Uncovering the relation of a scalar resonance to the Higgs boson

We consider the associated production of a scalar resonance with the standard model Higgs boson. We demonstrate via a realistic phenomenological analysis that couplings of such a resonance to the Higgs boson can be constrained in a meaningful way in future runs of the LHC, providing insights on its origin and its relation to the electroweak symmetry breaking sector. Moreover, the final state can provide a direct way to determine whether the new resonance is produced predominantly in gluon fusion or quark-anti-quark annihilation. The analysis focusses on a resonance coming from a scalar field with vanishing vacuum expectation value and its decay to a photon pair. It can however be straightforwardly generalised to other scenarios.

We consider the associated production of a scalar resonance with the standard model Higgs boson. We demonstrate via a realistic phenomenological analysis that couplings of such a resonance to the Higgs boson can be constrained in a meaningful way in future runs of the LHC, providing insights on its origin and its relation to the electroweak symmetry breaking sector. Moreover, the final state can provide a direct way to determine whether the new resonance is produced predominantly in gluon fusion or quark-anti-quark annihilation. The analysis focusses on a resonance coming from a scalar field with vanishing vacuum expectation value and its decay to a photon pair. It can however be straightforwardly generalised to other scenarios.

I. NEW SCALAR RESONANCES AT COLLIDERS.
Models with an additional (pseudo-)scalar singlet with a mass of several hundred GeV represent a well motivated class of extensions of the Standard Model (SM) of particle physics, including composite Higgs scenarios, supersymmetry, Coleman-Weinberg models, models addressing the strong CP problem, models of flavor, as well as generic Higgs portal setups (see e.g. [1][2][3][4][5][6][7][8][9]). A particularly promising channel to search for and analyze such a particle is its decay to two photons. Beyond being possibly sizable in certain scenarios, it offers a robust and clean way to detect a signal, emerging over a steeply falling background [10,11].
After its discovery, an important aspect of scrutinising any new resonance is in fact to measure its couplings, and hence determine its relation, to the known particle content of the Standard Model (SM). A crucial component of this task is to uncover its role in the arena of electroweak symmetry breaking (EWSB). As a first step in this direction, determination of the couplings of the new scalar to the SM-like Higgs boson is mandatory, which is the main focus of this article, employing its di-photon (γγ) decay channel. * For a γγ resonance originating from a scalar field S, neutral under the SM gauge group, the relevant effective Lagrangian for our study -augmenting the SM at * Electronic address: adrian.carmona@cern.ch † Electronic address: florian.goertz@cern.ch ‡ Electronic address: apapaefs@cern.ch * A specific motivation for the first version of this manuscript was provided by the apparent γγ resonance at Mγγ ∼ 750 GeV in AT-LAS [12,13] and CMS [14,15] data. This turned out not to be present in the 2016 data [16,17]. Consequently the article was generalized to other mass scales of a potential scalar resonance, which remains well motivated, taking into acount new limits on its cross section -see below. Comprehensive analyses studying constraints on (other) possible couplings of a di-photon resonance as well as detailed examinations of indirect footprints of new (high multiplicity) sectors, linked to its productions or decay appeared e.g. in .
Here, Q i L is the i-th generation left-handed SU (2) L fermion doublet, d j R and u j R are the right-handed SU (2) L fermion singlets for generation j, (y S q ) ij are the corresponding Yukawa-like couplings, c S B , c S W are the couplings of S to the U (1) Y and SU (2) L gauge fields B and W , c S G is the coupling to the gluon fields, H is the Higgs boson doublet, λ HS is the Higgs-Scalar portal coupling and λ S the new scalar quartic. Moreover, Λ denotes the scale of heavy new physics (NP), mediating the contact interactions of S with SM gauge bosons and fermions (the latter involving H to generate a gauge singlet). Note that we do not include terms with an odd number of S fields containing only scalars (as well as lepton fields). The corresponding interaction vertices will turn out irrelevant in general for the process we will consider, see below. † Beyond that, terms linear in S could also lead to the singlet mixing with the Higgs boson after EWSB, which would in fact affect its phenomenology. Although such effects could still be present at a non-negligible level, they are expected to be sub-leading and we neglect them for simplicity, see Appendix B. Furthermore, the analysis that follows is independent of the CP properties of S and its interactions and henceforth, for simplicity, we assume it to be CP-even with CP-conserving interactions.
With the potential for the Higgs doublet H taking the conventional form: The physical mass of the singlet thus reads M = µ 2 S + λ HS v 2 . The resulting trilinear interactions between the physical scalar resonances after EWSB are described by where h is the Higgs boson, which (due to the case of negligible scalar mixing) is basically fully embedded in H and describes excitations around its vev, such that in unitary gauge H 1/ √ 2(0, v+h) T , and S is a new scalar resonance, which can also be (approximately) identified as S = S. Moreover, v 246 GeV is the Higgs vev, M h 125 GeV is the measured Higgs boson mass and the portal coupling λ HS is to be determined, being a main scope of this paper. This coupling would be basically unconstrained by direct observation of a di-photon resonance. Nevertheless, loose indirect constraints can be derived, requiring vacuum stability not to be spoiled. They read and need to be imposed at least at the scale where the NP enters, i.e., the TeV scale (see, e.g. [1,46]). § Requiring λ H ∼ 0.13, to fit the observed Higgs mass, as well as λ S < (4π) 2 , we thus obtain −4.5 λ HS 4.5. We will see below that in general our analysis can put stronger bounds than these on λ HS . ¶ In the present study we consider measurement of the coupling λ HS at the LHC, where it can be probed via associated production of the new resonance with the SMlike Higgs boson: pp → hS. For this process, the interactions neglected in (1) play no role to good approximation: they would either not enter at leading order (LO), or, as is the case for the |H| 2,4 S interactions, contribute at most to a diagram with a (strongly suppressed) off-shell Higgs boson propagator, for details see the appendices A and B. ‡ The fact that S =0 guarantees the full absence of scalar mixing, that could otherwise occur even without linear terms in S. § While the first two conditions need to hold at all scales, for λ HS > 0 the last condition might be violated at higher scales, while still the electroweak vacuum remains stable [1]. Moreover, for the latter condition odd terms in S are assumed to vanish. ¶ Note that requiring a more conservative limit, such as λ S < O(10) (corresponding to, e.g., dλ S /λ S < 1 [1]), restricts λ HS to be not much larger than 1 and would remove a considerable portion of the parameter space where our analysis exhibits sensitivity. However, in any case, the limits presented here are complementary to such considerations.
In principle several decay modes of S can be considered. Here we focus on the process pp → hS → hγγ, where the new particle decays to a pair of photons. Given that the tentative cross section of the resonant di-photon production, pp → S → γγ, will be known and will be well-measured in the case of discovery, this allows constraints on the coupling λ HS to be imposed almost independently of the couplings to the initial-state partons and final-state photons, given that only a single production mode is relevant. We consider production via gluon fusion and quark-anti-quark annihilation, mediated through non-vanishing coefficients c S G , (y S d ) 22 or (y S d ) 33 , respectively, and show how these modes could be disentangled via appropriate measurements. We will implicitly assume not too large values of Γ(S → γγ), in such a way that photo-production is always subdominant.
Although we will focus on three specific benchmark masses of M = 600, 750, 900 GeV, our analysis could be applied to the general case of associated production of a Higgs boson with a scalar di-photon resonance of any mass. Moreover, several features of the final state studied here, such as the invariant mass of the final-state scalar or the total invariant mass of the process, will exhibit similar features when considering other decay modes.
The article is organised as follows: in Section II we examine the process of associated production of a scalar resonance and a Higgs boson, in Section III we describe the event generation and detector simulation setup and in Section IV we provide details of the analysis and results. Finally, we conclude in Section V.

II. ASSOCIATED PRODUCTION OF A SCALAR RESONANCE AND A HIGGS BOSON.
A. Production through gluon fusion.
The dominant diagrams at LO contributing to the production of the hS final state in gluon fusion and subsequent decay of the resonance S to a pair of photons via the interactions are shown in Fig. 1. In the analysis of the present article, we will consider the Higgs boson decaying to a bottomquark pair, since this maximizes the expected number of events, which would be modest in general. There exist both the s-channel S exchange, involving the portal coupling λ HS and depicted in the upper panel (a), as well as the 'direct' hS production, via t-channel gluon exchange, depicted in the lower panel (b). For the case of quark anti-quark annihilation, a new diagram arises from the contact interaction qqhS. Both contributing graphs are shown in Fig. 2. The new diagram (b) distinguishes the qq annihilation from the gluon fusion case. An important fact is that now the hS process is non-vanishing and significant even in the absence of the portal coupling λ HS . This indicates that one can employ this final state to exclude qq annihilation as the dominant production process (in the absence of a signal). In the following we will focus on the cases of q = b, q = s or q = c.
It will be useful in both scenarios to construct the ratio of the associated production process pp → hS, through all possible intermediate states, with the subsequent decay of the new resonance to a γγ final state, to that of the single production pp → S → γγ, where we consider xx = {gg, bb, ss , cc }. * * The ratio is a useful quantity since it removes the dependence on the product of couplings of the new resonance to the initialstate partons and final-state photons. Moreover, it can be used to absorb, at least approximately, theoretical and The t-channel diagram with the qqh interaction is suppressed due to a small Yukawa coupling. * * In the most general setup, the analysis of this article can constrain the sum of the squares of the couplings of S to all quark generations (for a given λ HS ), appropriately weighted by the parton density functions. The ratios ρ(gg), ρ(bb), ρ(ss) and ρ(cc) for gg, bb, ss and cc initial states respectively, defined between the associated production pp → hS → hγγ and the single production pp → S → γγ, as functions of the portal coupling λHS. The mass of the scalar resonance was taken to be M = 750 GeV and the width Γ = 1 GeV.
We show the dependence of the ratio ρ on the portal coupling λ HS in Fig. 3 for gg, bb, ss and cc initial states for the example di-photon resonance mass M = 750 GeV and width Γ = 1 GeV. ‡ ‡ A width of Γ 1 GeV can be † † For a similar idea investigated in the context of Higgs boson pair production, see [47]. ‡ ‡ We employ a single cut of Mγγ > 200 GeV at generation level in order to remove (SM-like) pp → hh → hγγ interference with the obtained for example, if c S γ ∼ O(10) and c G ∼ O(1) or (y S d ) 33 ∼ O(1), for Λ = 1 TeV, with a cross section in pp → γγ compatible with current constraints. Similar behaviour of the ratio ρ is obtained for different scalar S masses and widths.
Since the dominant matrix-element contribution to the gg-initiated hS process is proportional to the portal coupling, the process approximately vanishes as λ HS → 0, and hence ρ(gg) ρ 2gg λ 2 HS , where ρ 2gg ≈ 0.00133, for M = 750 GeV and Γ = 1 GeV, obtained by performing a quadratic fit of the gg curve in Fig. 3. As already discussed, this does not hold for the qq-initiated process due to the contact interaction diagram. This results in a non-negligible minimum for ρ(qq). A fit to the cross section, again for M = 750 GeV and Γ = 1 GeV, yields ρ(qq) ρ 2qq λ 2 HS + ρ 1qq λ HS + ρ 0qq with ρ 0bb ≈ 0.00828, ρ 1bb ≈ 0.00309, ρ 2bb ≈ 0.00086, corresponding to bb initial states, and ρ 0ss ≈ 0.01025, ρ 1ss ≈ 0.00371, ρ 2ss ≈ 0.00091, corresponding to ss initial states. The case of cc is similar to the ss case and therefore in the rest of the article we focus on the cases q = b and q = s. The positivity of the coefficient ρ 1qq indicates constructive interference between the contact interaction and resonant diagrams (for λ HS > 0). For an extended fit of the ratio ρ, including additional diagrams with the production of an intermediate Higgs boson due to S|H| 2,4 interactions, that turn out to be sub-dominant, see Appendix B.
The fact that for quark-anti-quark annihilation the hS process is non-vanishing for all values of the portal coupling λ HS indicates that one could employ this final state to exclude bb, ss or cc annihilation as the dominant production process. The analysis that will follow in the present article suggests however, that the di-photon decay of the S alone may not be sufficient for that purpose for the benchmark points that we consider.
We show in Fig. 4 the variation of the ratio ρ with the mass of the resonance, M , for the gg-iniated process and λ HS = 1, and for the bb-initiated process for λ HS = 0 (no portal) and λ HS = 1. We have fixed the width to Γ = 1 GeV. Interestingly, the pure qq-induced processes exhibit an increase of the ratio ρ with increasing mass -related to the new qqhS interaction growing with momentum -whereas the pure gg-induced process exhibits a slight decrease.
If the di-photon resonance is wide, the analyses performed for the hS final state will differ in the details due to changes in the kinematics. We show in Fig. 5 the variation of the ratio ρ with the width over the mass, Γ/M , at a fixed mass M = 750 GeV, for λ HS = 1, and for the bb-initiated process for λ HS = 0 (no portal) and λ HS = 1. One can observe that the central value of the signal, i.e., S(γγ) + h production. Only after this cut, we can identify a 'signal' contribution to the actual physical process -which is Higgs production in association with a photon pair -unambiguously with the process pp → hS → hγγ to good approximation, assuming the model (1).  ratio remains approximately constant in all cases, with only a slight decrease with increasing width. In both Figs. 4 and 5, we also provide, as coloured bands, the parton density function uncertainty for the MMHT14nlo68cl set [48] combined in quadrature with the scale variation between 0.5 and 2.0 times the default central dynamical scale implemented in MadGraph5 aMC@NLO. For a mass of 750 GeV, the total theoretical uncertainties due to scale and PDF variations are ∼ +40 −30 % for the gg-induced process, ∼ ±10% for the bb-induced and ∼ ±30% for the ss-induced cases (the latter not shown in the figure for simplicity). Assuming a total cross section for the production of a γγ resonance of mass M = 750 GeV of, say, σ(pp → S → γγ) = 5 fb (see below), one would expect a total of O(20) hS → hγγ events at the high-luminosity LHC (HL-LHC, assuming 3000 fb −1 of integrated luminosity) if the process is gluon-fusion initiated and O(200) events for bb-initiated production, for a portal coupling λ HS = 1. Moreover, the minimum expected number of events for the bb-initiated process is O(80), arising for λ HS −1.8 and for the ss-initiated process one expects a minimum of O(100) events for λ HS −2.0. We note here that the positions of the minima for the qq-initiated process will change after cuts due to the varying effect of the analysis on the different pieces contributing to the cross section.
The matrix-element level distribution of the diphoton invariant mass, Mγγ, in the gg → hS → hγγ process, normalised to unity, for the two different width scenarios, Γ = 1 GeV and Γ = 45 GeV, for M = 750 GeV.
The kinematic structure of the pp → hS → hγγ process can be well-described by examining the distribution of the invariant mass of the γγ state, M γγ , or the distribution of the invariant mass of the Higgs boson and di-photon combination, M hγγ . In Figs. 6 and 7 we show, respectively, these distributions for the gluon-   M + M h which we will refer to as "on-shell diphoton", exists irrespective of the mass of S. Note that both the three-body decay and on-shell di-photon regions exist even for Γ/M 1. The normalised distributions look identical for all values of the portal coupling, λ HS ( = 0), since the dominant contribution stems by far from the diagram shown in Fig. 1 (a).     Γ = 1 GeV, with the on-shell di-photon region dominating. For λ HS = 0, the "three-body decay" region disappears completely since the resonant s-channel diagram of Fig. 2 (a) vanishes. For large width the two regions merge into one and the effect of the vanishing three-body decay region for λ HS = 0 is not as evident as in the case of small width. The distributions for the ss-initiated process exhibit similar features, with different "mixtures" between the two regions arising from the differences between the strange and bottom quark parton density functions. We omit them for the sake of simplicity.   that we will consider as "benchmark" scenarios in our analysis, M = 600, 750, 900 GeV (see below) and Γ = 1 GeV. For the bb case we only show the λ HS = 1 distributions for simplicity. They all clearly demonstrate the existence of the main features described for the M = 750 GeV case, i.e. the "three-body decay" and "on-shell di-photon" regions.

III. EVENT GENERATION AND DETECTOR SIMULATION.
A. Event generation.
The signal model was generated via an implementation of the Lagrangian of Eq. 1 in FeynRules [49,50]. Via the UFO interface [51] this was used to generate parton-level events employing MadGraph5 aMC@NLO [52,53]. The background processes were also generated using MadGraph5 aMC@NLO, with appropriate generationlevel cuts to reduce the initial cross sections to a manageable level. All the events were passed through the HERWIG 7 [54][55][56][57][58] Monte Carlo for simulation of the parton shower, the underlying event and hadronization. As before, the MMHT14nlo68cl PDF set was employed. To remain conservative, we consider collisions at the LHC at a centre-of-mass energy of 13 TeV. The possible increase of energy to 14 TeV will increase rates in the considered processes by O(10%).
Since we expect to impose cuts on the di-photon mass window, M γγ , that are sufficiently far away from the Higgs boson resonance, we can immediately exclude any background processes containing h → γγ from the analysis. For this reason we do not include associated Higgs boson production with a vector boson or Higgs boson pair production, tth production, and so on. This implies that the relevant backgrounds are those with non-resonant γγ production, other processes that involve S → γγ, and reducible backgrounds. We thus consider the following processes: γ+jets, γγ+jets, events with at least one true b-quark at parton-level (b+jets), Zγγ with Z → bb, ttγγ including all the decay modes of the top quarks and the production of the resonance S in association with a nonresonant bb pair. § § All the multi-jet processes are generated without merging to the parton shower, in the fiveflavour scheme, with four outgoing partons at the matrixelement level.
The calculation of higher-order QCD corrections to these multi-leg processes, particularly when restricting the phase space with cuts, is numerically challenging at present. To remain conservative, we will assume that the corrections are large and apply K-factors of K = 2 to all the background processes. For the signal and the bbS associated production we do not apply any K-factors since the corrections are approximately absorbed into the ratio with the single inclusive production of the S resonance, see below. Throughout this article we assume that σ(pp → S → γγ) = 10, 5, 1 fb, corresponding to the benchmark masses M = 600, 750, 900 GeV, fixing the product c S G c S γ (or (y S d ) ii c S γ ), which drops out in the ratio ρ. The values of the cross sections are motivated by the current ATLAS [16] and CMS [17] limits on di-photon resonances.
Note that it turns out that the non-resonant bbS process is only relevant for gluon-fusion production of S, and we only report numbers for that in what follows.

B. Detector simulation.
In the hadron-level analysis that follows, performed without using any dedicated detector simulation software, we consider all particles within a pseudo-rapidity of |η| < 5 and p T > 100 MeV. We smear the momenta of all reconstructed objects (i.e. jets, electrons, muons and photons) according to HL-LHC projections [60,61]. We also apply the relevant reconstruction efficiencies. We simulate b-jet tagging by looking for jets containing Bhadrons, that we have set to stable in the simulation, and considering them as the b-jet candidates. The mistagging of c-jets to b-jets is performed by choosing cjet candidates (after hadronization) as those jets that lie within a distance ∆R < 0.4 from c-quarks (after the parton shower), with transverse momentum p T > 1 GeV. ¶ ¶ We apply a flat b-tagging efficiency of 70% and a mis-tag rate of 1% for light-flavour jets and 10% for charm-quarkinitiated jets. § § We also considered the hγγ process, including the loop-induced pieces [59], but found that it possess a negligible cross section. ¶ ¶ This procedure of associating jets to c-quarks is expected to be conservative.
We reconstruct jets using the anti-k t algorithm available in the FastJet package [62,63], with a radius parameter of R = 0.4. We only consider jets, photons and leptons with p T > 30 GeV within pseudo-rapidity |η| < 2.5 in our analysis. The jet-to-lepton mis-identification probability is taken to be P j→ = 0.0048×e −0.035p T j /GeV and the jet-to-photon mis-identification probability was taken to be P j→γ = 0.0093×e −0.036p T j /GeV [60,61], both flat in pseudo-rapidity. We demand all leptons and photons to be isolated, where an isolated object is defined to have i p T,i less than 15% of its transverse momentum in a cone of ∆R = 0.2 around it.

IV. DETAILED ANALYSIS.
We consider events with two reconstructed b-jets and two isolated photons as defined in Section III. Note that this final state has been previously considered in the context of searches for Higgs boson pair production, e.g. in [64][65][66][67][68]. We impose the following 'acceptance' cuts to all samples: • b-jets: transverse momenta p T,b1 > 30 GeV, p T,b2 > 30 GeV, all b-jets within |η| < 2.5, • photons: transverse momenta p T,γ1 > 30 GeV, p T,γ2 > 30 GeV, all photons within |η| < 2.5, • invariant mass of the two b-jets M bb ∈ [90, 160] GeV, • invariant mass of the two photons M γγ > M − 300 GeV, • veto events with leptons of p T > 25 GeV within |η| < 2.5, for each of the considered di-photon resonance masses, M . The cross sections after application of the acceptance cuts are given in Table I for two values of the widths Γ = 1 GeV and Γ = 45 GeV and for M = 750 GeV. For the case of qq we consider as examples λ HS = 1 and λ HS = 0. Throughout this analysis, the total signal cross section was calculated by using the ratio ρ (derived in Section II) as σ(pp → hS → hγγ) = ρ × σ(pp → S → γγ), where σ(pp → S → γγ) = 10, 5, 1 fb for M = 600, 750, 900 GeV, and including the decay h → bb. This cross section was employed as the normalisation of the signal event samples (before analysis cuts). The expected number of signal events, for λ HS ∼ O(1), after acceptance cuts is O(1)−O(10) at 3000 fb −1 of integrated luminosity. However, as already discussed, one should keep in mind that the cross section grows with λ 2 HS in both gg-and qq-initiated production.
The resulting di-photon invariant mass after acceptance cuts is shown in Fig. 12 for the example of M = 750 GeV. The M γγ observable can be used to separate the analysis into the two regions described in Section II: the "three-body decay" region ("TBD") and the "onshell di-photon" region ("OSγγ"). The separation is identical in both gg-and qq-initiated processess. We choose: M γγ < M − 50 GeV for the "TBD" region and M γγ > M −50 GeV for the "OSγγ" region for a di-photon resonance mass, M . We also show the distribution of the combined invariant mass of the two b-jet candidates and the di-photon system, M bbγγ in Fig. 13, which also clearly demonstrates the existence of the two regions. We apply further cuts to improve signal and background discrimination. As we did not attempt to fully  optimize the cuts in the present analysis, we apply a common set of cuts along with invariant mass cuts on the observables M γγ and M bbγγ that provide the main distinction between the two regions. The common cuts applied in each region are shown in Table II and the specific invariant mass cuts are shown in Table III. Effectively, the cuts aim to exploit the fact that the photons in the signal are harder than in the backgrounds and also feature tighter di-photon and bb mass windows, particularly in the "OSγγ" region for the former.  The additional common cuts applied along with the acceptance cuts in both the "three-body decay" and the "on-shell di-photon" region for the gg-and qq-initiated processess. The labels "1" and "2" correspond to the hardest and second hardest reconstructed objects (photons or b-jets), respectively.
We show the resulting cross sections after the application of these further cuts in Table IV, for the case of M = 750 GeV. A high efficiency is maintained for the signal, with high rejection factors for the background processes. We note again that the bbS associated production process is relevant only for the gluon-fusion scenario.
To obtain the 95% confidence-level exclusion regions for λ HS we use Poissonian statistics to calculate the probabilities. Since we have assumed that the production of a scalar di-photon resonance will have been observed, we have to construct a null hypothesis compatible with such an observation providing the expected number of events "TBD' "OSγγ"  Table II, in the "three-body decay" region ("TBD") and the "on-shell di-photon" region ("OSγγ"), for both the gg-and qq-initiated processess for a scalar diphoton resonance of mass M . The different choices for the mass windows were made according to the width of the resonance, Γ.  Tables II and III, for M = 750 GeV and Γ = 1, 45 GeV. All branching ratios, acceptances and tagging rates have been applied. We have assumed that the single production cross section for a di-photon scalar resonance of M = 750 GeV is σ(pp → S → γγ) = 5 fb.
at the LHC, that we will confront with the theory predictions in the parameter space to be tested. If these numbers differ by a certain significance, the corresponding point is expected to be excluded with this significance. In particular, any hypothesis has to be realistic and remain within the bounds of our model. Our underlying assumption is thus chosen to be that the scalar resonance S is produced purely in gluon fusion to good approximation and that there is no portal coupling, λ HS = 0, which means there is basically no h + S associated production. For further technical details on this statistical procedure, see Appendix C of Ref. [69]. We do not incorporate the effect of systematic uncertainties on the signal or backgrounds. To perform a combination of the two analysis regions, "TBD" and "OSγγ", we employ the "Stouffer method" [70], where the combined significance, Ω, is given, in terms of the individual significances Ω i , as: We show the resulting expected limits (assuming our null hypothesis is true) as a function of the integrated luminosity for the different benchmark scenarios that we consider in Figs. 14-25. For the case M = 750 GeV we show results for Γ = 45 GeV as well. For Γ = 1 GeV, we obtain more stringent constraints, limiting, for M = 600, 750, 900 GeV respectively, |λ HS | 2, 4, 5 for the gg-initiated process and λ HS ∈ [∼ −4, ∼ 1], [∼ −7, ∼ 3], [∼ −8, ∼ 4], both for the bb and ss-initiated processes, at the end of the HL-LHC run (3000 fb −1 ). The variation between the results for bb and ss-initiated processesvisible in the plots -is very small and can be attributed to the differences between the parton density functions for the strange and bottom quarks, as already mentioned.
The scenario with the larger width, M = 750 GeV, Γ = 45 GeV, clearly exhibits weaker constraints, with the gg-initiated processes yielding |λ HS | 5, the bb-initiated process λ HS ∈ [∼ −8, ∼ 5] and the ss-initiated process λ HS ∈ [∼ −8, ∼ 4]. It is conceivable that if further decay channels of the resonance S are discovered, the remaining unconstrained regions in the qq cases can be covered (in particular for narrow width), allowing determination of the initial state partons that produce the resonance.
The lower bound on λ HS for the qq-initiated cases is driven by the "TBD" region. This is understood by the fact that the "TBD" region always vanishes near λ HS ∼ 0, as it is dominated by the diagram with an on-shell schannel S, making the exclusion region resulting from it symmetric with respect to λ HS ∼ 0, whereas the "OSγγ" region possesses a symmetry point somewhere in λ HS < 0.
Mixed production of the di-photon resonance So far we have investigated production of the scalar resonance initiated purely either via gluon fusion or qq annihilation. We can generalize this to "mixed" production via gg and qq initial states simultaneously. We concentrate on the scenario of gg and bb for simplicity, with the extension to additional quark flavours being straightforward. In that case, the ratio of cross sections, ρ mixed , defined between the associated production and single production modes can be written as: where B 2 (λ HS ), G 2 (λ HS ) are functions of the portal coupling λ HS and B 1 , G 1 are constants with respect FIG . 14: The grey-shaded area shows the 95% confidencelevel exclusion region for the portal coupling λHS for the gginduced case and M = 600 GeV, Γ = 1 GeV, for the combination of the "TBD" (red) and "OSγγ" (green) regions as defined in the main text. We have assumed that the single production cross section σ(pp → S → γγ) = 10 fb.
FIG . 15: The grey-shaded area shows the 95% confidencelevel exclusion region for the portal coupling λHS for the bbinduced case and M = 600 GeV, Γ = 1 GeV, for the combination of the "TBD" (red) and "OSγγ" (green) regions as defined in the main text. We have assumed that the single production cross section σ(pp → S → γγ) = 10 fb.
to the portal coupling, all to be determined. Defining θ ≡ |[(y S d ) 33 ]/c S G |, the above expression can be re-written as: By considering the limits θ → 0 and θ → ∞, and dividing the numerator and denominator by G 1 , we can deduce that ρ mixed = ρ(bb, λ HS )r bg θ 2 + ρ(gg, λ HS ) r bg θ 2 + 1 , FIG . 16: The grey-shaded area shows the 95% confidencelevel exclusion region for the portal coupling λHS for the ssinduced case and M = 600 GeV, Γ = 1 GeV, for the combination of the "TBD" (red) and "OSγγ" (green) regions as defined in the main text. We have assumed that the single production cross section σ(pp → S → γγ) = 10 fb.
FIG . 17: The grey-shaded area shows the 95% confidencelevel exclusion region for the portal coupling λHS for the gginduced case and M = 750 GeV, Γ = 1 GeV, for the combination of the "TBD" (red) and "OSγγ" (green) regions as defined in the main text. We have assumed that the single production cross section σ(pp → S → γγ) = 5 fb.
where ρ(bb, λ HS ) and ρ(gg, λ HS ) are the ratios of cross sections as functions of the portal coupling as defined in Eq. 7, for the cases of pure production initiated either via bb or gg, respectively, and r bg is the ratio of pure single production cross sections for θ = 1: r bg = B 1 /G 1 . The former two functions have already been determined in the analysis of the pure cases. The constant r bg was calculated explicitly via Monte Carlo to be r bg ≈ 1.02, 0.69, 0.49 for M = 600, 750, 900 GeV respectively. The values are approximately equal for both the narrow width case (Γ = 1 GeV) and the larger width case (Γ = 45 GeV).
FIG . 18: The grey-shaded area shows the 95% confidencelevel exclusion region for the portal coupling λHS for the gginduced case and M = 750 GeV, Γ = 45 GeV, for the combination of the "TBD" (red) and "OSγγ" (green) regions as defined in the main text. We have assumed that the single production cross section σ(pp → S → γγ) = 5 fb.
FIG . 19: The grey-shaded area shows the 95% confidencelevel exclusion region for the portal coupling λHS for the bbinduced case and M = 750 GeV, Γ = 1 GeV, for the combination of the "TBD" (red) and "OSγγ" (green) regions as defined in the main text. We have assumed that the single production cross section σ(pp → S → γγ) = 5 fb.
Using Eq. 11, we can deduce an expression for the predicted number of signal events after the application of analysis cuts in the mixed production case: where N (bb, λ HS ) and N (gg, λ HS ) are the expected signal events for either pure bb or pure gg production for a given value of λ HS , at a specific integrated luminosity.
The predicted number of background events after a given set of analysis cuts is constant with θ, apart from the associated production of the di-photon resonance The grey-shaded area shows the 95% confidencelevel exclusion region for the portal coupling λHS for the bbinduced case and M = 750 GeV, Γ = 45 GeV, for the combination of the "TBD" (red) and "OSγγ" (green) regions as defined in the main text. We have assumed that the single production cross section σ(pp → S → γγ) = 5 fb.

FIG. 21:
The grey-shaded area shows the 95% confidencelevel exclusion region for the portal coupling λHS for the ssinduced case and M = 750 GeV, Γ = 1 GeV, for the combination of the "TBD" (red) and "OSγγ" (green) regions as defined in the main text. We have assumed that the single production cross section σ(pp → S → γγ) = 5 fb.
with a bb pair, which was found to be significant only in the gg-initiated scenario. This background scales as N assoc. = N assoc.,0 /(r bg θ 2 + 1), where N assoc.,0 is the expected number of events for the bbS associated production after analysis cuts in the case of pure gg-initiated production.
Using the expression of Eq. 12 and the event numbers for the two analysis regions "TBD" and "OSγγ" obtained for the pure production modes, we can derive constraints on the (θ, λ HS )-plane. These are shown in The grey-shaded area shows the 95% confidencelevel exclusion region for the portal coupling λHS for the ssinduced case and M = 750 GeV, Γ = 45 GeV, for the combination of the "TBD" (red) and "OSγγ" (green) regions as defined in the main text. We have assumed that the single production cross section σ(pp → S → γγ) = 5 fb.

FIG. 23:
The grey-shaded area shows the 95% confidencelevel exclusion region for the portal coupling λHS for the gginduced case and M = 900 GeV, Γ = 1 GeV, for the combination of the "TBD" (red) and "OSγγ" (green) regions as defined in the main text. We have assumed that the single production cross section σ(pp → S → γγ) = 1 fb.
respectively, at an integrated luminosity of 3000 fb −1 at 13 TeV. It can be seen that in the limits θ → 0 and θ 1, corresponding to gg-or bb-dominated production respectively, one can recover the pure production constraints of Figs. 14-25 obtained at 3000 fb −1 .

V. DISCUSSION AND CONCLUSIONS.
We have investigated the associated production of a diphoton scalar resonance with a Higgs boson and have employed the pp → hS → (bb)(γγ) final state to obtain con- The grey-shaded area shows the 95% confidencelevel exclusion region for the portal coupling λHS for the bbinduced case and M = 900 GeV, Γ = 1 GeV, for the combination of the "TBD" (red) and "OSγγ" (green) regions as defined in the main text. We have assumed that the single production cross section σ(pp → S → γγ) = 1 fb.

FIG. 25:
The grey-shaded area shows the 95% confidencelevel exclusion region for the portal coupling λHS for the ssinduced case and M = 900 GeV, Γ = 1 GeV, for the combination of the "TBD" (red) and "OSγγ" (green) regions as defined in the main text. We have assumed that the single production cross section σ(pp → S → γγ) = 1 fb. straints on the portal coupling with the SM Higgs boson λ HS , at the LHC. We have considered three benchmark scalar masses, M = 600, 750, 900 GeV, and we have assumed that the inclusive single production cross section is σ(pp → S → γγ) = 10, 5, 1 fb, respectively, compatible with current experimental constraints. To construct expected constraints we considered the null hypothesis (i.e. the supposed 'true' underlying theory) to correspond to gluon-fusion-initiated production with vanishing portal coupling, λ HS = 0. We then first analysed the case of either pure gluon-fusion-induced production or production via quark-anti-quark annihilation. For gluon-fusion FIG. 26: The 95% confidence-level exclusion region for the ratio of couplings to bb over the coupling to the gg initial states, θ = [(y S d ) 33 ]/c S G , versus the portal coupling λHS for M = 600 GeV, Γ = 1 GeV. The excluded region coming from the combination of the "TBD" (red) and "OSγγ" (green) regions is grey-shaded. We have assumed that the single production cross section σ(pp → S → γγ) = 10 fb.
FIG. 27: The 95% confidence-level exclusion region for the ratio of couplings to bb over the coupling to the gg initial states, θ = [(y S d ) 33 ]/c S G , versus the portal coupling λHS for M = 750 GeV, Γ = 1 GeV. The excluded region coming from the combination of the "TBD" (red) and "OSγγ" (green) regions is grey-shaded. We have assumed that the single production cross section σ(pp → S → γγ) = 5 fb.
production one can impose constraints on the portal coupling at the end of the HL-LHC run of |λ HS | 2, 4, 5 for M = 600, 750, 900 GeV, respectively, and Γ = 1 GeV, while |λ HS | 5 in the case of large width (Γ = 45 GeV) and M = 750 GeV. For quark-anti-quark annihilation, the production of an on-shell scalar and the Higgs boson is enhanced by a contact interaction ∼ qqhS, originating from the same D = 5 operator that mediates single S production. This implies that the cross section is nonnegligible even for vanishing portal coupling λ HS = 0.  33 ]/c S G , versus the portal coupling λHS for M = 900 GeV, Γ = 1 GeV. The excluded region coming from the combination of the "TBD" (red) and "OSγγ" (green) regions is grey-shaded. We have assumed that the single production cross section σ(pp → S → γγ) = 1 fb.
FIG. 29: The 95% confidence-level exclusion region for the ratio of couplings to bb over the coupling to the gg initial states, θ = [(y S d ) 33 ]/c S G , versus the portal coupling λHS for M = 750 GeV, Γ = 45 GeV. The excluded region coming from the combination of the "TBD" (red) and "OSγγ" (green) regions is grey-shaded. We have assumed that the single production cross section σ(pp → S → γγ) = 5 fb.
This fact can be exploited to exclude the whole plane of λ HS , thus excluding the hypothesised production via qq annihilation. For the case of bb we find that a region inside λ HS ∈ [∼ −4, ∼ 1], [∼ −7, ∼ 3], [∼ −8, ∼ 4] for M = 600, 750, 900 GeV, respectively, could remain unconstrained at the end of the HL-LHC in the narrow width scenario, while λ HS ∈ [∼ −8, ∼ 5] for large width and M = 750 GeV. For the case of ss annihilation we find the same unconstrained regions to good approximation for narrow width, while we obtain λ HS ∈ [∼ −8, ∼ 4] in the case of large width, where M = 750 GeV. We have also considered 'mixed' production via gluon fusion and bb annihilation and derived constraints on the plane of ratio of the corresponding couplings versus λ HS for an integrated luminosity of 3000 fb −1 .
Note that since a measurement of λ HS can straightforwardly be translated into a measurement of µ S , these numbers suggest that it is possible to exclude, for example, the scenario where the mass of S stems from EWSB (µ S = 0), which corresponds to λ HS = M 2 /v 2 ≈ 5.9, 9.3, 13.4 (still assuming linear terms in S to vanish). * * * Furthermore, if additional decay modes of the resonance S are discovered beyond γγ, it is conceivable that a combined analysis in various channels would be able to exclude all possible values of λ HS -for the case of qq production -thus providing an independent method of determining the production mode. Conversely, the analysis of the present article demonstrates that if the production mechanism is constrained via alternative means, it will be possible to obtain meaningful constraints on the interaction of a new scalar resonance and the Higgs boson, allowing determination of its relation to electroweak symmetry breaking.
interaction, linear in S, is non-vanishing: with κ HS = 1/2 λ HS + 3/(2Λ) c λH . In this scenario, terms will appear in the cross section ratio proportional to κ HS and κ 2 HS . Due to interference with the new hhS diagrams (note that also the 'nonsignal' S → hh interferes with the signal diagrams after an off-shell h → γγ), some dependence on the coupling of the resonance S to photons, c S γ , as well as the production couplings, (y S d ) 22 , (y S d ) 33 and c S G , is introduced in the denominator of the cross section ratio. Due to this, smaller values of |c S γ | and the production couplings enhance the effect of the new contributions proportional to κ HS and κ 2 HS . To take these effects into account and at the same time remain conservative, we set |c S γ | to the possible minimal value that produces a σ(pp → S → γγ) ∼ 5 fb for M = 750 GeV, as derived in Ref. [24]. This leads to |c S γ | {60, 100, 10} for ss, bb and gg production, respectively, where here and in the following we assume a normalisation of Λ = 1 TeV. Moreover, one can derive conservative lower bounds on |(y S d ) 22 |, |(y S d ) 33 | and |c S g |, by demanding σ(pp → S) 5 fb, leading to |(y S d ) 22 | 0.15, |(y S d ) 33 | 0.2 and |c S g | 0.25. We set these to their minimal values as well in what follows.
We can then parametrize the ratio ρ(xx ) = σ(xx → hγγ)/σ(xx → S → γγ), xx = bb, ss, gg, by § § § ρ(xx ) = δ x 1 + δ x 2 κ HS λ HS + δ x 3 κ HS + δ x 4 λ HS + δ x 5 κ 2 HS + δ x 6 λ 2 HS , where δ b,s,g i are coefficients to be determined. An estimate of the coefficients in this conservative scenario, obtained numerically, is given in Table V for width Γ = 1 GeV and M = 750 GeV. The contributions from diagrams with an off-shell Higgs boson are a priori not fully negligible for the bb-induced case, simply because of interference of the s-channel h → hS with the large matrix elements with the on-shell S in the final state. § § § For the case of vanishing κ HS , this more general definition coincides with Eq. 7 to good approximation.
To investigate the size of κ HS that renders the offshell Higgs boson contributions significant to the pp → hγγ process, we consider the ratio ρ(κ HS , λ HS )/ρ(κ HS = 0, λ HS ) = 1 + r, where r is a number that characterises the fractional change in the ratio ρ due to κ HS for a given value of λ HS . If we choose r = 1, which implies O(1) changes due to κ HS , and solve for κ HS for values λ HS ∼ O(1) we obtain: |κ HS | ∼ {O(10), O(10), O(10 2 )} for ss, bb and gg production, respectively.
Since the gauge-invariant terms in the Lagrangian generating this coupling also induce a Higgs-scalar mixing, they can not be arbitrarily large. Indeed, Higgs data can constrain |κ HS | 4 at 95% confidence-level [73][74][75]. ¶ ¶ ¶ Therefore, given the values calculated in the previous paragraph, for κ HS to eventually have a non-negligible effect on the analysis of the present article, this bound needs to be violated, and setting κ HS = 0 is a justified approximation. Despite the fact that we focussed on M = 750 GeV, the analysis is expected to give similar estimates for the other benchmark points considered in the main analysis of this article, M = 600, 900 GeV. ¶ ¶ ¶ Here, we have neglected the impact of the second operator in (B1), S|H| 4 , which breaks the correlation between Higgs-scalar mixing and the h 2 S interaction.