1)\cdot \varepsilon _2^\nu (\lambda _2)] T_{\mu \nu }^{(1)}\) | \(-s/2\) | 0 | 0 | \(-s/2\) |

\([\varepsilon _1^\mu (\lambda _1)\cdot \varepsilon _2^\nu (\lambda _2)] T_{\mu \nu }^{(2)}\) | \(s^2\) | \(-s^2(\sin \theta )^2 e^{2i\varphi }\) | \(-s^2(\sin \theta )^2 e^{-2i\varphi }\) | \(s^2\) |

The structure of Table 1 is remarkable. First, the amplitudes with equal photon helicities, \(T^{++}\) and \(T^{--}\), come in the combination \(-\frac{s}{2} A + s^2 B\). But, due to Eq. (13), this just cancels the LO contribution, and with it Rutherford’s 1 / *t* collinear divergence (and the exchanged one in 1 / *u*). There is no photon–photon annihilation with equal helicity into the Goldstone bosons at LO. Second, the opposite-helicity combinations \(T^{+-}\) and \(T^{-+}\) are non-vanishing at LO, but again Table 1 assigns a kinematic factor \(\sin ^2\theta \), which just cancels the angular dependence from the *t*- and *u*-channel \(\omega \) exchange diagrams, and thus once more the collinear divergences drop out. Therefore, polar angular integrals may easily be computed and partial-wave amplitudes to be introduced in Sect. 4 are well defined for all helicity combinations.

*N*and

*C*(for “neutral” and “charged”, respectively) to indicate, the

*zz*and \(\omega ^+\omega ^-\) final states (in the charge basis), as defined in Eqs. (12) and (14). We further shorten \(T_I^{\lambda _1\lambda _2}\equiv \langle {I,0|T|\lambda _1\lambda _2}\rangle \); explicitly, because of Eq. (21),

*hh*and \(\omega \omega \), then, because

*hh*has no charge, it decouples from \(\gamma \gamma \) too.

## 4 Scattering partial waves with \(\gamma \gamma \) states

*hh*final state is an isospin singlet, and it only couples with \(J=0\) and positive parity states (see Eq. (26)). Thus, the corresponding partial waves are

## 5 Unitarity requires \(\omega \omega \) resonances to be visible in \(\gamma \gamma \)

The \(\omega \omega \) partial waves decouple from the *hh* channel for \(a^2=b\), (see Eq. (37)). In keeping the more general \(a^2\ne b\) situation, the reaction matrix includes an inelastic \(\gamma \gamma \rightarrow hh\) coupling and is not block diagonal.

Because the electromagnetic interaction violates custodial isospin conservation (each \(\omega \) boson has a different electric charge), the \(\gamma \gamma \) state couples to both \(I=0\) and \(I=2\) (unlike *hh*, which is a singlet \(|{0,0}\rangle = |{hh}\rangle \)). Though each channel has its own separate strong dynamics, they both provide probability flow into the \(\gamma \gamma \) state as dictated by the corresponding Clebsch–Gordan coefficients.

### 5.1 \(\gamma \gamma \) scattering with *hh* channel decoupled

*hh*channel by setting \(a^2=b\) (and other parameters coupling \(\omega \omega \) and

*hh*in our earlier work, \(d=e=0\)). Then the amplitude matrix is three by three (we specify the \(I=0,2\) isospin index to make the three-channel nature of the matrix manifest; for each of them, the angular momentum index

*J*can also take the values 0 or 2). It can be given as

*s*and become strong, a unitarization scheme is mandatory to have a sensible amplitude [34, 35]. On the other hand, since \(\alpha \) remains small, it can be considered at leading order so that Eq. (41) will be satisfied to all orders in

*s*but only to LO in \(\alpha \), with no appreciable error. Imposing the unitarity condition to Eq. (40) and working to LO in \(\alpha \) returns

*K*(Eq. (44a)), \(B(\mu )\),

*D*and

*E*(Eq. (44b)) are given in Ref. [33] for each partial wave.

*s*plane, allowing for resonances in the second Riemann sheet below the RC, where it satisfies elastic unitarity. At low \(\sqrt{s}\) it matches the chiral expansion as can be seen by reexpanding it. And since its derivation follows from a fully prescribed dispersion relation, it can be written down to higher orders (should e.g. the NNLO chiral amplitude become known) without ambiguity.

*P*amplitudes, the second of Eq. (42) is the statement of Watson’s theorem, which sets its phase to that of \(\omega \omega \) rescattering. A solution strategy is, as in the case of

*A*(

*s*), to write a dispersion relation for the auxiliary “inverse” function \(w'(s)=P^{(0)2} /P(s)\) with contributions coming from the RC (right cut), the LC (left cut) and the infinite circle. To solve this dispersion relation the simplest ansatz for \(w'(s)\) is \(w'(s) = f(s) w(s)\) where

*f*(

*s*) is real for real

*s*and \(w(s) = (A^{(0)}(s))/A(s)\). The low-energy behavior of

*P*(

*s*) dictates, upon matching to it, \(f(s) = P^{(0)}/A^{(0)}(s)\). Then we can use the IAM result for

*w*(

*s*) and obtain

*N*/

*D*system of dispersion relations that satisfies elastic unitarity for all

*s*and has the right analytic properties, having only at hand one order of perturbation theory (here, the NLO) is

The *B* and *D* which appear in Eq. (50b) are the same that those in Eq. (44b). Note that, by means of perturbative unitarity, \(K=0\implies E=0\), thus simplifying the full N/D expression of Ref. [33].

### 5.2 Coupled \(\gamma \gamma \longleftrightarrow (\omega \omega , hh) \) scattering

*hh*channels feeding the \(\gamma \gamma \) state. Because the electromagnetic interaction violates isospin conservation, the reaction matrix must include both \(I=0,2\) subchannels of the \(\omega \omega \) system, and it is thus of dimension four. Assuming weak isospin conservation in the Goldstone dynamics, which puts zeros in row three and column three, and to order \(\alpha \), which makes the (4, 4) element vanish, it is

*s*chiral expansion, \(_sF^{(0)}\) and \(_sF^{(1)}\).

*s*coming from the WBGBs and

*h*rescatterings are taken into account in the IAM. For example, in expanding to one more order in Eq. (61) we find

*s*) amplitude, so that the N/D coupled-channel method is used instead for the unitarization of the WBGBs and

*h*scattering matrix elements (\(A_{I2}\), \(M_2\) and \(T_2\)). The matricial N/D formula, analogous to the elastic case of Eq. (49), is

*hh*states do not couple with \(\gamma \gamma \) for \(J=2\) [see Eq. (26)]. Thus, we need the matricial N/D method of Eq. (65) for unitarizing the \(\omega \omega \rightarrow \omega \omega \) partial waves, but the coupling with \(\gamma \gamma \) states can be computed by using the (scalar) equation (51).

## 6 Some numerical examples

We start by commenting on the perturbative partial-wave amplitudes very briefly. Referring to Eq. (38), we see that the NLO perturbative amplitudes \(P^{(0)}_{02}\), \(P^{(0)}_{22}\) and \(R^{(0)}_0\) are all constant, so we do not plot them. The two \(P^{(0)}\) amplitudes coming from the isoscalar \(\omega \omega \) state, quadratic in energy, are shown in Fig. 2. Therein and in what follows we have taken \(\alpha _\mathrm{EM}(Q^2=0)=\frac{1}{137}\) as the emitted photons are real. From the parameters of the EWSBS, we have taken all NLO coefficients to zero, and \(b=a^2\), so that the only slight separation from the SM is driven by \(a=0.95\); the further parameters of the photon sector are indicated in the figure.

All the amplitudes shown in the figure display the expected quadratic growth with energy (linearity in *s*). Eventually they must violate the unitarity bounds, for example by \(|A_{00}|> 1\), which occurs already below \(3~\mathrm{{TeV}}\) if we increase \(1-a\) or other parameters of the HEFT.

*P*amplitudes that are much smaller than the elastic

*A*amplitude, as demanded by the smallness of \(\alpha \). On the contrary, the third plot exposes values of

*P*of the same order of those of

*A*. This means that the a priori counting of Sect. 2.1 fails for the value of \((a_1-a_2+a_3)\) chosen around 0.3; this maximum value, still allowed by previous empirical constraints, is too large in comparison to the “natural” values of the \(a_i\), of order \(10^{-3}\). We do not employ such large values again later.

The approach that we have developed can be used to describe resonances that could be found in experimental data from the LHC and relate different channels. Figure 3 shows a narrow resonance, with \(\varGamma /M\sim 0.06\). This is useful to make contact with the large body of theoretical work following the \(\gamma \gamma \) statistical fluctuation in the CMS and ATLAS data (we next proceed to more phenomenologically viable resonances).

Although the electromagnetic interactions do not conserve weak isospin, our choice of \(P_{00}\) and \(P_{20}\) amplitudes implies that only the first is fed by a scalar resonance in the \(\omega \omega \) channel, as is patent in the figure.

Next, we provide an example of a typical broad resonance in Fig. 4.

Once more, the only non-vanishing parameter for the two-photon sector is \(\alpha =1/137\). The imaginary part of \(P_{00}\) (note it has again been multiplied by \(10^3\)) presents a clear resonating shape driven by that of *A*. We also show how well the unitarity relation of Eq. (42) is satisfied by our numerical program: the IAM is indeed up to the task, with unitarity satisfied exactly in *s* and to first order in \(\alpha \).

*hh*channel is coupled with larger LO strength. This makes the resonance of Fig. 3, which we take to exemplify the point, broader and somewhat less intense in the \(\gamma \gamma \) channel, resembling more and more the coupled-channel resonance described in [48].

An interesting feature is that the resonance moves toward higher energies. To understand it, we note that the pole of the IAM amplitude (with \(a^2-b\) set to 0) comes from a denominator \(\frac{1}{A^{(0)}-A^{(1)}}\). Upon activating the *hh* channel coupling with \(a^2-b\), the matrix amplitude in Eq. (57) has as denominator, in view of \(T^{(0)}=0\), a slightly more complicated one, \(\frac{1}{A^{(0)}-A^{(1)}+\frac{(M^{(0)}-M^{(1)})^2}{T^{(1)}}}\). The last term can shift the zeros of the denominator, and thus the resonance position, according to the sign of \(T^{(1)}\). The leading \((a^2-b)^2\) factor cancels in the ratio, so the effect is not huge, but there remains sensitivity to the next order, \((a^2-b)^3\) from the \(M^{(0)}M^{(1)}\) cross product. Both \(A^{(0)}\) and \(\mathrm{Re} T^{(1)}\) happen to be positive [33], so that \(\mathrm{Re} (\frac{(M^{(0)}-M^{(1)})^2}{T^{(1)}})\) has the same sign as \(A^{(0)}\) and reinforces it. Thus, \(A^{(1)}\) has to be larger to obtain a cancellation, and this requires a higher *s*, whence the resonance increases in energy.

*b*very close \(a^2\) to avoid the coupled-channel complication here (already shown to work in Fig. 7). With both methods we have varied \(a_4\) until a scalar resonance appears at \(2~\mathrm{{TeV}}\). The necessary value of this parameter is somewhat different, by 25%. The width and the minimum value of the \(P_{00}\) amplitude are not identical either (which could be perhaps arranged by varying some of the other NLO parameters, but we do not see the need at this stage). In conclusion, while both methods give qualitative similar results, their comparison gives us a warning that there is a systematic uncertainty in the choice of unitarization scheme of order one part in four.

Finally, in Fig. 9 we show \(P_{02}\), the \(\gamma \gamma \) amplitude with the initial \(\omega \omega \) in the isoscalar-tensor channel. The left plot is dedicated to showing a nonresonant tensor amplitude; for ease of comparison, we employ the same parameters as produced a scalar resonance in Fig. 4. An interesting remark is that at relatively low energies the ratio between the photon production amplitude and the elastic \(\omega \omega \) one, \(P_{02}/A_{02}\), is much more sizable than its scalar counterpart \(P_{00}/A_{00}\). This comes about because in Eq. (38a) there is an electromagnetic \(\alpha \) coupling as in all the photon amplitudes, but it is constant (*s* independent) as opposed to \(A_{02}\), which has an Adler zero at \(s=0\). Therefore, both amplitudes can be shown in the same plot by enhancing the photon one only by a factor \(10^2\), whereas in the scalar case we have been employing a factor \(10^3\).

As for the right plot of Fig. 9, we have increased \(a_4\) (from \(4\times 10^{-4}\) in the left plot to \(3\times 10^{-3}\)) so a tensor resonance appears below 3 TeV. The resonance is clearly visible, in the same position, in the \(\gamma \gamma \rightarrow \omega \omega \) \(P_{02}\) amplitude as in the elastic amplitude. (The latter is only two orders of magnitude larger, as just discussed). Coupled-channel unitarity is also clearly demonstrated as \(\mathrm{Im} P_{02} = P_{02} A^*_{02}\).

Many more example calculations are interesting, but we content ourselves with these examples until experimental data shows whether there is merit in pursuing further computations, and specifically which ones.

## 7 Conclusions

In this article we have coupled the EWSBS described with HEFT (for \(E<4\pi v\sim 3~\mathrm{{TeV}}\)) and the equivalence theorem (for energies \(E>M_h,\ M_W\)), in the regime of unitarity saturation and resonances, to the two-photon channel, which is a promising detection avenue for new physics.

We have developed the necessary unitarization formalism with two different, well explored methods (IAM and N/D), that are equivalent (up to NNLO) to the NLO perturbative amplitudes of [42] at low energies but that, unlike those of the HEFT, can be employed to describe any resonances of the EWSBS.

For example, in Figs. 3 and 4 we have shown that both a light, narrow, and a heavier, broader resonance feeding the \(\gamma \gamma \) spectrum can be parametrized in this approach, in terms of *a*, \(a_4\) and \(a_5\), which control the EWSBS. What the production cross section is for those particular resonances is work of phenomenological interest that we postpone to imminent work within an expanded collaboration.

Our formalism assumes that the symmetry-breaking dynamics in the *W*, *Z* and *h* sector is stronger than their electromagnetic coupling to \(\gamma \)s. Nevertheless we have also considered the NLO counterterms that arise in coupling \(\gamma \gamma \) to the EWSBS. As long as their values remain “natural”, our counting in Fig. 1 suggests that perturbation theory is valid in coupling \(\gamma \gamma \), and that the resonating \(\omega \omega \) (and/or *hh*) amplitudes can be separately computed first. Our theory satisfies Watson’s final-state theorem in that the phases of the photon–photon production amplitude coincide with those of the elastic EWSBS amplitudes.

A technical challenge that we have overcome is that of projecting the (earlier known) Feynman amplitudes into \(\gamma \gamma \)-helicity amplitudes of definite total angular momentum *J* and stemming from \(\omega \omega \) states of definite custodial isospin *I*. This was necessary and convenient as any resonance or new particle produced in the custodially invariant EWSBS sector will have specific *I*, *J* but the detection in the \(\gamma \gamma \) channel loses memory of *I*; a complete set of observables, however, includes the photon helicities \(\lambda _1\) and \(\lambda _2\) (though for many cross-section calculations one may sum them).

Even in the absence of new resonances, the set of projected amplitudes that we have provided can be useful to parametrize separations from the SM in a relatively low-energy regime below \(3~\mathrm{{TeV}}\), where the partial-wave series converges quickly.

We are currently collaborating with other authors in the preparation of a document with simple estimates for collider cross-sections of typical resonances as seen in the two-photon channel.

Finally, another natural final-state channel that couples with sufficient intensity to the EWSBS, and that may serve as an LHC probe thereof, is the \(t\bar{t}\) one. We are separately exploring it with the same methods and have recently shown that within HEFT this channel coupling admits a perturbative expansion in powers of \(M_t/\sqrt{s}\), obtaining the unitarized amplitudes needed for its description in the resonance region [62]. The calculation follows lines analogous to those here presented.

## Footnotes

- 1.
In Ref. [42], we also employed the exponential parametrization of the coset for the \(\gamma \gamma \) scattering. While intermediate results (i.e., the Feynman diagrams) are different, the on-shell amplitudes are exactly the same for the two parametrizations.

- 2.
Not to be confused with the \(SU(2)_L\) gauge coupling.

## Notes

### Acknowledgements

We thank for useful conversation and suggestions J.J. Sanz-Cillero, D. Espriu, and M.J. Herrero. Work supported by Spanish MINECO Grants FPA2011-27853-C02-01, FPA2014-53375-C2-1-P and BES-2012-056054 (RLD).

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