Exploring inert scalars at CLIC

We investigate the prospect of discovering the Inert Doublet Model scalars at CLIC. As signal processes, we consider the pair-production of inert scalars, namely e+e− → H+H− and e+e− → AH, followed by decays of charged scalars H± and neutral scalars A into leptonic final states and missing transverse energy. We focus on signal signatures with two muons or an electron and a muon pair in the final state. A number of selected benchmark scenarios that cover the range of possible collider signatures of the IDM are considered. For the suppression of SM background with the same visible signature, multivariate analysis methods are employed. For several bench-mark points discovery is already possible at the low-energy stage of CLIC. Prospects of investigating scenarios that are only accessible at higher collider energies are also discussed.


Introduction
A number of astrophysical observations based on gravitational interactions point to the existence of dark matter (DM) in the Universe, which can not be described with the Standard Model (SM). One of the simplest extensions of the SM, which can provide a dark matter candidate is the Inert Doublet Model (IDM) [1][2][3]. The scalar sector of the IDM consists of two SU(2) doublets where one is the SM-like Higgs doublet, Φ S , while the other is called inert or dark doublet, Φ D . After electroweak symmetry breaking the sector has five physical states: apart from the SM Higgs boson h it has two neutral ones, H and A, as well as two charged scalars, H ± . A discrete Z 2 symmetry prohibits the inert scalars from interacting with the SM fermions through Yukawa-type interactions and makes the lightest neutral scalar, chosen to be H in this work, a good dark matter candidate.
In this work we study the potential of CLIC running at three energy stages as a discovery machine for the IDM scalars. We consider neutral scalar (HA) and charged scalar (H + H − ) pair-production at center-of-mass energies of 380 GeV, 1.5 TeV and 3 TeV. We explore a set of benchmark points (BPs) proposed in [4], which satisfy all the recent experimental and theoretical constraints, provide the neutral H boson as the dark matter candidate (m H < m H ± , m A ), and span the inert scalar mass range from about 50 GeV to 1 TeV. Earlier analyses of the IDM at colliders were done in [2,.
The paper is organized as follows. The structure of the IDM scalar sector and the considered benchmark points are briefly described in section 2. In section 3 the analysis strategy and simulation tools are described. In section 4 results on the possible measurement of low-mass benchmark points at the first stage of CLIC, at 380 GeV, are presented, while section 5 discusses the prospects for heavy inert scalar production at high-energy CLIC. Finally, the conclusions are given in section 6.

JHEP07(2019)053 2 Inert Doublet Model
The scalar sector of the IDM consists of two scalar doublets, the SM Higgs doublet Φ S with SM-like Higgs boson h and the inert doublet Φ D with four inert scalars H, A, H ± . A discrete Z 2 symmetry is imposed under which the SM-like Higgs doublet Φ S and all the other SM fields are even, whereas the inert doublet Φ D is odd. As a result, inert scalars do not interact with the SM fermions through Yukawa-type interactions, and the most general renormalizable scalar potential for the IDM is given by (2.1) A more detailed discussion of the potential and physical parameters can be found in section 2 of [23].
Due to the exact Z 2 symmetry, the lightest neutral scalar H (or A) is stable and thereby may serve as a good dark matter candidate. In this work we choose H to be the lightest particle (choosing A instead of H as the lightest particle changes only the meaning of λ 5 → −λ 5 ). After fixing the SM-like Higgs boson mass to m h = 125.1 GeV and the vacuum expectation value of the SM-like doublet to v = 246 GeV (the SM value) we are left with 5 free parameters, which we take as m H , m A , m H ± , λ 2 , λ 345 , (2.2) where λ 345 = λ 3 + λ 4 + λ 5 determines the Higgs-DM coupling, while λ 2 corresponds to couplings within the dark sector.
To study the prospects of IDM scalar measurement at CLIC we consider a set of benchmark points proposed in [4], and listed in table 1 for the low-mass benchmarks accessible at 380 GeV, and in table 2 for the high-mass benchmarks accessible only at higher collider energies of 1.5 and 3 TeV. These benchmarks were selected from a larger set of points in the IDM parameter space, which were found to be in agreement with all the theoretical and current experimental constraints. Points corresponding to different assignment of masses and couplings were selected in the parameter range interesting in view of future linear e + e − collider searches.
We refer the reader to [4,23,29] for a detailed discussion of the theoretical and experimental constraints and the benchmark selection; comments on the impact of future XENON-nT measurements and prospects of testing the IDM model at the LHC can be found in [4]. We summarize the crucial points of the discussion presented there below.
Theoretical and experimental constraints. As theoretical constraints, positivity of the potential [31], the condition to be in the global inert vacuum [32] and perturbative unitarity [33,34] have been imposed. We also require agreement with electroweak precision tests [35] via oblique parameters [36][37][38][39], zero contributions to electroweak gauge boson widths from inert particles by kinematically forbidding decays W ± → AH ± , HH ± , Z → AH, H + H − [40], and agreement with recasts of LEP and LHC searches [6,20] [56]. The values of the λ 345 parameter for scenarios BP8 and BP22 were slightly modified with respect to [4], to be consistent with the most recent results on the relic density [54].
lower limit of 70 GeV for the charged scalar mass [41]. 1 We also set a hard upper cutoff on the charged scalar life-time to avoid constraints from long-lived charged particle searches; a more detailed study of these constraints has recently been presented in [43]. We make use of recent LHC findings to constrain the total width of the Higgs particle [44], its invisible branching ratio [45] and the branching ratio h → γγ [46]. 2 In order to not overclose the universe the relic density of dark matter candidate H was required to be at most within a two sigma range of the value recently published by the Planck experiment, i.e. Ω c h 2 ≤ 0.1224 [54]. Relic density has been calculated including all relevant channels, in particular decays into virtual gauge bosons. The proposed benchmarks have been selected 1 In recent work [42], an additional recast has been presented which uses 13 TeV LHC data. We found that our benchmark points are not constrained by the limits derived in that reference. 2 Additional tests of agreement with collider findings were performed using HiggsBounds [47][48][49][50][51] and HiggsSignals [52,53].

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No.  Table 2. Additional set of high mass IDM benchmark points considered in the presented study, taken from [4]. In all benchmarks m h = 125.1 GeV. For HP10 scenario (bold) H completely saturates DM relic density; the value of the λ 345 parameter for this scenario was slightly modified with respect to [4], to be consistent with the most recent results on the relic density [54].
(out of 13500 that passed all above criteria) to represent different signatures at e + e − colliders that differ e.g. in the mass spectrum or production cross sections, in order to cover a wide variety of possible collider signals. For the selected benchmark points, all of which correspond to masses of DM particles below 1 TeV, the Sommerfeld enhancement does not play a relevant role [55]. Finally the agreement with results from direct detection experiments [56] has been required. 3 As discussed in [4], constraints from indirect detection are weaker than direct detection constraints (see also e.g. discussion in [58]). The most stringent limits stem from indirect detection leading to bb final states [59]. These can be easily avoided by tuning the value of λ 345 , which is irrelevant for the collider phenomenology discussed here. It should also be noted that dark matter predictions for some points, especially BP23, highly depend on the input in the SM mass spectrum; e.g. variations

JHEP07(2019)053
within 3 standard deviations for bottom and Higgs masses can lead to large variations in the predictions (up to a factor 5), while these are of no importance for our studies. In the scan presented in [4], we made use of the publicly available tools: the 2HDMC code [60] and micrOMEGAs version 4.3.5 [61] to apply several of the constraints discussed above.
Prospects for detection at the LHC. In general, the IDM can be tested at the LHC through a variety of signatures, including mono-jet, mono-Z, mono-Higgs and Vector-Boson-Fusion + missing transverse energy signatures, as well as through multi-lepton and multi-jet final states, see discussions in [24,25,25,28,28,62]. In [42], a dedicated discussion suggests that current searches, e.g. for multi-lepton final states, would have to be significantly modified in order to access the current IDM parameter space even for low dark matter masses. On the other hand, vector boson fusion production of an invisibly decaying Higgs [63] can already significantly constraint the models parameter space for m H 62.5 GeV. Depending on the channel, masses of DM particle up to 200-300 GeV would be accessible at the HL-LHC [25,28,62]. For the di-jet plus missing transverse energy signature, large background significantly affects LHC sensitivity.
Production cross sections for the benchmark points proposed here depend on the signature that is considered and can reach up to O ( pb) at the 13 TeV LHC [64,65]. An increase in center-of-mass energy to 27 TeV, as in the HE-LHC setup [66], can lead to an increase of an order of magnitude in production cross sections. However, without detailed experimental analyses, projections of reachability are difficult to make. We therefore strongly encourage the experimental collaborations to investigate the benchmark points presented here at current and future LHC runs.

Analyses strategies and simulation setup
In this work, we consider the following tree-level production processes of inert scalars at e + e − collisions 4 For the calculation of cross-sections as well as detailed signal and background simulation, we make use of the Monte Carlo event generator WHizard 2.2.8 [67,68]. For the signal, we employ the IDM model implemented in SARAH [69]. Model parameter files for the considered benchmark scenarios were prepared using SPheno 4.0.3 [70,71]. When generating signal and background events samples for the presented analysis, energy spectra for CLIC [72], based on detailed beam simulations, were taken into account. For initial state radiation (ISR), the intrinsic WHizard implementation of the lepton ISR structure function includes all orders of soft and soft-collinear photons as well as up to the third order in hard-collinear photons. Leading-order cross-sections for the processes in (3.1) for 380 GeV collision energy, including initial state radiation, are presented in figure 1. In the scenarios considered in this paper the produced dark scalar A decays to a (real or virtual) Z boson and the (lighter) neutral scalar H, A → Z ( ) H, while the produced charged boson H ± decays predominantly to a (real or virtual) W ± boson and the neutral scalar H, H + → W ± ( ) H, where the DM candidate H escapes detection. Since both the production and decay processes are governed by the SM electroweak couplings, the inert masses are the only BSM parameters probed at CLIC.
Since isolated leptons (electrons and muons) can be identified and reconstructed with very high efficiency and purity [72], we concentrate on Z and W ± leptonic decays, leading to a signature of leptons and missing transverse energy. For the same flavour lepton pair signature, we restrict the analysis to the µ + µ − final state, allowing for almost perfect reconstruction of lepton kinematics, 5 while for different flavour lepton pairs µ + e − and e + µ − final states are considered. We refrain from including detector effects in the results presented here as for the considered final states they are expected to only marginally change the outcome of our study. Muon pair production can be a signature of the AH production process followed by the A decay: while the production of the different flavour lepton pair is the expected signature for H + H − production: Note that when both W bosons in (3.3) decay to muons, the charged Higgs pair production process will contribute to the signature (3.2) of HA production as well.

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During simulations we do not constrain the intermediate particles, but consider all processes leading to + ( − ) + / E ⊥ . Especially processes with additional neutrinos can contribute and need to be taken into account. This includes processes with tau (pair) production and their successive leptonic decays.
To be specific, for processes with muons in the final state, the following processes have been simulated: where the final state taus are then forced to decay to a muon and a neutrino. For the background the following Standard Model processes are considered: where the additional neutrino pair can be of any flavour (i = e, µ, τ ). As before, we generate all processes leading to the above final states, without constraining the intermediate particles states.
Similarly, for the electron-muon pair final state the following signal processes are considered: with the final state tau leptons decaying to an electron or a muon (to match the required final state signature, µ + e − or e + µ − ). For the background in this case the following Standard Model four-fermion processes are considered: As mentioned above, beam energy spectra and ISR were taken into account. Generator-level cuts corresponding to the expected detector acceptance were applied for both signal and background simulations: electrons or muons with energy of at least JHEP07(2019)053 5 GeV need to be emitted at least 100 mrad from the beam direction and their angular separation should also be at least 100 mrad. To reduce background from radiative Z-return events we also require that there are no ISR photons emitted at angles above 100 mrad with energies larger that 10 GeV.
For the considered final states we assume that only two charged leptons are reconstructed in the detector. The observed final state can be completely described by a small set of kinematic variables. To assure the best possible discrimination between signal and background events, resulting in highest expected significance of the possible observation, we make use of multivariate analyses. We apply the Boosted Decision Tree (BDT) classification algorithm, as implemented in TMVA toolkit [73], with the following eight input variables describing the kinematics of the dilepton final state: • total energy of the lepton pair, E ; • dilepton invariant mass, M ; • dilepton transverse momentum, p T ; • polar angle of the dilepton pair, Θ ; • Lorentz boost of the dilepton pair, β = p /E ; • reconstructed missing (recoil) mass M miss (calculated assuming nominal e + e − collision energy); • − production angle with respect to the beam direction, calculated in the dilepton center-of-mass frame, Θ ; • − production angle with respect to the dilepton pair boost direction, calculated in the dilepton center-of-mass frame, ∠ ( , ), where lepton pair denotes µ + µ − for AH channel and µ + e − or e + µ − for H + H − production. The first five variables refer to the dilepton pair system as a whole, while the last two correspond to the single lepton polar angle calculated in the two different reference frames. Please note that these eight variables are not independent, as the final state with two massless leptons and missing energy only can be completely described by five parameters (plus azimuthal angle, which is not relevant). However, using more input variables resulted in better signal selection efficiencies. The BDT algorithm is trained individually for each benchmark scenario and each running energy using the generated event samples after detector acceptance and pre-selection cuts.

Inert scalars at the first stage of CLIC
First we investigate the discovery prospects for the IDM benchmarks at the initial CLIC operation at √ s = 380 GeV with an expected integrated luminosity of 1 ab −1 [74]. The possibility to access benchmark points with m i ≥ 380 GeV, that are not accessible at the first stage, is investigated in section 5, where the second and third energy JHEP07(2019)053 4.1 Neutral dark scalar pair production e + e − → AH As described above, in this channel we focus on final states with muon pairs and missing transverse energy. As the DM particles escape detection, the signal process will lead to large missing energy and momentum. Furthermore, the invariant mass of the lepton pair, stemming from the decay of a real or virtual Z boson, should be relatively small (depending on the mass splitting between A and H, but not greater than m Z ). On the other hand, the dominant Standard Model background process proceeds via the s-channel Z/γ di-muon production, with most pairs produced either with high invariant mass (events without hard ISR) or with significant longitudinal boost (events with high-energy ISR photon). We display the lepton pair invariant mass distribution for signal and background processes in figure 2. As expected, we observe that the event distribution for the signal (for the benchmark scenario BP1; green points) is concentrated on a much smaller range in the P µµ z , M µµ plane than the SM background distribution (red points). For the 380 GeV analysis we therefore require an invariant mass of the produced lepton pair to be below 100 GeV, and the absolute value of the longitudinal momentum below 140 GeV. These pre-selection cuts significantly reduce background from direct two fermion production, e + e − → µ + µ − , hardly affecting the signal. Shown in figure 3 are the distributions of the muon pair energy, E µµ , total transverse momentum, p µµ T , polar angle, Θ µµ , and the difference of the lepton azimuthal angles, cos ∆φ µµ for three benchmark scenarios with virtual Z boson production and the SM background. As the experimental signature is expected to depend on the mass difference Events Figure 3. Distributions of the kinematic variables describing the leptonic final state considered in AH analysis: lepton pair energy, E µµ , total transverse momentum, p µµ T , pair production angle, cos Θ µµ and the difference of the lepton azimuthal angles, cos ∆φ µµ . Expected distributions for representative benchmarks BP1 (red histogram), BP2 (green) and BP7 (blue) are compared with expected background (black histogram). Samples simulated for CLIC running at 380 GeV are normalised to 1 ab −1 .
between A and H states, we present benchmark points corresponding to different mass splittings: of about 6 GeV (BP2), 35 GeV (BP1) and 58 GeV (BP7). For low mass differences, the contribution from H + H − channel to the muon pair production signature is also clearly visible (a tail of events with higher lepton pair energy and transverse momentum).
Distributions of variables presented in figure 3 can be used to select signal-enhanced samples of events. The following selection requirements are therefore further imposed on the lepton pair: • energy E µµ < 100 GeV, • transverse momentum p µµ T > 10 GeV, • production angle (polar angle of the Z boson) 30 • < Θ µµ < 150 • , • difference of the lepton azimuthal angles |∆ϕ µµ | < π 2 . Presented in figure 4 is the lepton pair invariant mass distribution after pre-selection and selection cuts. Signal samples for selected benchmark scenario and the background sample are normalised to 1 ab −1 . About 5400 background events are expected after all selection cuts, while 1810, 1290 and 540 signal events are expected for the BP1, BP2 and BP7 scenarios, respectively. This corresponds to about 21σ, 16σ and 7σ significance.
Higher signal significances are obtained making use of multivariate analyses after the application of pre-selection cuts. As an example, we show the BDT response distributions for BP1 (signal and SM background) in figure 5 for 1 ab −1 collected at CLIC 380 GeV center-of-mass energy. The optimal significance is obtained for a BDT response cut of about 0.12, corresponding to 71% signal selection efficiency and 2.2% background selection efficiency, with a resulting signal significance of about 27.7σ. In figure 6  using the above method is displayed as a function of the neutral inert scalar mass sum, m A + m H , and of the signal production cross-section for the considered final state, σ(e + e − → HHµ + µ − X inv ). The expected significance is mainly related to the AH production crosssection. A discovery, corresponding to 5 σ, at the initial stage of CLIC is expected for scenarios with the signal cross-section (in the µ + µ − channel, after pre-selection cuts on generator level) above about 0.5 fb, which corresponds to the neutral inert scalar mass sum below about 290 GeV. For the considered benchmark points we do not observe any sizable dependence of the expected significance on the mass splitting between the two neutral scalars, m A − m H (indicated by colour scale in figure 6).

Charged scalar pair production e + e − → H + H −
The selection of H + H − production events is more challenging than for the AH channel, as the two leptons in the final state no longer originate from a single (on-or off-shell) intermediate state. We therefore do not apply any additional pre-selection cuts (except for the detector acceptance cuts, as described in section 3). However, this also allows us to consider the electron-muon pairs in the final state, avoiding large SM background from the direct lepton pair production (e + e − → + − ; this channel contributes only via leptonic tau decays, suppressed by the corresponding branching fractions).
With only the detector acceptance cuts on the generator level, the expected background cross-section for the considered final state is about 500 fb, over two orders of magnitude higher than for the considered benchmark points. However, kinematic distributions are very different, as two massive scalars are produced in the signal case, reducing the kinematic space available for lepton pair production. In figure 7 distributions of the selected variables describing the leptonic final state for three benchmark scenarios (BP1, BP3 and BP6) are compared with Standard Model expectations. Clear differences between the signal and background distributions allow for efficient selection of signal-enhanced sample of events using the multivariate analysis. We follow the same procedure and the same set of input variables is used as for the AH analysis described above. The BDT classification algorithm is trained separately for each benchmark point to discriminate between signal and background processes. Examples of the BDT response distributions for the BP1 signal sample and SM background samples simulated for 1 ab −1 at 380 GeV CLIC are shown in figure 8. While due to a large SM background it is not possible to select the signaldominated sample based on the BDT response, the highest significance is obtained when JHEP07(2019)053   Figure 9. Expected significance of the deviations from the Standard Model predictions observed at 380 GeV CLIC for events with electron-muon pair in the final state (e + µ − or µ + e − ) as a function of twice the charged scalar mass (left) and the production cross-section for the considered signal channel (right), for different IDM benchmark points. Color indicates the mass splitting between the H ± and H scalars (right scale applies to both plots). selecting events with BDT response above 0.12. About 1700 signal events are expected in the final sample (BDT selection efficiency of 70%) with background contribution of about 8500 events (BDT selection efficiency of 1.7%), resulting in the significance of the observation of about 17σ.
As was the case for the AH channel, the expected significance of the eµ signal is mainly related to the production cross-section for the considered channel. This is shown in figure 9, where the expected significance for the electron-muon final state (e + µ − or µ + e − ) are plotted as a function of 2m H ± (left panel) and the production cross-section (right panel), for different IDM benchmark points. Discovery at the initial stage of CLIC is only possible for scenarios with signal cross-sections (in the electron-muon channel) above about 1 fb. This corresponds to charged scalar masses below roughly 150 GeV. We do not observe any sizable dependence of the expected significance on the mass splitting between the charged and neutral inert scalars, m H ± − m H (indicated by colour scale in figure 9), within the considered range of parameters. Reduced signal channel cross section and thus reduced signal sensitivity observed for one of the benchmark points in figure 9 (BP2 with m H ± = 112.8 GeV) is due to the significant contribution of cascade decays, H ± → W ± A → W ± Z H, which were not considered in the signal event selection.

Inert scalars at high-energy stages of CLIC
We now turn to the discovery prospects of the two high-energy stages at 1.5 TeV and 3 TeV with assumed integrated luminosities of 2.5 ab −1 and 5 ab −1 [74]. The same analysis procedure described in section 4 was applied to signal and background samples simulated for high-energy CLIC stages. As before, proper energy spectra for CLIC [72], based on detailed beam simulations, were taken into account, which is crucial for a correct description of signal and background at high collider energies. We applied the same generator-level cuts as before, but did not make use of any additional pre-selection cuts. Furthermore, we extend our study to include additional high-mass benchmark points not accessible at 380 GeV; these are listed in table 2.
In figure 10, we display the expected significances of the IDM signal in the AH and H + H − channel as a function of the inert scalar masses for subsequent CLIC running stages. For the AH channel (muon-pair production), increasing the running energy and integrated luminosity results in only a moderate extension of the discovery potential of CLIC. With 2.5 ab −1 at 1.  Figure 11. Significance of the deviations from the Standard Model predictions expected at different CLIC running stages, assuming the same integrated luminosity of 1 ab −1 , as a function of the signal cross-section in the considered channel, for: (left) events with two muons in the final state (µ + µ − ) and (right) events with electron-muon pair production (e + µ − or e − µ + ), for the IDM benchmark points in tables 1 and 2.
decreases much slower with the charged scalar mass and we can probe masses up to about 500 GeV at 1.5 TeV, compared to 150 GeV at the first CLIC stage (see figure 10 right panel).
The significance is mainly driven by the signal production cross section and is approximately proportional to the square-root of the integrated luminosity. For parameter points that are already accessible at Stage 1 the AH production cross sections decrease with the collision energy much faster than most of the backgrounds and the significance of observation decreases at Stage 2. Only for points with m A +m H 300 GeV, which are close to the production threshold at Stage 1, higher integrated luminosity and the production cross sections enhanced by up to a factor of 2 result in better sensitivity at center-of-mass energy of 1.5 TeV. Similarly, when going from 1.5 TeV to 3 TeV, the significance of observation increases only for scenarios with m A + m H 1.2 TeV. As we search for the signal contribution on top of a much larger background, we expect that the significance is (to a first approximation) proportional to the square-root of the integrated luminosity. In order to compare the CLIC sensitivity to the IDM benchmark scenarios at different energies, we scale the expected significance at high-energy stages to the integrated luminosity of 1 ab −1 assumed for 380 GeV running. This allows us to separate luminosity and cross-section contributions to the overall significance, and will also allow for projections of the discovery potential at arbitrary luminosities.
In figure 11, we show the scaled significance results, presented as a function of the signal production cross-section. For the AH channel, which leads to µ + µ − final states, a universal linear dependence on the signal cross-section is observed which does not seem to depend on the running energy. Significant (above 5σ) observation is possible for cross-sections roughly larger than 0.5 fb (for higher luminosities, these should be rescaled by 1 /L · ab).
For the H + H − channel, however, leading to e ± µ ∓ final states, the high-energy running of CLIC clearly gives better sensitivity to heavy IDM scenarios (for points with similar production cross-section and assuming same luminosity) than the initial CLIC stage (see figure 11 right). The relatively large differences between different BPs with similar cross-sections at the same center-of-mass energies originate from the mass scale of the heavy scalars.
Finally, we investigate the dependence of the signal significance on the mass difference between neutral/charged inert scalar and the DM candidate. In figure 12, we show the ratio of the expected significance (scaled to the integrated luminosity of 1 ab −1 ) to the signal cross-section in the considered channel, as a function of the corresponding scalar mass difference. This ratio indicates the expected significance for the particular mass splitting, assuming the reference signal channel cross section of 1 fb. For AH production (muonpair channel) at high-energy stages, the experimental sensitivity seems to be significantly better for low mass differences, below m Z , when the virtual Z boson is produced in the A boson decay, A → Z ( ) H. This is because signal events can be better separated from the SM backgrounds for such scenarios. One can also note that for high mass differences, m A − m H > m Z , the experimental sensitivity is clearly better for low-energy running.
The situation is similar for the H + H − production signal in the electron-muon channel. For high running energies, a better sensitivity is expected for low mass differences when the virtual W ± boson is produced in the charged scalar decay. However, it is also clear that the experimental sensitivity is much better at high-energy running than at the first CLIC stage and this observation does not depend on the considered dark scalar mass difference.
The results presented in figure 10 seem to indicate that many high mass IDM scenarios will remain inaccessible at CLIC, even at high energies. However, one has to stress that this is mainly due to the small branching ratios for the considered leptonic final states: 3.3% for AH → HHµ + µ − and 2.3% for H + H − → HHµ ± e ∓ νν. For scenarios where the signal cross sections in the dilepton channel are too small, it might be worthwhile to investigate semi-leptonic decays in the H + H − production channel. Due to the much larger branching ratios (28.6% for H + H − → HH ± νqq , with = e, µ) the expected number of H + H − signal events in the semi-leptonic final state is over an order of magnitude larger than for JHEP07(2019)053 the electron-muon signature. As a similar scaling is expected for the background processes (dominated by the W + W − production), we expect that the significance of the observation in the semi-leptonic channel should be increased by at least a factor of 3 (corresponding to the ten-fold increase of the integrated luminosity). An investigation in this channel could furthermore profit from a full reconstruction of the W ± that decays hadronically, which allows to use the reconstructed W boson mass and energy as additional discriminators in the BDT algorithm. However, a proper estimate of the expected significance for this case would require a much more detailed analysis, including parton showering, hadronisation and detector response simulation, and event reconstruction with particle flow algorithm and final state reconstruction using accurate jet clustering and lepton identification processes. This is beyond the scope of the work presented here.

Conclusions
In this work, we have studied prospects for discovery of inert scalars of the Inert Doublet Model at CLIC running at 380 GeV, 1.5 TeV and 3 TeV. A set of benchmark points, proposed in [4] and satisfying all experimental and theoretical constraints, has been considered. We focused on pair-production of charged dark scalars H + H − and production of the DM candidate with the second neutral scalar boson, HA, with subsequent decays to leptonic final states. Signal and background event samples were generated with WHizard 2.2.8, taking into account all processes that lead to the considered final states. Signatures for production of new scalars were searched for in the kinematic distributions for events with exclusive production of two muons or an electron and a muon. Significance of the possible observation was studied using multivariate analysis methods.
We found that most of the low-mass benchmark scenarios proposed in [4] can be observed with high significance in the di-muon channel already with 1 ab −1 collected at 380 GeV (the first stage of CLIC), provided that the sum of neutral inert scalar masses, m A + m H < 290 GeV. Similar constraints also apply to the observation of the charged scalar pair-production (electron-muon pair-production channel), which is however fulfilled for fewer scenarios.
Scenarios which are not kinematically accessible at the first stage of CLIC can be searched for at high-energy stages, at 1.5 TeV and 3 TeV. The signal production crosssection for both considered channels decreases significantly with energy, much faster than for the corresponding background processes. Signal cross sections for the considered final states are further reduced by the small branching fractions for the dilepton channels. We found that at 1.5 TeV the discovery reach is extended to the sum of scalar masses of about 550 GeV in the dimuon channel and for charged scalar masses up to about 500 GeV in the e ± µ ∓ channels. For the scenarios considered here, increasing the center-of-mass energy to 3 TeV does not significantly improve the sensitivity. Therefore, the observation of the inert scalar production in the leptonic channels will be challenging at high-energy CLIC and a significant signal is only expected for relatively low masses. However, higher significance and the discovery reach extending up to the kinematic limit could be expected for H + H − production in the semi-leptonic final state (isolated lepton and a pair of jets or one massive jet). This is in the line of future work.