###### Abstract

The Compact Linear Collider, CLIC, is a multi-TeV electron-positron collider
proposed for construction at CERN. A detector model, CLICdet, that is suited
for the experimental conditions at CLIC and is based on realistic performance,
has been developed. This paper describes the implementation of CLICdet in a
fast simulation tool for particle physics collider experiments, Delphes. The
geometry of the detector concept as well as performance parameters extracted
from full simulation studies are implemented in Delphes parameter cards for
CLICdet. Jet reconstruction for electron-positron colliders is added to the
Delphes analysis chain. Parameters for using Delphes to simulate the detector
effects of CLICdet are provided in three parameter cards, one for each energy
stage of CLIC. The effects of beam-induced background at the higher-energy
stages of CLIC are also incorporated. The results from the fast simulation
with Delphes are validated with respect to full detector simulation in a
number of relevant processes.

##  1 DELPHES for fast simulation of detector effects at CLIC

The detector model for CLIC, CLICdet, has been defined and described in [1]
and its performance is discussed in detail in [2]. Full simulation collision
events can be obtained by running the simulation and reconstruction chain on
hadron-level Monte Carlo events generated e.g. with Whizard [3] and Pythia6
[4]. Such a full detector simulation is performed for signal and background
events for most of the studies carried out in the framework of the CLICdp
collaboration, e.g. [5, 6]. In order to optimise the design of the detector
and to understand fully its impact on the proposed physics programme, the full
detector simulation and event reconstruction framework is essential. However,
for many studies including some of those of an exploratory nature or where the
relative sensitivity to physics parameters is paramount, a faster simulation
tool is extremely useful.

The fast simulation framework Delphes [7, 8, 9] is a modular system which
emulates the impact that the full detector simulation will have on hadron-
level simulation of collision events. These include the effects of detector
geometry, detector response and resolution as well as reconstruction
performance of physics objects. It can be used to approximate those kinematic
distributions for which the detector effects are well described by a
parametrisation. The simulation with Delphes requires much less CPU time than
a full detector simulation which models the interactions of particles with the
material of the detector. According to the specific detector, an input
parameter card must be provided. This document describes the parameter card
developed for the CLICdet detector model. It has been used for several studies
carried out in the framework of the recent report on the CLIC potential for
new physics [10].

A generic layout of a general-purpose detector for particle collider
experiments is implemented in Delphes, of which some geometry parameters can
be modified according to the given detector model. In addition, the
performance of identification and reconstruction based on certain objects and
particle types can be defined in the detector card. The geometry and
performance parameters inserted into the CLICdet card are described in Sec. 2.
The order in which detector effects are applied to generator level objects is
described in Sec. 3.1. Until recently, Delphes has been used predominantly for
studies at hadron colliders, in particular the LHC. Jet clustering at
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$
colliders offers additional features which have been implemented into the
Delphes framework in the context of the development of the parameter card.
This is described in Sec. 3.2. Sec. 4 describes the validation of the Delphes
card, comparing its results for important physics processes with full detector
simulation. Additional validation plots can be found in Appendix B. The
parameter card has been merged into the central Delphes repository and can be
accessed at [11]. User instructions are given in Appendix A.

##  2 Performance parameters of the CLICdet detector

###  2.1 The CLICdet detector

The current detector model for CLIC, CLICdet, is adapted to the high-energy
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$
collision environment at CLIC and optimised for particle flow reconstruction.
The CLICdet concept was developed based on the lessons learned from the
previous detector models for CLIC: CLIC_ILD and CLIC_SiD [12].

The CLICdet model comprises a low-mass silicon vertex and tracking system with
a central barrel and forward disks. CLICdet is optimised for particle flow in
particular by using highly granular calorimetry based on silicon-tungsten
electromagnetic calorimeters and scintillator-steel hadronic calorimeters. A
superconducting solenoid surrounding the calorimeters provides a magnetic
field of 4 T. Beyond the solenoid, an iron yoke is interleaved with muon
chambers. In the forward region, close to the beam pipe, CLICdet contains two
specialised forward electromagnetic calorimeters to measure luminosity and
beam properties as well as to extend the acceptance.

The two previous models differ from CLICdet in the following main aspects[1]:
The CLIC_SiD concept uses a 5 T magnetic field and a smaller radius tracking
system. CLIC_ILD uses a TPC as tracking system. In both cases, tungsten is
used as absorber in the barrel of the hadronic calorimeter. While the CLIC
Delphes card is based on the CLICdet performance, some of the validation
studies for which no CLICdet simulation was available are based on one of the
previous detector concepts.

The right-handed coordinate system has its origin at the centre of the
detector, the $z$ axis along the detector axis close to the beam direction,
and the $y$ axis pointing upwards. While most studies of
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$
collisions employ the polar angle $\theta$ coordinate, defined with respect to
the positive $z$ axis, this note mainly uses the pseudorapidity as it is the
standard for Delphes. The pseudorapidity $\eta$ is defined as
$\eta=-\ln{[\tan{(\theta/2)}]}$ and the distance $\Delta R$ in the
pseudorapidity-azimuth plane is defined as $\Delta
R=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}$.

Under the baseline staging scenario, CLIC is expected to deliver 1 ab-1 of
luminosity at $\sqrt{s}$ = 380 GeV, 2.5 ab-1 at $\sqrt{s}$ = 1.5 TeV, and 5
ab-1 at $\sqrt{s}$ = 3 TeV [13]. The luminosity will be shared evenly between
an electron beam polarisation of
$P(\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}})=-80\,\%$ and $+80\,\%$ at
the first stage. At the higher energy stages, 80 % (20 %) of the luminosity
will be collected with
$P(\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}})=-80\,\%$
($P(\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}})=+80\,\%$).

###  2.2 Performance parameters for the Delphes card

The performance of CLICdet has been thoroughly evaluated using full detector
simulation in [2]. The detector geometry of CLICdet for the full simulation is
implemented within the DD4Hep framework [14] and simulated in Geant4 [15, 16,
17]. The reconstruction software is implemented in the Linear Collider
framework MARLIN [18]. Track reconstruction is performed with the Conformal
Tracking method [19]. Based on the reconstructed tracks and calorimeter
deposits, particle flow clustering is performed using PandoraPFA [20, 21, 22].
Flavour tagging is applied using the LCFIPlus package [23].

Three cards for the three energy stages of CLIC are provided in Delphes.
Performance parameters for the CLICdet Delphes cards are taken from studies
for CLICdet described and referenced in the following.

####  2.2.1 Track reconstruction performance

The tracking efficiency is based on studies of the conformal tracking
algorithm in [2, Fig. 33 and 35]. Charged hadron efficiencies are based on the
results for pions. Efficiencies for muon tracks, electron tracks, and charged
hadron tracks are provided as a function of energy and pseudorapidity. In
addition, the momentum resolution from tracking is applied according to a log-
normal distribution with the resolution given as $\frac{\Delta
p_{\text{T}}}{p_{\text{T}}}=a\oplus b/(p\sin{\theta})$ binned in
pseudorapidity. The parameters used in the CLICdet cards are derived from [2,
Sec. 4.2.1, Fig. 31, 32].

####  2.2.2 Calorimeter performance

Delphes takes into account the granularity of the calorimeter in terms of
pseudorapidity and azimuthal angle. Based on the geometry of the calorimeters
described in [1], the $\Delta\eta$ and $\Delta\phi$ segmentations implemented
in the CLICdet cards are listed in Table 1.

Table 1: Calorimeter segmentations based on the CLICdet calorimeter cell sizes converted to $\Delta\phi\times\Delta\eta$. Part | $\eta_{max}$ | cell size [mm$\times$mm] | $\Delta\phi[^{\circ}]$ | $\Delta\eta$
---|---|---|---|---
ECAL barrel | 1.2 | 5 $\times$ 5 | 0.2 | 0.003
ECAL endcaps | 2.5 | 5 $\times$ 5 | 0.8 | 0.02
ECAL plug | 3.0 | 5 $\times$ 5 | 1.0 | 0.02
HCAL barrel | 0.8 | 30 $\times$ 30 | 1 | 0.02
HCAL ring | 0.9 | 30 $\times$ 30 | 1 | 0.02
HCAL endcaps | 3.5 | 30 $\times$ 30 | 6 | 0.1

The energy resolution in the calorimeters is implemented in the Delphes cards
in terms of absolute value, $\Delta E=\sqrt{n^{2}+s^{2}E+c^{2}E^{2}}$ with
noise term $n$, stochastic term $s$, and constant term $c$. In the CLICdet
cards, separate resolution parameters are provided for the inner barrel,
barrel, transition region and endcaps based on the results from [2, Fig. 37].
The resolution from neutral kaons is used for the HCAL, while the ECAL
resolution is determined from photons.

####  2.2.3 Isolated leptons and photons

Electron, photon, and muon candidates are selected from Particle Flow objects.
Identification efficiencies are then applied to mimic the behavior found in
full simulation [24], which includes the effects of track reconstruction and
particle flow analysis: for electrons with $E>3$ GeV, muons with $E>2$ GeV,
and photons with $E>2$ GeV, the identification efficiencies are given as a
function of energy and pseudorapidity [2, Fig. 38-40].

The isolation of an electron, muon or photon is determined according to the
jet content in a cone of $\Delta R=0.1$ surrounding the object. It is
considered as isolated if the ratio of the transverse momentum sum of the cone
to the transverse momentum of the isolated object does not exceed 0.2. These
settings are adapted from isolation criteria used in various full-simulation
studies with CLIC detector concepts, e.g. [25].

####  2.2.4 Heavy flavour tagging

Three working points for tagging of $b$ jets with efficiencies of 50 %, 70 %,
and 90 % are provided in the CLICdet cards. The corresponding mis-tagging
efficiencies for light flavour and charm jets are extracted from the double
spiral option of the vertex endcaps in [26] in bins of pseudorapidity and
energy.

The $c$-tagging performance of CLICdet described in [2, Sec.4.2.6] has not
been implemented yet in the CLICdet Delphes cards.

####  2.2.5 Tau lepton tagging

The $\tau$ tagging efficiencies for the CLIC TauFinder algorithm have been
studied in [27]. The CLICdet Delphes cards use efficiencies averaged for the
three processes studied there, resulting in the efficiencies reported in Table
2. Jets are tagged as $\tau$ jets according to these efficiencies, using a
mis-identification rate of 3 %. As the TauFinder uses a relatively simple,
cut-based approach it is expected that its performance can be significantly
improved with respect to these numbers by introducing a multivariate analysis.

Table 2: Efficiencies of the TauFinder for linear collider detectors adapted from [27]. $p_{T}(\tau)$ [GeV] | $\geq 5$ | $\geq 12.5$ | $\geq 25$ | $\geq 50$ | $\geq 75$ | $\geq 125$ | $\geq 250$
---|---|---|---|---|---|---|---
$\epsilon$ | 0.84 | 0.79 | 0.74 | 0.66 | 0.61 | 0.51 | 0.36

###  2.3 Effect of beam-induced backgrounds

The high bunch charge density at CLIC leads to strong radiation from the
incoming beams due to their interaction with the field of the other beam. This
results in beam-induced background processes, e.g.
$\mathup{}\mathup{}\to\mathrm{hadrons}$. To account for its effects, a jet
energy smearing is applied. It should be noted that the current implementation
of the jet energy smearing assumes massless jets.

In the reconstruction, timing information of the energy deposits in the
detector is used to reduce the impact of the
$\mathup{}\mathup{}\to\mathrm{hadrons}$ background. At the 380 GeV energy
stage, this allows almost complete mitigation of the effect [28]. At the
higher energy stages, some impact remains despite the timing cuts. It is
mimicked by applying an additional smearing of the jet energy. The impact of
$\mathup{}\mathup{}\to\mathrm{hadrons}$ background on the jet energy
resolution depends on the centre-of-mass energy, being most severe at
$\sqrt{s}\,=$ 3 TeV at CLIC. Therefore, separate cards are supplied for the
three stages of CLIC, as given in Appendix A. For the energy stage at
$\sqrt{s}\approx$ 380 GeV, denoted as Stage 1, no jet energy smearing is
applied. The jet energy is smeared according to the corresponding resolutions
at Stages 2 ($\sqrt{s}\approx$ 1.5 TeV) and 3 ($\sqrt{s}\approx$ 3 TeV).

##  3 Fast simulation and reconstruction

###  3.1 Application chain

The various components (Delphes modules) are called in the CLICdet card in the
following order: Particles are propagated through the magnetic field of 4 T
with the ParticlePropagator module. In the next step, charged hadrons,
electrons, and muons are retained or rejected based on random numbers drawn
according to the tracking efficiencies. Their momenta are then smeared
according to the corresponding momentum resolutions. The tracking efficiencies
as well as the momentum resolutions are given as a function of pseudorapidity
and transverse momentum, see Sec. 2.2.1. Energy deposits in the
electromagnetic and hadronic calorimeters are then generated from the given
expected energy fractions, taking into account the granularity of the
calorimeter towers and the resolutions described in Sec. 2.2.2. A particle
flow algorithm is applied to the tracks and calorimeter deposits using the
EFlowMerger module. Next, photon, charged hadron, electron, and muon
reconstruction efficiencies and isolation efficiencies are applied according
to Sec. 2.2.3. To prepare for the jet clustering, isolated electrons, muons,
and photons are removed from the list of objects passed to the jet finder
using an EFlowFilter module. Jet clustering with the VLC algorithm is applied
to the remaining objects as described in Sec. 3.2. In the CLICdet Delphes
cards for the higher energy stages (Stages 2 and 3, $\sqrt{s}>1$ TeV), the
effect of $\mathup{}\mathup{}\to\mathrm{hadrons}$ background is mimicked by
applying a JetMomentumSmearing module (see Sec. 2.3). Finally, the
JetFlavorAssociation, BTagging and TauTagging modules are run to first find
the true flavour of the jet and then to apply tagging efficiencies as well as
mis-tagging efficiencies for $b$-jets and hadronic $\tau$ candidates. Three
working points for $b$ tagging with tagging efficiencies of 50 %, 70 %, and 90
% are provided in bins of pseudorapidity and energy (Sec. 2.2.4). Tagging
efficiencies for hadronic $\tau$s are provided in bins of transverse momentum
(Sec. 2.2.5).

###  3.2 Linear collider jet clustering

Due to the lower levels of background at lepton colliders, rather large cone
size parameters $R$ can be used for jet clustering compared to typical values
at hadron colliders, leading to better mass resolutions for hadronic
resonances.

However, using larger $R$ implies that the jets can have a substructure of
smaller $R$ jets. At analysis level, this is usually crucial information, e.g.
for suppression of backgrounds with a different jet multiplicity or in order
to use the correct objects for invariant masses. Therefore, jet clustering is
often run in "exclusive mode" [29] as implemented in Fastjet [30, Sec. 3.3.2],
with a fixed number of jets. This forces the algorithm to stop when a certain
number of jets has been clustered, thus revealing the substructure information
directly.

A suitable jet clustering algorithm for high-energy lepton collisions is the
Valencia Linear Collider (VLC) algorithm [31, 32]. It is a sequential
recombination algorithm adapted to the linear collider environment.

The VLC algorithm as implemented in FastJet-contribs [33], the functionalities
for exclusive jet clustering, and the related observables have been added to
the Delphes framework as part of this work. In the CLICdet cards, the VLC is
applied with $\beta=\gamma=1.0$ and for all combinations of
$R=0.5,0.7,1.0,1.2,1.5$ and exclusive number of jets $N=2,3,4,5,6$ in order to
accommodate a range of jet multiplicities that might be useful for the
analyses conducted with these cards. A large value of $R$ should be chosen for
the first centre-of-mass stage of CLIC and slightly smaller $R$ for higher
centre-of-mass due to higher levels of beam-induced background.

##  4 Validation of the Delphes simulation

The performance of the Delphes cards is validated for the three energy stages
of CLIC separately by comparing various kinematic observables in important
physics processes obtained with the corresponding CLICdet Delphes card with
full simulation based on the CLIC_ILD, CLIC_SiD, and CLICdet models.

Studies in the CDR [12] use the CLIC_ILD and CLIC_SiD models. Therefore, large
event samples are available for these models and hence part of the validation
uses CLIC_ILD or CLIC_SiD detector simulation. The performance of these
detector models is similar to that of CLICdet for the relevant aspects.

The validation is based on the processes summarised in Table 3. For each of
the samples, the CLICdet Delphes card corresponding to the CLIC energy stage
has been used, i.e. for Stage N the card named
delphes_card_CLICdet_StageN.tcl.

Table 3: Overview of the samples used for validation of the CLICdet Delphes cards. Process |  $\sqrt{s}$ [GeV] | Stage | Detector | Performance of
---|---|---|---|---
$\mathup{{{H}}}\mathup{{{Z}}},\mathup{{{Z}}}\to\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}}$, | 350 | 1 | CLIC_o3_v14 | jets at low energy and low
$\mathup{{{H}}}\to$ incl. |  $\gamma\gamma\to$ had. background
$\mathup{{{H}}}\mathup{}\mathup{{\overline{{\upnu}}}},\mathup{{{H}}}\to\mathup{}\mathup{}$ | 1400 | 2 | CLIC_ILD | muons; dimuon mass resolution
$\mathup{{{\mathup{{{W}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{W}}}}}^{\scriptstyle{-}}}\to\ell\nu\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}}$ | 3000 | 3 | CLIC_ILD | jets and leptons at high energy
and high $\gamma\gamma\to$ had. background
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}\mathup{{{H}}},\mathup{{{H}}}\to\mathup{{{b}}}\mathup{{\overline{{\mathup{{{b}}}}}}}$, | 1400 | 2 |  CLIC_SID | jets and leptons with
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}\to$semileptonic |  $\gamma\gamma\to$ had. background

###  4.1 Higgsstrahlung with hadronic Z decay at 350 GeV

The process
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\to\mathup{{{Z}}}(\to\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}})\mathup{{{H}}}(\to\text{incl}.)$
is evaluated at $\sqrt{s}=$350 GeV. This is an important process for the CLIC
physics programme as the model-independent measurement of Higgs properties
relies on it. It allows the performance of the CLICdet card to be validated
for jets at low energy and with low levels of beam-induced
$\mathup{}\mathup{}\to\mathrm{hadrons}$ background. For the validation,
observables obtained by running Delphes with the CLICdet card of the first
stage are compared with full detector simulation for CLICdet, version
CLIC_o3_v14. In both cases, the same generator level events produced with
Whizard interfaced to Pythia6 for the parton shower and hadronisation are
used. Jets from VLC with $N$ = 4, $R$ = 1 and $\beta=\gamma=1$ are used unless
noted otherwise.

Figure 1: Comparison of jet transverse momentum (left) and pseudorapidity
(right) in Higgsstrahlung events with hadronic Z decay for full simulation of
CLICdet (blue) and Delphes (red).

Jet observables are shown in Fig. 1. Good agreement at the level of better
than 5 % is found for central jets over most of the transverse momentum range
above 10 GeV. Up to 10 % disagreement is found in the forward region
$1.5<|\eta|<2.5$. Larger disagreements exist only at low transverse momentum
$p_{\text{T}}<10$ GeV and high transverse momentum $p_{\text{T}}>150$ GeV.

Figure 2: Comparison of jet resolution thresholds $y_{23}$ (left) and $y_{34}$
(right) in Higgsstrahlung events with hadronic Z decay for full simulation of
CLICdet (blue) and Delphes (red).

Jet resolution threshold observables obtained from the exclusive clustering
mode are shown in Fig. 2. The variable $y_{23}$ ($y_{34}$) is the distance
measure associated with merging from 3 to 2 (4 to 3) jets [30, Sec. 3.3.2].
They are well-modelled apart from a slight shift to higher values for Delphes.

Figure 3: Comparison of the di-jet mass of jets assigned to the Z boson (left)
and the recoil mass (right) in Higgsstrahlung events with hadronic Z decay for
full simulation of CLICdet (blue) and Delphes (red).

The Higgsstrahlung process is further analysed by choosing the configuration
of two jets with an invariant mass closest to the mass of the Z boson,
$m_{\mathup{{{Z}}}}$, among the possible combinations obtained by clustering
the event with 2, 3, and 5 jets exclusively. Fig. 3 shows the good agreement
of the di-jet invariant mass for Z jets (left) and the recoil mass (right)
between full simulation and Delphes. Systematic differences mostly below 10 %
are seen in the low tail of the invariant mass distribution and in the slopes
of the recoil mass. Additional kinematic distributions can be found in
Appendix B.1.

###  4.2 Semi-leptonic top-quark pairs associated with a Higgs boson at 1.4
TeV

The process
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\to\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}\mathup{{{H}}}$
with semi-leptonic decay of the
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ pair and Higgs decays
to $\mathup{{{b}}}\mathup{{\overline{{\mathup{{{b}}}}}}}$ at 1.4 TeV collision
energy is simulated with the CLIC_SiD detector model [12], including the beam-
induced background from $\mathup{}\mathup{}\to\mathrm{hadrons}$. Jet
clustering is performed with the VLC algorithm with $R\,=\,1.0$ and $N$ = 6.
This process allows validation of the performance of leptons and jets,
including the effect of beam-induced background.

Figure 4: Comparison of electron transverse momentum (left) and pseudorapidity
(right) in semileptonic
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}\mathup{{{H}}}$ events
for full simulation of CLIC_SiD (blue) and Delphes (red).

In the full simulation analysis, optimised isolation criteria are used which
give good performance down to low momenta. A cut on the transverse momentum of
electrons and muons requiring at least 20 GeV selects the region where the
cone-based isolation with a transverse momentum ratio cut as implemented in
Delphes is applicable. The performance of the CLICdet Delphes card for
electrons is demonstrated in Fig. 4. Good agreement can be seen in the
transverse momentum of the electron, Fig. 4 (left), with the Delphes
distribution shifted to smaller values of transverse momentum. Fig. 4 (right)
shows the pseudorapidity distribution, which is well-modelled. The performance
of the fast simulation for muons and jets is presented in Appendix B.2.

###  4.3 Di-boson production WW at 3 TeV

The performance of fast simulation for muons and jets is validated using the
process
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}\to\ell\nu\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}}$
at 3 TeV. Jet clustering is performed with the VLC algorithm using $R$ = 0.7
and $N$ = 2. Beam induced $\mathup{}\mathup{}\to\mathrm{hadrons}$ background
is overlayed in the full simulation, and its effects are taken into account in
the fast simulation. Figure 5 demonstrates the validation of the muon
simulation. The transverse momentum and pseudorapidity are in agreement
between full and fast simulation.

Figure 5: Validation of fast simulation performance for the muon transverse
momentum (left) and pseudorapidity (right) for events with the final state
$\ell\nu\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}}$ at 3 TeV with
full simulation of CLIC_ILD (blue) and fast simulation with Delphes (red).

Figure 6: Validation of fast simulation performance for jet energy (left) and
di-jet invariant mass (right) using events with the final state
$\ell\nu\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}}$ at 3 TeV with
full simulation of CLIC_ILD (blue) and fast simulation with Delphes (red).

The performance of the Delphes simulation for jets at CLIC is presented in Fig
6. The energy is shifted to lower values for Delphes. The invariant mass of
the two jets, associated with the $\mathup{{{W}}}$ boson, is reconstructed
with a narrower width and shifted to lower values in Delphes compared to the
full simulation. As the jet clusters all of the typically soft particles from
the background distributed within its entire area, the effect on the mass due
to their angular spread is larger than the effect on the energy. The
description could be improved by smearing not only the energy but also the
mass. Alternatively, one could overlay $\mathup{}\mathup{}\to\mathrm{hadrons}$
events. However, highly granular timing information from the detector is
crucial to suppress contributions from
$\mathup{}\mathup{}\to\mathrm{hadrons}$. The current implementation cannot
estimate this suppression and its impact on the reconstruction of the signal
process since no timing information is available in the Delphes simulation.

Additional kinematic distributions of jets can be found in Appendix B.3.

###  4.4 Di-muon mass resolution in Higgs decays to muon pairs at 350 GeV and
1.4 TeV

The di-muon mass resolution is validated in $H\to\upmu\upmu$ decays at 350 GeV
and 1.4 TeV. The Higgs boson resonance peaks in the di-muon mass distributions
for the two energy stages are shown in Fig. 7. The corresponding mean and
width obtained with a Gaussian fit are presented in Table 4.

Figure 7: Validation of the di-muon peak obtained in $H\to\upmu\upmu$ decays
at in $\mathup{{{H}}}\mathup{{{Z}}}$ events at 350 GeV (left) and in
$\mathup{{{H}}}\nu\nu$ events at 1.4 TeV (right). Full detector simulation
events (blue) are compared with events produced by applying the CLICdet
Delphes card (red).

The resulting mean and width of the Gaussian di-muon peak are in good
agreement for
$\mathup{{{H}}}(\to\upmu\upmu)\mathup{{{Z}}}(\to\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}})$
events at 350 GeV and in agreement within 30 % for
$\mathup{{{H}}}(\to\mu\mu)\nu\nu$ events at 1.4 TeV. As shown in [2, Fig. 32],
tracking resolutions in the forward region are less well described by the
assumed parametrisation than in the central region. As the muons in
$\mathup{{{W}}}$-boson fusion $\mathup{{{H}}}(\to\upmu\upmu)\nu\nu$ events are
more forward in the detector than those in
$\mathup{{{H}}}(\to\upmu\upmu)\mathup{{{Z}}}(\to\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}})$
events, this explains the larger discrepancy between Delphes and full
simulation for the di-muon mass in the $\mathup{{{W}}}$-boson fusion
$\mathup{{{H}}}(\to\mu\mu)\nu\nu$ process.

Table 4: Resulting parameters of a Gaussian fit to the invariant di-muon mass in $\mathup{{{H}}}\to\upmu\upmu$ events Simulation |  $\mathup{{{H}}}(\to\upmu\upmu)\mathup{{{Z}}}(\to\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}})$ at 350 GeV |  $\mathup{{{H}}}(\to\mu\mu)\nu\nu$ at 1.4 TeV
---|---|---
|  $\mu$ [GeV] |  $\sigma$ [GeV] |  $\mu$ [GeV] |  $\sigma$ [GeV]
Full detector | $126.0\pm 0.03$ | $0.32\pm 0.03$ | $125.8\pm 0.003$ | $0.31\pm 0.003$
Delphes | $125.9\pm 0.04$ | $0.29\pm 0.03$ | $125.9\pm 0.004$ | $0.41\pm 0.005$

##  5 Conclusions

Three parameter cards for the detector model for CLIC, CLICdet, have been
prepared for the Delphes framework. They describe the CLICdet geometry and
performance for the three energy stages of CLIC. Performance parameters are
based on full simulation studies of the detector response and reconstruction
software. The cards have been validated by comparing with full simulation for
various processes. At the first energy stage of CLIC with low levels of beam-
induced background, the validation has found good agreement in particular for
single object observables. Also derived observables such as di-jet invariant
masses are well described in the regions. At the higher energy stages,
reasonable agreement has been found for transverse momentum, energy, and
pseudorapidity distributions of leptons and jets. The absence of the beam-
induced background in the fast simulation limits the ability to reproduce all
observables fully accurately. A smearing of the jet energy has been applied to
mimic the effects of $\mathup{}\mathup{}\to\mathrm{hadrons}$ background. This
could be improved in the future by applying a smearing also to the jet mass.

As Delphes has so far been used mainly for studies at hadron colliders, some
features of detectors at lepton colliders had not yet been implemented.
Exclusive jet clustering and the VLC algorithm have been added as part of this
study. In addition, implementations for $c$-tagging and for the very forward
calorimeters as well as a refinement of the isolation criteria could be added
in the future. The possibility of overlaying beam-induced
$\mathup{}\mathup{}\to\mathrm{hadrons}$ events would improve the description
of detectors at linear colliders. However, this would also require to add the
simulation of time information to exploit the timing selection capabilities of
the detector to suppress these background contributions.

Full simulation studies for future experiments remain indispensable for the
evaluation of the detector performance. Furthermore, they are necessary for
analyses using observables that are sensitive to beam-induced backgrounds and
rely on timing information, analyses based on tails of distributions that are
not well-modelled by fast simulation, as well as analyses using exotic
signatures such as long-lived particles.

In conclusion, a good description of the detector concept for CLIC has been
achieved in the Delphes fast simulation framework with the parameter cards
discussed in this note. These cards have been validated and have already been
used for CLIC physics studies. The implementation makes it possible to take
advantage of the flexibility and accessibility of Delphes for studying many
aspects of the CLIC physics potential.

## Acknowledgements

We thank Michele Selvaggi for helpful discussions. 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.

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##  Appendix A Instructions for using the CLICdet Delphes cards

There are three cards for the three energy stages of CLIC. The user chooses
the one appropriate for the relevant energy stage as indicated in Table 5.

Table 5: CLIC energy stages implemented in separate CLICdet DELPHES cards. Stage | Energy options | Card
---|---|---
Stage1 | 380 (350) GeV | delphes_card_CLICdet_Stage1.tcl
Stage2 | 1.5 (1.4) TeV | delphes_card_CLICdet_Stage2.tcl
Stage3 | 3 TeV | delphes_card_CLICdet_Stage3.tcl

To understand the output of running the card, the $b$-tagging and the correct
choice of jet branches is described below.

###  A.1 B-Tagging in the CLICdet DELPHES card

B-tagging is implemented in the card with three working points (WP), which tag
50%, 70%, and 90% of all b-quark-initiated jets, respectively. The
corresponding mis-tagging rates are implemented in the CLICdet card. Delphes
uses a three-bit mask to store the b-tagging information in the following way,
where

  * •

bit 0 is the tight WP with 50% b-tagging efficiency,

  * •

bit 1 is the medium WP with 70% b-tagging efficiency,

  * •

bit 2 is the loose WP with 90% b-tagging efficiency.

This leads to the combinations indicated in Table 6.

Table 6: B-tagging working points in bit-wise code with T for true and F for false. BTag value | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0
---|---|---|---|---|---|---|---|---
BTag binary | 111 | 110 | 101 | 100 | 011 | 010 | 001 | 000
(bit 0) & BTag | T | F | T | F | T | F | T | F
(bit 1) & BTag | T | T | F | F | T | T | F | F
(bit 2) & BTag | T | T | T | T | F | F | F | F

Loosely b-tagged jets are selected with BTag$\geq$4; for medium b-tagged jets,
BTag should be 2, 3, 6, or 7, and for the tight WP the BTag should be 1, 3, 5
or 7.

###  A.2 Taking into account the effect of beam-induced background

The levels of beam-induced $\mathup{}\mathup{}\to\mathrm{hadrons}$ events are
higher at the high energy stages. To account for their effects, a jet energy
smearing is applied in the two cards for Stage2 and Stage3.

The branches with the jet energy smearing applied are then named
JER_VLCjetR(r)N(njets) in the resulting root file.

##  Appendix B Supplementary validation studies

###  B.1 Di-jet mass in Higgsstrahlung with hadronic Z decay at 350 GeV

Using exclusive jet clustering with N = 4, the Z jets are assigned according
to the combination closest to the mass of the Z boson. The invariant mass of
the two remaining jets after assigning the Z jets is shown in Fig. 8(right) as
$m_{\text{jj}}(\mathup{{{H}}})$.

Figure 8: Comparison of the di-jet mass of remaining jets not assigned to the
Z boson in Higgsstrahlung events with hadronic Z decay for full simulation of
CLICdet (blue) and Delphes (red).

###  B.2 Semi-leptonic top-quark pairs associated with a Higgs boson at 1.4
TeV

The fast simulation for muons is validated in Fig. 9. Like for electrons (Fig.
4), the transverse momentum is described reasonably well, but with an increase
in contributions from Delphes at lower values.

Figure 9: Comparison of muon transverse momentum (left) and pseudorapidity
(right) in semileptonic
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}\mathup{{{H}}}$ events
for full simulation of CLIC_SiD (blue) and Delphes (red).

The jet performance is demonstrated in Fig. 10. It includes the jet energy
smearing mimicking the effects of $\mathup{}\mathup{}\to\mathrm{hadrons}$
background. The pseudorapidity distribution is well-modelled, while the
Delphes spectrum for transverse momentum shows an increase of events with low
as well as higher transverse momentum.

Figure 10: Comparison of jet transverse momentum (left) and pseudorapidity
(right) in semileptonic
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}\mathup{{{H}}}$ events
for full simulation of CLIC_SiD (blue) and Delphes (red).

###  B.3 Di-boson production WW at 3 TeV

The muon energy distribution is validated comparing Delphes to full detector
simulation with the CLIC_ILD model as shown in Fig. 11.

Figure 11: Validation of fast simulation performance for the muon energy for
events with the final state
$\ell\nu\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}}$ at 3 TeV with
full simulation of CLIC_ILD (blue) and fast simulation with Delphes (red).

The transverse momentum of the jets is underestimated by the Delphes
simulation, and it does not show all features of the CLIC_ILD detector
geometry in the pseudorapidity, as shown in Fig. 12.

Figure 12: Validation of fast simulation performance for the transverse
momentum of jets (left) and the pseudorapidity of jets (right) using events
with the final state
$\ell\nu\mathup{{{q}}}\mathup{{\overline{{\mathup{{{q}}}}}}}$ at 3 TeV with
full simulation of CLIC_ILD (blue) and fast simulation with Delphes (red).