Abstract
We investigate the potential of the channel monoHiggs + missing transverse energy (MET) in yielding signals of dark matter at the highluminosity Large Hadron Collider (LHC). As illustration, a Higgsportal scenario has been chosen, where an extension of the Standard Model with a real scalar gaugesinglet which serves as a dark matter candidate. The phenomenological viability of this scenario has been ensured by postulating the existence of dimension6 operators that enable cancellation in certain amplitudes for elastic scattering of dark matter in direct search experiments. These operators are found to have nonnegligible contribution to the monoHiggs signal. Thereafter, we carry out a detailed analysis of this signal, with the accompanying MET providing a useful handle in suppressing backgrounds. Signals for the Higgs decaying into both the diphoton and \(b{\bar{b}}\) channels have been studied. A cutbased simulation is presented first, optimizing over various event selection criteria. This is followed by a demonstration of how the statistical significance can be improved through analyses based on boosted decision trees and artificial neural networks.
1 Introduction
The Standard Model(SM) of particle physics has proven to be an extremely successful theory so far. Experimental studies have confirmed most of its predictions to impressive levels of accuracy [1,2,3,4,5]. It still remains an intense quest to look for physics beyond the standard model. Perhaps the most concrete and persistent reason for this is the existence of dark matter (DM) which constitutes to up to 23% of the energy density of the universe, and the belief that DM owes its origin to some hitherto unseen elementary particle(s). In such a situation, one would like to know if the DM particle interacts with those in the SM, and if so, what the signatures of such interactions will be. The literature is replete with ideas as to the nature of DM, a frequently studied possibility being one or more weakly interacting massive particle(s) (WIMP), with the DM particle(s) interacting with those in SM particles coupling strength of the order of the weak interaction strength.
The collider signal of a WIMP DM is commonly expected to consist in MET. In addition, signals of the monoX type(where X = jet, \(\gamma \) Z, h etc) are advocated as generic probes of WIMP dark matter [6,7,8,9,10,11,12]. It may be asked whether one can similarly have monoHiggs DM signals [13, 14], accompanied by hard MET caused by DM pairs (assuming that a \(Z_2\) symmetry makes the DM stable). Such analyses in the context of various supersymmetric [15, 16] and nonsupersymmetric [17,18,19,20] models have been performed in the past. However, the existing studies in this context leave enough scope for refinement, including (a) thorough analyses of the proposed signals as well as their SM backgrounds at the Large Hadron Collider (LHC), and (b) the viability of Higgs + DMpair production, consistently with already available direct search constraints. Such constraints already disfavour socalled ‘Higgsportal’ scenarios in their simplest versions [21,22,23,24,25]. However, there exist theoretical proposals [26,27,28] involving new physics, where the Higgsmediated contribution to spinindependent crosssection in direct search experiments undergo cancellations from additional contributing agents. Keeping this in mind, as also the fact that LHC is not far from its highluminosity phase, it is desirable to sharpen search strategies for monoHiggs + MET signals anyway, especially because it relates to the appealing idea that the Higgs sector is the gateway to new physics. However, such signals are understandably backgroundprone, and refinement of the predictions in a realistic LHC environment is a necessity. Furthermore, it needs to be ascertained how the additional terms in the lowenergy theory cancelling the Higgs contributions in direct search experiments affect searches at the LHC. We address both issues in the current study.
As for the additional terms cancelling the contributions of the 125GeV scalar to spinindependent inelastic scattering, scenarios with an extended Higgs sector have been studied earlier [27]. There is also a rich literature on Higgsportal models with fermionic dark matter, where different contributions to the direct detection cross sections lead to cancellations (socalled blindspots) [29,30,31,32]. Here, however, we take a modelindependent approach, and postulate the new physics effects to come from dimension6 and8 operators which are suppressed by the scale of new physics. These lead to the rather interesting possibility of partial cancellation between the coefficients of dimension4 and higher dimension operators. Thus at the same time, one obeys direct detection constraints, has notsosmall coupling between the Higgs and the DM, and matches the observed relic density.
We consider for illustration the \(\gamma \gamma \) and \(b {\bar{b}}\) decay modes of the monoHiggs, along with substantial MET. We have started with rectangular cutbased analyses for both final states. The diphoton events not only have the usual SM backgrounds but also can be faked to a substantial degree by \(e\gamma \)enriched dijet events. Following up on a cutbased analysis, we switch on to machine learning (ML) techniques to improve the signal significance, going all the way to using artificial neural networks (ANN) and boosted decision tree (BDT) for both \(\gamma \gamma \) and \(b \bar{b}\) final states.
We present the salient features of our present work in the following:

We have thoroughly studied the possibility of probing monoHiggs signature at the highluminosity (HL)LHC, and at the same time asserted the viability of such scenarios from the dark matter direct detection and relic density constraints. We have found out that simple Higgsportal dark matter can satisfy all the relevant constraints in the presence of high scale physics and it can be probed at HLLHC in the monoHiggs final state. In this context, our study contributes significantly beyond the analysis of [13].

We have gone beyond specific models [8, 15, 16, 20] and employed modelindependent effective theory approach to parametrize the highscale physics.

We have examined the cleanest final state () as well as the final state with maximum yield () and thus presented an exhaustive and comparative study.

In the context of LHC, we have performed a thorough background analysis following the experimental studies in this direction, which was not done in such detail in earlier theoretical studies.

Lastly, we have predicted significant improvements compared to the cutbased analysis by using advanced machinelearning techniques.
The plan of our paper is as follows. In Sect. 2 we discuss the outline of the modelindependent scenario that we have considered and we take into account all the relevant constraints on this scenario and find out viable and interesting parameter space which can give rise to substantial monoHiggs signature at the high luminosity LHC. In Sect. 3 we discuss in detail our signals and all the major background processes. In Sect. 4 we present our results of a rectangular cutbased analysis. In Sect. 5 we employ machinelearning tools to gain improved signal significance over our cutbased analysis. In Sect. 7 we summarize our results and conclude the discussion.
2 Outline of the scenario and its constraints
2.1 The theoretical scenario
We illustrate our main results in the context of a scenario of a scalar DM particle. Where scalar sector is augmented by the gaugesinglet \(\chi \), the potential can be generally written by
Here \(\Phi \) is the SM Higgs doublet and \({\lambda }_{\Phi \chi }\) and \(\lambda _{\chi }\) are the relevant quartic couplings. A \(Z_2\) symmetry is imposed, under which \(\chi \) is odd. This legitimizes the potential role of \(\chi \) as DM candidate, and also prevents the mixing between \(\phi \) and \(\chi \), which would otherwise bring in additional constraints on the scenario.
It is clear from above that the simplest operator involving the Higgs boson and a scalar DM \(\chi \) is the dimension4 renormalizable operator \(\Phi ^{\dagger } \Phi \chi ^2\). This operator gives rise to the dominant contribution to the monoHiggs + final state when Higgs is produced via gluon fusion, as can be seen from Fig. 1. We have illustrated with reference to the gluonfusion channel because (a) it is the dominant Higgs production mode, and (b) other widely studied modes involved some additional associated particles, while our focus here specifically on the monoHiggs final state. Of course similar final states can arise from the same operators, in quarkinitiated processes as well, but their contribution will be negligible compared to the gluoninitiated ones. This operator also takes part in the DMnucleon elastic scattering through tchannel Higgs exchange (see Fig. 2 (left)) and DM annihilation diagram with schannel Higgs mediation (Fig. 2 (right)). The standalone presence of this operator makes it difficult to satisfy both direct detection constraints and relic density requirements simultaneously, as will be discussed in the next section. However, one can go beyond dimension4 terms and construct higherdimensional operators involving (anti)quarks, Higgs and a pair of DM particles, which contribute to the monoHiggs + signal. At the same time such operators add credence to such a Higgsportal scenario by cancelling the contribution to spinindependent crosssections in direct search experiments.
One can write two SU(2)\(_L \times \) U(1)\(_Y\) gaugeinvariant and Lorentz invariant operators in this context, namely \({\mathscr {O}}_1\) and \({\mathscr {O}}_2\), which are of dimension6 and8 respectively, as follows:
is a dimension6 operator involving a quark–antiquark pair, the Higgs boson and a pair of DM particles. On the other hand, is of dimension8, involving derivatives of \(\Phi \) as well as \(\chi \). Both of these operators are multiplied by appropriate Wilson coefficients \(f_i(i = 1,2\)).
Our main purpose in this paper is to bring out the potential of monoHiggs signals for dark matter. With this in view, we have chosen a smallest set of operators guided by the requirement to satisfy the dark matter constraints and yet have a respectable collider signature. This is achieved by choosing a pair of operators that interfere destructively with the Higgsmediated amplitude in direct search experiment and choosing a collider channel that does not suffer this interference. For appropriate values of the Wilson coefficient a sizable collider signature can be obtained without violating the constraints.
Once the higher dimensional operators are introduced, they contribute to both spinindependent crosssection in direct searches and annihilation of \(\chi \) before freezeout. The appropriate Feynman diagrams are shown in Fig. 3. One should note that contributions in Fig. 3 arises when \(\Phi \) acquires a VEV. Such contributions will interfere with those coming from the diagrams shown in Fig. 2.
Let us also mention the operators and contribute only to the spinindependent crosssection in direct search due to the absence of \(\gamma _5\) in them. And finally, the presence of higher dimensional operators also opens up additional production channels leading to the monoHiggs signals via quark induced diagrams, whose generic representation can be found in Fig. 4.
2.2 Constraints from the dark matter sector and allowed parameter space
As the scenario under consideration treats \(\chi \) as a weakly interacting thermal dark matter candidate, it should satisfy the following constraints:

The thermal relic density of \(\chi \) should be consistent with the latest Planck limits at the 95% confidence level [33].

The \(\chi \)nucleon crosssection should be below the upper bound given by XENON1T experiment [34] and any other data as and when they come up.

Indirect detection constraints coming from both isotropic gammaray data and the gamma ray observations from dwarf spheroidal galaxies [35] should be satisfied at the 95% confidence level. This in turn puts an upper limit on the velocityaveraged \(\chi \)annihilation crosssection [36].

The invisible decay of the 125GeV scalar Higgs h has to be \(\le \) 19% [41].
It has been already mentioned that the simplest models using the SM Higgs as the dark matter portal is subject to severe constraints. The constraints are twofold: from the direct search results, especially those from Xenon1T [34], and from the estimates of relic density, the most recent one coming from Planck [33]. While the simultaneous satisfaction of both constraints restricts SM Higgsportal scenarios rather strongly, the same restrictions apply to additional terms in the Lagrangian as well. In our case, the coefficient of \(\Phi ^\dagger \Phi \chi ^2\), restricted to be ultrasmall from direct search data, cannot ensure the requisite annihilation rate in the dominant modes such as \(f \bar{f}\) and \(W^+W^\) (see Eq. (4) where the schannel annihilation crosssections are given in the centre of mass frame [42]).
where \(\beta _A = \sqrt{14m_A^2/s}\). It is thus imperative to have additional terms that might bring out cancellation of the Higgs contribution in direct search and thus make the quartic term less constrained. The signs of the trilinear coupling \(\lambda _{h\chi \chi }\) (which is basically \(\lambda _{\Phi \chi }v)\) and the Wilson coefficients \(f_i\)s have to be appropriately positive or negative to ensure destructive interference. It will become clearer if we look at the analytical expression for spinindependent DMnucleon scattering crosssection [43], in the presence of Higgsportal as well as dimension6 operator \({\mathscr {O}}_1\), given below.^{Footnote 1}
Here \(m_N\) is the mass of the nucleon and \(f_N\) is the effective Higgsnucleon coupling. A cancellation at the amplitude level will certainly produce small DMnucleon scattering crosssection. While such cancellation may apparently be inexplicable, it is important to phenomenologically examine its implication, in a modelindependent approach if possible. A similar approach has been taken in a number of recent works [26,27,28]. The higherdimensional operators listed in the previous subsection are introduced in this spirit.
In Fig. 5, we show regions of the parameter space consistent with the observed relic density (yellow points) as a function of dark matter mass \(m_{\chi }\). The black line in the figure represents the upper limit from Xenon1T on the spinindependent DMnucleon elastic scattering crosssection as a function of the mass of the DM particle. The region below this curve is our allowed parameter space. It is clearly seen from the figure that larger DM mass regions satisfy the direct detection bound easily, primarily because of the fact that the DMnucleon scattering crosssection decreases with increasing DM mass (see Eq. (5)) and also the experimental limit becomes weaker for larger DM mass.
In Fig. 6, we show the region of parameter space in the \(\frac{1}{v}\lambda _{h\chi \chi }  \frac{\Lambda }{\sqrt{f_1}}\) plane, which is consistent with both relic density and direct detection upper bound. One can see that only a narrow region is allowed, where the aforementioned cancellation in the DMnucleon scattering amplitude takes place. As the trilinear coupling decreases, the high scale gets pushed to a higher value, as expected. The broadness of the red curve owes itself to the \(2 \sigma \) band of the relic density as well as the the direct detection limit and most importantly, to the specified range of dark matter mass as described in Fig. 6. As discussed earlier, we can see from Eq. (5), the spinindependent scattering crosssection decreases with increasing DM mass \(m_{\chi }\). Also, from Fig. 5, it is clear that the upper limit on DMnucleon scattering crosssection becomes weaker as one goes higher in DM mass. Therefore, it follows that if one varies DM mass to larger values, the direct detection limit will be less stringent and larger band will be allowed. One can see that looking at the coloraxis in Fig. 6. All the relevant quantities in Figs. 5 and 6, such as relic density(\(\Omega h^2\)), spinindependent DMneucleon scattering crosssection (\(\sigma _{\chi N}\)) have been calculated using MicrOMEGAs5.0.8 [44], where we have implemented our model via Feynrules2.3 [45].
We use MicrOMEGA5.0.8 to also estimate the indirect detection crosssection for the benchmark points chosen in the upcoming sections. We find \(\langle \sigma v\rangle \) to be in the vicinity of \(10^{29}\) cm\(^3\) for all six benchmark points we have chosen. These indirect detection crosssections are way below the bound report from FermiLAT [35,36,37] (\(\langle \sigma v\rangle \gtrsim 10^{26}\) cm\(^3\)) and AMS [38] (\(\langle \sigma v\rangle \gtrsim 10^{28}\) cm\(^3\)).
We present in the next section, the collider analysis for a few benchmark points which satisfy all the aforementioned dark matter constraints. We mention here that our choice of benchmarks will be strongly guided by the phenomenological aspiration to probe the maximally achievable collider sensitivity. However, we have checked that all our benchmarks satisfy treelevel unitarity and vacuum stability using SARAH [39] and our own modification of 2HDME [40].
3 Signals and backgrounds
Having identified the regions of allowed parameter space we proceed towards developing strategies to probe such scenarios at the high luminosity LHC. Our study is based on a scalar DM \(\chi \) as mentioned earlier. One should note that a corresponding fermionic DM will not allow the production channels in Fig. 1 purely driven by dimension4 operator. Therefore one will have to depend on higherdimensional operators with the production rate considerably suppressed. As has been discussed earlier, we are looking for the monoHiggs + final state. Since the process will lead to substantial number of events with missing energy, the decay products of the Higgs constitute the visible system recoiling against the missing transverse momenta. The main contribution to production comes from the top two diagrams in Fig. 1.
In Fig. 7, we show the dependence of \(\sigma (p p \rightarrow h \chi \chi )\) on \(m_{\chi }\) for \(\lambda _{\Phi \chi } \approx 4\pi \) and \(\frac{\Lambda }{\sqrt{f_{1,2}}} \approx 5\) TeV. It is clear from this figure that a resonance takes place in the vicinity of \(\frac{m_h}{2}\), a behavior which can be intuitively understood from Fig. 1, where we have seen that, the first two diagrams make dominant contribution to the final state. It is worth mentioning that the effective operator \({\mathscr {O}}_1\) contributes close to 10% as much as the gluon fusion channel in \(h \chi \chi \) production, the contribution of \({\mathscr {O}}_2\) is about half of that of \({\mathscr {O}}_1\). Here also the assumption \(f_1 \approx f_2\) is made. While the choice of parameters in Fig. 7, as justified in the caption is on the optimistic side form the viewpoint of signals, they qualitatively capture the features of this scenario.
The next important task is to identify suitable visible final states which will recoil against the invisible \(\chi \chi \) system. The largest branching ratio of the 125 GeV scalar is seen in the \(b \bar{b}\) channel. However, while this assures one of a copious event rate, one is also deterred by the very large QCD backgrounds, whose tail poses a threat to the signal significance. While we keep the \(b \bar{b}\) channel within the purview of this study, we start with a relatively cleaner final state, namely a diphoton pair. Its branching ratio (2.27\(\times 10^{3}\)) considerably exceeds that of the fourlepton channel (7.2\(\times 10^{5}\)) which too is otherwise clean. As compared to the \(b \bar{b}\) channel, diphoton offers not only better fourmomentum reconstruction but also a cleaner MET identification. We suggest in the discussion below, some strategies to overcome this disadvantage largely making use of one feature of the signal, namely a substantial missing generated by the \(\chi \chi \) system.
3.1 channel
The diphoton channel is apparently one of the cleanest of Higgs signals. The absence of hadronic products is perceived as the main source of its cleanliness, together with the fact that there is a branching ratio suppression (though rather strong) at a single level only as opposed to the fourlepton final state. This channel has been under scrutiny from the earliest days of Higgsrelated studies at the LHC. In the present context we are focussing on events with at least two energetic photons and substantial . Searches for such events have been carried out by both CMS [47,48,49,50,51] and ATLAS [52,53,54].
As can be seen in Fig. 7, this channel is usable for \(m_{\chi } \lesssim 100\) GeV, and particularly in the resonant region. Moreover, the upper limit on the invisible decay of the Higgs prompts us to those benchmarks where \(m_{\chi } > m_h/2\). A set of such benchmark points, satisfying also all constraints related to dark matter, are listed in Table 1.
BP2 corresponds to the best possible scenario in terms of signal crosssection, with mass of the dark matter close to \(\frac{m_h}{2}\) and \(\lambda _{\Phi \chi }\) coupling satisfying the perturbativity limit. We move up in \(m_{\chi }\) in BP1, to illustrate the reach of signal for higher DM masses. BP3, on the other hand, has been chosen to explore the reach of the signal in terms of the quartic coupling \(\lambda _{\Phi \chi }\).
The apparent cleanliness of the signal, however, can be misleading. Various backgrounds as well as possibilities of misidentification or mismeasurement tend to vitiate the signal. In order to meet such challenges, the first step is to understand the backgrounds.
Backgrounds Contamination to the diphoton final state comes mainly from prompt photons that originate from the hard scattering process of the partonic system (e.g. \(q\bar{q} \rightarrow \gamma \gamma \) through Born process or \(gg \rightarrow \gamma \gamma \) through a oneloop process represented by “box diagram”) or nonprompt photons, that originate within a hadronic jet, either from hadrons that decay to photons or are created in the process of fragmentation, governed through the quark to photon and gluon to photon fragmentation function \(D^q_{\gamma }\) and \(D^g_{\gamma }\) [55,56,57,58,59]. Such nonprompt photons are always present in a jet, and can be misidentified as a prompt photon when most of the jet energy is carried by one or more of these photons. We shall refer to this effect as “jet faking photons”. Electrons with energy deposit in the electromagnetic calorimeter (ECAL), can be misidentified as a photon if the track reconstruction process fails to reconstruct the trajectory of the electron in the inner tracking volume, since both electron and photon deposit energy in the ECAL by producing an electromagnetic shower, with very similar energy deposit patterns (shower shapes). Therefore processes with energetic electrons in the final state can also contribute to the background. We shall refer to this type of misidentification as “electron faking photon”. It may be noted that the “jet faking photons ” also has a contribution from nonprompt electrons produced inside the jet, that fake a photon due to track misreconstruction. In the following we discuss the various SM processes that give rise to prompt and nonprompt backgrounds ordered according to their severity.
 QCD multijet::

Although the jet faking photon probability is small in the high \(p_T\) region of interest of this analysis(\(\sim 10^{5}\) as estimated from our MonteCarlo Analysis), the sheer enormity of the crosssection (\(\sim millibarns\) above our \(p_T\) thresholds) makes this the largest background to the diphoton final state.
To estimate this background as accurately as possible, we have first generated an \(e\gamma \)enriched dijet sample, which essentially means jets that contain photonlike(EM) objects within themselves. We mention here that, to achieve better statistics, we apply a generation level cut \(p_T > 30\) GeV and \(\eta  < 2.5\) on both jets. The most common source of jet faking a photon is through \(\pi ^0\) inside the jet, which decays into two photons. Other meson decays, electron faking photon and fragmentation photons contribute a lesser but nonnegligible amount. We have considered all QCD multijet final states which contain any one of the following objects: photon, electron, \(\pi ^0\) or \(\eta \) mesons (namely the EMobjects). Then we have categorized only those objects which have \(p_T > 5\) GeV and are within the rapidityrange \(\eta  < 2.7\), as ‘seeds’. Then energies and \(p_T\) of all the EMobjects within \(\Delta R < 0.09 \) around the seed are added with the energy and \(p_T\) of the seed. Thus, out of all those EMobjects within a jet, photon candidates are created. If in a QCD multijet event, there are at least two photon candidates with \(p_T \gtrsim \) 30 GeV, those events can in principle fake as a photon with high probability. However, one should also demand a strong isolation around those photon candidates following the isolation criteria described earlier to differentiate between these jet faking photons and actual isolated hard photons.
 \(\gamma +\) jets::

This background already has an isolated photon candidate. However, here too, the jets in the final state can fake as photon with a rather small probability(\(\sim 0.003\) as estimated from our MonteCarlo Analysis). But again the large crosssection (\(\approx 10^5\) pb) of this process makes suppression of the background challenging. For correct estimation of this background, we adopt the same method that has been applied for the multijet background discussed earlier. Like QCD multijet, this background is also generated with \(p_T > 30\) GeV and \(\eta  < 2.5\) on the photon and the jet for gaining better statistics.
 \(t \bar{t} + \gamma \)::

Another major background comes from \(t \bar{t} + \gamma \) production, when one or more of the leptons or jets from top decay are mistagged as photon. Although the crosssection is significantly lower compared to \(\gamma \) + jets background, a real source of large in this case makes it difficult to reduce this background. However, the isolation criterion as discussed above as well as an invariant mass cut help us suppress this background.
 diphoton::

As mentioned above, this background includes production of two photons in the final state through gluoninitiated box diagram, and also via quarkinitiated Born diagrams. Although this background gives rise to two isolated hard photons, it does not contribute much due to relatively low crosssection. Demanding a hard and using the fact that the invariant mass of the diphoton pair should peak around the Higgs mass, one can get rid of this background.
 \(V+\gamma \)::

A minor background arises from \(W/Z + \gamma \) channel, when one or more leptons or jets from W or Z decay are mistagged as photons. However, the associated with this process is not significant and one of the photons is not isolated. Therefore, this process contributes only a small amount to the total background.
 \(Z(\rightarrow \nu \bar{\nu }) h(\rightarrow \gamma \gamma )\)::

This is an irreducible background for our signal process. This gives rise to sizeable , with the invariant mass of the diphoton pair peaking around \(m_h\). However, this process has small enough crosssection compared to other backgrounds and proves to be inconsequential in the context of signal significance.
A few comments are in order before we delve deeper into our analysis. Some studies in the recent past have considered, Higgs production through higher dimensional operators, based on the diphoton signal [13]. However, the role of backgrounds from QCD multijets has not been fully studied there. Our analysis in this respect is more complete. Also, \(p p \rightarrow W^+ W^\), too, in principle lead to substantial + two ECAL hits with electrons from both the Ws being missed in the tracker. We can neglect, such fakes because (a) the event rate is doublesuppressed by electronic branching ratios, (b) demand on the invariant mass helps to reduce the number of events and (c) two simultaneous fakes by energetic electrons is relatively improbable.
Events for the signals and most of the corresponding backgrounds (excepting QCD multijet and \(\gamma \)+jet) and have been generated using Madgraph@MCNLO [46] and their crosssections have been calculated at the nexttoleading order (NLO). We have used the nn23lo1 parton distribution function. The QCD multijet and \(\gamma +\) jet backgrounds are generated directly using PYTHIA8 [60]. MLM matching with xqcut = 30 GeV is performed for backgrounds with multiple jets in the final state. PYTHIA8 has been used for the showering and hadronization and the detector simulation has been taken care of by Delphes3.4.1 [61]. Jets are formed by the builtin Fastjet [62] of Delphes.
3.2 channel
Signal The channel resulting in hadronic final states, poses a seemingly tougher challenge, as compared to the diphoton final state. However, the substantial rate in this channel creates an opportunity to probe the monoHiggs+MET signal, if backgrounds can be effectively handled. Searches in this channel have been carried out by both CMS [49, 63, 64] and ATLAS [65,66,67] experiments. We demand at least two energetic btagged jets, along with considerable .
It is clear from Fig. 7 that here too the resonance region (\(m_\chi \gtrsim \frac{m_h}{2}\)) offers the best signal prospect, as in the \(\gamma \gamma \) case. The enhancement in the resonant region enables one to probe \(\lambda _{\phi \chi } \lesssim 6\) in a cutbased analysis. We shall discuss later the possible improvements using machine learning techniques. This makes the \(b\bar{b}\) channel more attractive prima facie, as compared to the diphoton channel. the rates are large enough for probing up to \(\Lambda \approx \) 8 TeV, or, alternatively, DM masses up to 8 TeV. Although higher \(m_\chi \) implies lower yield (see Fig. 7), judicious demands on the \({\not {E}}_{T}\) lend discernibility to the final state. One thus starts by expecting to probe larger regions in the parameter space for the Higgs decaying into bpairs. The benchmark points listed in Table 2 are selected, satisfying all the aforementioned constraints, by keeping this in mind. The prospect of detectability at the LHC is of course the other guiding principle here. BP5 offers the best prospect, with the dark matter mass in the resonant region and the quartic coupling \(\lambda _{\phi \chi }\) close to its perturbativity limit. One should note that BP5 here has similar potential as BP2 in the case of the diphoton channel. BP4 is more favorable for relatively heavy dark matter particles for the same quartic coupling. On the other hand, BP4 can be explored for smaller \(\lambda _{\phi \chi }\) compared to BP4 and BP5, so long as \(m_\chi \) continues to remain in the resonant region.
Having chosen benchmark points for signal we proceed to analyse the corresponding backgrounds. Here too, for the backgrounds involving multiple jets in the final state, the MLM matching procedure has been used with xqcut = 30 GeV.
Background We list the dominant backgrounds for this channel in the following.
 \(t \bar{t} +\) single top::

The major background for the \( b \bar{b}\) channel comes from \(t \bar{t}\) production at the LHC when one of the resulting W’s decays leptonically. It is commonly known as semileptonic decay of \(t \bar{t}\) pair. This process has considerable production rate and it is also source of substantial . A minor contribution comes from the leptonic decay of both W’s from \(t \bar{t}\). We call this leptonic decay of \(t \bar{t}\). It also has in the final state from two neutrinos coming from leptonic W decay. However, a veto on \(p_T\) of leptons \(> 10\) GeV reduces this background at the selection level itself, whereas the semileptonic \(t \bar{t}\) background is less affected by such veto. The hadronic background where both W’s from \(t \bar{t}\) decay hadronically, has the largest crosssection among all \(t \bar{t}\) backgrounds. However, in this case the source of is essentially mismeasurement of jet energy. A full simulation shows that the hadronic \(t \bar{t}\) background plays a subdominant role. The single top background is also taken into account, but its contribution is rather small compared to the semileptonic and leptonic \(t \bar{t}\) because of its much smaller crosssection.
 \(V+\) jets::

The next largest contribution to the background, in our signal region comes from \(V+\) jets (\(V = W, Z)\) production. These processes have large crosssections (\(\approx 10^4\) pb) and also have significant sources of through the semileptonic decays of the weak gauge bosons. However, this background depends on the simultaneous mistagging of two light jets as bjets. The doublemistag probability is rather small for these backgrounds (\(\approx 0.04\%\) as estimated from our MonteCarlo simulation). It is worth mentioning here, that the contribution of \(W+\)jets is found to be subdominant compared to \(Z+\)jets. The main reason behind this is the presence of larger in the latter case and also the suppression of the former by the lepton veto.
 QCD \(b \bar{b}\)::

One major drawback of the \(b \bar{b}\) channel is the presence of QCD \(b \bar{b}\) production of events which has large crosssection (\(\approx 10^5\) pb). The nuisance value of this background, however,depends largely on coming from jetenergy mismeasurement. On applying a suitable strategy which we will discuss in the next section (see Table 5), we find that this background becomes subdominant to those from \(t \bar{t}\) and \(Z+\)jets processes. To gain enough statistics, we have applied a generation level cut on the bjets, i.e. \(p_T > 20\) GeV and \(\eta  < 4.7\) on the QCD \(b \bar{b}\) events.
 Diboson(WZ/ZZ)::

final state can also come from diboson production in the SM. However, the production crosssection in this case (\(\approx 1.3\) pb) is much smaller compared to the aforementioned backgrounds. Moreover, a hard lepton veto will significantly reduce WZ events and finally, a strong cut will help us control the ZZ as well as WZ background.
 \(Z(\rightarrow \nu \bar{\nu })h(\rightarrow b \bar{b})\)::

Similar to the \(\gamma \gamma \) case, this background too is irreducible. The and invariant mass of the \(b \bar{b}\) system are also similar to the signal processes. However the smallness of its crosssection (\(\approx 100\) fb) makes this background least significant among all the background processes discussed here.
The signal and background events (except QCD \(b \bar{b}\)) are generated using Madgraph@MCNLO [46] and showered through PYTHIA8 [60]. The QCD \(b \bar{b}\) background is generated directly using PYTHIA8. The detector simulation is performed by Delphes3.4.1 [61], the jet formation is taken care of by the builtin Fastjet [62] of Delphes.
We have used the CMS card in Delphes for the btagging procedure, which yields an average taggingefficiency of 70% per bjet approximately, in the \(p_T\) range of our interest (50–150 GeV). We have checked that this efficiency differs by not more than 5% on using the ATLAS specifications.s
4 Collider analysis: cutbased
4.1 channel
The discussion in the foregoing section convinces us that it is worthwhile to look at the channel because of the ‘clean’ diphoton final state. Our analysis strategy goes beyond the existing ones [13, 14] even at the level of rectangular cutbased studies, for example, we make the background analysis more exhaustive, detector information particularly that pertaining to the inner tracker is also studied in greater detail and of course we have subsequently upgraded our analysis using the methods based on gradient boosting as well as neural network. This will be described in detail in later sections.
We will discuss the results of our cutbased analysis for a few benchmarks presented in Table 1 which are allowed by all the constraints mentioned earlier. We will first identify variables which give us desired separation between the signal and backgrounds. We present in Fig. 8 the distribution of the transverse momenta of the leading and subleading photon for the signal and all the background processes. The signal photons are recoiling against the dark matter and therefore are boosted. On the other hand, in case of dijet events the photons are part of a jet, and it is unlikely that those photons will carry significant energy and \(p_T\) themselves. In case of \(\gamma \) + jet background at least one photon is isolated and it can carry considerable \(p_T\). Therefore the \(p_T\) distribution of the leading photon is comparable with the signal in this case, while for the subleading photon, which is expected to come from a jet, it falls off faster as expected.
In Fig. 9 (left), we plot the distribution for signal and background processes. distributions for dijet, \(\gamma \)+jet and diphoton show that these background events will have much less compared to the signal and Zh background. In the background events involving jets the source of missing energy is via the mismeasurement of jet energy or from the decay of some hadron inside the jet. \(\gamma \gamma \) background naturally shows lowest contribution owing to the absence of real or jets in the final state. On the other hand, the signal and Zh background have real source of . It is evident that observable will play a significant role in signalbackground discrimination. In Fig. 9 (right), we have shown the invariant mass of the diphoton pair. In case of signal and Zh background the \(m_{\gamma \gamma }\) distribution peaks at Higgs mass and for all other backgrounds no such peak is observed. This distribution is also extremely important for background rejection.
In Fig. 10 (left) we plot the distribution of \(\Delta R\) between the two photons for signal and background. We can see from the figure that in case of dijet and \(\gamma + \)jet background there is a peak at \(\Delta R \approx 0\), resulting from the cases where the two photons have come from a single jet. However the dijet and \(\gamma + \)jet events have a second peak too because of the events where each photon is part of a single jet and therefore the two photons are back to back. However, we do not really observe any peak in the \(\Delta R\) distribution of the signal or Zh background because in these cases the diphoton system is exactly opposite to the dark matter pair. Next we plot the jet multiplicity distribution for signal and background in Fig. 10 (right).
In Fig. 11 (left) and (right), we have plotted the \(\Delta \phi \) distribution between the and the leading and subleading photon respectively. We know that in case of signal and the Zh background the diphoton system is exactly opposite to the dark matter system. Therefore the photons from the Higgs boson decay tend to be mostly opposite to the . On the other hand in case of dijet or \(\gamma +\)jet events, the the is aligned with either of the jets for aforementioned reasons. This behaviour is visible from the Fig. 11.
We would like to remind the reader that no special strategy has been devised for the irreducible Zh background. This is because of its low rates compared to both the signal and the other background channels.
Results Having discussed the distributions of the relevant kinematical variables, we go ahead to analyse the signal and background events. Our basic event selection criteria here is, at least two photons with \(p_T > 10\) GeV and \(\eta  < 2.5\). We also impose veto on leptons (\(e, \mu )\) with \(p_T > 10\) GeV. For our analysis, we further apply the following cuts in succession for the desired signalbackground separation.
 Cut 1::

\(p_T\) of the leading(subleading) photon \(> 50(30)\) GeV
 Cut 2::

\(\Delta R\) between two photons \(> 0.3\)
 Cut 3::

\(\Delta \phi \) between leading(subleading) photon and
 Cut 4::

GeV
 Cut 5::

\(\Delta \phi \) between leading(subleading) jet and
 Cut 6::

115 GeV \(< m_{\gamma \gamma } < 135\) GeV
A clarification is in order on the way in which we apply the isolation requirement on each photon. We have imposed the requirement that the total scalar sum of transverse momenta of all the charged and neutral particles within \(\Delta R < 0.5\) of the candidate photon can not be greater than 12% of the \(p_T\) of the candidate photon i.e \(\frac{\sum _{i} p_T^i(\Delta R< 0.5)}{p_T^{\gamma }} < 0.12\). Thereafter we have estimated the isolation probability of a photon, defined by this criterion, as a function of \(p_T\). We have then multiplied this \(p_T\)weighted isolation probability with all the events surviving after applying the previously mentioned cuts (up to Cut6). We call this criterion Cut7 which goes somewhat beyond the rectangular cutbased strategy.
Table 3, indicates the cutefficiencies of various kinematic observables. We can see from this table that , \(m_{\gamma \gamma }\) and the isolation criterion turn out to be most important in the separation of the signal from the background. Having optimized cut values, we calculate the projected significance (\({\mathscr {S}})\) for each benchmark point for the 13 TeV LHC with 3000 fb\(^{1}\) in Table 4. The significance \({\mathscr {S}}\) is defined as
Where S and B are the number of signal and background events surviving the succession of cuts. This formula holds under the assumption that the signal as well as the background follow Poisson distribution, and the righthand side reduces to the familiar form \(\frac{S}{\sqrt{B}}\) in the limit \((S + B) \gg 1\) and \(S \ll B\) (see Equation 97 of [68]).
From Table 4, one can see that our cuts have improved the S/B ratio from \(10^{12}\) to of the order 0.05. And with our cutbased analysis it is possible to achieve \(\gtrsim 4 \sigma \) significance for BP2 which has the largest production crosssection. The other benchmarks with heavier dark matter mass or small quartic coupling do not perform very well. One should note that although we have identified the strongest classifier observables through our cutbased analysis and used them, it is possible to make use of the weaker classifiers too if we go beyond cutbased setup, which will precisely be our goal with machine learning in the next section. We hope to achieve some improvement over the results quoted in Table 4.
4.2 channel
We proceed to discuss the kinematic variables which yield significant signalbackground separation. In Fig. 12, we plot the \(p_T\) distribution of the leading and subleading bjets for the signal and all the background processes. We can see from the figures that for the signal, the \(p_T\) distribution of the b jets peak at a higher value compared to the QCD \(b \bar{b}\) and \(V+\)jets background. This behaviour is expected since the \(b \bar{b}\) system is recoiling against a massive DM particle in case of signal. The distribution of \(p_T\) of leading and subleading bjets from \(t \bar{t}\) backgrounds however peak at a similar region as the signal. The bjets in those cases come from the decay of top quarks and are therefore boosted. Guided by the distributions we put appropriate cut on the transverse momenta of the leading and subleading bjets in our cutbased analysis.
In Fig. 13 (left) we plot the distribution for signal and background processes. We can see that the QCD background produces the softest spectrum. The reason behind this is that there is no real source of for this final state. It is mainly the mismeasurement of the visible momenta of the jets, which leads to . Though large arising from such mismeasurement contributes mostly to the tail of the distribution, the sheer magnitude of the crosssection can still constitute a menace. However, a strong cut helps us reduce the \(b \bar{b}\) background to a large extent, as will be clear from the cutflow analysis in the next subsection. The peaks at much lower values in case of V+jets as well. However, \(t \bar{t}\) and Zh background also produce large enough , although less than our signal. Therefore a hard cut enhances the signal background separation. In Fig. 13 (right), we plot the invariant mass distribution of two bjets. In case of signal and Zh background it peaks at Higgs mass, whereas for all other backgrounds it falls off rapidly. It is evident that a suitable cut on the invariant mass of the bjet pair will also be effective in reducing the background.
In Fig. 14 (left) we show the jet multiplicity distribution of the signal and background processes. Jet multiplicity distribution here indicates the number of light jets in the process. We know that in \(t \bar{t}\) semileptonic case, which is one of the primary backgrounds, the number of light jets is expected to be more than the signal and other backgrounds, because it has two hard light jets coming from the W decay. This feature can be used to distinguish this background from signal. We also present the distribution of the invariant mass distribution of the leading and subleading light jet pair in Fig. 14 (right). In case of \(t \bar{t}\) semileptonic background, the two leading light jets come from W decay and therefore this \(m_{jj}\) distribution peaks at W mass. An exclusion of the \(m_{jj} \approx m_W\) region in the cutbased analysis as well as a suitable cut on the number of light jets help us control the severe \(t \bar{t}\) background.
In Fig. 15 (left) and (right), we plot the \(\Delta \phi \) distribution between the and the leading and subleading bjets respectively. In case of signal and the Zh background the \(b \bar{b}\) system is exactly opposite to the dark matter system. Therefore the bjets from the Higgs boson decay tend to be mostly opposite to the . On the other hand in case of QCD \(b \bar{b}\) events, the the is aligned with either of the bjets because the arises mainly due to the mismeasurement of the bjet energy in this case. This behaviour is visible from the Fig. 15 (left) and (right).
Results From the discussion on various kinematical observables, it is clear that we can choose suitable kinematical cuts on them to enhance the signalbackground separation. However, our basic event selection criteria in this case are, at least two btagged jets with \(p_T > 20\) GeV and \(\eta  < 4.7\). We also impose a veto on leptons (\(e, \mu )\) with \(p_T > 10\) GeV. In addition, the following cuts are applied for our analysis.
 Cut 1::

\(p_T\) of leading bjet \(> 50\) GeV and \(p_T\) of subleading bjet \(> 30\) GeV
 Cut 2::

GeV
 Cut 3::

80 GeV \(< M_{b_1 b_2} < 140\) GeV
 Cut 4::

and
 Cut 5::

Number of light jets (not btagged) \( < 3\)
 Cut 6::

Invariant mass of two leading light jet pair \(<70\) GeV or \(>90\) GeV
 Cut 7::

\(\Delta R\) between the leading bjet and the leading light jet \(> 1.5\)
We have applied these cuts on signal and background processes in succession. The cut efficiencies of various cuts for signal and backgrounds are given in Table 5.
From the cutflow efficiencies quoted in Table 5 we can see that Cut2 i.e. the cut is most essential in eliminating the major backgrounds such as \(t \bar{t}\) semileptonic and QCD \(b \bar{b}\). The other severe background \(V+\)jets is under control after applying the bveto at the selection level. Having optimized cut values, we calculate the projected significance (\({\mathscr {S}})\) for each benchmark point for the 13 TeV LHC with 3000 fb\(^{1}\) in Table 6. The formula used for calculating signal significance is given in Eq. (6).
From Table 6 one can see that our cuts have improved S/B ratio from \(10^{7}\)–\(10^{8}\) to the order 0.01. And with our cutbased analysis it is possible to achieve \(\sim 6\sigma \) signal significance for BP5 with the final state. As we have chosen BP5 to be exactly same as BP2 of the diphoton case, one can compare the reach of the two channels with this benchmark. At this level of analysis, we can see that clearly performs better than channel in this regard. BP4 for which dark matter mass is 120 GeV, performs fairly well even within the cutbased framework. We have used mainly the strong classifiers in our cut based analysis. However, it is well known in the machine learning literature that a large number of weak classifiers can also lead to a good classification scheme. To that end, we include a range of kinematical variables (weak classifiers) along with our strong classifiers to help us improve our reach in couplings and masses of the dark matter.
5 Improved analysis through machine learning
Having performed the rectangular cutbased analysis in the \(\gamma \gamma \) + and channel, we found that it is possible to achieve considerable signal significance at the HLLHC for certain regions of the parameter space. Those regions are highly likely to be detected in the future runs. However, there are some benchmarks, namely the ones with small quartic coupling \(\lambda _{\Phi \chi }\) which predict rather poor signal significance in a cutbased analysis. We will explore the possibility of probing those regions of parameter space with higher significance with machine learning. As we discussed in the previous section, we go beyond the rectangular cutbased approach here and use more observables, even the weaker classifiers, and take into account the correlation between the observables.
In cutbased analysis we apply rectangular cuts on the chosen observables. Therefore, the shape of the selected signal region is a ndimensional rectangular hypervolume. However, the actual shape of the signal region might be more complicated. In order to capture the relevant signal region, we have to adopt a more intricate scheme of selecting regions of the phase space. One such way is Boosted Decision Tree [69], which iteratively partitions the rectangular volume to select the relevant signal region. Alternatively, the Artificial Neural Network [70] attempts to encompass the relevant signal region with a set of hyperplanes, to as good accuracy as possible.
We have performed the analysis with BDT as well as ANN in order to make a comparison between the two as well as with the cutbased analysis. The usefulness of BDT and ANN has been widely demonstrated in [71,72,73,74,75] including studies in the Higgs sector [27, 76,77,78,79,80,81]. For ANN, we have used the toolkit Keras [82] with Tensorflow as backend [83]. For BDT, we have used the package TMVA [84]. Like the cutbased analysis here too, we present a comparison between the performances of and channels using ML.
5.1 channel
For our analysis in the channel, we have used 16 observables as feature variables. Those observables are listed in Table 7. For our BDT analysis, we have used 100 trees. A condition of minimum 2% events of the training sample has been set for leaf formation. Maximum depth of decision tree allowed is 2. An ANN has been constructed feeding these 16 variables in the input layer followed by 4 hidden layers with nodes 200, 150, 100 and 50 in them respectively. We have used rectified linear unit (RELU) as the activation function acting on the output of each layer. A regularization has been applied using 20% dropout. Finally there is a fully connected output layer with binary mode owing to softmax activation function. Categorical crossentropy was chosen as the loss function with adam as the optimizer [85] for network training with a batchsize 1000 for each epoch, and 100 such epochs. For training we use 80% of the data, while rest 20% was kept aside for test or validation of the algorithm.
We introduce a few new observables compared to the cutbased analysis, namely \(M_R\), \(M^T_R\), R [86] among others. These variables are collectively called the Razor variables. The definitions are as follows.
The \(M_R\) variable gives an estimate of the mass scale, which in the limit of massless decay products equals the mass of the parent particle. This variable contains both longitudinal and transverse information. \(M^T_R\) on the other hand, is derived only from the transverse momenta of the visible final states and . The ratio R between \(M_R\) and \(M^T_R\) captures the flow of energy along the plane perpendicular to the beam and separating the visible and missing momenta. We show the distribution of R for signal and background processes in Fig. 16, which indicates the variable R and correspondingly all the razor variables possess substantial discriminating power.
From the BDT analysis, we found out that the and \(m_{\gamma \gamma }\) play the most important role in distinguishing between signal and background, which was already expected from our cutbased analysis. \(\Delta R_{\gamma \gamma }\), \(p_T\) of the leading and subleading photons and the Razor variables are also good discriminating variables in this regard. However, there can be significant correlation between various important observables, which should be taken into account. We have calculated such correlations directly in BDT and used only those variables which have \(< 25 \%\) correlation between them, in our final BDT analysis. We have used the observable with highest ranking among the correlated (\(\gtrsim 75\%\)) ones. For example, we found that the razor variables (particularly \(M^T_R\)) are correlated with . One should also note that the significant background rejection (dijet, \(\gamma +\) jet) happens while two isolated photons are demanded as has been discussed in the cutbased analysis.
We apply the following cuts, after demanding at least two photons and leptonveto, and introduce the resulting training sample for the BDT as well as ANN analysis:

\(p_T\) of the leading photon > 30 GeV,

\(p_T\) of the subleading photon > 20 GeV,

GeV,

\(50~\text {GeV}< m_{\gamma \gamma } < 200\) GeV,

\(\Delta R_{\gamma \gamma } > 0.1\),

and

.
We emphasize that the cuts given above are weaker than those used in the cutbased analysis. This allows our algorithms to look at the larger phasespace, learn about the the background features better and then use them to come up with a better decision boundary that helps us cut down the background even more strongly.
In Fig. 17, we show the Receiver Operating Characteristic (ROC) curves for two mass points \(m_{\chi } = 64\) GeV and \(m_{\chi } = 70\) GeV from the ANN(left) and BDT(right) analyses. We can see from Fig. 17, that with increase in the dark matter mass, the discriminating power increases, the ROC curve for \(m_{\chi } = 70\) GeV performs better than \(m_{\chi } = 64\) GeV. The area under the ROC curve is 0.994(0.993) for \(m_{\chi } = 70\) GeV and 0.992(0.991) for \(m_{\chi } = 64\) GeV using ANN(BDT). In Table 8, we present the S/B ratio, the signal significance for BP1, BP2 and BP3 using both ANN and BDT. We have scanned along the ROC curves and presented results for selected true positive (\(\sim 0.55\)) and false negative rates (\(\sim 0.001\)) that will yield the best signal significance for respective benchmark points. The signal significance is calculated using the formula (6).
From Table 8, it is clear that machine learning improves the results of our cutbased analysis to a large extent. We notice that the S/B ratio has improved 4 to 8 fold depending upon the benchmark point and the machine learning methods that lead to improved significance. We also notice that ANN performs better than BDT in this case. With machine learning techniques at our disposal, regions with weaker dark matter couplings can be probed at the HLLHC in the final state, which was unattainable through an exclusive rectangular cutbased method.
5.2 channel
We proceed towards the analysis in channel with ML in this subsection. Here we have used 22 observables as feature variables which are listed in Table 9. Like the case, here too, we have used 100 trees, minimum 2% events of the training sample for leaf formation and maximum allowed depth of decision tree 2, for BDT analysis. An ANN was constructed with these 22 variables fed into the input layer and then a similar structure of the network has been used as described in the \(\gamma \gamma \) analysis. The input layer is followed by 4 hidden layers with nodes 200, 150, 100 and 50 respectively using rectified linear unit (RELU) as the activation function acted on the outputs of each layer. The final layer is a fully connected binary output layer with softmax as the activation function. A 20% dropout has been applied for regularization. For the loss function categorical crossentropy was chosen with adam as the optimizer [85] and the network was trained with a batchsize 1000 for each epoch, and 100 such epochs. For training, 90% of the data was used, while rest 10% was used for test or validation of the algorithm.
We can see from Table 9, that a number of new observables have been introduced for the ML analysis, compared to the cutbased approach. One such important addition is the significance which has been widely used in experimental analyses for the monoHiggs + DM search in the final state [66]. The significance is defined as the ratio of and the squareroot of scalar sum of \(p_T\) of all the visible final states (\(\sqrt{H_T}\)). This observable (albeit correlated with ) is particularly useful in reducing the \(b \bar{b}\) background as pointed out in [87]. We can see the distribution of significance for signal and background from Fig. 18 (left). We show the distribution of \(H_T\) for signal and backgrounds in Fig. 18 (right).
BDT analysis ranks , \(m_{bb}\), \(H_T\), \(N_{jets}\) observables highest in terms of signal and background separation, reinforcing our understanding from the cutbased analysis. Here too, the correlated observables are identified in BDT and only the most important ones among the correlated were retained for an effective BDT performance.
We apply the following cuts after demanding two bjets and leptonveto, and introduce the resulting training sample for the BDT as well as ANN analysis:

\(p_T\) of the leading bjet > 30 GeV,

\(p_T\) of the subleading bjet > 20 GeV,

GeV,

\(60~\text {GeV}< m_{bb} < 170\) GeV and

\(m_{jj} < 70\) GeV or \(m_{jj} > 90\) GeV.
Like in the case of channel, here too we use cuts weaker than those used in the cutbased analysis to better estimate the decision boundary in the phasespace. In Fig. 19, we present the ROC curve for two mass points for the dark matter particle \(\chi \), \(m_{\chi } = 64\) GeV and \(m_{\chi } = 120\) GeV using ANN(left) as well as BDT(right). We can see that as the mass of the dark matter increases, the discriminating power between signal and background increases. This can certainly be attributed to the larger in case of heavier dark matter mass which is clear from Fig. 13 (left). The area under the ROC curve is 0.95(0.93) in case of \(m_{\chi } = 120\) GeV and 0.93(0.91) for \(m_{\chi } = 64\) GeV using ANN(BDT). We have scanned along the ROC curves and presented results for selected true positive (\(\sim 0.20\)) and false negative rates (\(\sim 0.001\)) that will yield the best signal significance. In Table 10, we present the S/B ratio, the signal significance for the benchmark points given in Table 2, calculated using Eq. 6. We note that both BDT and ANN models give us about a factor 10 improvement in the S/B ratio and about 3 to 6 times improvement in the signal significance.
Table 10 shows that machine learning significantly improves the prospect of the channel, compared to the cutandcount analysis. Here too, ANN performs consistently better than BDT. There are certain regions of the parameter space in which the predictions for the signal are more optimistic owing to the large crosssections and several percent levels S/B ratio. All three benchmark points, BP4, BP5 and BP6 can be probed at significances much larger than \(5\sigma \). This indicates that we can probe larger masses and/or smaller couplings than the ones we have chosen for the benchmark points. But we must take note that the large significance reported in Table 10 are caused by a small statistical error of order 0.5% which is significantly smaller than the best estimates for systematic errors. Thus it becomes essential to repeat the analysis in the presence of systematic errors, which is done in the next section.
6 Analysis with systematic uncertainty
It is wellknown that the proposed future HLLHC will be extremely prone to large amount of systematic uncertainty coming from various sources. Therefore, it is imperative that we study the effect of systematics on our analysis. In order to check the effect of systematic uncertainty on our results, we have repeated the analysis, using BDT as well as ANN, in the presence of suitable chosen sets of values for systematic uncertainty in Tables 11, 12, 13 and 14. The signal significance in the presence of systematic uncertainty is given in Eq. (10) [68, 88] below:
Here x is the fractional systematic error. We note that the expression for the significance given in Eq. 6 and Eq. 10 depend upon three fractions: the signal to background ratio S/B, the fractional statistical uncertainty \(1/\sqrt{B}\) and the fractional systematic error x. To obtain a large significance, we need to have S/B much larger than combined statistical and systematic error.
For the channel, one expects to have a systematic uncertainty of up to 2.5% on integrated luminosity [90]. For this channel, we choose to repeat the analysis with \(x=1\%, \ 2.5\%\) and \(5\%\). For these choices of systematic uncertainty, we scan the ROC curve again looking for the best operating point and found that the earlier point still gives the best signal significance. These are presented in Tables 11 and 12. We see that BP1 and BP2 still remain above \(5\sigma \) for the ANN case with \(2.5\%\) systematic errors while in the case of BDT they remain above \(4\sigma \).
For the case of channel, one has additional uncertainty of about 15% coming from double btagging [89]. We choose to repeat the analysis with \(x=2.5\%, \ 5\%, \ 10\%\) and \(15\%\) and present the results in Tables 13 and 14. All the chosen values of systematic errors are about 5 to 30 times larger than the corresponding fractional statistical error and also comparable to or a few times larger than the S/B ratio. To improve the sensitivity with these choices of systematic errors we scan the ROC curve and find that with a false negative rate \(\sim 0.0001\) and true positive rate \(\sim 0.05\) we get maximum signal significance. We note that with the new operating point the S/B has almost doubled, reaching \(\sim 0.3\) for BP5. This leads to signal significance for BP5 to \(\sim 2\sigma \) for both ANN and BDT models with 15% systematics. We also note that one can improve the significance if one can reduce the large systematic uncertainty coming from double btagging along with the enhancement of S/B ratio by possibly choosing another operating point on the ROC curve. The former will requires a better understanding of the btagging while the latter requires a larger sample of the background events.
7 Summary and conclusion
In this work, we concentrate on the collider search for dark matter in the monoHiggs + final state. This channel along with monojet, monophoton, and monoV final states has garnered substantial interest among experimentalists and theorists alike for dark matter hunting. We have chosen the dark matter to be a WIMPlike scalar that interacts with the SM particles via Higgs mediation. Using a Higgs portal description and at the same time higher dimensional operators, we alleviate the tension between the constraints from direct searches and relic density. One is thus guided to optimal coupling strengths suitable to yield partial cancellation in the direct detection experiments and still reproduce the observed relic density. In addition, the presence of higherdimensional operators may enhance the production crosssection for the \(h+\chi \chi \) state.
Accounting for all the constraints we identify benchmark points that yield large production crosssections for \(h+\chi \chi \) state at the LHC. We would like to point out that the resonant Higgs mass region (\(m_\chi \sim m_h/2\)) is the best possible region to probe in the monoHiggs final state because of the large production crosssection. We choose the and final states.
For the diphoton channel, we go beyond the SM backgrounds with prompt photons usually considered in the literature and estimate the fake/nonprompt photons coming from QCD dijet and \(\gamma +\) jets events. We find that the tail of these backgrounds with large production crosssections can be detrimental for signal background separation in the diphoton channel however strong isolation of photons along with suitable cuts on kinematical observables enable us to get considerable signal significance at the high luminosity (3000 fb\(^{1}\)) LHC. For the \(b\bar{b}\) channel we simulate full backgrounds coming from QCD multijets, \(V+\)jets, and \(t\bar{t}\) final states. We find that the \(b\bar{b}\) channel fares better than the diphoton channel when the systematic uncertainties are ignored.
We consider (almost) an exhaustive list of kinematical variables to perform BDT and ANNbased analysis of the signal significance. We find that both these method leads to a significant improvement in the signal significance for all three benchmark points in the diphoton channel with ANN performing better than BDT. For the \(b\bar{b}\) channel the improvement is even more exciting if we consider only the statistical error.
Noting that the systematic uncertainties can be large [89, 90], we have also estimated their impact on our analysis. The experimental analysis on [89] demonstrates that uncertainty in the doubleb tagging has the highest impact(\(\approx 15\%\)) on the signal strength. For our analysis, we have considered a similar impact on our signal strengths to estimate its effect on the quoted significance and found out that for the most optimistic benchmark choice (BP5) in our analysis, considering the systematic uncertainty of 15% in the doubleb tagging can reduce our signal significance by up to 18 times. For analysis, a systematic uncertainly of 2.5% on integrated luminosity [90] impacts our signal significance by \(\lesssim 25\%\).
One can in principle extend this analysis to other Higgs decay modes such as WW, ZZ and \(\tau \tau \). However, such channels are either prima facie beset with low rates for the viable final states, or have challenges in the reconstruction of the Higgs peak. It may be useful to try the \(\tau \tau \) mode in particular as a confirmatory channel. It remains a challenge to see how much improvement occurs via neural network techniques. We plan to take this up in a followup study.
Data Availability
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The data set simulated for this paper is too large in size and requires a compute intensive pipeline to be useful to anyone else. Thus we choose not to share it.]
Notes
The spinindependent crosssection in direct search, however receives considerably lower contributions from \({\mathscr {O}}_2\), because of velocity suppression for a nonrelativistic DM candidate. We have included both the contributions from \({\mathscr {O}}_1\) and \({\mathscr {O}}_2\) in monoHiggs production at the LHC. However the estimates pertain to \(f_1 = f_2\) which may not be valid in some theoretical scenarios.
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Acknowledgements
We would like to thank Shamik Ghosh for useful discussions and help with the ANN codes. The work of JL and BM is partially supported by funding available from the Department of Atomic Energy, Government of India, for the Regional Centre for Acceleratorbased Particle Physics (RECAPP), HarishChandra Research Institute. The work of RKS is partially supported by SERB, DST, Government of India through the project EMR/2017/002778. JL would like to thank Saha Institute of Nuclear Physics, HBNI and Indian Institute of Science Education and Research, Kolkata for their hospitality where substantial part of this work was done. SB and DB would like to thank RECAPP, HRI for their hospitality, where initial part of the work was done.
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Bhowmik, D., Lahiri, J., Bhattacharya, S. et al. The monoHiggs + MET signal at the Large Hadron Collider: a study on the \(\gamma \gamma \) and \(b\bar{b}\) final states. Eur. Phys. J. C 82, 914 (2022). https://doi.org/10.1140/epjc/s10052022108286
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DOI: https://doi.org/10.1140/epjc/s10052022108286