# Exploring the Electromagnetically Interacting Dark Matter at the
International Linear Collider

Manish Kumar Sharma<EMAIL_ADDRESS>Department of Physics,
Birla Institute of Technology and Science Pilani, K K Birla Goa Campus,
NH-17B, Zuarinagar, Goa 403726, India  Saumyen Kundu<EMAIL_ADDRESS>pilani.ac.in Department of Physics, Birla Institute of Technology and Science
Pilani, K K Birla Goa Campus, NH-17B, Zuarinagar, Goa 403726, India  Prasanta
Kumar Das<EMAIL_ADDRESS>Department of Physics, Birla Institute
of Technology and Science Pilani, K K Birla Goa Campus, NH-17B, Zuarinagar,
Goa 403726, India

###### Abstract

Dark Matter being electrically neutral does not participate in electromagnetic
interactions at leading order. However, we discuss here fermionic dark matter
(DM) with permanent magnetic and electric dipole moment that interacts
electromagnetically with photon at loop-level through a dimension-5 operator.
We discuss the search prospect of the dark matter at the proposed
International Linear Collider (ILC) and constrain the parameter space in the
plane of the DM mass and the cutoff scale $\Lambda$. At the 500 GeV ILC with
$4$ ab-1 of integrated luminosity we probed the mono-photon channel and
utilizing the advantages of beam polarization we obtained an upper bound on
the cutoff scale that reaches up to $\Lambda=3.72$ TeV.

##  I Introduction

The existence of dark matter (DM) is well established from numerous
cosmological and astrophysical observations at different length scales of the
Universe. These observations such as the flatness of galactic rotation curves,
velocities of stars in dwarf galaxies, velocities of galaxies in clusters, hot
gas in galaxy clusters, collisions of galaxy clusters, and gravitational
lensing are all results of gravitational interaction with the visible sector,
and collisionless property Sofue and Rubin (2001); Chang and Necib (2021);
Zwicky (1937); Clowe  _et al._ (2006); Massey  _et al._ (2010). Experiments
like WMAP Bennett  _et al._ (2013) and PLANCK Aghanim  _et al._ (2020)
analyzing the anisotropy of the cosmic microwave background (CMB) also
reaffirm the existence of DM with quantified results that $\sim 27\%$ of the
total energy budget of the Universe is contributed by it. Despite all these
observations, the particle nature of the DM is still elusive. In a bid to
understand this, numerous theoretical models have been put forward and several
experimental efforts are being carried out and have been proposed. Of these
theoretical models, the WIMP paradigm is the most explored because of the
possibility of a weak scale mass and coupling which motivates different BSM
models. However, no conclusive evidence of WIMP DM observation has been
achieved. In fact, recent bounds from direct detection experiments both from
DM-electron as well as DM-nucleon scattering are pushing the DM parameter
space down to the neutrino floor Billard  _et al._ (2014a) or the neutrino fog
O’Hare (2021a) region.

So the community is looking beyond the WIMP paradigm, considering alternate
models based on the production mechanism (e.g., freeze-in DM) Bernal  _et al._
(2017), or interaction with the visible sector (e.g. strongly interacting DM)
Emken  _et al._ (2019), or interaction with the dark sector (e.g. self-
interacting DM) Mavromatos  _et al._ (2017), or with electric charge Stebbins
and Krnjaic (2019), or with fractional charge (e.g. millicharged DM) Davidson
_et al._ (2000) and so on. We have considered here fermionic DM that have zero
electric charge or have a very tiny charge so that they practically do not
interact with the photons (hence ‘dark’) at leading order, but have non-zero
intrinsic magnetic or electric or both the dipole moments. Due to this
intrinsic dipole moment, they interact with the photons at the 1-loop order or
higher Hambye and Xu (2021). This class of DMs, termed as _dipole dark matter_
(DDM), has been studied extensively in the literature in different
cosmological and astrophysical context Sigurdson  _et al._ (2004); Mahmoudi
_et al._ (2020); Barger  _et al._ (2011a); Geytenbeek  _et al._ (2017); Binh
_et al._ (2020); Aranda  _et al._ (2016); Hambye and Xu (2021); Lopes  _et
al._ (2014); Massó  _et al._ (2009); Chang  _et al._ (2021).

The DDMs are motivated by the possibility that the electromagnetic
interactions may improve the direct detection sensitivity Barger  _et al._
(2011b); Fitzpatrick and Zurek (2010); Banks  _et al._ (2010); Del Nobile  _et
al._ (2012); Sigurdson  _et al._ (2004); Massó  _et al._ (2009). Models of DM
as Dirac, Majorana, or pseudo-Dirac fermions with small mass-splitting are
studied in various literature that consider different electromagnetic form
factors for the fermionic DM candidates. In ref. Kopp  _et al._ (2014) Dirac
and Majorana DDMs with a leptophilic nature are discussed in the direct
detection context giving bounds on anapole and dipole moments. DDMs are also
studied in the context of energy transport in the solar interior Geytenbeek
_et al._ (2017), helioseismological study Lopes  _et al._ (2014), scattering
from the Sun Chatterjee and Laha (2023), $\gamma$-ray searches Del Nobile  _et
al._ (2012), CMB searches Mahmoudi  _et al._ (2020), extra dimension Flacke
and Maybury (2007), collider search Kadota and Silk (2014); Dienes  _et al._
(2023); Kling  _et al._ (2023); Fortin and Tait (2012) and so on. In this
work, we extend the collider probe of DDM at the proposed International Linear
Collider (ILC) as a complementary study to the direct and indirect detection
searches in an effort to present a detailed collider analysis presenting an
updated bound on the DDM parameter space.

The most promising channel for searching the weak-scale DM comprises of a
characteristic mono-$X$ signature, where the large missing transverse momentum
carried away by the DM pair is balanced by a visible sector particle $X$
(which can be either a photon, jet, $W$, $Z$ or Higgs, depending on the model)
emitted from an initial, intermediate or final state (see Refs. Kahlhoefer
(2017); Penning (2018) for reviews). However, in the busy environment of the
Large Hadron Collider (LHC), the mono-$X$ search does not provide satisfactory
results. A workaround can be looking at the far-forward detectors such as
FRASER (ForwArd Search ExpeRiment ) Dienes  _et al._ (2023), or FLArE (Forward
Liquid Argon Experiment) Kling  _et al._ (2023). Another good alternative can
be the lepton colliders like the International Linear Collider (ILC), Compact
Linear Collider (CLIC), or the muon collider because of their clean
environment. ILC with the advantage of polarizing both the incoming helps to
minimize the SM background contribution and naturally presents itself as a
very attractive choice to probe the DDM, particularly through the mono-photon
channel Chae and Perelstein (2013); Kadota and Silk (2014); Habermehl  _et
al._ (2020); Kalinowski  _et al._ (2021); Kundu  _et al._ (2023).

Here we assume that the DM is a Dirac fermion and it has magnetic dipole
moment ($\mu_{\chi}$) and electric dipole moment ($d_{\chi}$) form factors
through which it couples to photon via dimension-$5$ operator. The relevant
degrees of freedom in our analysis are the DM mass ($m_{\chi}$) and an
effective cutoff scale $\Lambda$ which determines the dark matter-photon
coupling i.e. the strength of the dimension-$5$ effective operator Del Nobile
_et al._ (2012); Kadota and Silk (2014). We make a comprehensive study of the
mono-photon signature of dark matter ($\chi$) at future $e^{+}e^{-}$ collider
(ILC: International Linear Collider) with the center-of-mass energy
$\sqrt{s}=500$ GeV and compare our results with others. The rest of the paper
is organized as follows. In Sec. II, we briefly describe the model Lagrangian
for the DM-photon in the fermion DM scenario. Sec. IV discusses the existing
constraints from different direct detection experiments. In subsection IV.5,
we present our cut-based analysis for the mono-photon$+\not{E}_{T}$ signal and
background, both with and without beam polarization. Our summary and
conclusion are given in subsection IV.5.

##  II The Model: Probing Dark Matter at ILC

Dark Matter, due to its electrically neutral nature, does not participate in
electromagnetic interaction at least in leading order. However, being a
fermion they might have interactions at loop-level through its electromagnetic
moments. A typical interaction may lead to an amplitude as

$i\mathcal{M}=i\overline{u}({\bf{p_{2}}})\mathcal{O}^{\mu}(l,q)u({\bf{p_{1}}})\epsilon_{\mu}$
(1)

where, $\mathcal{O}^{\mu}(l,q)$, the vertex matrix element which acts on the
spinors, is a Lorentz vector. Here $l=p_{2}+p_{1}$ and $q=p_{2}-p_{1}$. The
most general Lorentz invariant form of $\mathcal{O}^{\mu}(l,q)$ can be given
as Nowakowski  _et al._ (2005); Giunti and Studenikin (2015); Hambye and Xu
(2021)

$\displaystyle\mathcal{O}^{\mu}(l,q)=F_{Q}(q^{2})\gamma^{\mu}$
$\displaystyle+$ $\displaystyle
F_{M}(q^{2})i\sigma^{\mu\nu}q_{\nu}+F_{E}(q^{2})\sigma^{\mu\nu}\gamma^{5}q_{\nu}$
(2) $\displaystyle+$ $\displaystyle
F_{A}(q^{2})(q^{2}\gamma^{\mu}-q^{\mu}\not{q})\gamma^{5},$

where $q(=p_{2}-p_{1})$ is the photon momentum. For interactions with a real
photon ($q^{2}\to 0$), we have

$\displaystyle F_{Q}(0)=0,\;F_{M}(0)=\mu_{\chi}\text{ (say)},\;F_{E}(0)$
$\displaystyle=$ $\displaystyle d_{\chi}\text{ (say)},$ $\displaystyle
F_{A}(0)$ $\displaystyle=$ $\displaystyle a_{\chi}\text{ (say)}$ (3)

where the form factors are identified, respectively, as the electric charge,
magnetic dipole moment, electric dipole moment, and anapole of $\chi$ Giunti
and Studenikin (2015). We consider here only the magnetic and electric dipole
moment interactions as the anapole moment interactions are suppressed being
proportional to $q^{2}$ in the low-$q^{2}$ limit and are canceled by
$\mathcal{O}(q^{-2})$ term of the photon propagator Hambye and Xu (2021).

In this paper, we have considered a dark matter (DM) model where the DM
interacts with the photon field through its magnetic and electric dipole
moments. Since only Dirac fermions can have permanent dipole moments, our DM
particle ($\chi$) is a Dirac fermion, and its interaction with the photon is
given by Del Nobile  _et al._ (2012); Kadota and Silk (2014)

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\overline{\chi}\left(\mu_{\chi}+i\gamma_{5}d_{\chi}\right)\sigma_{\mu\nu}\chi
F^{\mu\nu}$ (4) $\displaystyle=$
$\displaystyle-\left(\frac{1}{\Lambda}\right)\frac{1}{2}\overline{\chi}\left(\mathcal{C}_{M}+i\gamma_{5}\mathcal{C}_{E}\right)\sigma_{\mu\nu}\chi
F^{\mu\nu}$ (5)

where $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$, is the
electromagnetic field strength tensor and
$\sigma^{\mu\nu}=\frac{i}{2}[\sigma^{\nu},\sigma^{\nu}]$, the spin tensor.
$(\mu_{\chi})$ and $(d_{\chi})$ are magnetic and electric dipole moments
respectively and $\Lambda=\mathcal{C}_{M}/\mu_{\chi}=\mathcal{C}_{E}/d_{\chi}$
represents the cutoff scale of the effective theory, and
$\mathcal{C}_{M},\mathcal{C}_{E}$ are dimensionless parameters which can take
the value $1$ or $0$. We have investigated three different scenarios of dark
matter having (i) both Electric and Magnetic, (ii) only Magnetic, and (iii)
only Electric dipole moments by considering the values of
$(\mathcal{C}_{M},\mathcal{C}_{E})$ as $(1,1),(1,0),(0,1)$ respectively.

Figure 1: The DM-nucleon spin-independent elastic scattering cross-section as
a function of the dark matter mass ($m_{\chi}$) is shown for different values
of the cutoff scale $\Lambda$. The shaded regions are ruled out by different
direct-detection experiments.

##  III Constraints from Direct Detection Experiments

The most stringent constraint on the DM parameter space comes from the direct
detection experiments. Therefore, it is important to discuss the outlook of
the DDM in direct detection experiments. In Fig. 1 we have shown the DM-
nucleon spin-independent elastic scattering cross-section
($\sigma_{SI}^{\chi-N}$) as a function of the DM mass ($m_{\chi}$) for
different values of the cutoff scale, $\Lambda$. The cross-sections are
estimated using the micrOMEGAs Belanger  _et al._ (2009, 2021) package
assuming equal form factors of the protons and neutrons. In the plot, we only
show the lines for the $\mathcal{C}_{M}=\mathcal{C}_{E}=1$ case as we noted
that the interaction with only the electric dipole moment (i.e.,
$\mathcal{C}_{M}=0$) vanishes resulting in unphysical cross-section. We
presented the current exclusion limits for DM-nucleon spin-independent elastic
scattering from some of the leading direct-detection experiments for the DM
mass range of $[1,1000]\,\mathrm{GeV}$ with the shaded regions. The purple and
cyan regions for respectively, Xenon1T Aprile  _et al._ (2018), and PandaX-4T
Meng  _et al._ (2021) for DM mass 6 GeV to 1000 GeV are the strongest ranging
from $10^{-44}\,\mathrm{cm}^{2}$ to $10^{-48}\,\mathrm{cm}^{2}$. Note that the
experimental limits from the DM-nucleon interactions dominate over the DM-
electron interactions. Therefore, we have not included the limits for DM-
electron scattering in the plot. For the lower mass region, we included the
constraints from DarkSide Agnes  _et al._ (2018), CRESST Angloher  _et al._
(2016), CDMSlite Agnese  _et al._ (2018) for which the cross-section ranges
from $10^{-38}\,\mathrm{cm}^{2}$ to $10^{-42}\,\mathrm{cm}^{2}$. The orange-
shaded region at the bottom of the plot arises due to the similarity of
nuclear recoil signature between DM of different masses and neutrino from
different sources Billard  _et al._ (2014b) and is termed as _neutrino floor_.
However, the strength of this neutrino background comes from the systematic
uncertainties on their fluxes and therefore it is not impermeable. In fact,
efforts are being made to penetrate this region of the DM parameter space
Ruppin  _et al._ (2014); Davis (2015); Sassi  _et al._ (2021). Terming this
region as _neutrino fog_ , ref. O’Hare (2021b) introduced an updated
definition, which is shown by the yellow-shaded region in the plot. So, we
find that while the ideal mass-window of DM observation is $\sim 30\,$GeV to
$\sim 100\,$GeV depending on the value of $\Lambda$ if the neutrino background
can be distinguished we might be able to probe WIMPs with mass up to $\sim
300\,$GeV.

(a) (b)

Figure 2: Feynman diagram for the mono-photon signal at an $e^{+}e^{-}$
collider. The photon can be radiated off either the electron or the positron
leg.

##  IV Collider Phenomenology

Assuming that the dark matter particles $\chi$ can be produced in the
$e^{+}e^{-}$ collision process, it can be observed through the mono-photon
signature. Fig. 2 shows the representative Feynman diagrams for such processes
$(e^{+}e^{-}\rightarrow\chi\bar{\chi}\gamma)$ where the photon can come from
the initial state leptons or the final state DM particle legs. Since the dark
matter particle $\chi$ escapes the detector contributing to missing energy, we
need a visible particle to identify the events. This purpose is fulfilled by
the initial state radiation (ISR) and final state radiation (FSR) photon.

In Fig. 3 we have shown the variation of the cross-section with respect to the
dark matter mass($m_{\chi}$) for all three cases of
$(\mathcal{C}_{M},\mathcal{C}_{E})$ corresponding to the machine c.o.m energy
$\sqrt{s}=500~{}\rm{GeV}$. Two background processes that potentially can
contribute to such mono-photon searches — the radiative neutrino pair
production $e^{+}e^{-}\to\nu\overline{\nu}\gamma$ and radiative Bhabha
scattering $e^{+}e^{-}\to e^{+}e^{-}\gamma$. The neutrino pair background is
the dominant irreducible and exactly mimics the mono-photon signals. It is
highly polarization-dependent and thus can be suppressed significantly by
polarizing the incoming beams. On the other hand, the Bhabha scattering
contributes to the mono-photon background only when the final state electron-
positron pair escapes detection. It is not affected by beam polarization but
can be reduced by improving the detection capability of $e^{+}e^{-}$-pairs in
the forward direction. In the next section, we describe at length the details
of event simulation and analysis for both unpolarized and polarized beam
cases.

Figure 3: Variation of mono-photon signal cross section (fb) with the DM mass
$m_{\chi}$ (GeV) for different cases of $(\mathcal{C}_{M},\mathcal{C}_{E})$ as
$(1,1),(1,0),(0,1)$ by red, green and blue lines respectively for
$\sqrt{s}=500$ GeV.

###  IV.1 Event Simulation

Table 1: Comparison of the irreducible SM backgrounds and signal cross sections (merged) for different choices of beam polarization at $\sqrt{s}=500~{}\rm{GeV}$. For the signal, we have chosen benchmark values of $m_{\chi}=50~{}\rm{GeV}$ and $\Lambda=3~{}\rm{TeV}$. The numbers in bold highlight the optimal polarization choice for a given operator type. Process | Unpolarized | Polarization | Polarized cross-section (fb)
---|---|---|---
type | CS (fb) | $P(e^{-},e^{+})$ | $(+,+)$ | $(+,-)$ | $(-,+)$ | $(-,-)$
$\nu\overline{\nu}+N\gamma$ | 5581.0 | $(\phantom{.}80,0\phantom{.})$ | 1629.8 | 1629.8 | 9571.4 | 9571.4
|  | $(80,30)$ | 1770.9 | 1478.3 | 12412.9 | 6737.3
$e^{+}e^{-}+N\gamma$ | 255608 | $(\phantom{.}80,0\phantom{.})$ | 255156 | 255156 | 258463 | 258463
|  | $(80,30)$ | 254348 | 254299 | 256423 | 252165
$C_{M}=C_{E}=1$ | 10.67 | $(\phantom{.}80,0\phantom{.})$ | 10.67 | 10.67 | 10.63 | 10.63
|  | $(80,30)$ | 8.09 | 13.22 | 13.16 | 8.09
$C_{M}=1,\;C_{E}=0$ | 5.81 | $(\phantom{.}80,0\phantom{.})$ | 5.81 | 5.81 | 5.82 | 5.82
|  | $(80,30)$ | 4.41 | 7.22 | 7.20 | 4.41
$C_{M}=0,\;C_{E}=1$ | 4.82 | $(\phantom{.}80,0\phantom{.})$ | 4.82 | 4.82 | 4.82 | 4.82
|  | $(80,30)$ | 3.66 | 5.97 | 5.98 | 3.67

We implemented our dipole DM model in FeynRules Alloul  _et al._ (2014) to
obtain the UFO library files which were then used in Whizard Kilian  _et al._
(2011) to estimate the cross-sections of the signal. The background processes
are also generated using Whizard with both the beamstrahlung and ISR effects.
These are two important beam-induced backgrounds that muddle the signal
photons and dilute the signals. Hence, careful modeling of these beamstrahlung
and ISR photons is imperative for a mono-photon analysis at $e^{-}e^{+}$
colliders. We implement beamstrahlung using a beam spectra file generated
using circe2 for ILC. For ISR modeling, we used the ‘recoil’ mode of the
handler with the ISR mass equal to the electron mass. Whizard gives us freedom
in selecting the generation level cuts that help in removing the infrared and
collinear singularities. To remove these divergences in the cross-section
calculations we restricted the event generation phase space with the following
generation-level cuts :

$p_{T}^{\gamma}>2\text{ GeV},\;\;\;E_{\gamma}>1\text{
GeV},\;\;\;|\cos{\theta_{\gamma}|}\leq 0.9975$ (6)

To avoid soft and collinear divergences in radiative Bhabha scattering, a few
more restrictions were applied:

$\displaystyle M_{e_{in}^{\pm},e_{out}^{\pm}}<-2\text{
GeV},\;\;M_{e_{out}^{\pm},e_{out}^{\pm}}>2\text{ GeV},$ $\displaystyle
M_{e^{\pm},\gamma_{i}}>2\text{ GeV},\;\;M_{e^{\pm},\gamma}>4\text{ GeV}$ (7)

where $\gamma_{i}$ and $\gamma$ denote matrix element (ME) photon and signal
photon, respectively. In order to take into account the proper summation of
higher-order corrections and avoid double counting, we have used the ME-ISR
merging technique introduced in Kalinowski  _et al._ (2020). We considered two
independent variables $q_{+}$ and $q_{-}$ as introduced in Kalinowski  _et
al._ (2020) and are defined as :

$\displaystyle q_{-}$ $\displaystyle=\sqrt{2E_{\rm
cm}E_{\gamma}}\sin\frac{\theta_{\gamma}}{2}$ $\displaystyle q_{+}$
$\displaystyle=\sqrt{2E_{\rm cm}E_{\gamma}}\cos\frac{\theta_{\gamma}}{2}$ (8)

where $E_{cm}$ = $\sqrt{s}$, $E_{\gamma}$ and $\theta_{\gamma}$ are the center
of mass energy, photon energy, and photon polar angle respectively. The
rejection cuts used in our analysis for ME photons are $q_{\pm}<1~{}\rm{GeV}$
or $E_{\gamma}<1~{}\rm{GeV}$ and for ISR photons, they are
$q_{\pm}>1~{}\rm{GeV}$ and $E_{\gamma}>1~{}\rm{GeV}$, respectively.

We have used Delphes3 de Favereau  _et al._ (2014) with ILCgen configuration
card for the fast detector simulations of the generated signal and background
events with the above-discussed cuts. It can be seen from Fig. 3 that there is
almost no change in the unpolarized signal cross-section up to a certain value
of dark matter mass and after that, it decreases rapidly due to phase space
suppression for all three cases of $(\mathcal{C}_{M},\mathcal{C}_{E})$ as
$(1,1),(1,0),(0,1)$.

###  IV.2 Effect of polarization

As per the ILC baseline design, we have studied the effect of polarization on
the incoming $e^{-}$ and $e^{+}$ beams on the signal and background cross-
sections by considering two cases of polarization configurations. First, we
examined the effect of polarization on only the electron beam. Then we studied
how the application of polarization on the positron beam also affects the
signal and background cross-sections. In Table 1, we presented the cross-
sections of the two backgrounds and the signals with all four possible
helicity orientations, namely, $(+,+),(+,-),(-,+)$ and $(-,-)$ in each case
Barklow  _et al._ (2015); Kundu  _et al._ (2023). From Table 1 we can see that
the neutrino background cross-section is highly polarization dependent, which
can best be reduced to $26.49\%$ by choosing $(+80,-30)$ as the polarization
choice of incoming beams, whereas the Bhabha background shows a polarization
independent behaviour. Also, the signal can be moderately enhanced by using
the same polarization choice for all cases of
$(\mathcal{C}_{M},\mathcal{C}_{E})$ values as $(1,1),\;(1,0),\;(0,1)$ by
$23.90\%$, $24.27\%$ and $23.86\%$ respectively with an integrated luminosity
of 4 ab-1. Hence, we will consider $(+80,-30)$ as our polarization choice for
further analysis in this paper.We have also considered the H20 scenario for
comparison which considers shared polarization states of $40\%$ for $(+,-)$
and $(-,+)$ and $10\%$ for $(+,+)$ and $(-,-)$ configuration at
$\sqrt{s}=500GeV$.

Figure 4: Normalized differential distributions after baseline selection cuts
for $E_{\gamma}$, $\cos\theta_{\gamma}$ for the unpolarized beam case for DM
mass of $m_{\chi}=1$ GeV and $\Lambda=3$ TeV. In each panel, the shaded blue
histograms are for the neutrino background and the shaded yellow histograms
are for the radiative Bhabha background. The unshaded red, green, and blue
histograms are for the three scenarios of $(\mathcal{C}_{M},\mathcal{C}_{E})$
as $(1,1),\;(1,0),\;(0,1)$, respectively.

###  IV.3 Event Selection

In this section, we discuss the cut-based analysis performed in the ROOT Brun
and Rademakers (1997) framework with the detector-simulated signal and
background datasets to get an optimized value of the signal-to-background
ratio. For that purpose we have considered here benchmark values of
$m_{\chi}=1$ GeV and $\Lambda=1$ TeV although, in the next section, we will
vary the DM mass to get $3\sigma$ sensitivity limit on the cutoff scale
$\Lambda$.

In the below, we described different selection cuts we have used in order to
minimize the contribution of the two backgrounds.

Baseline selection cuts — Keeping in mind the configuration of the generic
International Large Detector (ILD) and Silicon Detector (SiD) at ILC we set
the baseline selection criteria. We define our mono-photon signal events that
contain at least one photon that satisfies the following criteria (Table 2 )

$E_{\gamma}>5\text{ GeV}\;,\;|\eta_{\gamma}|<2.8$ (9)

where, the $E_{\gamma}$ and $\eta_{\gamma}$ are the photon energy and
pseudorapidity. We assume here that the tracking detector has a range of
$7^{\circ}-173^{\circ}$ in the polar angle. In the case of multiple photons in
an event, the one with the highest transverse momentum is considered the
signal photon. In Fig. 4 we have shown the normalized distributions of the
photon energy and pseudorapidity after the baseline-selection cut.

On top of the baseline -selection criteria we used some additional cuts

Cut 1 — We can reduce neutrino background by putting an appropriate constraint
on the value of $E_{\gamma}$ which can be worked out by analyzing the left
panel of Fig. 4. We can see that the neutrino background peaks at around
$E_{\gamma}=235$ GeV because of the radiative return of Z-resonance, whereas
the signal almost vanishes at $E_{\gamma}=212$ GeV. Thus, we can reduce both
backgrounds by requiring $E_{\gamma}<210$ GeV.

Cut 2 — The baseline cut on pseudorapidity $|\eta_{\gamma}|<2.8$ is to account
for the inability of the detector to detect collinear photons outside the
range of $7^{\circ}<\theta<173^{\circ}$. But in order to decrease the
background signal, we have applied a better cut on the quantity
$|\cos\theta_{\gamma}|<0.95$ which can be seen in the right panel of the Fig.
4. The Bhabha background peaks to a much higher value than the signal in the
region $|\cos\theta_{\gamma}|>0.95$ and should be excluded.

Cut 3 — We restrict $p_{T}>2$ GeV for the charged particles that are
reconstructed as electrons. This criterion was also used in Ref.Habermehl
(2018) and reduces the Bhabha event yield.

Cut 4 — The BeamCal veto is used to reject those events that have deposited
energy in the forward direction of the beamline in an electromagnetic
calorimeter which detects highly energetic electrons that escape the central
tracker and hence suppress the contribution from the Bhabha background events
Abramowicz  _et al._ (2010).

Table 2:  Event selection criteria applied to the signal and Standard Model(SM) backgrounds.  Description | Definition
---|---
Baseline selection | $E_{\gamma}>5\text{ GeV},\;\;|\eta_{\gamma}|<2.8,$
Cut-1 | $E_{\gamma}<210~{}\rm{GeV}$
Cut-2 | $|\cos\theta_{\gamma}|<0.95$
Cut-3 |  00Charged particle veto with $p_{T}>5$ GeV
Cut-4 | BeamCal veto

###  IV.4 Signal significance

There can be several sources of systematic uncertainties at the ILC like, beam
polarization, event estimation, integrated luminosity, and the shape of
luminosity spectrum Habermehl  _et al._ (2020); Kalinowski  _et al._ (2021);
Habermehl (2018) which should be considered for realistic uncertainty analysis
along with there correlations which are beyond the scope of our work and
hence, we have done a simplified analysis by considering only background
estimation uncertainty in a cut-and-count method by considering uncertainties
of $0.2\%$ and $1\%$ for neutrino-pair and Bhabha background respectively
Kalinowski  _et al._ (2020); Habermehl (2018); Blaising  _et al._ (2021). We
expect that the experiments will show sensitivities that will lie somewhere in
between the two scenarios we have considered, i.e., with no systematics and
with our choice of systematics as stated above.

Figure 5: $3\sigma$ sensitivity contours in the mono-photon channel for the
three combinations of the coupling values with _unpolarized_ (red lines),
_optimally polarized_ (blue lines) and _mixed polarized_ (red lines)
$e^{+}e^{-}$ beams at $\sqrt{s}=500$ GeV center-of-mass energy and with ${\cal
L}_{\rm int}=4$ ab-1 integrated luminosity. The solid (dashed) contours are
without (with) background systematics.

The signal significance is calculated using

$Sig=\frac{S}{\sqrt{S+\Sigma_{i}(B_{i}+\epsilon_{i}^{2}B_{i}^{2})}}$ (10)

where S denotes the number of signal events $B_{i}$ denotes $i$-th channel
background events for a given value of integrated luminosity, and the
corresponding systematic uncertainty in background estimation is given by
$\epsilon_{i}$.For the mixed polarization (H20) scenario which considers
shared polarization states of $40\%$ for $(+,-)$ and $(-,+)$ and $10\%$ for
$(+,+)$ and $(-,-)$ configuration at $\sqrt{s}=500$ GeV, significance for
individual polarization case has been calculated (assuming them to be
uncorrelated) using

$Sig_{H20}=\sqrt{{Sig^{2}_{(+,+)}}+{Sig^{2}_{(+,-)}}+{Sig^{2}_{(-,+)}}+{Sig^{2}_{(-,-)}}}.$
(11)

###  IV.5 Results and Summary

The $3\sigma$ sensitivity contours in the $m_{\chi}-\Lambda$ plane in the
mono-photon channel for the three combinations of the coupling values
$(\mathcal{C}_{M},\mathcal{C}_{E})$ as $(1,1),(1,0),(0,1)$ with _unpolarized_
(red lines), _optimally polarized_ (blue lines), and _mixed polarized_ (green
lines) at 500 GeV ILC with 4 ab-1 of integrated luminosity without (with)
systematics are shown in Fig. 5 with solid (dashed) lines. In the plots, we
have also shown the direct detection bound on $\Lambda$ as a function of
$m_{\chi}$ that follows from CRESST, CDMSlite, DarkSide, Xenon1T, and
PandaX-4T amongst which the most stringent bounds are given by Xenon1T and
PandaX-4T with the shaded regions. The dark yellow dashed lines represent the
values of the parameters that gives the correct Planck-observed relic density
and the shaded region above results in overabundance of DM and therefore is
forbidden. Hence, the unshaded region under the contour lines of our DDM model
shows the allowed region where the DDM with the correct relic density can be
observed.

Table 3:  $3\sigma$ upper limit on the cutoff scale at 500 GeV ILC with 4 ab-1 of integrated luminosity without (with) systematics for three different cases — with both EDM and MDM are non-zero, with only MDM, and with only EDM. Three beam polarization scenarios are considered namely, unpolarized, optimally polarized (Opt. Pol.), and mixed polarization case of H20 scenario (Mix. Pol.). | Upper limit on $\Lambda$ (TeV) for $m_{\chi}=1$ GeV at
---|---
Type | $\mathcal{L}_{int}=4\text{ ab}^{-1}$
| Unpolarized | Opt. Pol. | Mix. Pol.
$\mathcal{C}_{M}=1,~{}~{}\mathcal{C}_{E}=1$ | 3.30 (1.45) | 3.72 (1.60) | 3.49 (1.76)
$\mathcal{C}_{M}=1,~{}~{}\mathcal{C}_{E}=0$ | 2.95 (1.30) | 3.32 (1.43) | 3.12 (1.57)
$\mathcal{C}_{M}=0,~{}~{}\mathcal{C}_{E}=1$ | 2.77 (1.22) | 3.12 (1.34) | 2.92 (1.47)

In Table 3, we have displayed the $3\sigma$ upper limit on the cutoff scale
$\Lambda$ corresponding to $m_{\chi}=1$ GeV at 500 GeV ILC with 4 ab-1 of
integrated luminosity without (with) systematics for three different cases
mentioned above along with three beam polarization scenarios namely,
unpolarized, optimally polarized (Opt. Pol.) and mixed polarized case of H20
scenario (Mix. Pol.). We found that in all three coupling combinations of
$(\mathcal{C}_{M},\mathcal{C}_{E})$ as $(1,1),(1,0),(0,1)$, the best bounds
are given by the Optimally polarized case when we do not consider the
systematics and those are $3.14,2.59$ and $2.59$ TeV respectively. With
systematics, the H20 scenario is found to give better bounds of $1.48,1.22$
and $1.23$ TeV respectively in comparison to Unpolarized and Optimally
polarized cases.

Finally, we observed that among the three scenarios of incoming beam
polarization, namely, the unpolarized beam, optimally polarized beam, and
maximally polarized beam, the best constraint on the cut-off scale $\Lambda$
arises from the optimally polarized beam configuration. Specifically, for
$(\mathcal{C}_{M},\mathcal{C}_{E})$ equal to $(1,1)$ and without considering
systematics, we have determined this constraint to be $3.72$ TeV. However,
when taking systematics into consideration, we obtain a relatively relaxed
bound of $1.76$ TeV for $(\mathcal{C}_{M},\mathcal{C}_{E})$ set to $(1,1)$ in
the case of mixed polarization.

## References

  * Sofue and Rubin (2001) Y. Sofue and V. Rubin, Ann. Rev. Astron. Astrophys. 39, 137 (2001), arXiv:astro-ph/0010594 .
  * Chang and Necib (2021) L. J. Chang and L. Necib, Mon. Not. Roy. Astron. Soc. 507, 4715 (2021), arXiv:2009.00613 [astro-ph.CO] .
  * Zwicky (1937) F. Zwicky, Astrophys. J. 86, 217 (1937).
  * Clowe  _et al._ (2006) D. Clowe, M. Bradac, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones, and D. Zaritsky, Astrophys. J. Lett. 648, L109 (2006), arXiv:astro-ph/0608407 .
  * Massey  _et al._ (2010) R. Massey, T. Kitching, and J. Richard, Rept. Prog. Phys. 73, 086901 (2010), arXiv:1001.1739 [astro-ph.CO] .
  * Bennett  _et al._ (2013) C. L. Bennett, D. Larson, J. L. Weiland, N. Jarosik, G. Hinshaw, N. Odegard, K. M. Smith, R. S. Hill, B. Gold, M. Halpern, E. Komatsu, M. R. Nolta, L. Page, D. N. Spergel, E. Wollack, J. Dunkley, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, and E. L. Wright, The Astrophysical Journal Supplement Series 208, 20 (2013).
  * Aghanim  _et al._ (2020) N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Billard  _et al._ (2014a) J. Billard, L. Strigari, and E. Figueroa-Feliciano, Phys. Rev. D 89, 023524 (2014a), arXiv:1307.5458 [hep-ph] .
  * O’Hare (2021a) C. A. J. O’Hare, Phys. Rev. Lett. 127, 251802 (2021a).
  * Bernal  _et al._ (2017) N. Bernal, M. Heikinheimo, T. Tenkanen, K. Tuominen, and V. Vaskonen, Int. J. Mod. Phys. A 32, 1730023 (2017), arXiv:1706.07442 [hep-ph] .
  * Emken  _et al._ (2019) T. Emken, R. Essig, C. Kouvaris, and M. Sholapurkar, JCAP 09, 070 (2019), arXiv:1905.06348 [hep-ph] .
  * Mavromatos  _et al._ (2017) N. E. Mavromatos, C. R. Argüelles, R. Ruffini, and J. A. Rueda, in  _14th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories_, Vol. 1 (2017) pp. 639–666.
  * Stebbins and Krnjaic (2019) A. Stebbins and G. Krnjaic, JCAP 12, 003 (2019), arXiv:1908.05275 [astro-ph.CO] .
  * Davidson  _et al._ (2000) S. Davidson, S. Hannestad, and G. Raffelt, JHEP 05, 003 (2000), arXiv:hep-ph/0001179 .
  * Hambye and Xu (2021) T. Hambye and X.-J. Xu, JHEP 11, 156 (2021), arXiv:2106.01403 [hep-ph] .
  * Sigurdson  _et al._ (2004) K. Sigurdson, M. Doran, A. Kurylov, R. R. Caldwell, and M. Kamionkowski, Phys. Rev. D 70, 083501 (2004), [Erratum: Phys.Rev.D 73, 089903 (2006)], arXiv:astro-ph/0406355 .
  * Mahmoudi  _et al._ (2020) S. Mahmoudi, M. Haghighat, S. A. M. Vamegh, and R. Mohammadi, Eur. Phys. J. C 80, 402 (2020), arXiv:1805.11172 [hep-ph] .
  * Barger  _et al._ (2011a) V. Barger, W.-Y. Keung, and D. Marfatia, Physics Letters B 696, 74 (2011a).
  * Geytenbeek  _et al._ (2017) B. Geytenbeek, S. Rao, P. Scott, A. Serenelli, A. C. Vincent, M. White, and A. G. Williams, JCAP 03, 029 (2017), arXiv:1610.06737 [hep-ph] .
  * Binh  _et al._ (2020) D. T. Binh, V. H. Binh, and H. N. Long, (2020), arXiv:2006.09020 [hep-ph] .
  * Aranda  _et al._ (2016) A. Aranda, L. Barajas, and J. A. R. Cembranos, JCAP 03, 034 (2016), arXiv:1511.02805 [hep-ph] .
  * Lopes  _et al._ (2014) I. Lopes, K. Kadota, and J. Silk, Astrophys. J. Lett. 780, L15 (2014), arXiv:1310.0673 [astro-ph.SR] .
  * Massó  _et al._ (2009) E. Massó, S. Mohanty, and S. Rao, Phys. Rev. D 80, 036009 (2009).
  * Chang  _et al._ (2021) J. H. Chang, R. Essig, and A. Reinert, JHEP 03, 141 (2021), arXiv:1911.03389 [hep-ph] .
  * Barger  _et al._ (2011b) V. Barger, W.-Y. Keung, and D. Marfatia, Phys. Lett. B 696, 74 (2011b), arXiv:1007.4345 [hep-ph] .
  * Fitzpatrick and Zurek (2010) A. L. Fitzpatrick and K. M. Zurek, Phys. Rev. D 82, 075004 (2010), arXiv:1007.5325 [hep-ph] .
  * Banks  _et al._ (2010) T. Banks, J.-F. Fortin, and S. Thomas, (2010), arXiv:1007.5515 [hep-ph] .
  * Del Nobile  _et al._ (2012) E. Del Nobile, C. Kouvaris, P. Panci, F. Sannino, and J. Virkajarvi, JCAP 08, 010 (2012), arXiv:1203.6652 [hep-ph] .
  * Kopp  _et al._ (2014) J. Kopp, L. Michaels, and J. Smirnov, JCAP 04, 022 (2014), arXiv:1401.6457 [hep-ph] .
  * Chatterjee and Laha (2023) S. Chatterjee and R. Laha, Phys. Rev. D 107, 083036 (2023), arXiv:2202.13339 [hep-ph] .
  * Flacke and Maybury (2007) T. Flacke and D. W. Maybury, Int. J. Mod. Phys. D 16, 1593 (2007), arXiv:hep-ph/0601161 .
  * Kadota and Silk (2014) K. Kadota and J. Silk, Phys. Rev. D 89, 103528 (2014), arXiv:1402.7295 [hep-ph] .
  * Dienes  _et al._ (2023) K. R. Dienes, J. L. Feng, M. Fieg, F. Huang, S. J. Lee, and B. Thomas, Phys. Rev. D 107, 115006 (2023), arXiv:2301.05252 [hep-ph] .
  * Kling  _et al._ (2023) F. Kling, J.-L. Kuo, S. Trojanowski, and Y.-D. Tsai, Nucl. Phys. B 987, 116103 (2023), arXiv:2205.09137 [hep-ph] .
  * Fortin and Tait (2012) J.-F. Fortin and T. M. P. Tait, Phys. Rev. D 85, 063506 (2012), arXiv:1103.3289 [hep-ph] .
  * Kahlhoefer (2017) F. Kahlhoefer, Int. J. Mod. Phys. A 32, 1730006 (2017), arXiv:1702.02430 [hep-ph] .
  * Penning (2018) B. Penning, J. Phys. G 45, 063001 (2018), arXiv:1712.01391 [hep-ex] .
  * Chae and Perelstein (2013) Y. J. Chae and M. Perelstein, JHEP 05, 138 (2013), arXiv:1211.4008 [hep-ph] .
  * Habermehl  _et al._ (2020) M. Habermehl, M. Berggren, and J. List, Phys. Rev. D 101, 075053 (2020), arXiv:2001.03011 [hep-ex] .
  * Kalinowski  _et al._ (2021) J. Kalinowski, W. Kotlarski, K. Mekala, P. Sopicki, and A. F. Zarnecki, (2021), arXiv:2107.11194 [hep-ph] .
  * Kundu  _et al._ (2023) S. Kundu, A. Guha, P. K. Das, and P. S. B. Dev, Phys. Rev. D 107, 015003 (2023), arXiv:2110.06903 [hep-ph] .
  * Nowakowski  _et al._ (2005) M. Nowakowski, E. A. Paschos, and J. M. Rodriguez, Eur. J. Phys. 26, 545 (2005), arXiv:physics/0402058 .
  * Giunti and Studenikin (2015) C. Giunti and A. Studenikin, Rev. Mod. Phys. 87, 531 (2015), arXiv:1403.6344 [hep-ph] .
  * Belanger  _et al._ (2009) G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov, Comput. Phys. Commun. 180, 747 (2009), arXiv:0803.2360 [hep-ph] .
  * Belanger  _et al._ (2021) G. Belanger, A. Mjallal, and A. Pukhov, Eur. Phys. J. C 81, 239 (2021), arXiv:2003.08621 [hep-ph] .
  * Aprile  _et al._ (2018) E. Aprile _et al._ (XENON), Phys. Rev. Lett. 121, 111302 (2018), arXiv:1805.12562 [astro-ph.CO] .
  * Meng  _et al._ (2021) Y. Meng _et al._ (PandaX-4T), (2021), arXiv:2107.13438 [hep-ex] .
  * Agnes  _et al._ (2018) P. Agnes _et al._ (DarkSide), Phys. Rev. Lett. 121, 081307 (2018), arXiv:1802.06994 [astro-ph.HE] .
  * Angloher  _et al._ (2016) G. Angloher _et al._ (CRESST), Eur. Phys. J. C 76, 25 (2016), arXiv:1509.01515 [astro-ph.CO] .
  * Agnese  _et al._ (2018) R. Agnese _et al._ (SuperCDMS), Phys. Rev. D 97, 022002 (2018), arXiv:1707.01632 [astro-ph.CO] .
  * Billard  _et al._ (2014b) J. Billard _et al._ , JINST 9, P01013 (2014b), arXiv:1305.2360 [astro-ph.IM] .
  * Ruppin  _et al._ (2014) F. Ruppin, J. Billard, E. Figueroa-Feliciano, and L. Strigari, Phys. Rev. D 90, 083510 (2014), arXiv:1408.3581 [hep-ph] .
  * Davis (2015) J. H. Davis, JCAP 03, 012 (2015), arXiv:1412.1475 [hep-ph] .
  * Sassi  _et al._ (2021) S. Sassi, A. Dinmohammadi, M. Heikinheimo, N. Mirabolfathi, K. Nordlund, H. Safari, and K. Tuominen, Phys. Rev. D 104, 063037 (2021), arXiv:2103.08511 [hep-ph] .
  * O’Hare (2021b) C. A. J. O’Hare, (2021b), arXiv:2109.03116 [hep-ph] .
  * Alloul  _et al._ (2014) A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, and B. Fuks, Comput. Phys. Commun. 185, 2250 (2014), arXiv:1310.1921 [hep-ph] .
  * Kilian  _et al._ (2011) W. Kilian, T. Ohl, and J. Reuter, Eur. Phys. J. C 71, 1742 (2011), arXiv:0708.4233 [hep-ph] .
  * Kalinowski  _et al._ (2020) J. Kalinowski, W. Kotlarski, P. Sopicki, and A. F. Zarnecki, Eur. Phys. J. C 80, 634 (2020), arXiv:2004.14486 [hep-ph] .
  * de Favereau  _et al._ (2014) J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, and M. Selvaggi (DELPHES 3), JHEP 02, 057 (2014), arXiv:1307.6346 [hep-ex] .
  * Barklow  _et al._ (2015) T. Barklow, J. Brau, K. Fujii, J. Gao, J. List, N. Walker, and K. Yokoya, (2015), arXiv:1506.07830 [hep-ex] .
  * Brun and Rademakers (1997) R. Brun and F. Rademakers, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 389, 81 (1997), new Computing Techniques in Physics Research V.
  * Habermehl (2018) M. Habermehl,  _Dark Matter at the International Linear Collider_ , Ph.D. thesis, Hamburg U., Hamburg (2018).
  * Abramowicz  _et al._ (2010) H. Abramowicz _et al._ , JINST 5, P12002 (2010), arXiv:1009.2433 [physics.ins-det] .
  * Blaising  _et al._ (2021) J.-J. Blaising, P. Roloff, A. Sailer, and U. Schnoor (CLICdp), (2021), arXiv:2103.06006 [hep-ex] .