Mono-Higgs Signature in the Scotogenic Model with Majorana Dark Matter

We study the phenomenology of scotogenic model in the case of Majorana Dark Matter (DM) candidate. This scenario gives important consequences since the parameter space of the model is almost unconstrained compared to the Inert Higgs Doublet Model (or the scotogenic model with scalar DM), and hence, offers new opportunities for discovery at future high energy collider, e.g. the HL-LHC. As an example, we focus on the production of the Standard Model (SM) Higgs boson in association with a pair of dark scalars. Owing to its clean signature, the $\gamma\gamma$ decay channel of the SM Higgs boson is investigated in great detail at both the HL-LHC (at $\sqrt{s}=14$ TeV) and the future FCC-hh (at $\sqrt{s}=100$ TeV). After revisiting the LHC constraints from run-II on the parameter space of the model, and selecting benchmark points satisfying all the theoretical and experimental constraints, we found that scalars with mass up to $140$ GeV ($160$ GeV) can be probed at the LHC (FCC-hh) with a $3$ ab$^{-1}$ of integrated luminosity assuming $5\%$ of uncertainty.


Introduction
The observation of neutrino oscillations in solar, atmospheric, reactor and accelerator experiments remains one clear indication that the Standard Model (SM) is not a complete framework of fundamental physics. The smallness of the observed neutrino masses tells that at the non-renormalizable level we might not have a straightforward answer to the mechanism that bestows neutrinos with mass. One popular mechanism for generating tiny neutrino mass is the so called seesaw mechanism [1][2][3]. However, realistic models based on the seesaw mechanism involve high mass scales that are hard to be probed at collider experiments. Neutrino mass generation through loop diagrams is interesting and give naturally small masses due to loop-suppression factors. Therefore, these models can be probed at present and future colliders. In these class of models, the smallness of neutrino mass has been addressed within frameworks at one-loop [4,5], two loops [6][7][8][9][10], three loops [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26], and four loops [27].
Additionally, experimental evidence of dark matter (DM) has driven many years of investigation shedding light on possible particle and electroweak-size interaction explanations that can reproduce the observed DM relic abundance in the Universe. This paradigm is interesting since it can be tested at colliders such as the Large Hadron Collider (LHC). One of the simplest extensions of the SM consists in incorporating an additional Inert Higgs Doublet Φ with a discrete Z 2 symmetry under which the new scalar is odd, Φ → −Φ, and the other SM fields even [28]. In this case, the lightest odd particle would act as DM candidate. This model, known as the Inert Higgs Doublet Model (IHDM), a aahriche@ictp.it b aarhrib@gmail.com c adil.jueid@sjtu.edu.cn d snasri@uaeu.ac.ae e lagrange2001@gmail.com contains one CP-even Higgs identified as the SM Higgs, an other CP-even Higgs H 0 , one CP-odd A 0 and a pair of charged Higgs H ± , and consequently has a rich phenomenology [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]. For example, the model provides mono-jet, mono-Higgs, mono-Z, mono-photon signatures that can be tested at the LHC and future colliders. It appears from the above phenomenological studies that the IHDM is strongly constrained from direct and indirect DM searches both for low and intermediate DM masses [39,49]. For DM lighter than 62.5 GeV, LHC data also puts severe constraints on the invisible decay of the SM Higgs which in turn translate into constraints on a combination of the scalar parameters of the potential [39,50]. Moreover, collider bounds on the IHDM are obtained as a reinterpretation of neutralinos and charginos pair production both from LEP II [51] and from LHC [52]. From LEP II data, Ref. [51] sets an upper bound on the pseudo-scalar mass, m A 0 (resp m H 0 ) , below 100 GeV (resp 80 GeV) consistent with mass splittings ∆m(A 0 , H 0 ) ≥ 8 GeV. While from LHC data, Ref [52] limits have been derived using a dilepton plus missing energy signature which excludes masses for the exotic scalar up to 62.5 GeV. A recent study [50] showed that the LHC at 13 TeV and 3000 fb −1 luminosity could exclude exotic scalar masses below 83 GeV using the mono-jet channel.
However, If one focuses on a degenerate spectrum of exotic scalars, which is a natural outcome of accidental symmetries in the scalar potential [53], the region of scalar masses above M Z /2 remains unconstrained for splittings between the exotic scalar and the charged scalar mass below 5 GeV. It was also found that LHC searches are not strong enough to probe the degenerate window due to lepton p T requirements. In the light of current collider experimental bounds and the viable region of parameter space in the IHDM, and in order to address the DM nature, one has to go beyond this minimal extension of the SM. For instance, extending the IHDM by three right handed Majorana fermions may provide a possible solution to the problem of over-constrained quartic couplings and, on the other hand, give rise to small neutrino masses generated through one-loop diagrams. In the present work, we build on a recent phenomenological analysis in the framework of scotogenic model [30] performed by some of us [54] 1 . The scotogenic model is a SM minimal extension where the SM neutrinos obtain naturally small masses at the one-loop order. In order to achieve this, the scalar potential has to be augmented by an inert complex scalar doublet with a small mixing quartic coupling to the SM Higgs. Due to the new Yukawa couplings, the scalar potential has an enhanced SU (2) symmetry acting only on the exotic scalar and the new right-handed neutrino fields 2 . Because of this global symmetry, the quartic coupling λ 5 between Φ and the SM Higgs, that is responsible for the mass splitting between the CP-odd and CP-even neutral scalars, does not run and thus can naturally be very close to zero, which naturally yields small mass for the active neutrino. In contrast to the λ 5 term in the potential, there is another coupling between Φ and the SM Higgs, λ 4 , which has non-vanishing β-function even if the coupling is chosen to be zero at some very high energy scale 3 . This region of parameter space corresponds also to a spectrum of a compressed exotic scalar/pseudo-scalar spectrum that leads to interesting collider signatures which are difficult to probe in the IHDM with current and near-future data.
In [54,55], the DM candidate is considered to be the lightest Majorana fermion N 1 , which implies significant difference in the parameter space compared to both IDHM or the scotogenic model with scalar DM candidate. For instance, in case of scalar DM candidate, the null results from searches in direct detection imply that the coupling combination λ L = λ 3 + λ 4 + λ 5 is extremely suppressed to be suppressed, while for the fermionic DM case this constraint do not affect the scalar potential's parameters. In the fermionic DM case, the CP-odd and CP-even scalars decay predominantly into SM neutrino and the Majorana fermion N i , and therefore they can not be seen at colliders, i.e., they behave as dark scalars. In other words, both the IHDM and the scotogenic model provide identical signatures at colliders but with different event yields since they have different parameter space.
As pointed above, the production of dark scalars can lead to several signatures dubbed as mono-X. The most known of and studied in the literature is the mono-jet signature. However, within the framework of scotogenic model, the mono-jet signature is only sensitive to the masses of the particles produced in the final state and not to the scalar couplings such as e.g. λ L . The reason for this is that the mono-jet cross section gets the most important contribution from diagrams with the exchange of Z-boson and involving gauge couplings only. Therefore, alternatives to the mono-jet channel need to be exploited. In this regard, we focus the scope of this work on the mono-Higgs channel in the diphoton final state at both the HL option of the LHC at 14 TeV and a Future Circular Collider (FCC-hh) at 100 TeV. This signature is an excellent probe of new physics and DM [61][62][63][64]. We stress out that searches of DM in events with Higgs and missing transverse energy have been carried out by the ATLAS and the CMS collaborations [65][66][67][68][69][70][71] using τ + τ − , γγ and bb decay channels of the Higgs boson. These searches yielded null results which were used to put severe constraints on simplified models of DM production at hadron colliders. However, these limits do not apply to our model due to the smallness of the corresponding production cross sections of DM particles in association with a Higgs boson. In this work, we will follow closely the analysis setup of reference [66].
The outline of the paper is as follow. In the second section we review the scotogenic model and all the theoretical and the experimental bounds in the entire degenerate window, where all exotic scalars have approximate equal masses.
We then carry out a complete comparison of this model to the latest LHC run II data, and expose the available parameter space in the third section. In the fourth and the fifth sections, we present a full sensitivity analysis to a mono-Higgs signature within this framework.

Model
In this model, the SM is extended by one SU (2) L inert Higgs doublet and three singlet Majorana fermions N i ∼ (1, 1, 0), i = 1, 2, 3. These new particles are odd under a Z 2 symmetry, whereas the SM particles are even. In this setup, the most general gauge-invariant, and renormalizable scalar potential that is invariant under CP-and Z 2 -symmetries has the form The electroweak symmetry breaking occurs due to the non vanishing Vacuum Expectation Value (VEV) acquired by the SM Higgs doublet, through its neutral component, while the Z 2 -odd inert doublet Φ does not develop a VEV as its quadratic term has positive curvature. The SM Higgs and the inert doublets can be parametrized as The Lagrangian that involves the Majorana fermions can be written as whereL i is the left-handed lepton doublet and = iσ 2 is an antisymmetric tensor. Note that the absence ofL i HN j in the Lagrangian (3) is due to the imposed discrete Z 2 symmetry. The parameters λ 1 and µ 2 1 in (1) can be eliminated in favor of the SM Higgs mass and its VEV (υ = 246 GeV), which is considered at one-loop level à la DR scheme [72].
After EWSB, three degrees of freedom are absorbed by the longitudinal gauge bosons and we are left with two CP-even scalars (h 0 , H 0 ), one CP-odd scalar A 0 and a pair of charged scalars H ± . Their tree-level masses are given by: The neutrino mass can be obtained at the one loop level via the diagram in Fig. 2.1. The neutrino mass matrix elements [30,73] are given à la Casas-Ibarra form [74] by Fig. 1 Feynman diagram responsible for the neutrino mass.
In this model, the smallness of neutrino mass is a consequence of the tiny mass splitting in the inert neutral sector. In other words, the following ratio := |λ5|υ 2 is much smaller than unity. Then, after the expansion over , the parameter G k in (5) is given by with According to the Casas-Ibarra parameterization, the coupling h can be written as where D √ is an orthogonal rotation matrix (m 1,2,3 are the neutrino eigenmasses), and Uν is the Pontecorvo-Maki-Nakawaga-Sakata (PMNS) mixing matrix [75].
The parameters of the model are subject to constraints from the measurements of the mixing angles and mass-squared differences [76] which we implement in our analysis.

Constraints
The parameters of the scalar potential have to satisfy a number of theoretical and experimental constraints. On the theoretical side, we should require perturbativity of all the quartic couplings of the scalar fields. In addition, the scalar potential has to be bounded from below in all directions of the field space. For that, the necessary and sufficient conditions are given by [77] However, these constraints do not ensure the vacuum stability since the inert vacuum may not be the global minimum of the potential, and to guarantee this feature we should also impose the condition [78]. Another set of constraints comes from the tree-level perturbative unitarity which which should be preserved at high energies in variety of processes involving scalars or gauge bosons. At high energies, using the equivalence theorem, we replace the longitudinal W and Z bosons by the corresponding charged and neutral Goldstone bosons respectively. Therefore, we are left only with pure scalar scattering amplitudes. Computing the decay amplitudes for these processes, one finds a set of 4 matrices with quartic couplings as their entries. The eigenvalues for those matrices have to be smaller than 4π [79,80].
Electroweak precision tests (EWPT) is a common approach to constrain physics beyond SM by using the global fit through the oblique S, T and U parameters [81]. In the scotogenic model, the new gauge-inert interactions will induce non-vanishing contributions to the oblique parameters ∆T and ∆S [82] 4 . To study the impact of the EWPT 4 The corrections to the U -parameter in the IHDM are very small. Therefore, we assume that ∆U = 0 in the present analysis. on the mass splitting between the pseudo-scalar (A 0 ) and the charged Higgs boson (H ± ), we have to minimize the function with Sexp = 0.06 ± 0.09, Texp = 0.10 ± 0.07 are the experimental values of the S and the T parameters, σ S,T are their corresponding errors, and ρ ST = +0.91 is their correlation. The constraints from EWPT can easily be satisfied in regions of the parameter space where the mass splitting between the neutral and the charged components of the inert doublet is small (for light scalars) or where the scalars are very heavy regardless the values of their mass splittings.
In this model, constraints from neutrino masses and mixings imply extremely small values of λ 5 . Therefore, the only parameter that is directly affected by EWPT constraint is λ 4 . This is can easily be seen from the left panel of Fig.   2 where we display the 1-, 2and 3-sigma allowed regions plotted in the (m H ± , λ 4 ) plane. We can see that, for e.g.  (left) and from Higgs signal strength measurements in the charged Higgs mass and λ 4 (right). The red, yellow, and blue contours correspond to the 68%, 95%, and 99.7% allowed regions respectively. Two different benchmark points corresponding to (m H ± , m A 0 ) = (95, 100) GeV, (95, 160) GeV are shown as black markers in the right panel. In the right panel, the white marker corresponds to the allowed value of λ 3 chosen in the rest of the analysis. Due to the constraints from neutrino mass and mixing parameters, we have taken λ 5 = 0.
Moreover, the gauge bosons decay widths are well measured [83], and must not be modified by any new interactions.
Therefore, one needs to impose the conditions the decay channels of W and Z gauge bosons into inert particles closed.
The new Yukawa interactions in (3) lead to lepton flavor violating (LFV) decay processes that arise at one-loop level with the exchange of charged Higgs H ± and Majorana fermions N k particles. The branching ratio of the decays α → β + γ and α → β β β are given in the literature [84], and should be in agreement with the available experimental constraints [83].
with the superscript i refers to the production channel of the SM Higgs boson. In the right panel of Fig. 2, we plot the allowed regions from Higgs boson signal strength measurements in the (λ 3 , m H ± ) plane. We can see that µγγ constrains strongly the 2D parameter space. There are two notable regions for m H ± < 125 GeV; the first one is centered around λ 3 0 while the second one is a small segment corresponding to λ 3 ∈ [0, 9] and m H ± < 100 GeV.
This unsurprising because even for large positive values of λ 3 , the Rγγ ratio in our model can still agree with data within the +10% of experimental uncertainty reported in e.g. the recent CMS analysis [88].
In this study, we assume that the lightest right-handed Majorana neutrino is a DM candidate as was done in ref [54]. 1.01m N1 and m N3 1.02m N1 .
In this case, co-annihilation becomes important and, therefore, is included in our analysis. Co-annihilation with inert scalars, which give rise to final states such as ± γ, is sub-leading due to the smallness of the electromagnetic coupling This can happen despite the absence of a tree level HN 1 N 1 coupling which arises at the one-loop order. We estimated spin-independent (SI) scattering cross section of N 1 off a nucleon N and subject it to constraints from the searches performed by Xenon1T [91]. One notices that constraints from direct detection are easily satisfied in our model due to the smallness of HN 1 N 1 coupling. We refer the reader to [54] for more details about the DM constraints in our model.

LEP constraints
Multiple searches for supersymmetric particles at e + e − collisions has been carried out by several collaborations [92-95] for center-of-mass energies of 183-209 GeV. The searches focused on charginos and neutralinos pair production in events with two or three leptons and large transverse missing energy. Several interpretations in terms of models containing charged and neutral scalars have been made. Ref. [51] made a comprehensive re-interpretation of neutralino pair production (χ 0 1 χ 0 2 ) to constrain the production of H 0 A 0 in the IHDM and got a limit m A 0 > 100 GeV for large mass splitting. Pair production of charginos (χ + 1 χ − 1 ) was analyzed to put constraints on H ± H ∓ production in a DM model with TeV scale colored particles [96] and in the compressed IHDM [53].
In this section, we study the impact of LEP searches on the parameter space of our model. For instance, LEP put strong bounds on the pair production cross section of lightest neutralino. However, the LEP limits on neutralino pair production do not apply in the considered scenario of the scotogenic model, because the tiny value of the coupling λ 5 (of order 10 −8 -10 −10 ) required by the smallness of neutrino masses forbids off-shell decays, such as A 0 → H 0 Z → H 0 , and therefore yields an undetected final state. However, limits from chargino pair production can be applied to our model. Two processes can be used for such constraints; e + e − → H + H − and e + e − → H 0 A 0 . The latter contribute, if λ 4 < 0 and ∆ H ± H 0 = m A 0 ,H 0 − m H ± > me,µ, through off-shell decays. This contribution is proportional to (∆ H ± H 0 ) 5 and, hence, is very small. Therefore, charged Higgs pair production is the only process to which the exclusion limits from LEP searches can be used to constrain our model. We consider the results of searches carried by Opal [94] at √ s = 208 GeV and L = 680 pb −1 of integrated luminosity to derive conservative limits on the model parameter space i.e. by assuming that the efficiency of the selection is 100% 5 The pair production of the charged particle depends on new Yukawa couplings and the gauge couplings. The first contribution is proportional to in the case of degenerate Majorana fermions. Because, for λ 4 ≥ 0, the Charged Higgs boson decays with 100% branching ratio into N k , the limits from charginos searches can be used to constrain both the charged Higgs boson and the mass splitting ∆ H ± defined by We consider two scenarios for the new Yukawa couplings; 1) where the Yukawa matrix is chosen as follows 75 25.14 − i0.57 −1.12 − i2.49 40.87 + i0. 24 3.70 + i0.62 1.10 + i3.88 −44.20 + i0.14    (13) which we called first scenario and 2) the second scenario where the h ek couplings take the highest values allowed by all the theoretical and experimental constraints (h e1 = −0.026 + i0.042, h e2 = 2.22 − i0.081, h e3 = 0.32 − i0.0098).
In the second scenario, the most important contribution comes from |h e2 |.
We estimate the r 95% ratio defined by where, in the second line of eq. (14), we used BR(H ± → ± N k ) = 1. A point in the parameter space is excluded if the corresponding r 95% is larger than 1. In Fig. 3, we depict the exclusions from charginos pair production on the (m H ± , ∆ H ± ) plan. As can be seen from the left panel of Fig. 3, all points are allowed by LEP searches. However, in the second scenario, one notices that the model is excluded for m H ± < 100 GeV. A small window corresponding to ∆ H ± < 5 GeV and 90 GeV < m H ± < 100 GeV is still allowed by these constraints.
The LHC searches used in this analysis are displayed in Table 1. The model parameter space can be affected by the LHC searches displayed in Table 1   In the degenerate window, since the decay A 0 → H 0 Z 0 is kinematically forbidden, the scalar/pseuodoscalar can be produced in association with a charged scalar which subsequently decays to a charged lepton and a right-handed neutrino. While the scalar and pseudoscalar may only decay invisibly; we obtain a signal with a single lepton and large missing E miss T . In this channel the most sensitive LHC search comes from the work in [97] that searches for SUSY in a final state with one isolated lepton. In the case where the exotic scalars are pair produced, in the degenerate region, their decays lead only to missing E miss T and one can tag this channel with a mono-jet from initial state radiation. In these cases, LHC searches with photons and jets are the most sensitive, with the largest amount of missing E miss T when the scalar/pseudoscalar mass approaches the right-handed neutrino mass, and this is where the bulk of the exclusion lies in as can be seen from Figure 4 after the inclusion of all relevant LHC searches given in Table 1. Following the results of the re-interpretation of LHC searches of new physics that we have shown in Fig. 4, we choose the following benchmark points for the mono-Higgs study;

Mono-Higgs signature
In this section, we describe different aspects of our analysis. First we discuss the contribution to the signal process as well as the possible backgrounds and the corresponding cross sections. Then, we discuss in depth the phenomenological setup used in our analysis and event selection.

Signal and backgrounds
In this model, mono-Higgs production proceeds through two different processes, i.e and The corresponding Feynman diagrams are depicted in of the total cross section. This is unsurprising since this contribution occurs at the tree level and is enhanced for large values of λ L . Using simple power counting, one notices that the total cross section behaves as The contribution of diagram 5-d is proportional to the squared of the HN i N j coupling which is one-loop induced [54] and it is expected to be very small. In this regard, we define the ratio R by which gives a rough estimate of the relative contribution of diagram 5-d to the signal cross section where only the leading contribution to SSH production ( |λ L | 4 ) is included. We show this ratio in Fig. 6 as function of the mass splitting ∆m N H 0 = m H 0 − m N k with a color map showing |λ 3 |. One can see that this ratio can only be important for very small values of λ L , i.e |λ L | < 0.1. Given that this region is not interesting from phenomenological point of view as it yields very small cross sections (see Fig. 7), we conclude that the contribution of diagram (5-d) can be safely neglected.
The cross sections for the mono-Higgs production are depicted in the left panel of Fig. 7. As expected, one can see that the cross section is pretty small for the LHC at √ s = 14 TeV with the maximum being σmax 53 fb for m H 0 = 100 GeV which increases by about an order of magnitude at the FCC-hh with 100 TeV. Since the mass splitting ∆ H ± H 0 can be as large as 100 GeV, the dark neutral (pseudo)-scalar does not always decay exclusively into an invisible final state. Therefore, in order to estimate correctly the number of events in a signal benchmark point, one has to scale correctly the corresponding cross section by BR(H 0 → invisible) 2 . We show the Dark scalar branching ratios as a function of m H 0 in Fig. 7 (right). We can see that, unless m H 0 > 190 GeV, the invisible decays of H 0 have always a branching fraction larger than 90%.
The γγ decay channel represents a very clean signature of the mono-Higgs final state boson despite the smallness of the corresponding branching ratio (which is about 0.23%). In this case, the following backgrounds have to be considered gg → H → γγ: this is the dominant background. The missing energy is due to the mis-identification of soft QCD radiation. However, it can be substantially suppressed by requiring high missing transverse energy as we will show later on.
pp → ZH: where the Z-boson decays to a pair of neutrinos is an irreducible background. The suppression of this background can be achieved by applying specific selection criteria, e.g on the transverse mass of the (Higgs,  m H ± = 95 GeV, λ 5 0 pp → W ± H: where the W ± -boson decays into ± ν where the charged lepton escapes the detection, i.e not passing the selection threshold. At the LHC, the charged lepton efficiency is high and, therefore, we expect that this background will have small contribution. pp → V γγ: where the V = Z-boson decays invisibly and the V = W -boson decays leptonically. The Zγγ background is irreducible contrarily to the W γγ. The contribution of the latter can be reduced by imposing a lepton veto in the selection procedure. Both the two backgrounds have weaker γ spectrum and, therefore, their contribution can be weakened by strong requirements on the p γ T and the invariant mass of the γγ spectrum. pp → V γ: this background is similar to V γγ. Table 2 Cross sections for processes contributing to the Higgs+E miss T background. The numbers outside (inside) the brackets refers to the rates at 14 (100) TeV. Details about the computation are explained in the text. Here, σ × BR refers to σ(gg → H) × BR(H → γγ) for gg → H, and to σ(pp → ZH) × BR(H → γγ) × BR(Z →ν ν ) for ZH, to σ(pp → W ± H) × BR(H → γγ) × BR(W ± → ± ν ) for the case of W ± H and to σ(pp → W ± + nγ) × BR(W ± → ± ν ) + σ(pp → Z + nγ) × BR(Z →ν ν ) for V + nγ, n = 1, 2. to reduce the contribution of this background either by defining some kinematical variables or use azimuthal separation between the reconstructed Higgs candidate and the missing transverse energy. These features will be discussed briefly in the next subsection.  Fig. 8 The acceptance times the efficiency (A × ) for the signal after each step of the event selection as a function of the dark Higgs mass for √ s = 14 TeV (left) and √ s = 100 TeV (right). We show A × after the "2 Photons" selection step (blue), for the events passing the "Photon PT" selection (red), after the photon isolation selection denoted by "Ratio Tag" (purple) and for events in which the invariant mass of the diphoton system falls in the interval mγγ ∈ [110, 160] GeV (rose). The efficiency for the two signal regions are shown in green (ATLAS signal region) and in black (tight selection).

Phenomenological setup and Event selection
The cross sections of the background processes are depicted in Table 2   Events for both the signal and the backgrounds were generated using Madgraph5_aMC@NLO and Pythia8 [130] at LO in QCD. Background events involving the Higgs boson were generated and decayed with Pythia8 while V γ and V γγ events were generated using Madgraph5_aMC@NLO including the leptonic decays of the massive electroweak gauge bosons. The γγ+jets events were generated with Pythia and normalized to their rate at NLO. Since the rate of this process is huge, and most of the events will be vetoed in the initial selection, events are generated with a p min T,γ = 70 GeV, and |η γ | < 2.5. Events for gg → H were scaled by a K-factor of 3.2 using the results of Sushi while V H events were scaled by a factor of 1.6. All the background events were showered with Pythia. Delphes3 was used for fast detector simulation [131].
The analysis of events was carried out at the detector level using implemented efficiencies, and mis-identification rates in Delphes where the parameters are tuned for the ATLAS experiment and extrapolated for a future FCChh [132]. Events pass a preselection stage with all the objects (leptons, jets, photons and missing E T ) are kept. The Acceptance times the efficiency (A × ) is depicted in Fig. 8  The cutflow for the event selection is shown in Tables 3 and 4. Events are selected if they contain at least two photons with p γ T > 25 GeV and |η γ | < 2.37. This selection is denoted by "Photon PT" in Fig. 8 and Tables 3 and 4. Besides, we do not impose any requirement on the multiplicity, hardness and flavor compositions of jets or the multiplicity of charged leptons. The photons that pass the initial selection will be subject to further isolation cuts (as in [66]), and the photon candidates are ordered by their transverse momentum. The two leading photons are used to reconstruct a Higgs candidate. Further, The ratio of the transverse momentum to the invariant mass p γ T /mγγ is required to be larger than 0.35 (0.25) for the leading (sub-leading) photon. Furthermore, a cut on the invariant mass of the diphoton system is imposed; namely events are selected if 110 GeV < mγγ < 160 GeV. But in some cases, events in the γγ and γ+jets backgrounds contain large fake transverse missing energy, which is due to the fact that, in such events, the vertex with larger p 2 T (where the sum runs over all the tracks) is not the primary vertex but the one coming from pile-up. 6 Both ATLAS and CMS collaborations used sophisticated methods to reduce the contribution of pile-up to missing transverse energy. The ATLAS collaboration has defined a new variable S E miss T defined by where i correspond to all the objects (photons, jets, and leptons) used to construct the missing transverse energy. , pγγ )|). By requiring that such quantity is larger than 2.1, only events where the reconstructed Higgs and missing transverse energy are back-to-back are selected. Therefore, the contribution from e.g. γγ backgrounds is significantly reduced.
We compared between the approaches used by ATLAS and CMS to reduce the contribution from γγ+jets backgrounds, on our benchmark points; and we found that they produce results that agree with each other. We will follow the ATLAS selection criteria throughout this study. We define two signal regions; the mono-Higgs signal region (denoted by ATLAS signal region in this paper) and a tight signal region. The kinematical quantities and selection rules are displayed in Table 5. In the two signal regions, we require that the invariant mass of the diphoton system falls inside the interval [115,135] GeV. Furthermore, we require that events contain no lepton (either electron or muon) with p T > 10 GeV and |η | < 2.5.

Tight selection
At √ s = 14 TeV, we can see from Table 3 that the signal-to-background ratio (S/B) can go from 10 −5 (after the first selection) to about 1 in the mono-Higgs signal region. Besides, the efficiency of the signal for m H 0 = 100 GeV is A × 25% in the ATLAS signal region. For the FCC-hh at 100 TeV, the signal-to-background ratio can go up to 0.24 in the tight signal region. If one requires, in addition to the tight selection rules, that p γ T > 60 GeV (for the leading photon) and p γ T > 50 GeV (for the sub-leading photon), the significance can increase to around 20 but the statistics goes down by about an order of magnitude.

Results and Discussion
In Figs. 9 and 10, we display the normalized distributions for some key observables used in the signal-to-background optimization. We can see that the p γ T of the leading photon is stronger for the signal than in the backgrounds with a slightly high peak value for the signal case. The transverse momentum of the diphoton system (top right panel of Figs. 9 and 10) is a good discriminator. This is can be understood as follows; the Higgs candidate (reconstructed from the two photons) is produced in association with heavy particles (resulting in a hard missing transverse energy spectrum) and therefore the corresponding recoil imply a harder p T than in the backgrounds (especially SM Higgs backgrounds and γγ+jets). The same observation applies to the E miss variables; in the ATLAS paper, γγ and γ+jets events can still have some contribution to these variables (in the hard region) due to the presence of pile-up events (which are not taken into account in our analysis). However, the number of events is still not very important; e.g. about 10 events for S E miss T > 7 at √ s = 13 TeV and L = 36.5 fb −1 . We can assign the differences in the modeling to an additional systematic uncertainty (see below).
To quantify the discovery potential of the signal, we estimate the significance defined by [133,134]  the same PDF set (used in the calculation of the cross section) with the variations due to alternative PDFs 7 . The size of the envelope spanned by all the variations define the uncertainty due to PDF+αs. In the signal region, such uncertainties are very small for SM Higgs backgrounds and can be of order 1.2-2.5% [66]. An additional component of the theory uncertainty comes from the calculations of H → γγ branching ratio which is of order 1.73% [136].
The uncertainties on the non-resonant backgrounds can be larger than on the resonant backgrounds. In the analysis of [66], they were estimated directly from data and were of order 0.1-9.8% in the 110 < mγγ /GeV < 160 region. On the other hand, there are three major experimental uncertainties in the γγ + E miss T final state; Luminosity, Photon identification efficiency and pileup reweighting. The total uncertainty on the background contribution including both resonant and non-resonant processes was estimated by the ATLAS collaboration to be about 15% 8 . In this work, we compute the signal significance taking into account statistical uncertainty only or statistical uncertainty in addition to a systematic uncertainty of order 5%, 10% and 20%. In Fig. 11, we plot the significance of the signal process at L = 3000 fb −1 as a function of the dark Higgs mass m H 0 for both the HL-LHC and FCC-hh. We show the significance in the ATLAS mono-Higgs signal region (left panels) and in the signal region defined in our paper by the tight selection (right panels). We can see that masses up to 140 (160) GeV can be probed at the LHC (FCC-hh) if one assumes a 5% of total error. 7 according to the recommendations of PDF4LHC, the central PDF set is Nnpdf30 while the two alternatives are Ct10 and The ATLAS collaboration reported on the total error without specifying the contribution of systematic uncertainties. However, the CMS collaboration [71] reported both the contribution of the statistical error which is the dominant one and the systematic error to the total uncertainty. In the signal region (defined as the High-p miss T in the CMS paper), the total systematic uncertainty is 1.6%.

Conclusions and Outlook
In this work, we carried out a complete study of the mono-Higgs signature in the scotogenic model in the limit of degenerate scalars with a focus on the γγ final state at both the LHC-HL and FCC-hh. After revisiting the collider constraints from LEP and LHC run-II, we have shown that a considerable region of the parameter space is still allowed which is already excluded in general scenarios. Using the most significant benchmark points, we have shown that this model can be probed at the LHC-HL and the FCC-hh in the H(→ γγ) + E miss T channel with 3 ab −1 of integrated luminosity. The final state we considered has a small rate compared to the other production mechanisms of Majorana DM in the model, however, due to the cleanliness of the γγ decay channel and the high efficiency of photon identification at hadron colliders, we have shown that it can be used to probe the model with the compressed spectrum. In summary, we have found that scalar masses up to 150 (160) GeV can be probed at the LHC (FCC-hh) assuming a 5% systematic uncertainty. We stress out, however, that these results can be significantly improved by the use of multivariate techniques such as boosted decision tree or neural networks and, by including other decay channels of the SM Higgs boson with larger branching fractions. We point out that the importance of the Mono-Higgs signature to probe the scalar coupling λ L which can't be probed using e.g. mono-jet searches of DM.
In the limit of compressed spectrum, i.e., for λ 5 0; and for m H ± = 95 GeV, the dark scalars decay exclusively to a SM lepton (charged lepton or neutrino) and a Majorana fermion. Therefore, the mono-Higgs analysis itself  Fig. 12-a). The ATLAS and the CMS collaborations have been searched for these decays channels at √ s = 8 ⊕ 13 TeV (see e.g. [137,138]). The null results were used to put severe limits on the LFV Higgs decays, i.e BR(H → µτ ) < 0.25% and BR(H → eτ ) < 0.61% [138]. Possibly observing one or more of these decays channels can be used to constrain one or several combinations of the new Yukawa coupling (These processes are also quadratically dependent on g HH ± H ± ∝ λ 3 which can still have large values).
-Precision measurement of lepton pair production at Lepton Colliders. In the scotogenic model, a pair of charged leptons (e + e − → + i − j ) can be produced with decent rate at center-of-mass energies below or above the Z-boson pole ( Fig. 12-b). Of these processes, the ones with i = j are particularly interesting since they have almost zero cross section in the SM. Therefore, measurement of both inclusive as well as differential rates in + i − j can be used to extract several combinations of the new Yukawa couplings.
-Direct production at lepton colliders. The production of inert scalars and Majorana fermions are the most sensitive channels on the new Yukawa couplings (They can also be used as a model discriminators, see e.g. [139]). Production of Majorana fermions (in association with photons, leptons or Z-bosons) either in the prompt mode or from the decays of dark scalars is possible in the scotogenic model (in Fig. 12-c we display an example of a Feynman diagram for N j N k γ production).
description of the analyses used in the reinterpretation effort. These analyses, implemented in Checkmate, are listed in Table 1.
-atlas_conf_2016_050 [97]: the ATLAS collaboration has been searched for new phenomena in the final state consisting of 1 + (b)jets + E miss T at 13.3 fb −1 of luminosity. These searches were focused on the supersymmetric partner of the top quark, and also on DM production in association with a pair of top quarks. Upper bounds on the stop quark mass (for different assumptions regarding its decay branching ratios) were put. Furthermore, limits on DM simplified models were obtained and presented on a plane of DM mass and pseudo-scalar mediator mass for a coupling g DM = 3.5. These searches were used to probe supersymmetric models with gauge-mediated supersymmetry breaking, where neutralinos decay into a photon and a gravitino. Limits were put on the mass of a degenerate gluino state; i.e. mg > 1800 GeV for a large range of neutralino (the Next-to-Lightest Supersymmetric Particle -NLSP-which is a mixture of higgsino and bino) masses and mg > 2000 GeV for high neutralino mass. -atlas_1712_02332 [102]: The final state consisting of (2-6) jets +E miss T at the luminosity 36.1 fb −1 recorded by the ATLAS detector, was used to search for squarks and gluinos. 95% CL lower limits on gluino masses (mg > 2.03 TeV) and squark masses (mq > 1.55 TeV) were placed.  -cms_sus_16_039 [107]: Using the data recorded by CMS at 13 TeV and luminosity 35.9 fb −1 , the final state of multileptons is considered to search for neutralinos and charginos that are weakly produced. In simplified SUSY models, these negative searches were interpreted as exclusions on the mass interval 180-1150 GeV.
-cms_sus_16_048 [108]: Using the same CMS dataset, the final state consisting of two low-momentum, oppositely charged leptons with missing transverse momentum is used to search for new physics. Negative searches results implied exclusions on the wino-like masses up to 230 GeV for 20 GeV mass difference relative to the lightest neutralino, and the higgsino-like masses are excluded up to 168 GeV for the same mass difference. In addition, the top squark masses up to 450 GeV are excluded for a mass difference of 40 GeV relative to the lightest neutralino.
Several processes in the scotogenic model are sensitive to these searches. These processes lead to different final states; + − + E miss T , 1γ + E miss T , mono-jet, and 1 + jets + E miss T among others. First, Charged Higgs boson pair production will lead to a final state composed primarily of 2 isolated charged leptons and a large E miss T . In some cases, where one charged lepton escapes the detection, this final state can be triggered as a 1 + jets + E miss T where the jets are produced in initial state radiation. For small mass splittings (∆ H ± = m H ± −m N k ), the missing transverse energy triggered by the Majorana fermion is even larger and thus gives high sensitivity. Production of a CP-odd (CP-even) dark scalar in association with a charged Higgs boson (pp → H 0 H ± ) leads exclusively to 1 + jets + E miss T . We also considered the mono-V process with V = W, Z which contributes to a final state composed of multi-jets (n ≥ 2) and large transverse missing energy. On the other hand, mono-photon and mono-jet processes are also possible in this model. For mono-jet production, we generated S 0 S 0 + n jets (S 0 = H 0 , A 0 ) using Madgraph5_aMC@NLO [117] with jet multiplicity up to 3 jets. We matched these inclusive samples using the MLM matching scheme [143]. Pythia 8.155 [144] was used for showering and hadronization. We have added by hand the PDG codes of the three Majorana fermions (which should be considered as invisible particles) to the HCAL modules of the Delphes card.