Coupling of $t\bar t$ and $\gamma\gamma$ with a strongly interacting Electroweak Symmetry Breaking Sector

We report the coupling of an external $\gamma\gamma$ or $t\bar t$ state to a strongly interacting EWSBS satisfying unitarity. We exploit perturbation theory for those coupling of the external state, whereas the EWSBS is taken as strongly interacting. We use a modified version of the IAM unitarization procedure to model such a strongly interacting regime. The matrix elements $V_LV_L\to V_LV_L$, $V_LV_L\leftrightarrow hh$, $hh\to hh$, $V_LV_L\leftrightarrow\{\gamma\gamma,t\bar t\}$, and $hh\leftrightarrow\{\gamma\gamma,t\bar t\}$ are all computed to NLO in perturbation theory with the Nonlinear Effective Field Theory of the EWSBS, within the Equivalence Theorem. This allows us to describe resonances of the electroweak sector that may be found at the LHC and their effect on other channels such as $\gamma\gamma$ or $t\bar t$ where they may be discovered.


Introduction
The new boson with a mass of 125 GeV found at the LHC [1,2] would complete the Standard Model (SM) in its minimal version. The experimental collaborations at CERN are looking for deviations of its behaviour from that of the SM Higgs particle [3,4]. The electroweak symmetry breaking happens at a scale of v = 246 GeV. New scalar resonances are constrained up to 600 − 700 GeV [5]. The constraint over new vector bosons reaches even higher in energy [6]. Thus, the study of the TeV scale is the goal of LHC Run-II. And it is expectable that new physics related with the Electroweak Symmetry Breaking Sector (EWSBS), if it exists, will be found at this scale.
In our Refs. [26][27][28][29][30], we simplify the study of the HEFT by means of the Equivalence Theorem [31], which is valid for  √ s) counting for coupling with tt. Right: chiral (α, s) counting for coupling with γγ. Note that the HEFT is perturbative in M t /v and α EM , respectively, but requires unitarization in s/(4πv) 2 (all orders along the OY axis included) to reach the resonance region.
Since the possible resonance region is above 500 GeV, this approximation is safe, and allows us to identify the longitudinal W ± L , Z L with the pseudo-Goldstone bosons of symmetry breaking ω a (a = 1, 2, 3) in their scattering amplitudes. For instance, We have found [26][27][28][29][30] that, for any parameter choice separating from the SM, the theory becomes strongly interacting at sufficiently high energy, and resonances may appear. Dispersion relations (as unitarization procedures) are used to deal with this non-perturbative regime. This also happens for the ChPT theory for hadron physics [32,33]. In Ref. [34,35], unitarization procedures are tested for the hadron case of ChPT.
In this work, based on our recent works [36,37], we couple γγ and tt states to the unitarized scattering amplitudes of our refs. [28,30]. The couplings between the EWSBS and {tt, γγ} states are perturbative in M t /v (negligible if compared with √ s/v) and α EM , respectively. However, the EWSBS itself is strongly interacting (see Fig. 1). For the tt case, account must be taken that we are in the regime This allows us to neglect diagrams like the 1-loop top mass renormalization, since they contribute to the scattering amplitudes with higher orders of √ s/v. Note also that, in the high energy limit s M 2 Z,W ∼ M 2 h where the Equivalence Theorem can be applied, we can take M 2 W = M 2 Z = M 2 h = 0 consistently. Note that we compute {ω, ω} ↔ {ωω, hh, γγ, tt}. γγ and tt can both appear as initial and final states. Of course, γγ and tt as final states are being tested at the LHC experiments, and physics of two photon decays has been pursued since the dawn of particle physics [38]. But we should not forget the physics of γγ → ωω, since the photon can be a parton of the proton [39] or the electron in pp and e − e + colliders, respectively. The CMS collaboration [40] is currently setting bounds to anomalous quartic gauge couplings by analysing γγ → W + W − . Photon colliders driven by lepton beams due to Compton backscattering could also become a future application of this work [41,42]. CONF12

The Electroweak Chiral Lagrangian
First we quote the HEFT Lagrangian [29,30,37] which includes only the low-energy dynamics of the quark sector and the four light modes: three would-be Goldstone Bosons ω a (WBGBs) and the Higgs-like particle h. A global symmetry breaking pattern S U(2) L ×S U(2) R → S U(2) C is considered.
The U(x) ∈ S U(2) C isospin can be parametrized by using the so-called spherical parametrization [30] by means of Note that i = 1, 2, 3 in the isospin basis, which is related to the charge one by where the dots represent terms of higher order in (ω a /v).
The Higgs potential is expanded as The SM is recovered for Note that these terms are subleading in our approximation (Eq. 1), so that we could neglect the whole potential V provided that this behaviour holds by whatever beyond SM theory happens to succeed. This is the case in most models of interest, and is a reasonable hypothesis since the constraints of these couplings have so far been found to be close to the SM values.
By expanding Eq. (3) without considering (yet) the Yukawa part, we obtain where A µ = sin θ W W µ,3 + cos θ W B µ . We are not considering couplings with external transverse gauge bosons. Note that these neglected states do not appear in inner loops since this would be a higher order correction in α EM . The Yukawa part of the Lagrangian of Eq. (3), once the Yukawa-coupling matrix is diagonalized [37], can be written as so that Eq. (8) can be written as Since this is an EFT, in order to renormalize the tree level Lagrangian in Eq. (7) at the one-loop level, counterterms of dim. 8 are needed. For the EWSBS (strong) interactions, the minimal set of counterterms is {a 4 , a 5 , d, e, g} (see Refs. [27,30]), For the γγ coupling [29,36] we consider {a 1 , a 2 , a 3 , c γ }, though only a 1 + a 2 − a 3 appears and none is strictly needed, And for the Yukawa coupling with top quarks [37],

Partial waves
The perturbative {ω i ω j , hh} → {ω k ω l , hh} scattering amplitudes can be found in our Ref. [30]. For the representation of the ωω states, we will use the isospin basis |I, M I (I is the isospin and M I , its projection). Because of isospin symmetry, scattering amplitudes factorizes in this basis, and their values do not depend on M I . Unitarization procedures (based on dispersion relations) are most easily applicable over a partial wave decomposition [30] which, for {ω i ω j , hh} states, are computed by means of Since the hh state is an isospin singlet I

CONF12
where A (0) I J (s) ∼ s corresponds to the LO term and A (1) I J (s) ∼ s 2 , to the NLO computation and counterterms.

Unitarization procedures
As pointed out in Fig. 1, our key assumption will be a strongly interacting EWSBS while couplings with γγ and tt states remain perturbative. The EWSBS {ω i ω j , hh} scattering partial waves are unitarized by means of the same unitarization procedures (IAM, N/D, Improved-K matrix) that were exposed on our Refs. [28,30]. We have two possibilites when unitarizing the EWSBS [30]. ω i ω j states couple with hh if I J = 00, 20 and some of this conditions are verified: a 2 b, d 0 or e 0. Otherwise, the hh channel decouples. For instance, the single channel IAM is [30] A IAM Note that if coupling with hh channel happens, the coupled-channel versions of the unitarization procedures are required. Indeed, the strongly interacting regime could be triggered by ωω → hh even if ωω → ωω is weak [28]. Couplings with γγ and tt are unitarized by means of a modification of the IAM and N/D procedures that uses as input the unitarized partial waves {ω i ω j , hh} → {ω i ω j , hh}. This modification can be found in our Refs. [36,37]. For the single-channel version of the IAM, it can be used whereP, in this case, is the unitarized ωω → {γγ, tt} amplitude; P (0) , the corresponding perturbative one;Ã, the unitarized A(ωω → ωω); and A (0) (s), the tree level matrix element A(ωω → ωω). This guarantees that the phase ofP coincides with that ofÃ IAM . For other channels, see Refs. [28,30].
For computing the position of poles on the second Riemann sheet (resonances) in Fig. 3, we have used Cauchy's Theorem via numerical integration, following the procedure that was already explained in our Ref. [30].
Note the presence of a pole (broad resonance) on the second Riemann sheet in the same position in all the three channels (elastic scattering ωω → ωω and couplings with tt and γγ). This was expected since the pole comes from the inner EWSBS (strongly coupled) dynamics, not from the physics of the (weak) coupling with γγ and tt states.
Poles in the second Riemann sheet, if there are any, should come from the unitarization procedures applied to the inner dynamics of the EWSBS (Eq. 18 and other expressions from our Ref. [30]). Couplings with γγ and tt do not give rise to poles in the second Riemann sheet, according to Eq. (19) (and similar expressions in Refs. [36,37]), since perturbative partial waves (in particular, P (0) and A (0) in Eq. 19) are just polynomial in s/v 2 . Thus, the parameters associated with γγ (a 1 , a 2 , a 3 , c γ ) and tt (c 1 , c 2 , g t and g t ) should control the γγ and tt coupling strength, but not the physics of the inner dynamics of the EWSBS. This is the picture which emerges from the chiral counting of Fig. 1. (However, if the parameters which control the couplings with γγ and tt states were unnaturally large, either M t /v or α EW would be non-perturbative and a full multi-channel unitarization involving {γγ, tt} should be carried out [37].)
Our formalism assumes that the inner EWSBS dynamics (in the W ± L , Z L and h sector) is stronger than their electromagnetic coupling with γs (∼ α EM ) and the electroweak couplings with tt (∼ M t /v). Hence, any new resonance (within the range M W , M h < E < 3 TeV) in {ωω, hh} ↔ {γγ, tt} channels would come from the inner EWSBS dynamics. This assumption could be broken for unnaturally large values of the NLO parameters related with γγ and tt couplings. Such a situation would require the usage of a full coupled channel unitarization, including strongly γγ or tt rescattering.
Even in the case that there are no new resonances, our scattering amplitudes can be a useful tool to parametrize separations from the SM in the regime M h , M W < E < 3 TeV. Currently, HEFTs themselves are being widely used in this way by CERN collaborations [23].
Finally, we are working within an expanded collaboration in the computation of simple estimates for collider cross-sections of typical resonances, and even for releasing a Monte Carlo module for the experimentalists.