###### Abstract

The Inert Doublet Model (IDM) is a simple extension of the Standard Model,
introducing an additional Higgs doublet that brings in four new scalar
particles. The lightest of the IDM scalars is stable and is a good candidate
for a dark matter particle. The potential of discovering the IDM scalars in
the experiment at the Compact Linear Collider (CLIC), an $e^{+}e^{-}$ collider
proposed as the next generation infrastructure at CERN, has been tested for
two high-energy running stages, at 1.5 TeV and 3 TeV centre-of-mass energy.
The CLIC sensitivity to pair-production of the charged IDM scalars was studied
using the full detector simulation with Geant4 for selected high-mass IDM
benchmark scenarios and the semi-leptonic final state. To extrapolate full
simulation results to a wider range of IDM benchmark scenarios, the CLIC
detector model defined in the Delphes fast simulation framework was modified
to take into account the $\gamma\gamma\to$ had. beam-induced background.
Results of the study indicate that heavy charged IDM scalars can be discovered
at CLIC for most of the considered benchmark scenarios, up to masses of the
order of 1 TeV.

##  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 concept of 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 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 Sec. 3. Details of the event selection procedure
are presented in Secs. 4 and 5, and final results are discussed in Sec. 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 the simplest extensions of the
SM. 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±, 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 $\lambda_{345}$.111For more information about
the model, please refer to [25, 26] and references therein. 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].  No. | $M_{H}$ [GeV] | $M_{A}$ [GeV] | $M_{H^{\pm}}$ [GeV] | $Z$ | $W$ | DM | $\lambda_{2}$ | $\lambda_{345}$ | $\Omega_{H}h^{2}$
---|---|---|---|---|---|---|---|---|---
on-shell | on-shell |  $>$50%
BP23 | 62.69 | 162.397 | 190.822 | $\checked$ | $\checked$ | $\checked$ | 2.63894 | 0.0056 | 0.064038
BP18 | 147 | 194.647 | 197.403 |  |  |  | 0.387 | -0.018 | 0.0017718
BP21 | 57.475 | 288.031 | 299.536 | $\checked$ | $\checked$ | $\checked$ | 0.929911 | 0.00192 | 0.11946
HP1 | 176 | 291.36 | 311.96 | $\checked$ | $\checked$ |  | 1.4895 | -0.1035 | 0.00072156
HP2 | 557 | 562.316 | 565.417 |  |  | $\checked$ | 4.0455 | -0.1385 | 0.072092
HP3 | 560 | 616.32 | 633.48 |  |  |  | 3.3795 | -0.0895 | 0.001129
HP4 | 571 | 676.534 | 682.54 | $\checked$ | $\checked$ |  | 1.98 | -0.471 | 0.00056347
HP5 | 671 | 688.108 | 688.437 |  |  |  | 1.377 | -0.1455 | 0.024471
HP6 | 713 | 716.444 | 723.045 |  |  |  | 2.88 | 0.2885 | 0.035152
HP7 | 807 | 813.369 | 818.001 |  |  |  | 3.6675 | 0.299 | 0.032393
HP8 | 933 | 939.968 | 943.787 |  |  | $\checked$ | 2.9745 | -0.2435 | 0.09639
HP9 | 935 | 986.22 | 987.975 |  |  |  | 2.484 | -0.5795 | 0.0027958
HP10 | 990 | 992.36 | 998.12 |  |  | $\checked$ | 3.3345 | -0.051 | 0.12478
HP11 | 250.5 | 265.49 | 287.226 |  |  |  | 3.90814 | -0.150071 | 0.00535
HP12 | 286.05 | 294.617 | 332.457 |  |  |  | 3.29239 | 0.112124 | 0.00277
HP13 | 336 | 353.264 | 360.568 |  |  |  | 2.48814 | -0.106372 | 0.00937
HP14 | 326.55 | 331.938 | 381.773 |  |  |  | 0.0251327 | -0.0626727 | 0.00356
HP15 | 357.6 | 399.998 | 402.568 |  |  |  | 2.06088 | -0.237469 | 0.00346
HP16 | 387.75 | 406.118 | 413.464 |  |  |  | 0.816814 | -0.208336 | 0.0116
HP17 | 430.95 | 433.226 | 440.624 |  |  |  | 3.00336 | 0.082991 | 0.0327
HP18 | 428.25 | 453.979 | 459.696 |  |  |  | 3.87044 | -0.281168 | 0.00858
HP19 | 467.85 | 488.604 | 492.329 |  |  |  | 4.12177 | -0.252036 | 0.0139
HP20 | 505.2 | 516.58 | 543.794 |  |  |  | 2.53841 | -0.354 | 0.00887

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:

$\displaystyle e^{+}e^{-}$ $\displaystyle\to$
$\displaystyle\mathrm{H}\;\;\mathrm{A}\;,$ $\displaystyle e^{+}e^{-}$
$\displaystyle\to$ $\displaystyle\mathrm{H}^{+}\mathrm{H}^{-}.$

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

$\displaystyle e^{+}e^{-}$ $\displaystyle\to$
$\displaystyle\mathrm{H}\;\;\;\mathrm{A}\;\;\;\to\mathrm{H}\;\;\mathrm{H}\;\;\mathrm{Z}\;,$
$\displaystyle e^{+}e^{-}$ $\displaystyle\to$
$\displaystyle\mathrm{H}^{+}\mathrm{H}^{-}\to\mathrm{W}^{+}\mathrm{W}^{-}\mathrm{H}\;\;\mathrm{H}\;.$
(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± allows also for better background suppression for events with
virtual W± production. The cross sections for the H+H- production at 1.5 TeV
and 3 TeV CLIC are presented in Figure 1. They depend mostly on the scalar
masses; the influence of couplings $\lambda_{2}$ and $\lambda_{345}$ is
marginal.

$\sqrt{s}=$1.5 TeV
$\sqrt{s}=$3 TeV

Figure 1:  Leading-order cross sections for the charged inert scalar
production, $e^{+}e^{-}\to\mathrm{H}^{+}\mathrm{H}^{-}$, 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. Figure taken from
[27].

##  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.

### 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 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
$\gamma\gamma\to$ 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 BX 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% to 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± 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.222Lepton identification purity is mainly affected by signal
events with hadronic tau decays. Purity of up to 99% is obtained when only
electron and muon decays of W boson are considered.

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± boson. Isolated photons were also identified. A photon with
transverse momentum $p_{T}^{\gamma}$ was considered isolated, if
$p_{T}^{cone}/p_{T}^{\gamma}\leq 0.2$, 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.

### 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 $p_{T}$.333These isolation criteria differ from the default CLICdet
cards settings described in [40].  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± 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 $\gamma\gamma\to$ 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 $\gamma\gamma\to$ had. events added to each physical
event is $20\cdot n_{had}$, where 20 comes from the number of BX that survive
the primary 10 ns window cut and $n_{had}$ is the expected number of
$\gamma\gamma\to$ 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 spacial overlap of detector
deposits are neglected.

Figure 2 presents the impact of $\gamma\gamma\to$ 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 Tab. 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 $\gamma\gamma\to$ 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.

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).

##  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).

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.  variable | cut @ 1.5 TeV | cut @ 3 TeV
---|---|---
$E_{jj}$ | $<750\,\mathrm{GeV}$ | $<1.3\,\mathrm{TeV}$
$M_{jj}$ | $>3\,\mathrm{GeV}$ | $>3\,\mathrm{GeV}$
$\theta_{jj}$ | $0.2<\theta_{jj}<\pi-0.2$ | $0.3<\theta_{jj}<\pi-0.3$
$E_{\ell}$ | $<600\,\mathrm{GeV}$ | $<1\,\mathrm{TeV}$
$p_{T}^{\ell}$ | $<550\,\mathrm{GeV}$ | $<800\,\mathrm{GeV}$
$\theta_{\ell}$ | $0.25<\theta_{\ell}<\pi-0.25$ | $0.5<\theta_{\ell}<\pi-0.5$
$M_{miss}$ | $>400\,\mathrm{GeV}$ | -

The following variables were used in the procedure:

  * •

$E_{jj}$ \- energy of a dijet system,

  * •

$M_{jj}$ \- invariant mass of a dijet system444The cut on a dijet system mass
was introduced to suppress significant contributions of leptonic events
misidentified as two-jet events in the fast simulation analysis, when the
overlay background was not taken into account. This cut is not required when
the overlay background contribution is included nor in the full simulation
analysis. It was kept for the consistency.,

  * •

$\theta_{jj}$ \- 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± boson production (BP21) and one where the produced W± 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 $\gamma\gamma\to$ 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.

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. 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 in the Appendix. Signal scenarios with off-shell W± production are marked with a star.  channel ($m_{H^{\pm}}$ [GeV]) |  $\sigma$ [fb] | exp. ev. after pres. | eff.
---|---|---|---
BP21 (300) | 8.09 | 9110 | 56%
BP23 (191) | 12.5 | 14200 | 57%
HP17∗ (633) | 2.43 | 1600 | 33%
HP20∗ (441) | 1.32 | 1500 | 57%
HP3∗ (544) | 0.629 | 749 | 60%
qq$\ell\nu$ | 7000 | 1020000 | 7.3%
qq$\ell\ell$ | 2715 | 244000 | 4.5%
$\ell\ell$ | 1406 | 140000 | 5%
qqqq | 1940 | 14100 | 0.36%
qq$\ell\nu\ell\nu$ | 14.9 | 2370 | 8%
qqqq$\ell\nu$ | 169 | 14400 | 4.3%
qq$\ell\nu\nu\nu$ | 66.8 | 56100 | 42%
qq$\nu\nu$ | 1460 | 124000 | 4.2%
qq | 2860 | 48000 | 0.84%
tot. backg. | 17638.9 | 1660000 | 4.7%
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 in the Appendix. Scenarios with off-shell W± production are marked with a star. channel ($m_{H^{\pm}}$ [GeV]) |  $\sigma$ [fb] | exp. ev. after pres. | eff.
---|---|---|---
BP21 (300) | 4.21 | 6460 | 38%
BP23 (191) | 5.77 | 8490 | 37%
HP17∗ (633) | 1.68 | 2020 | 30%
HP20∗ (441) | 1.51 | 2640 | 44%
HP3∗ (544) | 1.78 | 3150 | 44%
qq$\ell\nu$ | 8670 | 707000 | 2%
qq$\ell\ell$ | 3180 | 203000 | 1.6%
$\ell\ell$ | 1670 | 83000 | 1.2%
qqqq | 902 | 14900 | 0.41%
qq$\ell\nu\ell\nu$ | 20.4 | 6610 | 8.1%
qqqq$\ell\nu$ | 148 | 17300 | 2.9%
qq$\ell\nu\nu\nu$ | 96.8 | 72500 | 19%
qq$\nu\nu$ | 2330 | 212000 | 2.3%
qq | 1270 | 36400 | 0.72%
tot. backg. | 18286.7 | 1350000 | 1.8%

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 $qql\nu$ 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 Sec. 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± 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 Figure 4, for the
two selected signal scenarios, BP21 and HP17, and the two CLIC running stages.

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$\sigma$.

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± 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.

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
Figure 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, $2m_{H^{\pm}}$.

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.

Results based on full simulation are compared with those based on Delphes,
with and without the $\gamma\gamma\to$ had. background included. All results
in Figure 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 Figure 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 Figure 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 $\gamma\gamma\to$ 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:

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

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

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 $\gamma\gamma\to$ 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.

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 Sec. 5), are
presented in Figure 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 $\gamma\gamma\to$ had. events.

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.

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
Figure 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 mas 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).

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.

##  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± 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 considered,
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 $\gamma\gamma\to$ 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].

## Acknowledgements

The work was carried out in the framework of the CLIC detector and physics
(CLICdp) collaboration. We thank collaboration members for fruitful
discussions, valuable comments and suggestions. This work benefited from
services provided by the ILC Virtual Organisation, supported by the national
resource providers of the EGI Federation. This research was done using
resources provided by the Open Science Grid, which is supported by the
National Science Foundation and the U.S. Department of Energy’s Office of
Science. The work was partially supported by the National Science Centre
(Poland) under OPUS research project no. 2017/25/B/ST2/00496 (2018-2021).

## References

  * [1] Serguei Chatrchyan  “Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC”  In _Phys. Lett. B_ 716, 2012, pp. 30–61  DOI: 10.1016/j.physletb.2012.08.021
  * [2] Georges Aad  “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC”  In _Phys. Lett. B_ 716, 2012, pp. 1–29  DOI: 10.1016/j.physletb.2012.08.020
  * [3] N. Aghanim  “Planck 2018 results. VI. Cosmological parameters” [Erratum: Astron.Astrophys. 652, C4 (2021)]  In _Astron. Astrophys._ 641, 2020, pp. A6  DOI: 10.1051/0004-6361/201833910
  * [4] B. Abi  “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm”  In _Phys. Rev. Lett._ 126.14, 2021, pp. 141801  DOI: 10.1103/PhysRevLett.126.141801
  * [5] Roel Aaij  “Tests of lepton universality using $B^{0}\to K^{0}_{S}\ell^{+}\ell^{-}$ and $B^{+}\to K^{*+}\ell^{+}\ell^{-}$ decays”  In _Phys. Rev. Lett._ 128.19, 2022, pp. 191802  DOI: 10.1103/PhysRevLett.128.191802
  * [6] Roel Aaij  “Test of lepton universality in beauty-quark decays”  In _Nature Phys._ 18.3, 2022, pp. 277–282  DOI: 10.1038/s41567-021-01478-8
  * [7] T. Aaltonen  “High-precision measurement of the W boson mass with the CDF II detector”  In _Science_ 376.6589, 2022, pp. 170–176  DOI: 10.1126/science.abk1781
  * [8] Tania Robens  “Constraining extended scalar sectors at current and future colliders”  In _21st Hellenic School and Workshops on Elementary Particle Physics and Gravity_ , 2022  arXiv:2203.17016 [hep-ph]
  * [9] Georges Aad  “Combined Measurement of the Higgs Boson Mass in $pp$ Collisions at $\sqrt{s}=7$ and 8 TeV with the ATLAS and CMS Experiments”  In _Phys. Rev. Lett._ 114, 2015, pp. 191803  DOI: 10.1103/PhysRevLett.114.191803
  * [10] Morad Aaboud  “Measurement of the Higgs boson mass in the $H\rightarrow ZZ^{*}\rightarrow 4\ell$ and $H\rightarrow\gamma\gamma$ channels with $\sqrt{s}=13$ TeV $pp$ collisions using the ATLAS detector”  In _Phys. Lett. B_ 784, 2018, pp. 345–366  DOI: 10.1016/j.physletb.2018.07.050
  * [11] Albert M Sirunyan  “A measurement of the Higgs boson mass in the diphoton decay channel”  In _Phys. Lett. B_ 805, 2020, pp. 135425  DOI: 10.1016/j.physletb.2020.135425
  * [12] Albert M Sirunyan  “Combined measurements of Higgs boson couplings in proton–proton collisions at $\sqrt{s}=13\,\text{Te}\text{V}$”  In _Eur. Phys. J. C_ 79.5, 2019, pp. 421  DOI: 10.1140/epjc/s10052-019-6909-y
  * [13] Georges Aad  “Combined measurements of Higgs boson production and decay using up to $80$ fb-1 of proton-proton collision data at $\sqrt{s}=$ 13 TeV collected with the ATLAS experiment”  In _Phys. Rev. D_ 101.1, 2020, pp. 012002  DOI: 10.1103/PhysRevD.101.012002
  * [14] Albert M Sirunyan  “Evidence for Higgs boson decay to a pair of muons”  In _JHEP_ 01, 2021, pp. 148  DOI: 10.1007/JHEP01(2021)148
  * [15] J. Blas  “Higgs Boson Studies at Future Particle Colliders”  In _JHEP_ 01, 2020, pp. 139  DOI: 10.1007/JHEP01(2020)139
  * [16] “2020 Update of the European Strategy for Particle Physics (Brochure)”, 2020  URL: https://cds.cern.ch/record/2721370
  * [17] Philip Bambade et al.  “The International Linear Collider: A Global Project”, 2019  arXiv:1903.01629 [hep-ex]
  * [18] T.K. Charles  “The Compact Linear Collider (CLIC) - 2018 Summary Report”, 2018  DOI: 10.23731/CYRM-2018-002
  * [19] A. Abada  “FCC-ee: The Lepton Collider: Future Circular Collider Conceptual Design Report Volume 2”  In _Eur. Phys. J. ST_ 228.2, 2019, pp. 261–623  DOI: 10.1140/epjst/e2019-900045-4
  * [20] “CEPC Conceptual Design Report: Volume 1 - Accelerator”, 2018  arXiv:1809.00285 [physics.acc-ph]
  * [21] Mei Bai  “C3: A ”Cool” Route to the Higgs Boson and Beyond”  In _2022 Snowmass Summer Study_ , 2021  arXiv:2110.15800 [hep-ex]
  * [22] Dominik Arominski  “A detector for CLIC: main parameters and performance” CLICdp-Note-2018-005, 2018  arXiv:1812.07337 [physics.ins-det]
  * [23] Jan Kalinowski et al.  “Exploring Inert Scalars at CLIC”  In _JHEP_ 07, 2019, pp. 053  DOI: 10.1007/JHEP07(2019)053
  * [24] Nilendra G. Deshpande and Ernest Ma  “Pattern of Symmetry Breaking with Two Higgs Doublets”  In _Phys. Rev. D_ 18, 1978, pp. 2574  DOI: 10.1103/PhysRevD.18.2574
  * [25] Agnieszka Ilnicka, Maria Krawczyk and Tania Robens  “Inert Doublet Model in light of LHC Run I and astrophysical data”  In _Phys. Rev. D_ 93.5, 2016, pp. 055026  DOI: 10.1103/PhysRevD.93.055026
  * [26] Jan Kalinowski et al.  “Benchmarking the Inert Doublet Model for $e^{+}e^{-}$ colliders”  In _JHEP_ 12, 2018, pp. 081  DOI: 10.1007/JHEP12(2018)081
  * [27] Aleksander Filip Zarnecki et al.  “Searching Inert Scalars at Future e+e- Colliders”  In _International Workshop on Future Linear Colliders_ , 2020  arXiv:2002.11716 [hep-ph]
  * [28] Wolfgang Kilian, Thorsten Ohl and Jurgen Reuter  “WHIZARD: Simulating Multi-Particle Processes at LHC and ILC”  In _Eur. Phys. J. C_ 71, 2011, pp. 1742  DOI: 10.1140/epjc/s10052-011-1742-y
  * [29] Torbjörn Sjöstrand et al.  “An Introduction to PYTHIA 8.2”  In _Comput. Phys. Commun._ 191, 2015, pp. 159–177  DOI: 10.1016/j.cpc.2015.01.024
  * [30] S. Agostinelli  “Geant4 — a simulation toolkit”  In _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 506.3, 2003, pp. 250–303  DOI: https://doi.org/10.1016/S0168-9002(03)01368-8
  * [31] M Frank, F Gaede, C Grefe and P Mato  “DD4hep: A Detector Description Toolkit for High Energy Physics Experiments”  In _Journal of Physics: Conference Series_ 513.2 IOP Publishing, 2014, pp. 022010  DOI: 10.1088/1742-6596/513/2/022010
  * [32] M.A. Thomson  “Particle flow calorimetry and the PandoraPFA algorithm”  In _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 611.1 Elsevier BV, 2009, pp. 25–40  DOI: 10.1016/j.nima.2009.09.009
  * [33] J.. Marshall and M.. Thomson  “The Pandora Software Development Kit for Pattern Recognition”  In _Eur. Phys. J. C_ 75.9, 2015, pp. 439  DOI: 10.1140/epjc/s10052-015-3659-3
  * [34] C Grefe, S Poss, A Sailer and A Tsaregorodtsev  “ILCDIRAC, a DIRAC extension for the Linear Collider community”  In _J. Phys.: Conf. Ser._ 513.CLICdp-Conf-2013-003, 2013, pp. 032077. 5  URL: https://cds.cern.ch/record/1626585
  * [35] “Physics and Detectors at CLIC: CLIC Conceptual Design Report, CERN-2012-003”  CERN, 2012
  * [36] Frank Gaede and J. Engels  “Marlin et al - A Software Framework for ILC detector R&D”  In _EUDET Report_ , 2007  URL: https://www.eudet.org/e26/e27/e584/eudet-report-2007-11.pdf
  * [37] Jan Franciszek Klamka  “Searching for Inert Doublet Model scalars at high energy CLIC”, University of Warsaw, 2020  URL: https://cds.cern.ch/record/2728552
  * [38] M. Boronat et al.  “Jet reconstruction at high-energy electron–positron colliders”  In _Eur. Phys. J. C_ 78.2, 2018, pp. 144  DOI: 10.1140/epjc/s10052-018-5594-6
  * [39] J. Favereau et al.  “DELPHES 3, A modular framework for fast simulation of a generic collider experiment”  In _JHEP_ 02, 2014, pp. 057  DOI: 10.1007/JHEP02(2014)057
  * [40] Emilia Leogrande, Philipp Roloff, Ulrike Schnoor and Matthias Weber  “A DELPHES card for the CLIC detector” CLICdp-Note-2018-007, 2019  arXiv:1909.12728 [hep-ex]
  * [41] Matthias Artur Weber  “Jet Performance at CLIC”, CLICdp-Note-2018-004, 2018  URL: https://cds.cern.ch/record/2648827
  * [42] A. Hoecker  “TMVA - Toolkit for Multivariate Data Analysis”, 2007  arXiv:physics/0703039 [physics.data-an]
  * [43] Jean Pierre Delahaye et al.  “Muon Colliders”, 2019  arXiv:1901.06150 [physics.acc-ph]