1 Introduction

Since the 2012 discovery of a particle with properties of the Standard Model (SM) Higgs boson [1, 2], no direct observation of any new physics, not described within the SM, has been made at the Large Hadron Collider (LHC), nor in any other high energy physics experiment. Still, many indications suggest the existence of physics Beyond the Standard Model (BSM). This includes in particular numerous astrophysical and cosmological observations pointing to the existence of dark matter (DM) in our Universe, constituting to about 27% of the total energy budget [3]. The SM alone is not able to explain the fact that neutrinos have mass, deviations from the SM predictions are observed in precision low-energy observables such as \(g_\mu -2\) [4] and some B meson decays [5, 6]. Also the recent high-precision measurement of the W boson mass with the CDF II detector [7] is in significant tension with the SM expectation.

The scalar sector of the SM leaves lots of room for new physics and its extensions are considered in a wide range of BSM models (see eg. [8] and references therein). At the same time it is still the least tested part of the theory. So far only the mass of the new particle has been determined with high precision [9,10,11]. While the Higgs boson couplings to other SM particles are being tested with increasing precision [12,13,14], its self-coupling, needed to confirm the shape of the scalar potential and fully validate the model, is basically not constrained. This will be a very challenging task for the LHC and its planned upgrade to higher luminosity, but also for the proposed future colliders [15].

A consensus has formed in recent years in the particle physics community, confirmed by the 2020 Update of the European Strategy for Particle Physics [16], that an electron-positron Higgs factory is the highest-priority next collider. Within the variety of proposals [17,18,19,20,21], the Compact Linear Collider (CLIC) at CERN [18] gives the best prospects for the direct BSM searches at the energy frontier. CLIC is planned to be built in three energy stages, starting from 380 GeV, extending to 1.5 TeV and finally to 3 TeV, with corresponding total integrated luminosities of 1000 \(\text {fb}^{-1}\), 2500 \(\text {fb}^{-1}\) and 5000 \(\text {fb}^{-1}\), respectively. The first running stage, focused on the Higgs boson measurements (Higgs factory), will also allow for the precision measurements of top quark pair-production (including the threshold scan). The subsequent stages will mainly focus on direct searches for BSM physics, but additional Higgs and top-quark measurements will also become possible. A dedicated detector concept for CLIC (CLICdet) was proposed, optimised for the Particle Flow analysis at TeV energies, as well as for the treatment of beam-induced backgrounds expected at CLIC due to high bunch rate and beam intensity [22].

In this paper we investigate the CLIC potential for discovering new heavy scalar particles pair-produced in \(e^{+}e^{-}\) collisions. We consider the Inert Doublet Model (IDM), a simple extension of the SM, introducing four new scalars with the lightest neutral one being a good candidate for a dark matter (DM) particle. This is a challenging scenario, as two DM particles are always produced and the observed SM final state can be very soft. It was already considered in [23], looking at events with two leptons in the final state. The study was performed on the generator level, including the CLIC luminosity spectra and cuts reflecting the expected detector acceptance. A leptonic signature allows for very efficient background supression. However, the CLIC sensitivity to IDM scalar pair-production in this channel is limited to masses below \(\sim \)500 GeV due to the small branching ratio and to the fact that the production cross section decreases with scalar masses and collision energy.

Considered in this paper is the charged IDM scalar pair-production with semi-leptonic final states, as the expected event rates are order of magnitude higher than those for the leptonic final states. The analysis is based on the CLICdet model and the Geant4-based simulation tools to simulate the detector response for a limited number of IDM benchmark scenarios, and is extended to other benchmark scenarios with a fast simulation technique based on Delphes, as described in Sect. 3. Details of the event selection procedure are presented in Sects. 4 and 5, and final results are discussed in Sect. 6. When describing details of the analysis we focus mainly on the full detector simulation results. The same analysis approach has also been used for fast simulation results, with small modifications which are described in detail.

2 Inert doublet model

The Inert Doublet Model (IDM) [24, 25] extends the SM by only one additional doublet in the scalar sector, making it one of its simplest extensions. The scalar potential in this model contains the SM-like Higgs doublet, \(\phi _{S}\), and the so-called inert (or dark) doublet, \(\phi _{D}\), that contains four new scalar fields: H\(^{\pm }\), A and H. Due to the additional \(Z_{2}\) symmetry, under which the inert doublet is odd, the new scalars do not interact with SM fermions (on tree-level) and the lightest of the IDM scalars (H) is stable, hence it is a good DM candidate.

After electroweak symmetry breaking, the model contains seven free parameters. Fixing the SM-like Higgs boson mass \(m_{h}\) and the Higgs field vacuum expectation value to the SM values, the following set of physical parameters is selected [25]: three scalar masses, \(m_{H^{\pm }}\), \(m_{A}\) and \(m_{H}\), and two couplings, \(\lambda _{2}\) and \(\lambda _{345}\).Footnote 1 Two sets of IDM benchmark points were proposed in [26], based on the scan over the whole 5-dimensional IDM parameter space, taking into account all existing theoretical and experimental constraints. The points were selected to cover all interesting areas of parameter space and to respect a wide range of dark scalar masses and mass splittings.

In this paper, 23 out of 41 benchmark points presented in [26] are considered: all high-mass benchmark points (HP) and three low-mass scenarios (BP) with the highest scalar masses: BP18, BP21 and BP23. All benchmark points considered in the study, together with associated model parameters, are shown in Table 1. More information about the benchmarks, as well as the exact constraints on the model parameters, can be found in [26].

Table 1 Benchmark points considered in study, accessible at \(e^+e^-\) colliders with \({\mathcal {O}}\left( \mathrm {TeV}\right) \) center-of-mass energies. \(M_h=125.1\,\mathrm {GeV}\) for all points. BP21 and HP10 provide 100% dark matter relic density [26]
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Production of IDM scalars at lepton colliders is dominated by production of neutral or charged scalar pairs via couplings of dark scalars to SM gauge bosons:

$$\begin{aligned} e^{+}e^{-}\rightarrow & {} {\mathrm {H}}\;\;{{\mathrm {A}}}\;, \\ e^{+}e^{-}\rightarrow & {} {\mathrm {H}}^{+}{\mathrm {H}}^{-}. \end{aligned}$$

For the neutral scalar pair production, the produced dark scalar A decays to a (real or virtual) Z boson and the (lighter) neutral scalar H, \({\mathrm {A}} \rightarrow {\mathrm {Z}} {\mathrm {H}}\), while the produced charged boson H\(^\pm \) decays predominantly to a (real or virtual) W\(^\pm \) boson and the neutral scalar H, \({\mathrm {H}}^\pm \rightarrow \mathrm {W}^\pm {\mathrm {H}}\). Decays involving SM states only are forbidden due to the \(Z_{2}\) symmetry. The two possible production channels can be thus written as:

$$\begin{aligned} e^{+}e^{-}\rightarrow & {} {\mathrm {H}}\;\;\; {\mathrm {A}} \;\;\; \rightarrow {\mathrm {H}} \; \; {\mathrm {H}}\;\;{\mathrm {Z}}\;, \nonumber \\ e^{+}e^{-}\rightarrow & {} {\mathrm {H}}^{+}{\mathrm {H}}^{-} \rightarrow \mathrm {W}^{+}\mathrm {W}^{-}{\mathrm {H}}\;\;{\mathrm {H}}\;. \end{aligned}$$
(1)

The sensitivity of CLIC to neutral and charged IDM scalar pair-production, for leptonic decays of the produced Z and \(W^\pm \) bosons, was studied in details in [23]. For CLIC running at 380 GeV, discovery of the IDM scalars is possible for most of the benchmark scenarios where dark scalar production is kinematically allowed at this stage, for \(m_A+m_H<290\) GeV and \(2m_{H^\pm }<310\) GeV. However, for CLIC running at higher centre-of-mass energies, at 1.5 TeV and 3 TeV, the discovery-reach increases only to about 500 GeV, as it is limited by the production cross section decreasing fast with collision energy.

Considered in the presented study is the H\(^+\)H\(^-\) production at high energy stages of CLIC with the semi-leptonic final state, offering higher decay rates and hence also higher statistics. Gauge bosons produced in the final state can be both on- or off-shell, depending on the dark scalar mass difference, \(m_{H^{\pm }} - m_{H}\). Reconstruction of the invariant mass of the hadronicaly decaying W\(^\pm \) allows also for better background suppression for events with virtual W\(^\pm \) production. The cross sections for the H\(^+\)H\(^-\) production at 1.5 TeV and 3 TeV CLIC are presented in Fig. 1. They depend mostly on the scalar masses; the influence of couplings \(\lambda _2\) and \(\lambda _{345}\) is marginal.

Fig. 1
figure 1

Figure taken from [27]

Leading-order cross sections for the charged inert scalar production, \(e^+e^-\rightarrow {\mathrm {H}}^+{\mathrm {H}}^-\), for collision energy of 1.5 TeV (left) and 3 TeV (right). The yellow band represents all scenarios selected in the model scan [26] while the blue dots represent the selected benchmark scenarios. Beam energy spectra are not included.

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3 Signal and background simulation

The CLIC potential for the discovery of charged IDM scalars was studied using two complementary approaches. First, five selected benchmark scenarios were studied with full detector response simulation based on Geant4. Then, using the full simulation results to verify and tune the fast simulation model, the analysis was extended to the set of 23 benchmark points to estimate the CLIC sensitivity also in other parts of the parameter space. Only the semi-leptonic final state was considered in generating the signal samples.

3.1 Full detector description

Fully simulated event samples were generated with Whizard 2.7.0 [28], and Pythia6 [29] was used for hadronisation. The Monte Carlo particles were propagated through the Geant4 [30] detector model for CLICdet [22] (CLIC_o3_v14), described by the DD4hep toolkit [31]. The reconstruction of physical objects in the detector was based on PandoraPFA  [32, 33]. The iLCDirac interface [34] was used for the job handling and submission to the grid resources.

Due to high beam intensities and small intervals between subsequent bunches, beam-induced backgrounds are important and have to be properly taken into account in the event reconstruction at CLIC. In the context of the detector performance, the most important contribution comes from the photons radiated due to the beam-beam interactions. Hadrons produced in soft \(\gamma \gamma \) collisions overlay on the \(e^{+}e^{-}\) events and significantly bias object reconstruction. These effects are taken into account in the full simulation, together with a reconstruction procedure aimed at reducing the impact of this background. This turns out to be crucial for this particular analysis, taking into account that in part of the IDM scenarios the gauge boson is produced off-shell. Its low mass results in low energy and momenta of its decay products, what makes their reconstruction more vulnerable to the influence of soft particles from the overlay events.

To model the real beam interactions at CLIC, each full simulation event includes 30 bunch crossings (BX) on top of the physical event produced in hard interaction, 10 before and 20 after it (the physical event is placed in the 11th BX). For each of the additional BX, on average 1.3 and 3.2 \(\gamma \gamma \rightarrow \) hadrons events are overlaid at 1.5 TeV and 3 TeV, respectively. To suppress this background, dedicated cuts on the timing of detector hits and reconstructed objects are applied. First, only hits from 10 ns following the physical event are accepted for event reconstruction (which roughly corresponds to 20 BXs for 0.5 ns bunch separation at CLIC) [22]. Subsequently, cuts are applied to the reconstructed Particle Flow Objects (PFOs), as described in the CLIC Conceptual Design Report (CDR) [35]. The cuts, defined separately for 1.5 TeV and 3 TeV CLIC running, depend on the reconstructed object type (track, neutral hadron or photon), transverse momentum and the polar angle. Reconstructed PFOs are accepted for final event reconstruction, if their reconstructed arrival time differs from a nominal time of a physical event by less than the corresponding time threshold \(t_{\mathrm {cut}}\).

Particle flow approach assumes reconstruction and identification, by combining information from all subdetectors, of each visible particle in an event. The high calorimeter granularity is essential for this goal. Full simulation studies [22] indicate that, even for complex events, muons above 10 GeV can be identified with more than 98% efficiency for all energies and polar angles, while electrons are correctly identified in 85–90% of cases at energies of 20 GeV and higher. After the reconstruction and selection of PFOs in an event, isolated leptons (electrons and muons) were selected with the IsolatedLeptonFinder Marlin processor [36], using the so-called polynomial isolation criterion. For candidate PFOs with energy above the minimum value of 5 GeV, the cut on the energy in a cone surrounding the track was applied, given by the polynomial dependence on the track energy [37]. We also used the so-called lepton dressing option of the IsolatedLeptonFinder, correcting for possible electron bremsstrahlung in the tracking detectors by merging close-by photons and electrons. For the considered charged IDM scalar pair-production with the semi-leptonic final state, W\(^\pm \) decays into tau leptons were also included. However, identification of hadronic tau decays (tau-jet tagging) was not used and only events with isolated electron or muon were accepted. This results in about 20% loss of signal selection efficiency. The final efficiency for isolated lepton identification for signal events, for the considered semi-leptonic final state, is about 65–70%. It decreases to about 50–60% for scenarios with the lowest scalar mass difference. The purity of the isolated lepton selection (probability of the selected PFO matching closely the generator level lepton) is about 97–98% for 1.5 TeV CLIC and about 94–96% for 3 TeV running.Footnote 2

PFOs not classified as isolated leptons were further used for jet clustering. Jets were reconstructed in the exclusive mode, with two jets in the final state, using the Valencia Linear Collider (VLC) algorithm [38]. This algorithm was designed to be least sensitive to the influence of the overlay events, and is assumed to be the best choice for the jet reconstruction at CLIC high-energy stages. Parameters of the algorithm were set to \(\gamma =\beta =1\), \(R=0.9\) for 1.5 TeV CLIC running and \(R=1.2\) for 3 TeV. The choice of the R parameter was based on the shape and placement of the peak corresponding to an on-shell W\(^\pm \) boson. Isolated photons were also identified. A photon with transverse momentum \(p_{T}^{\gamma }\) was considered isolated, if \(p_{T}^{cone}/p_{T}^{\gamma } \le 0.2\), where \(p_{T}^{cone}\) is the total transverse momentum of other particles in a cone, which is defined by the squared radius \((\Delta R)^{2} = (\Delta \theta )^{2} + \sin ^{2}(\theta ) \cdot (\Delta \phi )^{2} < 0.1\), surrounding photon direction.

3.2 Fast detector simulation

To extend the study to a larger number of benchmark scenarios, the realistic fast simulation toolkit Delphes  [39] was used, version 3.4.2, with CLICdet detector model cards [40], based on the full simulation results [41]. The object reconstruction implemented in the Delphes model follows the Particle Flow approach of the full simulation reconstruction. However, particle identification is based on the true MC information, taking into account finite identification efficiency only. Isolation criteria are also simplified compared to the full reconstruction: an electron, muon or photon was considered isolated, if the total \(p_{T}\) of other particles in the cone of radius \(\Delta R=0.5\) surrounding it was less than 0.12 of the candidate particle \(p_{T}\).Footnote 3 The jets were reconstructed using VLC algorithm in the exclusive mode again, with the parameters \(\gamma =\beta =1\) and \(R=1.0\) (\(R=1.2\)) for 1.5 TeV (3 TeV). As for the full detector simulation, selected values of R correspond to the best reconstruction of the W\(^\pm \) peak.

As a part of the presented study, the CLICdet cards for Delphes were also modified to take into account the beam-induced background. The PileUpMerger module of Delphes, designed to include pile-up events in hadron colliders, was used to add \(\gamma \gamma \rightarrow \) had. events to the event record. Overlay background events generated with Pythia6 cannot be used directly, as timing cuts used to select reconstructed PFOs are not implemented in the CLICdet model. Hence, a dedicated pre-processing of the overlay events was implemented to take into account the influence of these cuts and make the evaluation of the CLIC sensitivity to IDM scalars more realistic.

The average number of \(\gamma \gamma \rightarrow \) had. events added to each physical event is \(20\cdot n_{had}\), where 20 comes from the number of BXs that survive the primary 10 ns window cut and \(n_{had}\) is the expected number of \(\gamma \gamma \rightarrow \) had. events in a single BX (at the given CLIC energy stage). This gives on average 26 background events at 1.5 TeV and 64 at 3 TeV [35], where the number of events is drawn from the Poisson distribution. To model the PFO timing cuts used in the full reconstruction, particles in an overlay event-file are pre-selected based on their event number. Considering that the BX separation at CLIC is 0.5 ns, timing cut of \(t_\mathrm {cut}\) applied to a given particle category (particle type and transverse momentum range) is modelled by accepting particles from the first \(N_{BX} = t_{\mathrm {cut}}/0.5\) ns events out of every 20 events. While neglecting the time resolution, this procedure assures that the number and proportions of particles of different types and kinematic properties passing the timing cuts are preserved with respect to the full simulation. However, the possible impact of timing cuts on the reconstruction of particles coming from the physical events is not taken into account. Also, particles from the overlay events are rejected before the PFOs reconstruction, so possible effects due to spatial overlap of detector deposits are neglected.

Figure 2 presents the impact of \(\gamma \gamma \rightarrow \) had. events produced in the fast simulation on the distributions of variables describing jets. The histograms resulting from Delphes with and without influence of the overlay are compared to the outcome of the full simulation. Distributions are shown for the leading background channel, qq\(\ell \nu \) (dominated by W\(^+\)W\(^-\) production channel) and for the example signal scenario, HP17, with low scalar mass difference, \(m_{H^\pm }-m_H \approx 10\) GeV (see Table 1). W mass peak is clearly visible in the di-jet mass distribution for the background, while the signal is dominated by low di-jet masses. Without the overlay event contribution, di-jet invariant mass distribution for signal is peaked, as expected, at around 10 GeV. However, overlay background contribution strongly affects the measurement of the soft final state: the di-jet mass distribution gets significantly wider and is shifted towards higher values. The improvement in the agreement between the fast and full simulation methods after including the \(\gamma \gamma \rightarrow \) had. process in Delphes is clearly visible. Good agreement is obtained for the maxima of the signal distributions, although contribution of events in high mass tails is still underestimated. For background events, the maximum of the single jet invariant mass distribution is also well reproduced with fast simulation procedure, when overlay events are taken into account. Only the di-jet mass distribution for background events show systematic differences between fast and full simulation. We decided to apply the described selection, reflecting the timing cuts used in the full simulation, as only marginal improvement could be achieved by fine-tuning the procedure and cuts applied to the overlay events.

Fig. 2
figure 2

Histograms of the masses of a single jet (upper row) and of a dijet system (bottom row), for the qq\(\ell \nu \) background (left) and HP17 signal (right) samples at 3 TeV. Results of the Delphes simulation without (blue histogram) and with (green) the overlay event contribution are compared with results of the full detector simulation (red)

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4 Event preselection

As a first step, only events with exactly one isolated lepton, electron or muon, expected from a leptonic W boson decay, and a pair of jets (di-jet system) resulting from a hadronic decay of a second W boson were selected, corresponding to the considered signal signature. Processes with tau lepton production were also included, both for signal and background samples. However, most of these events were rejected at this stage, as tau jet tagging was not used and only leptonic tau decays could match the required event topology. Furthermore, to avoid possible bias due to hard initial state radiation, events containing at least one isolated photon with energy \(E_{\gamma }>10\) GeV were rejected. Also, the total transverse momentum of PFOs not contributing to the required final state (two jets and a lepton), \(p_{T}^{utg}\) (untagged transverse momentum) had to be smaller than 20 GeV. This cut was imposed to reject events with significant deposits excluded from the reconstructed final state in the VLC clustering (activities in the forward direction assigned to the beam jets).

Fig. 3
figure 3

Histograms of the masses (left) and the energies (right) of a dijet system at 1.5 TeV (upper) and 3 TeV (bottom). The red line denotes HP17, and the blue one BP21. The black histogram is the sum of all background channels. Histograms are normalized to the number of events expected in the real experiment. Solid line shows the full simulation histograms and the dashed line the corresponding fast simulation ones

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As a next step, kinematic variables describing the event were calculated and a simple cut-based preselection was applied. The criteria used in the analysis for the two considered CLIC running stages are presented in Table 2.

Table 2 Preselection cuts applied to the kinematic variables calculated for selected signal and background events at \(\sqrt{s} = 1.5\) TeV and \(\sqrt{s} = 3\) TeV
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The following variables were used in the procedure:

  • \(E_{jj}\)—energy of a dijet system,

  • \(M_{jj}\)—invariant mass of a dijet system,Footnote 4

  • \(\theta _{jj}\)—polar angle of a dijet system,

  • \(E_{\ell }\)—energy of an isolated lepton,

  • \(p_{T}^{\ell }\)—transverse momentum of an isolated lepton,

  • \(\theta _{\ell }\)—polar angle of an isolated lepton,

  • \(M_{miss}\)—missing mass, which is an invariant mass of the missing four-momentum, \(P_{miss}\), calculated by subtracting four-momenta of a lepton and two jets from the four-momentum (\(\vec {p}=\vec {0}\), \(E = \sqrt{s}\)) of the initial state.

Figure 3 shows the distributions of two of the variables used for the preselection cuts at 1.5 TeV and 3 TeV CLIC. Presented are histograms for the combined SM backgrounds and for the two example signal scenarios: one with on-shell W\(^\pm \) boson production (BP21) and one where the produced W\(^\pm \) is far off-shell (HP17). Distributions obtained with full and fast detector simulation and reconstruction are compared. Despite some discrepancies, general agreement in the shapes of the distributions is observed, which is not the case for Delphes without including the \(\gamma \gamma \rightarrow \) had. overlay background. In particular, good agreement is obtained in the dijet mass distributions, which is very sensitive to the overlay contribution. All histograms are normalised to the numbers of events expected at CLIC, for nominal integrated luminosity. Differences in normalisation between the fast and full simulation reflect the differences in preselection efficiencies. Higher background level estimates resulting from full simulation are mainly due to the contribution from false lepton identification (events without a final state lepton on the parton level), which is not modeled in Delphes.

Table 3 Results of the event preselection at \(\sqrt{s}=1.5\) TeV CLIC running stage for the benchmark points and background channels considered in the full simulation study. Shown are cross sections \(\sigma \), numbers of events expected after preselection cuts for an integrated luminosity of 2 ab\(^{-1}\) and the preselection efficiency. For signal scenarios, the mass scale of the charged IDM scalar is indicated in parenthesis, for other model parameters see Table 1. Signal scenarios with off-shell W\(^\pm \) production are marked with a star
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Table 4 Results of the preselection at \(\sqrt{s}=3\) TeV CLIC running stage for the benchmark points and background channels considered in the full simulation study. Shown are cross sections \(\sigma \), numbers of events expected after preselection cuts for an integrated luminosity of 4 ab\(^{-1}\) and the preselection efficiency. For signal scenarios, the mass scale of the charged IDM scalar is indicated in parenthesis, for other model parameters see Table 1. Scenarios with off-shell W\(^\pm \) production are marked with a star
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Results of the preselection for all background channels and signal scenarios considered for the full simulation are presented in Tables 3 and 4, for 1.5 TeV and 3 TeV CLIC stages, respectively. Presented are the generator level cross sections, \(\sigma \), number of events expected after preselection, assuming \(-80\)% electron beam polarisation and total integrated luminosities of 2 ab\(^{-1}\) (1.5 TeV) and 4 ab\(^{-1}\) (3 TeV), and the corresponding preselection efficiencies. The largest contribution to the SM background after preselection comes from qq\(\ell \nu \) production (dominated by pair production of W bosons), mainly due to the largest generator level cross section and the matching event topology. The highest preselection efficiency (weakest background suppression) is obtained for the qq\(\ell \nu \nu \nu \) final state, which matches the expected signal topology as well and also results in large reconstructed missing mass from the three escaping neutrinos. Fortunately, this background channel has a relatively small cross section.

5 Multivariate analysis

After the cut-based preselection, Boosted Decision Trees (BDTs) were used for the multivariate analysis, as implemented in the TMVA toolkit [42]. The classifier consisted of 1000 decision trees and a principal component analysis was applied on the input data in the preprocessing phase. Decision trees were randomised, which means that every decision tree uses only a randomly chosen subset of input variables (six in case of this study). The following variables were used as an input to the BDT algorithm, in addition to the set used in preselection (refer to Sect. 4):

  • \(E_{j_1}\), \(E_{j_2}\)—energies of the two jets,

  • \(p_{T}^{j_1}\), \(p_{T}^{j_2}\)—transverse momenta of the two jets,

  • \(\theta _{j_1}\), \(\theta _{j_2}\)—polar angles of the two jets,

  • \(\Delta \theta _{Wj}^{*}\)—polar angle between the leading (higher \(p_T\)) jet and the dijet direction (flight direction of hadronically decaying \(W^{\pm }\)), calculated in the dijet centre-of-mass frame,

  • \(\Delta \phi _{Wj}^{*}\)—azimuthal angle between the jet and the dijet direction, calculated in the dijet centre-of-mass frame,

  • \(p_{T}^{utg}\)—untagged transverse momentum,

  • \(E_{T}^{miss}\)—missing transverse energy (MET), calculated as the transverse component of \(P_{miss}\).

In the full simulation study, BDTs were trained separately for each of the considered signal scenarios. A more conservative approach would be to use many different signal scenarios as an input to the BDT training, as it is very unlikely that one of the analysed benchmark points is realised by nature. Unfortunately, the data-set available for the full simulation analysis was limited to only five benchmark scenarios, two with on-shell and three with off-shell W\(^\pm \) production (these two classes of scenarios should be trained separately). A single scenario optimisation approach was chosen to avoid possible bias of sensitivity towards one of the signal scenarios (or to particular region in the parameter space). The same approach was used when comparing full simulation results for these 5 scenarios with results based on the Delphes fast simulation.

Example distributions of the BDT response are presented in Fig. 4, for the two selected signal scenarios, BP21 and HP17, and the two CLIC running stages.

Fig. 4
figure 4

BDT response distributions for the BP21 (left) and the HP17 (right) scenarios at \(\sqrt{s} = 1.5\) TeV (upper plots) and at \(\sqrt{s} = 3\) TeV (bottom plots). Benchmark scenarios and SM background expectations are normalised to the integrated luminosity of 2 ab\(^{-1}\) and 4 ab\(^{-1}\), for 1.5 TeV and 3 TeV, respectively. All of the presented distributions were obtained with full detector simulation and maximal achievable significance with them exceeds 5\(\sigma \)

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The final event selection is optimised separately for each scenario by finding the cut on the BDT response that maximises the statistical significance of expected deviations from the SM predictions, calculated as \(S/\sqrt{S+B}\), where S and B are the numbers of signal and background events after the cut, respectively. Although it is not possible to select a signal dominated sample even with a tight BDT response cut, significant deviations from SM predictions can be observed in the measured event distributions. For example, signal for BP21 scenario can be observed at 3 TeV CLIC with statistical significance of about 18\(\sigma \): about 2640 signal events, 41% of preselected sample, pass the BDT selection for optimal response cut of about 0.10, while background contribution is suppressed by a factor of about 70, to 19,000 event. Resulting signal to background ratio is \(S/B \approx 0.14\): 14% excess in the number of selected events, relative to the SM predictions, is expected in the considered IDM scenario.

When estimating the CLIC sensitivity to the charged IDM scalar production for all of the 23 benchmark scenarios considered, based on the fast detector response simulation with Delphes, a conservative approach was used and the BDTs were trained on all available samples, separately for scenarios with on- and off-shell W\(^\pm \) boson production. However, the optimal cut on the BDT response was found separately for each signal scenario. This is justified by the fact that, in the actual experiment, the measured and predicted BDT response distributions should be compared and not just the number of events after the cut. For both fast and full simulation analyses, an additional requirement was imposed when looking for the optimal BDT response cut. Only cut values resulting in a total signal selection efficiency of at least 10% were allowed, to reduce the impact of statistical fluctuations in the generated MC samples.

Fig. 5
figure 5

Statistical significance expected from the full simulation study (red points), compared with the fast simulation results after including the overlay background (green points). The significance obtainable from the fast simulation analysis without the influence of the overlay are shown for comparison as blue crosses. The results are presented for considered benchmark points as a function of \(2m_{H^{\pm }}\), for 1.5 TeV (left) and 3 TeV (right) CLIC running, and were produced with the selection optimised for a particular scenario. The red line shows the 5\(\sigma \) threshold

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There are visible correlations between input variables, resulting from their definitions. The highest correlations occur between \(E_{jj}\) and \(E_{j_{i}}\), \(\theta _{W}\) and \(\theta _{j_{i}}\) (\(i=1,2\)), as well as \(E_{\ell }\) and \(p_{T}^{\ell }\), reaching from 93 to 97%. In general, correlations depend on the considered sample (signal scenario or background). Only for few variable pairs high correlations are observed for all signal scenarios and SM background. In most cases, two variables highly correlated in one training data-set (e.g. the signal sample in case of the full simulation analysis) are not necessarily that much correlated in another one. Differences in variable correlations were also observed between fast and full simulation studies. Because of the randomisation procedure mentioned above, correlations between variables do not affect the efficiency of the BDT training. By keeping all considered variables as an input to the BDT, a consistent approach can be used for all scenarios, for both fast and full simulation. It was verified that removing any of these variables results in worse discrimination-power between signal and background events, and a reduced sensitivity to IDM scalar production for at least some of the scenarios.

6 Results

The main goal of the analysis was to establish the statistical significance of deviations from the SM predictions expected at high energy CLIC stages for each of the considered benchmark scenarios. The significance, calculated for the 5 scenarios included in the full detector simulation analysis, is shown in Fig. 5 as a function of the total mass of the produced IDM scalars, which is twice the charged IDM scalar mass for the considered process, \(2 m_{H^\pm }\).

Results based on full simulation are compared with those based on Delphes, with and without the \(\gamma \gamma \rightarrow \) had. background included. All results in Fig. 5 were obtained with the BDT selection optimised for an individual benchmark scenario. While the expected significance is decreasing with increasing scalar mass, the pair production of charged IDM scalars can be discovered at CLIC (significance above 5\(\sigma \)) for all benchmark points considered in the full simulation. The results based on the fast simulation tend to be more optimistic. However, when overlay background is included in the Delphes simulation, the agreement with the full simulation results is significantly improved.

The impact of the proposed procedure to include overlay contribution on the agreement between fast and full simulation is also presented in Fig. 6. It shows the ratios of the significance obtained using the fast simulation, with and without overlay background included, to the full simulation (ratios of the significances presented in Fig. 5), as a function of the IDM scalar mass difference, \(m_{H^{\pm }} - m_{H}\), which determines the virtuality of the W bosons produced in the \(H^\pm \) decays. The two points with the highest mass splittings (BP21 and BP23) correspond to the points with the smallest charged scalar mass \(m_{H^\pm }\), shown in Fig. 5. For large mass differences, \(m_{H^\pm } - m_H > m_W\), good agreement is observed between fast and full simulation results. If the overlay background is not included in Delphes, significant discrepancies between fast and full simulation arise for low mass differences, \(m_{H^\pm } - m_H < m_W\). While the observed deviations are significantly reduced after the \(\gamma \gamma \rightarrow \) had. background is taken into account, some discrepancies still remain. Therefore, in order to further account for the possible systematic effects, the correction factor \(\Delta \) was introduced to describe the remaining discrepancy between significance, \({\mathbf {S}}\), resulting from fast and full simulation study:

$$\begin{aligned} \mathbf {S}_\text {full} = \mathbf {S}_\text {fast} \cdot \left[ 1 - \Delta (m_{H^{\pm }}-m_H) \right] \; , \end{aligned}$$
(2)

where the correction \(\Delta \), as suggested by Fig. 6, is assumed to depend only on the scalar mass splitting. The following functional form was chosen to describe the correction:

$$\begin{aligned} \Delta (\delta m) = a + \frac{b}{1+\left( \delta m / c \right) ^6} \; , \end{aligned}$$
(3)

where parameters a, b and c were fitted to the points corresponding to the fast simulation with overlay background included, at both energy stages simultaneously. The resulting value of parameter c is consistent with the mass of the W boson, indicating that the differences between the full and fast simulation are relevant only when virtual boson production is expected, for \(m_{H^{\pm }} - m_{H}\) \( < m_W\). When mass splitting is large and on-shell W production is expected for signal scenarios, similarly as for the dominant background channel (qq\(\ell \nu \)), possible differences between fast and full simulation do not affect signal-background separation and the results for the two simulation approaches are very similar.

The correction given by Eq. 2 was used to scale the significance results obtained with modified Delphes. To take other possible systematic effects and the arbitrary choice of the correction function into account, we assume 100% uncertainty on \(\Delta \). Final results obtained for all considered benchmark scenarios with the Delphes fast simulation, including overlay events and training the BDTs on all available scenarios (as described in Sect. 5), are presented in Fig. 7 together with the uncertainties. The expected statistical significance of IDM charged scalar production is shown as a function of the scalar mass differences, \(m_{H^{\pm }} - m_{H}\), as well as of the total mass of the produced IDM scalars, \(2\; m_{H^\pm }\), for the two high energy stages of CLIC. For most of the considered scenarios, pair-production of charged IDM scalars can be observed at CLIC with high significance, attaining even more than 40\(\sigma \), and for the scalar masses up to 1 TeV. Only for two of the benchmark points the expected significance is below 5\(\sigma \) and for three of them it is over 5\(\sigma \) within less than the systematic uncertainty. There is also no visible dependence of the discovery reach on the dark scalar mass difference but for scenarios with very small values of \(m_{H^{\pm }} - m_{H}\) which are clearly much more challenging due to the influence of \(\gamma \gamma \rightarrow \) had. events.

Fig. 6
figure 6

Ratios of the expected observation significances resulting from fast and full simulation studies. Open circles correspond to Delphes   simulation results with no overlaid \(\gamma \gamma \rightarrow \) had. events, while full circles show the ratios after inclusion of this background, for CLIC running at 1.5 TeV (orange) and 3 TeV (blue points). The red curve corresponds to the dependence described by Eq. (3) used to correct for the remaining difference between full simulation results and Delphes with overlay at both energy stages. See the text for details

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Fig. 7
figure 7

Expected statistical significance of IDM charged scalar pair-production observation as a function of the IDM scalar mass difference, \(m_{H^{\pm }} - m_{H}\) (left) and of the total mass of the produced IDM scalars, \(2m_{H^{\pm }}\) (right). Results of the Delphes fast simulation study are presented for CLIC running at 1.5 TeV (orange circles) and 3 TeV (blue points). Error bars indicate the systematic uncertainty, estimated from the observed difference between fast and full simulation results, see text for details. The red horizontal lines indicate the 5\(\sigma \) threshold

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The significance of the observation is expected to depend mainly on the signal production cross section, which is determined by the charged scalar mass (see Fig. 1). To study the impact of other model parameters, results presented in Fig. 7 were scaled to the signal production cross section times the branching ratio in the semi-leptonic channel of 1 fb. Also, for comparison of the experimental sensitivity at 1.5 TeV and 3 TeV, the same integrated luminosity of 4 ab\(^{-1}\) was assumed for both CLIC high energy running stages. Scaled significance values, after applying correction of Eq. (2), are presented in Fig. 8. Results show that the best signal-background separation can be obtained for scenarios with scalar mass difference 20 GeV < \(m_{H^{\pm }} - m_{H}\) < 50 GeV. For smaller mass differences, the impact of overlay events limits the experimental sensitivity, while for higher mass differences backgrounds dominated by processes with real W\(^\pm \) production are more difficult to suppress. For a given production cross section, there is no additional systematic dependence of the sensitivity on the charged scalar mass, \(m_{H^\pm }\). The sensitivity to charged IDM scalar production at 1.5 TeV CLIC seems to be slightly better than at 3 TeV, which is most likely due to the smaller boost of the produced IDM scalars and W bosons, resulting in better reconstruction of the di-jet invariant mass (see Fig. 3).

Fig. 8
figure 8

Expected statistical significance of IDM charged scalar pair-production observation, assuming the semi-leptonic channel cross section of 1 fb: as a function of the IDM scalar mass difference, \(m_{H^{\pm }} - m_{H}\) (left) and of the total mass of the produced IDM scalars, \(2m_{H^{\pm }}\) (right). Results of the Delphes fast simulation study are presented for CLIC running at 1.5 TeV (orange circles) and 3 TeV (blue points) with 4 ab\(^{-1}\) of integrated luminosity assumed for both stages. Error bars indicate the systematic uncertainty, estimated from the observed difference between fast and full simulation results, see text for details. The red horizontal lines indicate the 5\(\sigma \) threshold

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7 Summary

Prospects for observing production of new heavy scalar particles were studied for CLIC high energy running stages. Pair-production of heavy charged scalars was studied in the framework of the Inert Doublet Model, based on the benchmark scenarios proposed in [26]. The expected final state, resulting from the charged scalar decays, consists of two (real or virtual) W\(^\pm \) bosons and a large missing energy-momentum from the two escaping neutral scalars H, dark matter candidates. For low mass scenarios and collision energies up to 500 GeV, production rates are high and the di-lepton final state can be used as a discovery channel [27]. However, this is no longer the case for scalar masses accessible only at higher collision energies. For high mass scenarios considered in this study, the semi-leptonic final state was examined, offering higher decay rates and hence also higher statistical significance than the leptonic channels studied previously. The CLIC potential was studied for 5 selected IDM scenarios with the full detector simulation based on Geant4 and for a complete set of 23 high mass IDM benchmark scenarios using the Delphes fast simulation framework. The CLICdet model for Delphes was extended to take into account the \(\gamma \gamma \rightarrow \) had. overlay events. This beam-induced background is crucial for the analysis, in particular for signal scenarios with low scalar mass differences, when the virtual W boson decay products are very soft. Results of the study indicate that charged IDM scalars with masses up to 1 TeV can be detected at high energy running stages of CLIC. For low scalar masses, the expected significance of the observation reaches levels of up to about 40 standard deviations. Presented results indicate that semi-leptonic final state can be an important discovery channel also at other high energy lepton colliders, as 1 TeV ILC [17] or Muon Collider [43].