Uncovering the relation of a scalar resonance to the Higgs boson
CCERN-TH-2016-133, MCnet-16-19
Uncovering the relation of a scalar resonance to the Higgs boson
Adri´an Carmona, ∗ Florian Goertz, † and Andreas Papaefstathiou ‡ CERN, Theoretical Physics Department,CH-1211 Geneva 23, Switzerland
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 theHiggs boson can be constrained in a meaningful way in future runs of the LHC, providing insightson its origin and its relation to the electroweak symmetry breaking sector. Moreover, the final statecan provide a direct way to determine whether the new resonance is produced predominantly ingluon fusion or quark-anti-quark annihilation. The analysis focusses on a resonance coming from ascalar field with vanishing vacuum expectation value and its decay to a photon pair. It can howeverbe straightforwardly generalised to other scenarios.
I. NEW SCALAR RESONANCES ATCOLLIDERS.
Models with an additional (pseudo-)scalar singlet witha mass of several hundred GeV represent a well moti-vated class of extensions of the Standard Model (SM)of particle physics, including composite Higgs scenarios,supersymmetry, Coleman-Weinberg models, models ad-dressing the strong CP problem, models of flavor, as wellas generic Higgs portal setups (see e.g. [1–9]). A partic-ularly promising channel to search for and analyze sucha particle is its decay to two photons. Beyond beingpossibly sizable in certain scenarios, it offers a robustand clean way to detect a signal, emerging over a steeplyfalling background [10, 11].After its discovery, an important aspect of scrutinisingany new resonance is in fact to measure its couplings, andhence determine its relation, to the known particle con-tent of the Standard Model (SM). A crucial componentof this task is to uncover its role in the arena of elec-troweak symmetry breaking (EWSB). As a first step inthis direction, determination of the couplings of the newscalar to the SM-like Higgs boson is mandatory, whichis 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 effec-tive Lagrangian for our study – augmenting the SM at ∗ Electronic address: [email protected] † Electronic address: fl[email protected] ‡ Electronic address: [email protected] ∗ A specific motivation for the first version of this manuscript wasprovided by the apparent γγ resonance at M γγ ∼
750 GeV in AT-LAS [12, 13] and CMS [14, 15] data. This turned out not to bepresent in the 2016 data [16, 17]. Consequently the article was gen-eralized to other mass scales of a potential scalar resonance, whichremains well motivated, taking into acount new limits on its crosssection - see below. Comprehensive analyses studying constraintson (other) possible couplings of a di-photon resonance as well asdetailed examinations of indirect footprints of new (high multiplic-ity) sectors, linked to its productions or decay appeared e.g. in[18–45]. dimension D ≤ L eff ⊃ ∂ µ S ∂ µ S − µ S S (1) − ( y Sd ) ij S Λ ¯ Q iL Hd jR − ( y Su ) ij S Λ ¯ Q iL ˜ Hu jR + h . c . − S Λ 116 π (cid:2) g (cid:48) c SB B µν B µν + g c SW W Iµν W Iµν + g S c SG G aµν G aµν (cid:3) − λ HS | H | S − λ S S . Here, Q iL is the i -th generation left-handed SU (2) L fermion doublet, d jR and u jR are the right-handed SU (2) L fermion singlets for generation j , ( y Sq ) ij are the corre-sponding Yukawa-like couplings, c SB , c SW are the couplingsof S to the U (1) Y and SU (2) L gauge fields B and W , c SG is the coupling to the gluon fields, H is the Higgs bosondoublet, λ HS is the Higgs-Scalar portal coupling and λ S the new scalar quartic. Moreover, Λ denotes the scaleof heavy new physics (NP), mediating the contact inter-actions of S with SM gauge bosons and fermions (thelatter involving H to generate a gauge singlet). Notethat 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 ir-relevant in general for the process we will consider, seebelow. † Beyond that, terms linear in S could also leadto the singlet mixing with the Higgs boson after EWSB,which would in fact affect its phenomenology. Althoughsuch effects could still be present at a non-negligible level,they are expected to be sub-leading and we neglect themfor simplicity, see Appendix B. Furthermore, the analysisthat follows is independent of the CP properties of S andits interactions and henceforth, for simplicity, we assumeit to be CP-even with CP-conserving interactions.With the potential for the Higgs doublet H taking theconventional form: V = λ H | H | − µ H | H | , (2) † The full list of potential D ≤ a r X i v : . [ h e p - ph ] A ug and assuming that H is the only scalar that gets a vac-uum expectation value (vev), |(cid:104) H (cid:105)| = v/ √
2, triggeringEWSB, we obtain the condition ‡ λ HS µ H − λ H µ S < , µ H < , (3)The physical mass of the singlet thus reads M = (cid:112) µ S + λ HS v .The resulting trilinear interactions between the physi-cal scalar resonances after EWSB are described by L = − M h v h − λ HS v hS , (4)where h is the Higgs boson, which (due to the case ofnegligible scalar mixing) is basically fully embedded in H and describes excitations around its vev, such that inunitary gauge H (cid:39) / √ , v + h ) T , and S is a new scalarresonance, which can also be (approximately) identifiedas S = S . Moreover, v (cid:39)
246 GeV is the Higgs vev, M h (cid:39)
125 GeV is the measured Higgs boson mass andthe portal coupling λ HS is to be determined, being amain scope of this paper. This coupling would be basi-cally unconstrained by direct observation of a di-photonresonance. Nevertheless, loose indirect constraints canbe derived, requiring vacuum stability not to be spoiled.They read λ S > , λ H > , λ HS < λ H λ S , (5)and need to be imposed at least at the scale where the NPenters, i.e. , the TeV scale (see, e.g. [1, 46]). § Requiring λ H ∼ .
13, to fit the observed Higgs mass, as well as λ S < (4 π ) , we thus obtain − . (cid:46) λ HS (cid:46) .
5. We willsee below that in general our analysis can put strongerbounds than these on λ HS . ¶ In the present study we consider measurement of thecoupling λ HS at the LHC, where it can be probed viaassociated production of the new resonance with the SM-like Higgs boson: pp → hS . For this process, the interac-tions 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 | , S interactions, contribute atmost to a diagram with a (strongly suppressed) off-shellHiggs boson propagator, for details see the appendices Aand B. ‡ The fact that (cid:104)S(cid:105) =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 > 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 < λ HS to be notmuch larger than 1 and would remove a considerable portion of theparameter space where our analysis exhibits sensitivity. However,in any case, the limits presented here are complementary to suchconsiderations. In principle several decay modes of S can be consid-ered. Here we focus on the process pp → hS → hγγ ,where the new particle decays to a pair of photons. Giventhat the tentative cross section of the resonant di-photonproduction, pp → S → γγ , will be known and will bewell-measured in the case of discovery, this allows con-straints on the coupling λ HS to be imposed almost inde-pendently of the couplings to the initial-state partonsand final-state photons, given that only a single pro-duction mode is relevant. We consider production viagluon fusion and quark-anti-quark annihilation, mediatedthrough non-vanishing coefficients c SG , ( y Sd ) or ( y Sd ) ,respectively, and show how these modes could be disen-tangled via appropriate measurements. We will implic-itly assume not too large values of Γ( S → γγ ), in such away that photo-production is always subdominant.Although we will focus on three specific benchmarkmasses of M = 600 , ,
900 GeV, our analysis couldbe applied to the general case of associated production ofa Higgs boson with a scalar di-photon resonance of anymass. Moreover, several features of the final state studiedhere, such as the invariant mass of the final-state scalaror the total invariant mass of the process, will exhibitsimilar features when considering other decay modes.The article is organised as follows: in Section II weexamine the process of associated production of a scalarresonance and a Higgs boson, in Section III we describethe event generation and detector simulation setup and inSection IV we provide details of the analysis and results.Finally, we conclude in Section V.
II. ASSOCIATED PRODUCTION OF ASCALAR RESONANCE AND A HIGGS BOSON.A. Production through gluon fusion.
The dominant diagrams at LO contributing to the pro-duction of the hS final state in gluon fusion and subse-quent decay of the resonance S to a pair of photons viathe interactions L eff ⊃ − S Λ 116 π (cid:2) e ( c SB + c SW ) F µν F µν + g s c G G aµν G aµν (cid:3) = − S Λ 116 π (cid:2) e c Sγ F µν F µν + g s c G G aµν G aµν (cid:3) , (6)are shown in Fig. 1. In the analysis of the present article,we will consider the Higgs boson decaying to a bottom-quark pair, since this maximizes the expected number ofevents, which would be modest in general. There existboth the s -channel S exchange, involving the portal cou-pling λ HS and depicted in the upper panel (a), as well asthe ‘direct’ hS production, via t -channel gluon exchange,depicted in the lower panel (b). gg S (cid:0) b(cid:0) bh (a) gg Sh (cid:0) b(cid:0) b (b) FIG. 1: The diagrams contributing to the process gg → hS → ( b ¯ b )( γγ ) at the LHC at LO. B. Production through quark-anti-quarkannihilation.
For the case of quark anti-quark annihilation, a newdiagram arises from the contact interaction q ¯ qhS . (cid:107) Bothcontributing graphs are shown in Fig. 2. The new dia-gram (b) distinguishes the q ¯ q annihilation from the gluonfusion case. An important fact is that now the hS pro-cess is non-vanishing and significant even in the absenceof the portal coupling λ HS . This indicates that one canemploy this final state to exclude q ¯ q annihilation as thedominant 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 ratioof the associated production process pp → hS , throughall possible intermediate states, with the subsequent de-cay of the new resonance to a γγ final state, to that ofthe single production pp → S → γγ , ρ ( xx (cid:48) ) = σ ( xx (cid:48) → hS → hγγ ) σ ( xx (cid:48) → S → γγ ) , (7)where we consider xx (cid:48) = { gg, b ¯ b, s ¯ s , c ¯ c } . ∗∗ The ratio isa useful quantity since it removes the dependence on theproduct of couplings of the new resonance to the initial-state partons and final-state photons. Moreover, it canbe used to absorb, at least approximately, theoretical and (cid:107)
The t-channel diagram with the q ¯ qh interaction is suppressed dueto a small Yukawa coupling. ∗∗ In the most general setup, the analysis of this article can constrainthe sum of the squares of the couplings of S to all quark generations(for a given λ HS ), appropriately weighted by the parton densityfunctions. S (cid:0) b(cid:0) bhqq (a)
Sh (cid:0) b(cid:0) bqq (b)
FIG. 2: The diagrams contributing to the process q ¯ q → hS → ( b ¯ b )( γγ ) at the LHC at LO. experimental systematic uncertainties. †† λ HS σ ( pp → h S → h γγ ) / σ ( pp → S → γγ ) b ¯ bggs ¯ sc ¯ c FIG. 3: The ratios ρ ( gg ), ρ ( b ¯ b ), ρ ( s ¯ s ) and ρ ( c ¯ c ) for gg , b ¯ b , s ¯ s and c ¯ c initial states respectively, defined between the asso-ciated production pp → hS → hγγ and the single production pp → S → γγ , as functions of the portal coupling λ HS . Themass of the scalar resonance was taken to be M = 750 GeVand the width Γ = 1 GeV. We show the dependence of the ratio ρ on the portalcoupling λ HS in Fig. 3 for gg , b ¯ b , s ¯ s and c ¯ c initial statesfor the example di-photon resonance mass M = 750 GeVand width Γ = 1 GeV. ‡‡ A width of Γ (cid:46) †† For a similar idea investigated in the context of Higgs boson pairproduction, see [47]. ‡‡ We employ a single cut of M γγ >
200 GeV at generation level inorder to remove (SM-like) pp → hh → hγγ interference with the obtained for example, if c Sγ ∼ O (10) and c G ∼ O (1) or( y Sd ) ∼ O (1), for Λ = 1 TeV, with a cross section in pp → γγ compatible with current constraints. Similarbehaviour 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 cou-pling, the process approximately vanishes as λ HS → ρ ( gg ) (cid:39) ρ gg λ HS , where ρ gg ≈ . M = 750 GeV and Γ = 1 GeV, obtained by perform-ing a quadratic fit of the gg curve in Fig. 3. As alreadydiscussed, this does not hold for the q ¯ q -initiated processdue to the contact interaction diagram. This results ina non-negligible minimum for ρ ( q ¯ q ). A fit to the crosssection, again for M = 750 GeV and Γ = 1 GeV, yields ρ ( q ¯ q ) (cid:39) ρ q ¯ q λ HS + ρ q ¯ q λ HS + ρ q ¯ q with ρ b ¯ b ≈ . ρ b ¯ b ≈ . ρ b ¯ b ≈ . b ¯ b ini-tial states, and ρ s ¯ s ≈ . ρ s ¯ s ≈ . ρ s ¯ s ≈ . s ¯ s initial states. The case of c ¯ c is similar to the s ¯ s case and therefore in the rest ofthe article we focus on the cases q = b and q = s . Thepositivity of the coefficient ρ q ¯ q indicates constructive in-terference between the contact interaction and resonantdiagrams (for λ HS > ρ , including additional diagrams with the production ofan intermediate Higgs boson due to S| H | , 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 cou-pling λ HS indicates that one could employ this final stateto exclude b ¯ b , s ¯ s or c ¯ c annihilation as the dominant pro-duction process. The analysis that will follow in thepresent article suggests however, that the di-photon de-cay of the S alone may not be sufficient for that purposefor the benchmark points that we consider.We show in Fig. 4 the variation of the ratio ρ with themass of the resonance, M , for the gg -iniated process and λ HS = 1, and for the b ¯ b -initiated process for λ HS = 0(no portal) and λ HS = 1. We have fixed the width toΓ = 1 GeV. Interestingly, the pure q ¯ q -induced processesexhibit an increase of the ratio ρ with increasing mass– related to the new q ¯ qhS interaction growing with mo-mentum – whereas the pure gg -induced process exhibitsa slight decrease.If the di-photon resonance is wide, the analyses per-formed for the hS final state will differ in the details dueto changes in the kinematics. We show in Fig. 5 thevariation of the ratio ρ with the width over the mass,Γ /M , at a fixed mass M = 750 GeV, for λ HS = 1, andfor the b ¯ b -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 iden-tify a ‘signal’ contribution to the actual physical process - whichis Higgs production in association with a photon pair - unambigu-ously with the process pp → hS → hγγ to good approximation,assuming the model (1).
400 500 600 700 800 900 1000 M [GeV] σ ( pp → h S → h γγ ) / σ ( pp → S → γγ ) gg , λ SH =1 b ¯ b , λ SH =1 b ¯ b , λ SH =0 FIG. 4: The ratios ρ ( gg ) and ρ ( b ¯ b ) for gg and b ¯ b initial states,defined between the associated production pp → hS → hγγ and the single production pp → S → γγ , as functions of themass of the resonance, for Γ = 1 GeV. The bands display theparton density function uncertainty for the MMHT14nlo68cl setcombined in quadrature with the scale variation between 0.5and 2.0 times the default central dynamical scale implementedin
MadGraph5 aMC@NLO . Γ /M σ ( pp → h S → h γγ ) / σ ( pp → S → γγ ) gg , λ SH =1 b ¯ b , λ SH =1 b ¯ b , λ SH =0 FIG. 5: The ratios ρ ( gg ) and ρ ( b ¯ b ) for gg and b ¯ b initial states,defined between the associated production pp → hS → hγγ and the single production pp → S → γγ , as functionsof the width of the resonance over the mass, Γ /M . Thebands display the parton density function uncertainty for the MMHT14nlo68cl set combined in quadrature with the scalevariation between 0.5 and 2.0 times the default central dy-namical scale implemented in
MadGraph5 aMC@NLO . The scalarresonance mass was chosen to be M = 750 GeV. ratio remains approximately constant in all cases, withonly a slight decrease with increasing width.In both Figs. 4 and 5, we also provide, as colouredbands, the parton density function uncertainty forthe MMHT14nlo68cl set [48] combined in quadraturewith the scale variation between 0.5 and 2.0 timesthe default central dynamical scale implemented in
MadGraph5 aMC@NLO . For a mass of 750 GeV, the totaltheoretical uncertainties due to scale and PDF variationsare ∼ +40 − % for the gg -induced process, ∼ ±
10% for the b ¯ b -induced and ∼ ±
30% for the s ¯ s -induced cases (thelatter not shown in the figure for simplicity).Assuming a total cross section for the production ofa γγ resonance of mass M = 750 GeV of, say, σ ( pp → S → γγ ) = 5 fb (see below), one would expect a totalof O (20) hS → hγγ events at the high-luminosity LHC(HL-LHC, assuming 3000 fb − of integrated luminosity)if the process is gluon-fusion initiated and O (200) eventsfor b ¯ b -initiated production, for a portal coupling λ HS =1. Moreover, the minimum expected number of events forthe b ¯ b -initiated process is O (80), arising for λ HS (cid:39) − . s ¯ s -initiated process one expects a minimumof O (100) events for λ HS (cid:39) − .
0. We note here that thepositions of the minima for the q ¯ q -initiated process willchange after cuts due to the varying effect of the analysison the different pieces contributing to the cross section.
200 300 400 500 600 700 800 900 1000110012001300140015001600 M γγ [GeV] -4 -3 -2 -1 ( / σ ) d σ ( gg → h S → h γγ ) / d M γγ [ G e V − ] Γ =1
GeV
Γ =45
GeV
FIG. 6: The matrix-element level distribution of the di-photon invariant mass, M γγ , 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 distri-bution of the invariant mass of the γγ state, M γγ , orthe distribution of the invariant mass of the Higgs bo-son and di-photon combination, M hγγ . In Figs. 6 and 7we show, respectively, these distributions for the gluon-fusion-initiated process, for two widths, Γ = 1 GeV,Γ = 45 GeV. For the sake of clarity, here we only showdistributions for a scalar mass of M = 750 GeV, but themain features remain unaltered as long as the scalar isheavier than the Higgs boson, M > M h . The distribu-tions clearly show the existence of two regions: a regionin which the intermediate s -channel propagator for the S scalar is on-shell and the final-state S ( γγ ) is off-shell,and a region in which the s -channel internal propagatoris instead off-shell and the final-state S ( γγ ) is on-shell.The existence of the former region, M γγ (cid:46) M − M h , M hγγ ∼ M which henceforth we will call “three-body de-cay” since the intermediate S is decaying approximatelyon-shell, is made possible by the fact that the mass of
500 600 700 800 900 1000 1100 1200 1300 1400 1500 M hγγ [GeV] -4 -3 -2 -1 ( / σ ) d σ ( gg → h S → h γγ ) / d M h γγ [ G e V − ] Γ =1
GeV
Γ =45
GeV
FIG. 7: The matrix-element-level distribution of the com-bined Higgs boson and di-photon invariant mass, M hγγ , inthe gg → hS → hγγ process, normalised to unity, for the twodifferent width scenarios, Γ = 1 GeV and Γ = 45 GeV, for M = 750 GeV. the particle produced in association with S , the Higgsboson, is smaller than the masses of S we are consider-ing, M > M h = 125 GeV. The other region, M γγ ∼ M , M hγγ (cid:38) M + M h which we will refer to as “on-shell di-photon”, exists irrespective of the mass of S . Note thatboth the three-body decay and on-shell di-photon regionsexist even for Γ /M (cid:28)
1. The normalised distributionslook identical for all values of the portal coupling, λ HS ( (cid:54) = 0), since the dominant contribution stems by far fromthe diagram shown in Fig. 1 (a).
200 300 400 500 600 700 800 900 1000110012001300140015001600 M γγ [GeV] -4 -3 -2 -1 ( / σ ) d σ ( b ¯ b → h S → h γγ ) / d M γγ [ G e V − ] Γ =1
GeV, λ HS =1Γ =45 GeV, λ HS =1Γ =1 GeV, λ HS =0Γ =45 GeV, λ HS =0 FIG. 8: The matrix-element-level distribution of the di-photon invariant mass, M γγ , in the b ¯ b → hS → hγγ process,normalised to unity, for two width scenarios, Γ = 1 ,
45 GeVand two values of the portal coupling, λ HS = 0 ,
1, for M = 750 GeV. In Figs. 8 and 9 we show the di-photon invariant massand the combined Higgs boson and di-photon invari-ant mass for the b ¯ b -initiated process, respectively, for M = 750 GeV. Evidently the two regions observed forthe gg case are clearly still present for λ HS (cid:54) = 0 and
500 600 700 800 900 1000 1100 1200 1300 1400 1500 M hγγ [GeV] -4 -3 -2 -1 ( / σ ) d σ ( b ¯ b → h S → h γγ ) / d M h γγ [ G e V − ] Γ =1
GeV, λ HS =1 .
0Γ =45
GeV, λ HS =1 .
0Γ =1
GeV, λ HS =0 .
0Γ =45
GeV, λ HS =0 . FIG. 9: The matrix-element level distribution of the com-bined Higgs boson and di-photon invariant mass, M hγγ , inthe b ¯ b → hS → hγγ process, normalised to unity, for twowidth scenarios, Γ = 1 ,
45 GeV and two values of the portalcoupling, λ HS = 0 ,
1, for M = 750 GeV. Γ = 1 GeV, with the on-shell di-photon region dominat-ing. For λ HS = 0, the “three-body decay” region disap-pears completely since the resonant s -channel diagram ofFig. 2 (a) vanishes. For large width the two regions mergeinto one and the effect of the vanishing three-body decayregion for λ HS = 0 is not as evident as in the case of smallwidth. The distributions for the s ¯ s -initiated process ex-hibit similar features, with different “mixtures” betweenthe two regions arising from the differences between thestrange and bottom quark parton density functions. Weomit them for the sake of simplicity.
500 600 700 800 900 1000 1100 1200 1300 1400 1500 M hγγ [GeV] -4 -3 -2 -1 ( / σ ) d σ ( gg → h S → h γγ ) / d M h γγ [ G e V − ] gg, M =600 GeV gg, M =750
GeV gg, M =900
GeV
FIG. 10: The matrix-element level distribution of the com-bined Higgs boson and di-photon invariant mass, M hγγ , inthe gg → hS → hγγ process, normalised to unity, for threebenchmark mass scenarios, M = 600 , ,
900 GeV, andΓ = 1 GeV.
Figures 10 and 11 show the distributions of the com-bined invariant mass of the Higgs boson and di-photon, M hγγ , for the pure gg -initiated and pure b ¯ b -initiatedcases respectively, for the three values of the scalar mass
500 600 700 800 900 1000 1100 1200 1300 1400 1500 M hγγ [GeV] -4 -3 -2 -1 ( / σ ) d σ ( gg → h S → h γγ ) / d M h γγ [ G e V − ] b ¯ b, M =600 GeV b ¯ b, M =750 GeV b ¯ b, M =900 GeV
FIG. 11: The matrix-element level distribution of the com-bined Higgs boson and di-photon invariant mass, M hγγ , inthe b ¯ b → hS → hγγ process, normalised to unity, for threebenchmark mass scenarios, M = 600 , ,
900 GeV, andΓ = 1 GeV, for λ HS = 1. that we will consider as “benchmark” scenarios in ouranalysis, M = 600 , ,
900 GeV (see below) andΓ = 1 GeV. For the b ¯ b case we only show the λ HS = 1distributions for simplicity. They all clearly demon-strate the existence of the main features described forthe M = 750 GeV case, i.e. the “three-body decay” and“on-shell di-photon” regions. III. EVENT GENERATION AND DETECTORSIMULATION.A. Event generation.
The signal model was generated via an implementa-tion of the Lagrangian of Eq. 1 in
FeynRules [49, 50].Via the UFO interface [51] this was used to generateparton-level events employing
MadGraph5 aMC@NLO [52,53]. The background processes were also generated us-ing
MadGraph5 aMC@NLO , with appropriate generation-level cuts to reduce the initial cross sections to a man-ageable level. All the events were passed through the
HERWIG 7 [54–58] Monte Carlo for simulation of the par-ton shower, the underlying event and hadronization. Asbefore, the
MMHT14nlo68cl
PDF set was employed. Toremain conservative, we consider collisions at the LHC ata centre-of-mass energy of 13 TeV. The possible increaseof energy to 14 TeV will increase rates in the consideredprocesses by O (10%).Since we expect to impose cuts on the di-photon masswindow, M γγ , that are sufficiently far away from theHiggs boson resonance, we can immediately exclude anybackground processes containing h → γγ from the anal-ysis. For this reason we do not include associated Higgsboson production with a vector boson or Higgs boson pairproduction, t ¯ th production, and so on. This implies thatthe relevant backgrounds are those with non-resonant γγ production, other processes that involve S → γγ , andreducible backgrounds. We thus consider the followingprocesses: γ +jets, γγ +jets, events with at least one true b -quark at parton-level ( b +jets), Zγγ with Z → b ¯ b , t ¯ tγγ including all the decay modes of the top quarks and theproduction of the resonance S in association with a non-resonant b ¯ b pair. §§ All the multi-jet processes are gener-ated without merging to the parton shower, in the five-flavour scheme, with four outgoing partons at the matrix-element level.The calculation of higher-order QCD corrections tothese multi-leg processes, particularly when restrictingthe phase space with cuts, is numerically challenging atpresent. To remain conservative, we will assume thatthe corrections are large and apply K -factors of K = 2to all the background processes. For the signal and the b ¯ bS associated production we do not apply any K -factorssince the corrections are approximately absorbed into theratio with the single inclusive production of the S reso-nance, see below. Throughout this article we assumethat σ ( pp → S → γγ ) = 10 , , M = 600 , ,
900 GeV, fixingthe product c SG c Sγ (or ( y Sd ) ii c Sγ ), which drops out in theratio ρ . The values of the cross sections are motivated bythe current ATLAS [16] and CMS [17] limits on di-photonresonances.Note that it turns out that the non-resonant b ¯ bS pro-cess is only relevant for gluon-fusion production of S , andwe only report numbers for that in what follows. B. Detector simulation.
In the hadron-level analysis that follows, performedwithout using any dedicated detector simulation soft-ware, we consider all particles within a pseudo-rapidityof | η | < p T >
100 MeV. We smear the momenta ofall reconstructed objects (i.e. jets, electrons, muons andphotons) according to HL-LHC projections [60, 61]. Wealso apply the relevant reconstruction efficiencies. Wesimulate b -jet tagging by looking for jets containing B -hadrons, that we have set to stable in the simulation,and considering them as the b -jet candidates. The mis-tagging of c -jets to b -jets is performed by choosing c -jet candidates (after hadronization) as those jets that liewithin a distance ∆ R < . c -quarks (after the par-ton shower), with transverse momentum p T > ¶¶ We apply a flat b-tagging efficiency of 70% and a mis-tagrate of 1% for light-flavour jets and 10% for charm-quark-initiated jets. §§ We also considered the hγγ process, including the loop-inducedpieces [59], but found that it possess a negligible cross section. ¶¶ This procedure of associating jets to c -quarks is expected to beconservative. We reconstruct jets using the anti- k t algorithm avail-able in the FastJet package [62, 63], with a radius pa-rameter of R = 0 .
4. We only consider jets, photons andleptons with p T >
30 GeV within pseudo-rapidity | η | < . P j → (cid:96) = 0 . × e − . p Tj / GeV and the jet-to-photon mis-identification probability wastaken to be P j → γ = 0 . × e − . p Tj / GeV [60, 61], bothflat in pseudo-rapidity. We demand all leptons and pho-tons to be isolated, where an isolated object is defined tohave (cid:80) i p T,i less than 15% of its transverse momentumin a cone of ∆ R = 0 . IV. DETAILED ANALYSIS.
We consider events with two reconstructed b -jets andtwo isolated photons as defined in Section III. Note thatthis final state has been previously considered in the con-text of searches for Higgs boson pair production, e.g.in [64–68]. We impose the following ‘acceptance’ cutsto all samples: • b -jets: transverse momenta p T,b >
30 GeV, p T,b >
30 GeV, all b -jets within | η | < . • photons: transverse momenta p T,γ >
30 GeV, p T,γ >
30 GeV, all photons within | η | < . • invariant mass of the two b -jets M b ¯ b ∈ [90 , • invariant mass of the two photons M γγ > M −
300 GeV, • veto events with leptons of p T >
25 GeV within | η | < . M .The cross sections after application of the acceptancecuts are given in Table I for two values of the widthsΓ = 1 GeV and Γ = 45 GeV and for M = 750 GeV.For the case of q ¯ q we consider as examples λ HS = 1and λ HS = 0. Throughout this analysis, the total signalcross section was calculated by using the ratio ρ (derivedin Section II) as σ ( pp → hS → hγγ ) = ρ × σ ( pp → S → γγ ), where σ ( pp → S → γγ ) = 10 , , M = 600 , ,
900 GeV, and including the decay h → b ¯ b . This cross section was employed as the normalisationof the signal event samples (before analysis cuts). Theexpected number of signal events, for λ HS ∼ O (1), afteracceptance cuts is O (1) −O (10) at 3000 fb − of integratedluminosity. However, as already discussed, one shouldkeep in mind that the cross section grows with λ HS inboth gg - and q ¯ q -initiated production.The resulting di-photon invariant mass after accep-tance cuts is shown in Fig. 12 for the example of M =750 GeV. The M γγ observable can be used to separatethe analysis into the two regions described in Section II: process acceptance σ [fb] gg → h ( b ¯ b ) S ( γγ ), Γ = 1 GeV, λ HS = 1 0.00054 gg → h ( b ¯ b ) S ( γγ ), Γ = 45 GeV, λ HS = 1 0.00055 b ¯ b → h ( b ¯ b ) S ( γγ ), Γ = 1 GeV, λ HS = 1 0.00266 b ¯ b → h ( b ¯ b ) S ( γγ ), Γ = 45 GeV, λ HS = 1 0.00254 b ¯ b → h ( b ¯ b ) S ( γγ ), Γ = 1 GeV, λ HS = 0 0.00184 b ¯ b → h ( b ¯ b ) S ( γγ ), Γ = 45 GeV, λ HS = 0 0.00172 s ¯ s → h ( b ¯ b ) S ( γγ ), Γ = 1 GeV, λ HS = 1 0.00366 s ¯ s → h ( b ¯ b ) S ( γγ ), Γ = 45 GeV, λ HS = 1 0.00370 s ¯ s → h ( b ¯ b ) S ( γγ ), Γ = 1 GeV, λ HS = 0 0.00291 s ¯ s → h ( b ¯ b ) S ( γγ ), Γ = 45 GeV, λ HS = 0 0.00249at least one b -quark + jets 0.31300 γ + jets 0.11259 γγ + jets 0.15766 Zγγ → ( b ¯ b ) γγ t ¯ tγγ gg → b ¯ bS ( γγ ), Γ = 1 GeV 0.00058 gg → b ¯ bS ( γγ ), Γ = 45 GeV 0.00063TABLE I: The expected cross sections at 13 TeV pp collisionenergy for all the considered processes after acceptance cutsfor M = 750 GeV and Γ = 1 ,
45 GeV. All branching ratios,acceptances and tagging rates have been applied. We haveassumed that the single production cross section for a di-photon scalar resonance of M = 750 GeV is σ ( pp → S → γγ ) = 5 fb.
500 600 700 800 900 1000 M γγ [GeV] -4 -3 -2 -1 ( / σ ) d σ ( pp → h S → b ¯ b γγ ) / d M γγ [ G e V − ] Three-body decay On-shell di-photon gg , Γ=1
GeV gg , Γ=45
GeV b ¯ b , Γ=1
GeV, λ HS =0 . b ¯ b , Γ=1
GeV, λ HS =1 . b ¯ b , Γ=45
GeV, λ HS =0 . b ¯ b , Γ=45
GeV, λ HS =1 . FIG. 12: The distribution of the di-photon invariant mass, M γγ , for the gg → hS → b ¯ bγγ process after acceptance cuts,normalised to unity, for two width scenarios, Γ = 1 GeV andΓ = 45 GeV, while M = 750 GeV. the “three-body decay” region (“TBD”) and the “on-shell di-photon” region (“OS γγ ”). The separation isidentical in both gg - and q ¯ q -initiated processess. Wechoose: M γγ < M −
50 GeV for the “TBD” region and M γγ > M −
50 GeV for the “OS γγ ” region for a di-photonresonance mass, M . We also show the distribution of thecombined invariant mass of the two b -jet candidates andthe di-photon system, M b ¯ bγγ in Fig. 13, which also clearlydemonstrates the existence of the two regions.We apply further cuts to improve signal and back-ground discrimination. As we did not attempt to fully
500 600 700 800 900 1000 1100 1200 1300 1400 1500 M b ¯ bγγ [GeV] -4 -3 -2 -1 ( / σ ) d σ ( pp → h S → b ¯ b γγ ) / d M b ¯ b γγ [ G e V − ] gg , Γ=1
GeV gg , Γ=45
GeV b ¯ b , Γ=1
GeV, λ HS =0 . b ¯ b , Γ=1
GeV, λ HS =1 . b ¯ b , Γ=45
GeV, λ HS =0 . b ¯ b , Γ=45
GeV, λ HS =1 . FIG. 13: The distribution of the combined di-photon + b ¯ b invariant mass, M b ¯ bγγ , for the gg → hS → b ¯ bγγ process afteracceptance cuts, normalised to unity, for two width scenarios,Γ = 1 GeV and Γ = 45 GeV, while M = 750 GeV. optimize the cuts in the present analysis, we apply a com-mon set of cuts along with invariant mass cuts on the ob-servables M γγ and M b ¯ bγγ that provide the main distinc-tion between the two regions. The common cuts appliedin each region are shown in Table II and the specific in-variant mass cuts are shown in Table III. Effectively, thecuts aim to exploit the fact that the photons in the sig-nal are harder than in the backgrounds and also featuretighter di-photon and b ¯ b mass windows, particularly inthe “OS γγ ” region for the former. observable cut p T,γ >
200 GeV p T,γ >
120 GeV∆ R ( γ, γ ) ∈ [2 . , . M b ¯ b ∈ [100 , R ( b, ¯ b ) ∈ [0 . , . R ( γγ, b ) < . gg - and q ¯ q -initiated pro-cessess. The labels “1” and “2” correspond to the hardestand second hardest reconstructed objects (photons or b -jets),respectively. We show the resulting cross sections after the appli-cation of these further cuts in Table IV, for the case of M = 750 GeV. A high efficiency is maintained for the sig-nal, with high rejection factors for the background pro-cesses. We note again that the b ¯ bS associated productionprocess is relevant only for the gluon-fusion scenario.To obtain the 95% confidence-level exclusion regionsfor λ HS we use Poissonian statistics to calculate the prob-abilities. Since we have assumed that the production ofa scalar di-photon resonance will have been observed, wehave to construct a null hypothesis compatible with suchan observation providing the expected number of events “TBD’ “OS γγ ” Γ = 1 GeV M γγ ∈ M − − GeV ∈ M ± M b ¯ bγγ ∈ M ±
30 GeV -Γ = 45 GeV: M γγ ∈ M − − GeV ∈ M ±
40 GeV M b ¯ bγγ ∈ M ±
40 GeV -TABLE III: The additional invariant mass cuts applied alongwith the further cuts of Table II, in the “three-body decay” re-gion (“TBD”) and the “on-shell di-photon” region (“OS γγ ”),for both the gg - and q ¯ q -initiated processess for a scalar di-photon resonance of mass M . The different choices for themass windows were made according to the width of the reso-nance, Γ. process “TBD” “OS γγ ” gg → h ( b ¯ b ) S ( γγ ), Γ = 1 GeV, λ HS = 1 0.00007 0.00020 gg → h ( b ¯ b ) S ( γγ ), Γ = 45 GeV, λ HS = 1 0.00006 0.00014 b ¯ b → h ( b ¯ b ) S ( γγ ), Γ = 1 GeV, λ HS = 1 0.00007 0.00105 b ¯ b → h ( b ¯ b ) S ( γγ ), Γ = 45 GeV, λ HS = 1 0.00038 0.00074 b ¯ b → h ( b ¯ b ) S ( γγ ), Γ = 1 GeV, λ HS = 0 0 0.00064 b ¯ b → h ( b ¯ b ) S ( γγ ), Γ = 45 GeV, λ HS = 0 0.00018 0.00055 s ¯ s → h ( b ¯ b ) S ( γγ ), Γ = 1 GeV, λ HS = 1 0.00005 0.00134 s ¯ s → h ( b ¯ b ) S ( γγ ), Γ = 45 GeV, λ HS = 1 0.00005 0.00107 s ¯ s → h ( b ¯ b ) S ( γγ ), Γ = 1 GeV, λ HS = 0 0 0.00101 s ¯ s → h ( b ¯ b ) S ( γγ ), Γ = 45 GeV, λ HS = 0 0.00003 0.00072at least one b -quark + jets O (10 − ) O (10 − ) γ + jets 0.00025 O (10 − ) γγ + jets 0.00182 0.00003 Zγγ → ( b ¯ b ) γγ × − − t ¯ tγγ × − − gg → b ¯ bS ( γγ ), Γ = 1 GeV (cid:46) O (10 − ) 0.00020 gg → b ¯ bS ( γγ ), Γ = 45 GeV O (10 − ) 0.00018TABLE IV: The expected cross sections at 13 TeV pp collisionenergy in fb for all the considered processes after the applica-tion of further cuts as in Tables II and III, for M = 750 GeVand Γ = 1 ,
45 GeV. All branching ratios, acceptances andtagging rates have been applied. We have assumed that thesingle production cross section for a di-photon scalar reso-nance of M = 750 GeV is σ ( pp → S → γγ ) = 5 fb. at the LHC, that we will confront with the theory predic-tions in the parameter space to be tested. If these num-bers differ by a certain significance, the correspondingpoint is expected to be excluded with this significance.In particular, any hypothesis has to be realistic and re-main within the bounds of our model. Our underlyingassumption is thus chosen to be that the scalar resonance S is produced purely in gluon fusion to good approxi-mation and that there is no portal coupling, λ HS = 0,which means there is basically no h + S associated pro-duction. For further technical details on this statisticalprocedure, see Appendix C of Ref. [69]. We do not in-corporate the effect of systematic uncertainties on thesignal or backgrounds. To perform a combination of thetwo 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: Ω = 1 √ k k (cid:88) i =1 Ω i . (8)We show the resulting expected limits (assuming ournull hypothesis is true) as a function of the integratedluminosity for the different benchmark scenarios that weconsider in Figs. 14-25. For the case M = 750 GeV weshow results for Γ = 45 GeV as well. For Γ = 1 GeV,we obtain more stringent constraints, limiting, for M =600 , ,
900 GeV respectively, | λ HS | (cid:46) , , gg -initiated process and λ HS ∈ [ ∼ − , ∼ , [ ∼ − , ∼ , [ ∼ − , ∼ b ¯ b and s ¯ s -initiated processes,at the end of the HL-LHC run (3000 fb − ). The variationbetween the results for b ¯ b and s ¯ s -initiated processes –visible in the plots – is very small and can be attributedto the differences between the parton density functionsfor 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 | (cid:46)
5, the b ¯ b -initiatedprocess λ HS ∈ [ ∼ − , ∼
5] and the s ¯ s -initiated process λ HS ∈ [ ∼ − , ∼ S are discovered, the remainingunconstrained regions in the q ¯ q cases can be covered (inparticular for narrow width), allowing determination ofthe initial state partons that produce the resonance.The lower bound on λ HS for the q ¯ q -initiated cases isdriven by the “TBD” region. This is understood by thefact that the “TBD” region always vanishes near λ HS ∼
0, as it is dominated by the diagram with an on-shell s -channel S , making the exclusion region resulting from itsymmetric with respect to λ HS ∼
0, whereas the “OS γγ ”region possesses a symmetry point somewhere in λ HS < Mixed production of the di-photon resonance
So far we have investigated production of the scalar res-onance initiated purely either via gluon fusion or q ¯ q an-nihilation. We can generalize this to “mixed” productionvia gg and q ¯ q initial states simultaneously. We concen-trate on the scenario of gg and b ¯ b for simplicity, with theextension to additional quark flavours being straightfor-ward. In that case, the ratio of cross sections, ρ mixed , de-fined between the associated production and single pro-duction modes can be written as: ρ mixed = σ ( b ¯ b → hS → hγγ ) + σ ( gg → hS → hγγ ) σ ( b ¯ b → S → γγ ) + σ ( gg → S → γγ )= B ( λ HS )[( y Sd ) ] + G ( λ HS )( c SG ) B [( y Sd ) ] + G ( c SG ) , (9)where B ( λ HS ), G ( λ HS ) are functions of the portalcoupling λ HS and B , G are constants with respect0 FIG. 14: The grey-shaded area shows the 95% confidence-level exclusion region for the portal coupling λ HS for the gg -induced case and M = 600 GeV, Γ = 1 GeV, for the com-bination of the “TBD” (red) and “OS γγ ” (green) regions asdefined in the main text. We have assumed that the singleproduction cross section σ ( pp → S → γγ ) = 10 fb.FIG. 15: The grey-shaded area shows the 95% confidence-level exclusion region for the portal coupling λ HS for the b ¯ b -induced case and M = 600 GeV, Γ = 1 GeV, for the com-bination of the “TBD” (red) and “OS γγ ” (green) regions asdefined in the main text. We have assumed that the singleproduction cross section σ ( pp → S → γγ ) = 10 fb. to the portal coupling, all to be determined. Defining θ ≡ | [( y Sd ) ] /c SG | , the above expression can be re-writtenas: ρ mixed = B ( λ HS ) θ + G ( λ HS ) B θ + G . (10)By considering the limits θ → θ → ∞ , and dividingthe numerator and denominator by G , we can deducethat ρ mixed = ρ ( b ¯ b, λ HS ) r bg θ + ρ ( gg, λ HS ) r bg θ + 1 , (11) FIG. 16: The grey-shaded area shows the 95% confidence-level exclusion region for the portal coupling λ HS for the s ¯ s -induced case and M = 600 GeV, Γ = 1 GeV, for the com-bination of the “TBD” (red) and “OS γγ ” (green) regions asdefined in the main text. We have assumed that the singleproduction cross section σ ( pp → S → γγ ) = 10 fb.FIG. 17: The grey-shaded area shows the 95% confidence-level exclusion region for the portal coupling λ HS for the gg -induced case and M = 750 GeV, Γ = 1 GeV, for the com-bination of the “TBD” (red) and “OS γγ ” (green) regions asdefined in the main text. We have assumed that the singleproduction cross section σ ( pp → S → γγ ) = 5 fb. where ρ ( b ¯ b, λ HS ) and ρ ( gg, λ HS ) are the ratios of crosssections as functions of the portal coupling as definedin Eq. 7, for the cases of pure production initiated ei-ther via b ¯ b or gg , respectively, and r bg is the ratio ofpure single production cross sections for θ = 1: r bg = B /G . The former two functions have already been de-termined in the analysis of the pure cases. The con-stant r bg was calculated explicitly via Monte Carlo to be r bg ≈ . , . , .
49 for M = 600 , ,
900 GeV re-spectively. The values are approximately equal for boththe narrow width case (Γ = 1 GeV) and the larger widthcase (Γ = 45 GeV).1
FIG. 18: The grey-shaded area shows the 95% confidence-level exclusion region for the portal coupling λ HS for the gg -induced case and M = 750 GeV, Γ = 45 GeV, for the com-bination of the “TBD” (red) and “OS γγ ” (green) regions asdefined in the main text. We have assumed that the singleproduction cross section σ ( pp → S → γγ ) = 5 fb.FIG. 19: The grey-shaded area shows the 95% confidence-level exclusion region for the portal coupling λ HS for the b ¯ b -induced case and M = 750 GeV, Γ = 1 GeV, for the com-bination of the “TBD” (red) and “OS γγ ” (green) regions asdefined in the main text. We have assumed that the singleproduction cross section σ ( pp → S → γγ ) = 5 fb. Using Eq. 11, we can deduce an expression for the pre-dicted number of signal events after the application ofanalysis cuts in the mixed production case: N mixed = r bg θ r bg θ + 1 N ( b ¯ b, λ HS ) + 1 r bg θ + 1 N ( gg, λ HS ) , (12)where N ( b ¯ b, λ HS ) and N ( gg, λ HS ) are the expected sig-nal events for either pure b ¯ b or pure gg production for agiven value of λ HS , at a specific integrated luminosity.The predicted number of background events after agiven set of analysis cuts is constant with θ , apart fromthe associated production of the di-photon resonance FIG. 20: The grey-shaded area shows the 95% confidence-level exclusion region for the portal coupling λ HS for the b ¯ b -induced case and M = 750 GeV, Γ = 45 GeV, for the com-bination of the “TBD” (red) and “OS γγ ” (green) regions asdefined in the main text. We have assumed that the singleproduction cross section σ ( pp → S → γγ ) = 5 fb.FIG. 21: The grey-shaded area shows the 95% confidence-level exclusion region for the portal coupling λ HS for the s ¯ s -induced case and M = 750 GeV, Γ = 1 GeV, for the com-bination of the “TBD” (red) and “OS γγ ” (green) regions asdefined in the main text. We have assumed that the singleproduction cross section σ ( pp → S → γγ ) = 5 fb. with a b ¯ b pair, which was found to be significant onlyin the gg -initiated scenario. This background scales as N assoc . = N assoc., / ( r bg θ + 1), where N assoc., is the ex-pected number of events for the b ¯ bS associated produc-tion after analysis cuts in the case of pure gg -initiatedproduction.Using the expression of Eq. 12 and the event num-bers for the two analysis regions “TBD” and “OS γγ ”obtained for the pure production modes, we can deriveconstraints on the ( θ, λ HS )-plane. These are shown inFigs. 26-28 and 29, for masses M = 600 , ,
900 GeVand Γ = 1 GeV and for M = 750 GeV and Γ = 45 GeV,2 FIG. 22: The grey-shaded area shows the 95% confidence-level exclusion region for the portal coupling λ HS for the s ¯ s -induced case and M = 750 GeV, Γ = 45 GeV, for the com-bination of the “TBD” (red) and “OS γγ ” (green) regions asdefined in the main text. We have assumed that the singleproduction cross section σ ( pp → S → γγ ) = 5 fb.FIG. 23: The grey-shaded area shows the 95% confidence-level exclusion region for the portal coupling λ HS for the gg -induced case and M = 900 GeV, Γ = 1 GeV, for the com-bination of the “TBD” (red) and “OS γγ ” (green) regions asdefined in the main text. We have assumed that the singleproduction cross section σ ( pp → S → γγ ) = 1 fb. respectively, at an integrated luminosity of 3000 fb − at13 TeV. It can be seen that in the limits θ → θ (cid:29) gg - or b ¯ b -dominated production respec-tively, one can recover the pure production constraints ofFigs. 14-25 obtained at 3000 fb − . V. DISCUSSION AND CONCLUSIONS.
We have investigated the associated production of a di-photon scalar resonance with a Higgs boson and have em-ployed the pp → hS → ( b ¯ b )( γγ ) final state to obtain con- FIG. 24: The grey-shaded area shows the 95% confidence-level exclusion region for the portal coupling λ HS for the b ¯ b -induced case and M = 900 GeV, Γ = 1 GeV, for the com-bination of the “TBD” (red) and “OS γγ ” (green) regions asdefined in the main text. We have assumed that the singleproduction cross section σ ( pp → S → γγ ) = 1 fb.FIG. 25: The grey-shaded area shows the 95% confidence-level exclusion region for the portal coupling λ HS for the s ¯ s -induced case and M = 900 GeV, Γ = 1 GeV, for the com-bination of the “TBD” (red) and “OS γγ ” (green) regions asdefined in the main text. We have assumed that the singleproduction 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 benchmarkscalar masses, M = 600 , ,
900 GeV, and we have as-sumed that the inclusive single production cross sectionis σ ( pp → S → γγ ) = 10 , , i.e. the supposed ‘true’ underlying theory) to correspond togluon-fusion-initiated production with vanishing portalcoupling, λ HS = 0. We then first analysed the case ofeither pure gluon-fusion-induced production or produc-tion via quark-anti-quark annihilation. For gluon-fusion3 FIG. 26: The 95% confidence-level exclusion region for theratio of couplings to b ¯ b over the coupling to the gg initialstates, θ = [( y Sd ) ] /c SG , versus the portal coupling λ HS for M = 600 GeV, Γ = 1 GeV. The excluded region coming fromthe combination of the “TBD” (red) and “OS γγ ” (green) re-gions is grey-shaded. We have assumed that the single pro-duction cross section σ ( pp → S → γγ ) = 10 fb.FIG. 27: The 95% confidence-level exclusion region for theratio of couplings to b ¯ b over the coupling to the gg initialstates, θ = [( y Sd ) ] /c SG , versus the portal coupling λ HS for M = 750 GeV, Γ = 1 GeV. The excluded region coming fromthe combination of the “TBD” (red) and “OS γγ ” (green) re-gions is grey-shaded. We have assumed that the single pro-duction cross section σ ( pp → S → γγ ) = 5 fb. production one can impose constraints on the portal cou-pling at the end of the HL-LHC run of | λ HS | (cid:46) , , M = 600 , ,
900 GeV, respectively, and Γ = 1 GeV,while | λ HS | (cid:46) M = 750 GeV. For quark-anti-quark annihilation,the production of an on-shell scalar and the Higgs bosonis enhanced by a contact interaction ∼ q ¯ qhS , originatingfrom the same D = 5 operator that mediates single S production. This implies that the cross section is non-negligible even for vanishing portal coupling λ HS = 0. FIG. 28: The 95% confidence-level exclusion region for theratio of couplings to b ¯ b over the coupling to the gg initialstates, θ = [( y Sd ) ] /c SG , versus the portal coupling λ HS for M = 900 GeV, Γ = 1 GeV. The excluded region coming fromthe combination of the “TBD” (red) and “OS γγ ” (green) re-gions is grey-shaded. We have assumed that the single pro-duction cross section σ ( pp → S → γγ ) = 1 fb.FIG. 29: The 95% confidence-level exclusion region for theratio of couplings to b ¯ b over the coupling to the gg initialstates, θ = [( y Sd ) ] /c SG , versus the portal coupling λ HS for M = 750 GeV, Γ = 45 GeV. The excluded region comingfrom the combination of the “TBD” (red) and “OS γγ ” (green)regions is grey-shaded. We have assumed that the single pro-duction cross section σ ( pp → S → γγ ) = 5 fb. This fact can be exploited to exclude the whole planeof λ HS , thus excluding the hypothesised production via q ¯ q annihilation. For the case of b ¯ b we find that a re-gion inside λ HS ∈ [ ∼ − , ∼ , [ ∼ − , ∼ , [ ∼ − , ∼ M = 600 , ,
900 GeV, respectively, could remainunconstrained at the end of the HL-LHC in the narrowwidth scenario, while λ HS ∈ [ ∼ − , ∼
5] for large widthand M = 750 GeV. For the case of s ¯ s annihilation wefind the same unconstrained regions to good approxima-tion for narrow width, while we obtain λ HS ∈ [ ∼ − , ∼ M = 750 GeV. We havealso considered ‘mixed’ production via gluon fusion and b ¯ b annihilation and derived constraints on the plane ofratio of the corresponding couplings versus λ HS for anintegrated luminosity of 3000 fb − .Note that since a measurement of λ HS can straight-forwardly be translated into a measurement of µ S , thesenumbers suggest that it is possible to exclude, for ex-ample, the scenario where the mass of S stems fromEWSB ( µ S = 0), which corresponds to λ HS = M /v ≈ . , . , . S to van-ish). ∗∗∗ Furthermore, if additional decay modes of theresonance S are discovered beyond γγ , it is conceivablethat a combined analysis in various channels would beable to exclude all possible values of λ HS - for the case of q ¯ q production - thus providing an independent method ofdetermining the production mode. Conversely, the anal-ysis of the present article demonstrates that if the pro-duction mechanism is constrained via alternative means,it will be possible to obtain meaningful constraints on theinteraction of a new scalar resonance and the Higgs bo-son, allowing determination of its relation to electroweaksymmetry breaking. Acknowledgments
We would like to thank Peter Richardson, Jernej Ka-menik, Jos´e Zurita, Tania Robens and Eleni Vryonidoufor useful comments and discussion. AC acknowledgessupport by a Marie Sk(cid:32)lodowska-Curie Individual Fellow-ship of the European Community’s Horizon 2020 Frame-work Programme for Research and Innovation under con-tract number 659239 (NP4theLHC14). The research ofF.G. is supported by a Marie Curie Intra European Fel-lowship within the 7th European Community FrameworkProgramme (grant no. PIEF-GA-2013-628224). AP ac-knowledges support by the MCnetITN FP7 Marie CurieInitial Training Network PITN-GA-2012-315877 and aMarie Curie Intra European Fellowship within the 7thEuropean Community Framework Programme (grant no.PIEF-GA-2013-622071).
Appendix A: The full set of operators.
In this appendix, we provide the full set of operators forthe SM in the presence of an additional dynamical scalarsinglet S , up to D = 5. The fact that S does not trans-form under the SM gauge group makes the constructionstraightforward: in the case that it is CP even (with CP-conserving interactions), each additional operator will ∗∗∗ This is somewhat similar to the case of testing the presence/sizeof the µ term in the SM Higgs potential in Higgs pair production[71]. consist of a gauge invariant SM operator, multiplied bypowers of S (and potentially derivatives). ††† This meansthat in particular that, schematically,
L ⊃ L SM + L SM · S .If all operators in the SM would be marginal, i.e. , fea-ture D = 4, this would already correspond to the fullLagrangian at D ≤
5, connecting the SM with the newresonance. In turn, the only operators that are missing(up to pure NP terms) are those containing the singlerelevant ( i.e. , D <
4) operator of the SM, | H | . This canbe multiplied by up to three powers of S as well as by ∂ S (the latter being equivalent via integration by partsto S ∂ | H | ). This finally leads to the Lagrangian ‡‡‡ L eff = L SM + 12 ∂ µ S ∂ µ S − µ S S − λ (cid:48) S √ v S − λ S S − λ (cid:48) HS v | H | S − λ HS | H | S − S Λ (cid:2) c λS S + c HS | H | S + c λH | H | (cid:3) − ( y Sd ) ij S Λ ¯ Q iL Hd jR − ( y Su ) ij S Λ ¯ Q iL ˜ Hu jR + h . c . − S Λ 116 π (cid:2) g (cid:48) c SB B µν B µν + g c SW W Iµν W Iµν + g S c SG G aµν G aµν (cid:3) . (A1)Note that we have used equations of mo-tion/field redefinitions to eliminate the operators S / Λ( ∂ µ S ) , S / Λ | D µ H | , S / Λ ∂ | H | and those of type S / Λ ¯ qD µ γ µ q . The couplings in the third line of (A1) and λ (cid:48) S,HS , which are those that have been neglected in (1),do not contribute to the process under consideration togood approximation: They either do not enter at leadingorder, or (in the case of the | H | , S interactions) at mostcontribute to a diagram with a (strongly suppressed)off-shell Higgs boson propagator, discussed in AppendixB. The same holds true for the S term, that was keptfor the discussion in Section I. Appendix B: Fits for the ratio ρ including a linearterm. In addition to the diagrams of Figs. 1 and 2, therecan be contributions to the final state under considera-tion originating from hhS interactions (for example, anintermediate s -channel Higgs boson to hS or an S → hh intermediate state), provided the following triple scalar ††† We neglect the only D = 5 operator consisting only of SM buildingblocks – the Weinberg operator – since the tiny neutrino massessuggest that it is suppressed by a very large scale Λ ∼ M GUT . ‡‡‡ See also [40, 72]. We do not include leptons, as they do not enterthe examined processes to the order considered. Moreover, notethat (induced) tadpoles in S can always be removed via a fieldredefinition. S , is non-vanishing: − κ HS vh S ⊂ − λ (cid:48) HS v | H | S − c λH S Λ | H | , (B1)with κ HS = 1 / λ (cid:48) HS + 3 / (2Λ) c λH .In this scenario, terms will appear in the cross sectionratio proportional to κ HS and κ HS . Due to interferencewith the new hhS diagrams (note that also the ‘non-signal’ S → hh interferes with the signal diagrams afteran off-shell h → γγ ), some dependence on the coupling ofthe resonance S to photons, c Sγ , as well as the productioncouplings, ( y Sd ) , ( y Sd ) and c SG , is introduced in the de-nominator of the cross section ratio. Due to this, smallervalues of | c Sγ | and the production couplings enhance theeffect of the new contributions proportional to κ HS and κ HS . To take these effects into account and at the sametime remain conservative, we set | c Sγ | to the possible min-imal value that produces a σ ( pp → S → γγ ) ∼ M = 750 GeV, as derived in Ref. [24]. This leads to | c Sγ | (cid:38) { , , } for s ¯ s , b ¯ b and gg production, respec-tively, where here and in the following we assume a nor-malisation of Λ = 1 TeV. Moreover, one can derive con-servative lower bounds on | ( y Sd ) | , | ( y Sd ) | and | c Sg | , bydemanding σ ( pp → S ) (cid:38) | ( y Sd ) | (cid:38) . | ( y Sd ) | (cid:38) . | c Sg | (cid:38) .
25. We set these to theirminimal values as well in what follows.We can then parametrize the ratio ρ ( xx (cid:48) ) = σ ( xx (cid:48) → hγγ ) /σ ( xx (cid:48) → S → γγ ) , xx (cid:48) = b ¯ b, s ¯ s, gg , by §§§ ρ ( xx (cid:48) ) = δ x + δ x κ HS λ HS + δ x κ HS + δ x λ HS + δ x κ HS + δ x λ HS , (B2)where δ b,s,gi are coefficients to be determined. i δ bi δ si δ gi . × − . × − < O (10 − )2 1 . × − − . × − O (10 − )3 8 . × − . × − < O (10 − )4 3 . × − . × − < O (10 − )5 2 . × − . × − < O (10 − )6 8 . × − . × − . × − TABLE V: The ρ fit coefficients for the general case of in-cluding an interaction term proportional to L ⊃ − κ HS vh S for width Γ = 1 GeV and M = 750 GeV. An estimate of the coefficients in this conservative sce-nario, obtained numerically, is given in Table V for widthΓ = 1 GeV and M = 750 GeV. The contributions fromdiagrams with an off-shell Higgs boson are a priori notfully negligible for the b ¯ b -induced case, simply becauseof interference of the s -channel h → hS with the largematrix elements with the on-shell S in the final state. §§§ For the case of vanishing κ HS , this more general definition coincideswith Eq. 7 to good approximation. To investigate the size of κ HS that renders the off-shell 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 characterisesthe fractional change in the ratio ρ due to κ HS for agiven 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 ) } for s ¯ s , b ¯ b and gg production, respectively.Since the gauge-invariant terms in the Lagrangian gen-erating this coupling also induce a Higgs-scalar mixing,they can not be arbitrarily large. Indeed, Higgs data canconstrain | κ HS | (cid:46) ¶¶¶ Therefore, given the values calculated in the previousparagraph, for κ HS to eventually have a non-negligibleeffect on the analysis of the present article, this boundneeds to be violated, and setting κ HS = 0 is a justi-fied approximation. Despite the fact that we focussed on M = 750 GeV, the analysis is expected to give similarestimates for the other benchmark points considered inthe main analysis of this article, M = 600 ,
900 GeV. ¶¶¶
Here, we have neglected the impact of the second operator in (B1), S| H | , which breaks the correlation between Higgs-scalar mixingand the h S interaction. [1] A. Falkowski, C. Gross, and O. Lebedev, JHEP , 057(2015), arXiv:1502.01361 [hep-ph] .[2] B. Bellazzini, C. Cski, and J. Serra, Eur. Phys. J. C74 ,2766 (2014), arXiv:1401.2457 [hep-ph] .[3] B. Bellazzini, R. Franceschini, F. Sala, and J. Serra,JHEP , 072 (2016), arXiv:1512.05330 [hep-ph] .[4] U. Ellwanger, C. Hugonie, and A. M. Teixeira, Phys.Rept. , 1 (2010), arXiv:0910.1785 [hep-ph] .[5] R. Hempfling, Phys. Lett. B379 , 153 (1996), arXiv:hep-ph/9604278 [hep-ph] .[6] C. Englert, J. Jaeckel, V. V. Khoze, and M. Spannowsky,JHEP , 060 (2013), arXiv:1301.4224 [hep-ph] .[7] T. Gherghetta, N. Nagata, and M. Shifman, Phys. Rev. D93 , 115010 (2016), arXiv:1604.01127 [hep-ph] .[8] K. Tsumura and L. Velasco-Sevilla, Phys. Rev.
D81 ,036012 (2010), arXiv:0911.2149 [hep-ph] .[9] M. Bauer, T. Schell, and T. Plehn, Phys. Rev.
D94 ,056003 (2016), arXiv:1603.06950 [hep-ph] .[10] V. Khachatryan et al. (CMS), Eur. Phys. J.
C74 , 3076(2014), arXiv:1407.0558 [hep-ex] .[11] G. Aad et al. (ATLAS), Phys. Rev.
D90 , 112015 (2014),arXiv:1408.7084 [hep-ex] .[12]
Search for resonances decaying to photon pairs in 3.2 fb − of pp collisions at √ s = 13 TeV with the ATLAS detector ,Tech. Rep. ATLAS-CONF-2015-081 (CERN, 2015).[13] Search for resonances in diphoton events with the ATLASdetector at √ s = 13 TeV , Tech. Rep. ATLAS-CONF-2016-018 (CERN, 2016).[14] Search for new physics in high mass diphoton events inproton-proton collisions at 13TeV , Tech. Rep. CMS-PAS-EXO-15-004 (CERN, 2015).[15]
Search for new physics in high mass diphoton events in . − of proton-proton collisions at √ s = 13 TeV andcombined interpretation of searches at and
13 TeV,Tech. Rep. CMS-PAS-EXO-16-018 (CERN, 2016).[16]
Search for scalar diphoton resonances with 15.4 fb − ofdata collected at √ s =13 TeV in 2015 and 2016 withthe ATLAS detector , Tech. Rep. ATLAS-CONF-2016-059(CERN, 2016).[17] V. Khachatryan et al. (CMS), (2016), arXiv:1609.02507[hep-ex] .[18] R. Franceschini, G. F. Giudice, J. F. Kamenik, M. Mc-Cullough, A. Pomarol, R. Rattazzi, M. Redi, F. Riva,A. Strumia, and R. Torre, JHEP , 144 (2016),arXiv:1512.04933 [hep-ph] .[19] A. Angelescu, A. Djouadi, and G. Moreau, Phys. Lett. B756 , 126 (2016), arXiv:1512.04921 [hep-ph] .[20] S. Knapen, T. Melia, M. Papucci, and K. Zurek, Phys.Rev.
D93 , 075020 (2016), arXiv:1512.04928 [hep-ph] .[21] S. Di Chiara, L. Marzola, and M. Raidal, (2015),arXiv:1512.04939 [hep-ph] .[22] S. D. McDermott, P. Meade, and H. Ramani, Phys. Lett.
B755 , 353 (2016), arXiv:1512.05326 [hep-ph] .[23] J. Ellis, S. A. R. Ellis, J. Quevillon, V. Sanz, and T. You,JHEP , 176 (2016), arXiv:1512.05327 [hep-ph] .[24] R. S. Gupta, S. Jger, Y. Kats, G. Perez, and E. Stamou,(2015), arXiv:1512.05332 [hep-ph] .[25] A. Falkowski, O. Slone, and T. Volansky, JHEP , 152(2016), arXiv:1512.05777 [hep-ph] .[26] A. Alves, A. G. Dias, and K. Sinha, Phys. Lett. B757 ,39 (2016), arXiv:1512.06091 [hep-ph] . [27] F. Goertz, J. F. Kamenik, A. Katz, and M. Nardecchia,(2015), arXiv:1512.08500 [hep-ph] .[28] M. Son and A. Urbano, (2015), arXiv:1512.08307 [hep-ph] .[29] J. Gao, H. Zhang, and H. X. Zhu, (2015),arXiv:1512.08478 [hep-ph] .[30] A. Salvio and A. Mazumdar, Phys. Lett.
B755 , 469(2016), arXiv:1512.08184 [hep-ph] .[31] J. Gu and Z. Liu, Phys. Rev.
D93 , 075006 (2016),arXiv:1512.07624 [hep-ph] .[32] A. Djouadi, J. Ellis, R. Godbole, and J. Quevillon, JHEP , 205 (2016), arXiv:1601.03696 [hep-ph] .[33] A. Salvio, F. Staub, A. Strumia, and A. Urbano, JHEP , 214 (2016), arXiv:1602.01460 [hep-ph] .[34] C. Gross, O. Lebedev, and J. M. No, (2016),arXiv:1602.03877 [hep-ph] .[35] F. Goertz, A. Katz, M. Son, and A. Urbano, (2016),arXiv:1602.04801 [hep-ph] .[36] J. Bernon, A. Goudelis, S. Kraml, K. Mawatari, andD. Sengupta, (2016), arXiv:1603.03421 [hep-ph] .[37] G. Panico, L. Vecchi, and A. Wulzer, (2016),arXiv:1603.04248 [hep-ph] .[38] J. F. Kamenik, B. R. Safdi, Y. Soreq, and J. Zupan,(2016), arXiv:1603.06566 [hep-ph] .[39] M. Chala, C. Grojean, M. Riembau, and T. Vantalon,(2016), arXiv:1604.02029 [hep-ph] .[40] R. Franceschini, G. F. Giudice, J. F. Kamenik, M. Mc-Cullough, F. Riva, A. Strumia, and R. Torre, (2016),arXiv:1604.06446 [hep-ph] .[41] D. Buttazzo, A. Greljo, and D. Marzocca, Eur. Phys. J. C76 , 116 (2016), arXiv:1512.04929 [hep-ph] .[42] L. Berthier, J. M. Cline, W. Shepherd, and M. Trott,JHEP , 084 (2016), arXiv:1512.06799 [hep-ph] .[43] P. S. B. Dev, R. N. Mohapatra, and Y. Zhang, JHEP , 186 (2016), arXiv:1512.08507 [hep-ph] .[44] P. Roig and J. J. Sanz-Cillero, (2016), arXiv:1605.03831[hep-ph] .[45] S. Dawson and I. M. Lewis, (2016), arXiv:1605.04944[hep-ph] .[46] J. Zhang and S. Zhou, Chin. Phys. C40 , 081001 (2016),arXiv:1512.07889 [hep-ph] .[47] F. Goertz, A. Papaefstathiou, L. L. Yang, and J. Zurita,JHEP , 016 (2013), arXiv:1301.3492 [hep-ph] .[48] P. Motylinski, L. Harland-Lang, A. D. Martin, andR. S. Thorne, in
International Conference on High En-ergy Physics 2014 (ICHEP 2014) Valencia, Spain, July2-9, 2014 (2014) arXiv:1411.2560 [hep-ph] .[49] N. D. Christensen and C. Duhr, Comput.Phys.Commun. , 1614 (2009), arXiv:0806.4194 [hep-ph] .[50] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr,and B. Fuks, Comput.Phys.Commun. , 2250 (2014),arXiv:1310.1921 [hep-ph] .[51] C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mat-telaer, and T. Reiter, Comput. Phys. Commun. ,1201 (2012), arXiv:1108.2040 [hep-ph] .[52] J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, andT. Stelzer, JHEP , 128 (2011), arXiv:1106.0522[hep-ph] .[53] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, et al. , JHEP , 079 (2014), arXiv:1405.0301 [hep-ph]. [54] M. Bahr, S. Gieseke, M. Gigg, D. Grellscheid, K. Hamil-ton, et al. , Eur.Phys.J. C58 , 639 (2008), arXiv:0803.0883[hep-ph] .[55] S. Gieseke, D. Grellscheid, K. Hamilton, A. Papaefs-tathiou, S. Platzer, et al. , (2011), arXiv:1102.1672 [hep-ph] .[56] K. Arnold, L. d’Errico, S. Gieseke, D. Grellscheid,K. Hamilton, et al. , (2012), arXiv:1205.4902 [hep-ph].[57] J. Bellm, S. Gieseke, D. Grellscheid, A. Papaefstathiou,S. Platzer, et al. , (2013), arXiv:1310.6877 [hep-ph] .[58] J. Bellm et al. , Eur. Phys. J.
C76 , 196 (2016),arXiv:1512.01178 [hep-ph] .[59] V. Hirschi and O. Mattelaer, (2015), arXiv:1507.00020[hep-ph] .[60]
Performance assumptions for an upgraded ATLAS detec-tor at a High-Luminosity LHC , Tech. Rep. (2013).[61]
Performance assumptions based on full simulation for anupgraded ATLAS detector at a High-Luminosity LHC ,Tech. Rep. (2013).[62] M. Cacciari, G. P. Salam, and G. Soyez, Eur.Phys.J.
C72 , 1896 (2012), arXiv:1111.6097 [hep-ph] .[63] M. Cacciari and G. P. Salam, Phys.Lett.
B641 , 57(2006), arXiv:hep-ph/0512210 [hep-ph] .[64] U. Baur, T. Plehn, and D. L. Rainwater, Phys.Rev.
D69 ,053004 (2004), arXiv:hep-ph/0310056 [hep-ph] .[65] J. Baglio, A. Djouadi, R. Grber, M. M. Mhlleitner, J. Quevillon, and M. Spira, JHEP , 151 (2013),arXiv:1212.5581 [hep-ph] .[66] W. Yao, in Proceedings, Community Summer Study2013: Snowmass on the Mississippi (CSS2013): Min-neapolis, MN, USA, July 29-August 6, 2013 (2013)arXiv:1308.6302 [hep-ph] .[67] V. Barger, L. L. Everett, C. B. Jackson, and G. Shaugh-nessy, Phys. Lett.
B728 , 433 (2014), arXiv:1311.2931[hep-ph] .[68] N. Chen, C. Du, Y. Fang, and L.-C. L, Phys. Rev.
D89 ,115006 (2014), arXiv:1312.7212 [hep-ph] .[69] A. Papaefstathiou, K. Sakurai, and M. Takeuchi, JHEP , 176 (2014), arXiv:1404.1077 [hep-ph] .[70] S. Stouffer, E. Suchman, L. DeVinnery, S. Star, andR. W. J. Williams, Princeton University Press (1949).[71] F. Goertz, (2015), arXiv:1504.00355 [hep-ph] .[72] B. Gripaios and D. Sutherland, (2016), arXiv:1604.07365[hep-ph] .[73] D. Carmi, A. Falkowski, E. Kuflik, T. Volansky, andJ. Zupan, JHEP , 196 (2012), arXiv:1207.1718 [hep-ph] .[74] K. Cheung, P. Ko, J. S. Lee, and P.-Y. Tseng, JHEP10