# EFT analysis of leptophilic dark matter at future electron-positron
colliders in the mono-photon and mono-$Z$ channels

Saumyen Kundu<EMAIL_ADDRESS>Department of Physics, Birla
Institute of Technology and Science Pilani, K K Birla Goa Campus, NH-17B,
Zuarinagar, Goa 403726, India  Atanu Guha<EMAIL_ADDRESS>Department of
Physics, Chungnam National University, Daejeon 34134, Republic of Korea
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  P. S. Bhupal Dev<EMAIL_ADDRESS>Department of
Physics and McDonnell Center for the Space Sciences, Washington University,
St. Louis, Missouri 63130, USA

###### Abstract

We consider the possibility that dark matter (DM) only interacts with the
Standard Model leptons, but not quarks at tree level, and analyze the future
lepton collider prospects of such leptophilic DM in the monophoton and
mono-$Z$ (both leptonic and hadronic) channels. Adopting a model-independent
effective field theory framework, we consider all possible dimension-six
operators of scalar-pseudoscalar (SP), vector-axial vector (VA), and tensor-
axial tensor (TAT) types for a fermionic DM and derive the collider
sensitivities on the effective cutoff scale $\Lambda$ as a function of the DM
mass. As a concrete example, we take the beam configurations of the
International Linear Collider with $\sqrt{s}=1$ TeV and $8$ ab-1 integrated
luminosity, including the effect of beam polarization, and show that it can
probe leptophilic DM at $3\sigma$ level up to $\Lambda$ values of $6.6$,
$8.8$, and $7.1$ TeV for the SP-, VA- and TAT-type operators, respectively.
This is largely complementary to the direct and indirect searches for
leptophilic DM and can potentially provide the best-ever sensitivity in the
low-mass DM regime.

Dark Matter, Effective Field Theory, Leptophilic Operator, International
Linear Collider, Lepton Collider

##  I Introduction

There is overwhelming evidence for the existence of dark matter (DM) in our
Universe from various astrophysical and cosmological observations, such as
galactic rotation curves, velocities of stars in dwarf galaxies, velocities of
galaxies in clusters, hot gas in galaxy clusters, collisions of galaxy
clusters, gravitational lensing, cosmic microwave background (CMB) and large-
scale structure measurements (see e.g. Refs. [1, 2] and references therein).
However, apart from its gravitational interaction and the fact that it
constitutes 26.8% of the total energy budget of the Universe [3], very little
is known about the true nature and properties of DM. On the theory front,
there exists a plethora of particle DM candidates [4] and identifying the
correct one is a pressing issue in beyond the Standard Model (BSM) physics. On
the experimental front, a number of ongoing and planned efforts in direct [5],
indirect [6] and collider [7] searches for DM are poised to shed more light on
the DM mystery [8].

Most of the current DM searches are traditionally motivated by the so-called
weakly interacting massive particle (WIMP) miracle (see, e.g., Ref. [9]) – the
simple observation that an electrically neutral, cosmologically stable WIMP
$\chi$ with mass around the electroweak scale and couplings of the order of
weak coupling strength to the SM sector can successfully reproduce the
observed DM relic density of $\Omega_{\chi}h^{2}=0.120\pm 0.001$ [3]. The WIMP
DM mass can actually be in a wide range varying from $\mathcal{O}(1)$ MeV to
$\mathcal{O}(100)$ TeV, although the possibility of an electroweak scale mass
is much more appealing as a host of well-motivated BSM scenarios, from the
supersymmetric extensions to the extra dimension models, can easily
accommodate such a WIMP DM. The robustness of the WIMP paradigm comes from the
fact that the interactions of the WIMP with the SM keep it in thermal
equilibrium at high temperatures in the early Universe. Later, as the Universe
expands and cools, the rate of interaction of WIMPs with the SM sector falls
below the Hubble expansion rate, and as a result, the WIMP DM decouples
(“freezes out”) from the thermal plasma and explains the observed relic
density in the present Universe.

However, the WIMP paradigm is under significant strain, as the stringent
exclusion limits from DM direct detection experiments, such as LUX [10],
XENON1T [11] and PANDAX-4T [12] have pushed the WIMP parameter space down to
an awkward corner fast approaching the neutrino floor [13, 14], below which it
will be extremely challenging to disentangle the DM signal from the coherent
neutrino background. Therefore, it is important to explore other DM ideas that
could potentially explain the absence of a signal in these experiments, while
having promising prospects at other experiments. It is interesting to note
that a majority of the existing experimental constraints crucially rely on the
WIMP interactions with nucleons and therefore can be largely weakened if the
WIMP predominantly interacts with the SM leptons, but not quarks at tree
level. Such “leptophilic” DM could indeed arise naturally in many BSM
scenarios [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, 53, 54, 55, 56, 57, 58, 59], some of which could even explain various
experimental anomalies, such as the muon anomalous magnetic moment [60, 61],
DAMA/LIBRA annual modulation [62], anomalous cosmic ray positron excess as
observed in ATIC [63], PAMELA [64, 65], Fermi-LAT [66, 67], AMS-02 [68, 69,
70], DAMPE [71] and CALET [72], the Galactic Center gamma-ray excess [73], 511
keV gamma-ray line [74], IceCube ultrahigh-energy neutrino excess [75], and
XENON1T electron excess [76]. Dedicated searches for leptophilic DM in direct
detection experiments [77, 78, 79], beam dump experiments [80, 81], and
gravitational wave detectors [53], as well as using terrestrial [82] and
celestial objects like Sun [83, 84], supernovae [85] and neutron stars [86,
87, 88, 89, 90] have also been discussed.

In this paper, we focus on the leptophilic DM searches at colliders, which are
complementary to the direct and indirect detection searches. In order to keep
our discussion as general as possible, we will adopt an effective field theory
(EFT) approach, which has been widely used in the context of collider searches
for DM following the early works of Refs. [29, 91, 92, 93, 94, 95, 96, 97, 98]
(see Ref. [99] for a review). The same interactions responsible for WIMP DM
pair annihilation in the early Universe leading to their thermal freeze-out
guarantee their direct production at colliders, as long as kinematically
allowed. This will give 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. [99, 100] for reviews). Specifically, the monojet signature
has become emblematic for LHC DM searches [101, 102, 103, 104]. However, for a
leptophilic DM with loop-suppressed interactions to the SM quarks, hadron
colliders like the LHC are not expected to provide a better model-independent
limit than the existing constraints from indirect searches, such as from
AMS-02 [105, 106], at least within the EFT framework with contact
interactions.

On the other hand, lepton colliders provide an ideal testing ground for the
direct production of leptophilic DM and its subsequent detection via either
monophoton [107, 108, 109, 110, 95, 111, 112, 98, 113, 114, 115, 116] or
mono-$Z$ [117, 118, 119, 120] signatures. In this paper, we go beyond the
existing literature and perform a comprehensive and comparative study of both
monophoton and mono-$Z$ signatures of leptophilic DM at future $e^{+}e^{-}$
colliders in a model-independent, EFT approach. Our analysis is generically
applicable to all future $e^{+}e^{-}$ colliders, such as the International
Linear Collider (ILC) [121], Compact Linear Collider [122], Circular Electron
Positron Collider [123] and the electron-positron Future Circular Colliders
[124], but for concreteness, we have taken the $\sqrt{s}=1$ TeV ILC as our
case study for numerical simulations. We also assume the DM to be fermionic
and limit ourselves to the dimension-six operators, but taking into
consideration all possible dimension-six operators of scalar-pseudoscalar
(SP), vector-axial vector (VA) and tensor-axialtensor (TAT)-type as applicable
for the most general DM-electron coupling. Within the minimal EFT approach,
the only relevant degrees of freedom in our analysis are the DM mass and an
effective cutoff scale $\Lambda$ which determines the strength of the four-
Fermi operators. This enables us to derive model-independent ILC sensitivities
on leptophilic DM in the $(m_{\chi},\Lambda)$ plane in both monophoton and
mono-$Z$ (leptonic and hadronic) channels, after taking into account all
relevant backgrounds and systematic uncertainties. We consider both
unpolarized and polarized beam options [121, 125], and find that with the
proper choice of polarizations for the $e^{-}$ and $e^{+}$ beams (which
depends on the operator type), the DM sensitivities could be significantly
enhanced. We also find that the hadronic mono-$Z$ channel provides the best
sensitivities for the SP and TAT-type operators, while the monophoton channel
gives better sensitivity for the VA-type operators. For example, for a $1$ GeV
DM mass, the monophoton channel can probe $\Lambda$ values at $3\sigma$
confidence up to $6.6$ TeV, $8.8$ TeV and $7.1$ TeV for the SP, VA and TAT-
type operators respectively at $\sqrt{s}=1$ TeV ILC with 8 ${\rm ab}^{-1}$
integrated luminosity, whereas the mono-$Z$ hadronic (leptonic) channel can
probe $\Lambda$ values up to $5.5\;(4.3)$ TeV, $4.2\;(2.7)$ TeV and
$6.7\;(5.2)$ TeV respectively. We compare our results with the existing
literature wherever applicable and also with the current constraints from
direct and indirect detection, as well as from relic density considerations.
We find that the $\sqrt{s}=1$ TeV ILC sensitivities could surpass the existing
constraints for light DM with masses below about 300 GeV, above which the
collider reach is weakened mainly due to the kinematic suppression.

The rest of the paper is organized as follows. In Sec. II, we briefly describe
the EFT approach and list all possible dimension-six operators for the DM-
electron coupling in our leptophilic DM scenario. In Sec. III, we present our
cut-based analysis for the monophoton+$\not{E}_{T}$ signal and background,
both with and without beam polarization. In Sec. IV, we repeat the cut-based
analysis for the mono-$Z$+$\not{E}_{T}$ channel, with the leptonic case of
$Z\to\ell^{+}\ell^{-}$ (with $\ell=e,\mu$) in subsection IV.1 and the hadronic
case of $Z\to{\rm jets}$ in subsection IV.2. Our conclusions are given in Sec.
V.

##  II Effective operators

Our goal in this paper is to perform a model-independent collider analysis for
the leptophilic DM scenario in an EFT approach, with the primary assumptions
that (i) the DM particle $\chi$ couples directly only to the SM leptons but
not to the quarks (hence leptophilic), and (ii) the energy scale of the
associated new physics is large compared to the collider energies under
consideration, thus allowing us to integrate out the heavy mediators and
parametrize the DM-SM interactions using effective higher-dimensional
operators. For concreteness, we assume that the DM particles are Dirac
fermions, and therefore, the leading order DM-SM interactions are the
dimension-six four-Fermi interactions shown in Fig. 1, with the most general
effective Lagrangian given by [29]

$\displaystyle\mathcal{L}_{\rm
eff}=\frac{1}{\Lambda^{2}}\sum_{j}\left(\overline{\chi}\Gamma^{j}_{\chi}\chi\right)\left(\overline{\ell}\Gamma^{j}_{\ell}\ell\right)\,,$
(1)

where $\Lambda$ is the cutoff scale for the EFT description and the index $j$
corresponds to different Lorentz structures, as shown below. Since our main
focus is on $e^{+}e^{-}$ colliders, we will just set $\ell=e$ in Eq. (1) and
assume this to be the only leading-order coupling, but our discussion below
could be easily extended to other cases, e.g. future muon colliders [126] by
setting $\ell=\mu$.

Figure 1: Feynman diagram for the effective four-Fermi interactions between
the DM and electrons induced by the dimension-six operator (1).

A complete set of Lorentz-invariant operators consists of scalar (S), pseudo-
scalar (P), vector (V), axial-vector (A), tensor (T) and axial-tensor (AT)
currents. We classify them as follows:

$\displaystyle\text{Scalar-pseudoscalar (SP) type}:$
$\displaystyle\Gamma_{\chi}=c^{\chi}_{S}+ic^{\chi}_{P}\gamma_{5}\,,$
$\displaystyle\Gamma_{e}=c^{e}_{S}+ic^{e}_{P}\gamma_{5}\,,$
$\displaystyle\text{Vector-axial vector (VA) type}:$
$\displaystyle\Gamma_{\chi}^{\mu}=\left(c^{\chi}_{V}+c^{\chi}_{A}\gamma_{5}\right)\gamma^{\mu}\,,$
$\displaystyle\Gamma_{e\mu}=\left(c^{e}_{V}+c^{e}_{A}\gamma_{5}\right)\gamma_{\mu}\,,$
$\displaystyle\text{Tensor-axial Tensor (TAT) type}:$
$\displaystyle\Gamma_{\chi}^{\mu\nu}=\left(c^{\chi}_{T}+ic^{\chi}_{AT}\gamma_{5}\right)\sigma^{\mu\nu}\,,$
$\displaystyle\Gamma_{e\mu\nu}=\sigma_{\mu\nu}\,,$ (2)

where $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$ is the spin
tensor and $c_{j}^{\chi,e}$ are dimensionless, real couplings. Note that for
the TAT type, we did not write the AT term for electron separately, because
the relation
$\sigma^{\mu\nu}\gamma_{5}=\frac{i}{2}\epsilon^{\mu\nu\alpha\beta}\sigma_{\alpha\beta}$
implies that the ${\rm AT}\otimes{\rm AT}$ coupling is equivalent to ${\rm
T}\otimes{\rm T}$, and similarly, ${\rm T}\otimes{\rm AT}={\rm AT}\otimes{\rm
T}$. Also if we had considered Majorana DM, then vector- and TAT-type
interactions are forbidden, i.e. $c_{V}^{\chi}=c_{T}^{\chi}=c_{AT}^{\chi}=0$.
For simplicity, in Eq. (1) we have used a common cutoff scale $\Lambda$ for
all Lorentz structures. Furthermore, in our subsequent numerical analysis, we
will consider one type of operator at a time, by setting the corresponding
couplings $c_{j}^{\chi,e}=1$ without loss of generality and all other
couplings equal to zero, unless otherwise specified. For instance, setting
$c_{S}^{\chi}=c_{P}^{\chi}=c_{S}^{e}=c_{P}^{e}=1$ and all other couplings
equal to zero gives us the (S+P)-type operator, which we will simply refer to
as the SP-type in the following discussion. Similarly, we will denote the
$c_{V}^{\chi}=c_{A}^{\chi}=c_{V}^{e}=c_{A}^{e}=1$ case simply as the VA-type,
and $c_{T}^{\chi}=c_{AT}^{\chi}=1$ as the TAT-type for presenting our
numerical results in the $(m_{\chi},\Lambda)$ plane. For other choices of the
absolute values of the couplings, our results for the sensitivity on $\Lambda$
can be easily scaled accordingly.

We do not discuss any specific realization of these effective operators since
we are making a model-independent analysis. In the context of a given
ultraviolet (UV)-complete theory or even a simplified model, the suppression
or cutoff scale $\Lambda$ of the effective theory can be understood in terms
of the mass $m_{\rm med}$ of a mediator (which couples to electrons as well as
DM) as

$\displaystyle\frac{1}{\Lambda^{2}}=\frac{g_{e}g_{\chi}}{m_{\rm med}^{2}}\,,$
(3)

where $g_{e}$ and $g_{\chi}$ are the mediator couplings to electron and DM
respectively, and the mediator is assumed to be heavier than the energy scale
of interest. As a concrete example, let us consider the axial-vector-type
operator, which can be obtained from a theory with a massive spin-1 particle
$V^{\mu}$ with axial couplings to leptons and DM,

$\displaystyle\mathcal{L}=\frac{1}{2}m^{2}_{V}V_{\mu}V^{\mu}+g_{Ae}\overline{e}\gamma_{\mu}\gamma^{5}eV^{\mu}+g_{A\chi}\overline{\chi}\gamma_{\mu}\gamma^{5}\chi
V^{\mu}\,.$ (4)

In the context of DM pair production at lepton colliders via the
$e^{+}e^{-}\to\chi\overline{\chi}$ process, $V^{\mu}$ acts as the mediator of
the interaction between the electrons and DM particles. For instance, if this
is an $s$-channel process, then the corresponding matrix element will be

$\displaystyle\mathcal{M}$
$\displaystyle\sim\left(\frac{g_{Ae}~{}g_{A\chi}}{m^{2}_{V}-s}\right)\left(\overline{e}\gamma^{\mu}\gamma^{5}e\right)\left(\overline{\chi}\gamma_{\mu}\gamma^{5}\chi\right)$
$\displaystyle\simeq\left(\frac{g_{Ae}~{}g_{A\chi}}{m^{2}_{V}}\right)\left[1+\frac{s}{m^{2}_{V}}+{\cal
O}\left(\frac{s^{2}}{m_{V}^{4}}\right)\right]\left(\overline{e}\gamma^{\mu}\gamma^{5}e\right)\left(\overline{\chi}\gamma_{\mu}\gamma^{5}\chi\right)\,,$
(5)

in the limit where the center-of-mass energy is smaller than the mediator
mass, i.e. $s\ll m_{V}^{2}$. In this case, the leading term in Eq. (5) will be
of the form $g_{Ae}g_{A\chi}/m^{2}_{V}$, which can be simply identified as
$1/\Lambda^{2}$ in the contact interaction description of EFT [cf. Eq. (3)].
On the other hand, if $m^{2}_{V}\ll s$, i.e., in the light mediator scenario,
Eq. (5) should be written as

$\mathcal{M}\simeq-\left(\frac{g_{Ae}~{}g_{A\chi}}{s}\right)\left[1+\frac{m^{2}_{V}}{s}+{\cal
O}\left(\frac{m_{V}^{4}}{s^{2}}\right)\right]\left(\overline{e}\gamma^{\mu}\gamma^{5}e\right)\left(\overline{\chi}\gamma_{\mu}\gamma^{5}\chi\right)\,,$
(6)

causing the cross section to fall as $1/s$ [127, 128, 115]. For,
$m_{V}^{2}\simeq s$, i.e. for an on shell mediator, the cross section
increases due to the Breit-Wigner resonance, but this resonant behavior is not
realized in the EFT framework [127, 128].

In this work we have assumed the heavy mediator scenario (i.e., $m_{\rm
med}^{2}\gg s$). Then from Eq. (3), we have $\Lambda=m_{\rm
med}/\sqrt{g_{e}g_{\chi}}>\sqrt{s}$, as long as the couplings are of ${\cal
O}(1)$ or less. Hence, we will impose a theoretical limit of
$\Lambda>\sqrt{s}$ for the EFT validity with the couplings set to unity.

For relatively larger DM mass, we must also have $\Lambda>2m_{\chi}$ in order
to describe DM pair annihilation by the EFT. In fact, using the lower limit of
$\Lambda=2m_{\chi}$ induces 100% error in the EFT prediction for $s$-channel
UV completions. Therefore, we will use $\Lambda>{\rm
max}\\{\sqrt{s},3m_{\chi}\\}$ as a conservative lower bound [129] to ensure
the validity of our EFT approach.

As opposed to the lepton colliders, the validity of the EFT approach for
mono-$X$ searches at the hadron colliders is problematic [130]. Instead,
simplified models [131, 132] are more appropriate for the interpretation of
the LHC results. Therefore, it becomes difficult to directly compare the
results from lepton and hadron colliders.

As for the specific model realizations of the other operators listed in Eq.
(2), the vector-type operator can be easily realized by a similar spin-1
mediator as in Eq. (4), with vector instead of axial-vector couplings.
Similarly, the SP-type operators can be realized by a scalar/pseudoscalar
mediator, whereas the TAT-type operators can be realized by a spin-2 mediator.
A detailed discussion of these model realizations is beyond the scope of the
current work.

##  III monophoton channel

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

Since DM behaves as missing energy in the collider environment, we need a
visible particle along with the DM pair to identify the DM production process.
In the context of leptophilic DM at lepton colliders, monophoton and mono-$Z$
are the most promising channels, where the extra $\gamma/Z$ comes from
initial-state radiation (ISR) from either the electron or positron leg. In
this section, we analyze the monophoton signal and backgrounds in detail and
present our numerical sensitivity results for $\sqrt{s}=1$ TeV ILC with 8 ab-1
integrated luminosity. In the next section, we will repeat the same analysis
for the mono-$Z$ case.

For the monophoton signal $e^{+}e^{-}\to\chi\overline{\chi}\gamma$ (as shown
in Fig. 2), the $\chi$’s will contribute to the missing transverse energy at
the detector. The dominant irreducible SM background to this process comes
from neutrino pair production with an associated ISR photon, i.e.
$e^{+}e^{-}\to\nu\overline{\nu}\gamma$. Since neutrinos are practically
indistinguishable from WIMPs on an event-by-event basis, the majority of this
background survives the event selection cuts. However, as we will show later,
this background is highly polarization dependent and therefore can be
significantly reduced by the proper choice of polarized beams, without
affecting the signal much.

Apart from the neutrino background, any SM process with a single photon in the
final state can contribute to the total background if all other visible
particles escape detection. The SM processes containing either jets or charged
particles are relatively easy to distinguish from a DM event, so their
contribution to the total background is negligible [111]. The only exception
is the “Bhabha scattering” process associated with an extra photon (either
from initial or final-state radiation), i.e., $e^{+}e^{-}\to
e^{+}e^{-}\gamma$, which has a large cross section, is polarization
independent, and can significantly contribute to the total background whenever
the final-state electrons and positrons go undetected, e.g. along beam pipes.
In our following analysis, we consider both neutrino and radiative Bhabha
backgrounds.

###  III.1 Cross sections

The cross sections for the monophoton signal
$e^{+}e^{-}\to\chi\overline{\chi}\gamma$ at $\sqrt{s}=1$ TeV ILC are estimated
using CalcHEP [133] with proper implementation of ISR and beamstrahlung
effects, which significantly affect the width and position of the neutrino $Z$
resonance. For this purpose, the EFT Lagrangian (1) is implemented in
FeynRules [134] to generate the model library files suitable for CalcHEP. To
avoid collinear and infrared divergences, we limit the phase space in the
event generation with the following cuts on the outgoing photon:

$p_{T}^{\gamma}>2\text{ GeV},\;\;E_{\gamma}>1\text{
GeV},\;\;|\cos\theta_{\gamma}|\leq 0.9975\,.$ (7)

Figure 3: Variation of monophoton signal cross section with the DM mass (left)
and the cutoff scale (right) at $\sqrt{s}=1$ TeV ILC. The solid, dashed and
dotted lines are for the SP-, VA- and TAT-type operators respectively. In the
left panel, the red, green and blue curves correspond to different values of
the cutoff scale $\Lambda=1$, 3, and 5 TeV, respectively, whereas in the right
panel, they correspond to different values of the DM mass $m_{\chi}=100$, 250,
and 450 GeV respectively.

The _radiative_ neutrino-pair and the _radiative_ Bhabha background events are
generated using WHIZARD [135] (to better handle the singularities) with the
same set of cuts as in Eq. (7) to the matrix element (ME) photon (i.e.,
excluding the ISR and beamstrahlung photons). Also, some additional cuts are
implemented on the invariant masses for the Bhabha process to take care of the
soft and collinear divergences:

$\displaystyle M_{e^{\pm}_{\rm in},e^{\pm}_{\rm out}}<-2\text{
GeV},~{}M_{e^{\pm}_{\rm out},e^{\pm}_{\rm out}}>2\text{ GeV},$ $\displaystyle
M_{e^{\pm},\gamma_{i}}>2\text{ GeV},~{}M_{e^{\pm},\gamma}>4\text{ GeV}\,,$ (8)

where $\gamma_{i}$ are the ME photons and $\gamma$ is the signal photon.

To properly merge the ISR and ME photons in the events we also incorporated
the ME-ISR merging technology as discussed in Ref. [136]. This takes into
account the higher-order corrections (which contribute to about 20% of cross
section) while avoiding double counting of multiple photon emission. We
considered the values of the independent variables $q_{-}$ and $q_{+}$ to be
$1$ GeV [136] where $q_{\pm}$ are defined as

$\displaystyle q_{-}$ $\displaystyle=\sqrt{2E_{\rm
cm}E_{\gamma}}\cdot\sin{\theta_{\gamma}/2}\,,$ (9) $\displaystyle q_{+}$
$\displaystyle=\sqrt{2E_{\rm cm}E_{\gamma}}\cdot\cos{\theta_{\gamma}/2}\,,$
(10)

where $E_{\rm cm}\equiv\sqrt{s}$, and $E_{\gamma}$ and $\theta_{\gamma}$ are
the photon energy and polar angle respectively. We rejected all events with
$q_{\pm}<1$ GeV or $E_{\gamma}<1$ GeV for any ME photons, and with $q_{\pm}>1$
GeV and $E_{\gamma}>1$ GeV for any ISR photons. After generating the signal
and background events, we performed a fast detector simulation using Delphes3
[137] with the ILCgen configuration card.

The variations of the unpolarized signal cross section as a function of the DM
mass and the cutoff scale are shown in Fig. 3—left and right panels,
respectively— for all three operator types, namely, SP (solid), VA (dashed)
and TAT (dotted) types. We find that the cross section is the smallest
(largest) for the SP (TAT)-type operator at any given DM mass. In the left
panel, the sudden drop in the cross section as $m_{\chi}$ approaches
$\sqrt{s}/2$ is due to phase-space suppression. Otherwise, for smaller DM
masses, the cross section for a given operator type and a given cutoff scale
is almost independent of the DM mass. In the right panel, we see that for a
given DM mass the cross section drops as $\Lambda^{-4}$, as expected. As for
the background, we find that the neutrino background cross section at
$\sqrt{s}=1$ TeV is 6.63$\times 10^{3}$ fb, while the radiative Bhabha
background is 1.07$\times 10^{5}$ fb before ME-ISR merging. After ME-ISR
merging these cross sections reduce to 5.08$\times 10^{3}$ fb and 7.68$\times
10^{4}$ fb, respectively for neutrino-pair and Bhabha background. On the other
hand, the DM signal cross section is found to be much smaller, as shown in
Table 1 for a benchmark DM mass of $m_{\chi}=100$ GeV and the cutoff scale
$\Lambda=3$ TeV.

Table 1: Comparison of the SM backgrounds and signal cross sections for the monophoton channel for the unpolarized beam, as well as for different choices of beam polarization at $\sqrt{s}=1~{}\rm{TeV}$. For the signal, we have chosen benchmark values of $m_{\chi}=100~{}\rm{GeV}$ and $\Lambda=3~{}\rm{TeV}$. The numbers in bold highlight the optimal polarization choice for a given operator type. |  |  | Polarized cross section (fb)
---|---|---|---
Process type | Unpolarized cross section (fb) | Polarization $P(e^{-},e^{+})$ | $(+,+)$ | $(+,-)$ | $(-,+)$ | $(-,-)$
|  | $(80,0)$ | $1.15\times 10^{3}$ |  $1.15\times 10^{3}$1 |  1$9.05\times 10^{3}$1 |  1$9.05\times 10^{3}$
$\nu\overline{\nu}\gamma$ | $5.08\times 10^{3}$ | $(80,20)$ |  $1.31\times 10^{3}$1 |  1$9.71\times 10^{2}$1 |  1$1.08\times 10^{4}$1 |  1$7.25\times 10^{3}$
|  | $(80,30)$ |  $1.40\times 10^{3}$1 |  1$8.87\times 10^{2}$1 |  1$1.18\times 10^{4}$1 |  1$6.30\times 10^{3}$
|  | $(80,0)$ |  $3.02\times 10^{4}$1 |  1$3.02\times 10^{4}$1 |  1$2.99\times 10^{4}$1 |  1$2.99\times 10^{4}$
$e^{-}e^{+}\gamma$ | $2.97\times 10^{4}$ | $(80,20)$ |  $3.13\times 10^{4}$1 |  1$2.99\times 10^{4}$1 |  1$3.08\times 10^{4}$1 |  1$2.93\times 10^{4}$
|  | $(80,30)$ |  $3.02\times 10^{4}$1 |  1$3.01\times 10^{4}$1 |  1$3.16\times 10^{4}$1 |  1$2.93\times 10^{4}$
|  | $(80,0)$ | 31.6 | 31.6 | 31.6 | 31.6
SP | $31.6$ | $(80,20)$ | 36.6 | 26.5 | 26.5 | 36.6
|  | $(80,30)$ | 39.1 | 24.0 | 24.0 | 39.1
|  | $(80,0)$ | 76.5 | 76.5 | 8.5 | 8.5
VA | $42.5$ | $(80,20)$ | 61.2 | 91.8 | 6.8 | 10.2
|  | $(80,30)$ | 53.5 | 99.4 | 6.0 | 11.1
|  | $(80,0)$ | 45.0 | 45.0 | 45.0 | 45.0
TAT | $45.0$ | $(80,20)$ | 52.2 | 37.8 | 37.8 | 42.2
|  | $(80,30)$ | 55.8 | 34.2 | 34.2 | 55.8

###  III.2 Effect of polarization

One important advantage of lepton colliders is that the incoming beams can be
polarized. This helps to reduce the neutrino background considerably, as shown
in Table 1. To utilize the full advantage of the beam polarization, we
investigate the effect of different choices of polarization on the signal and
background. At the ILC, the baseline design foresees at least 80% electron-
beam polarization at the interaction point, whereas the positron beam can be
polarized up to 30% for the undulator positron source (up to 60% may be
possible with the addition of a photon collimator) [121]. For comparison, we
show our results for three different nominal absolute values of polarization:
$|P(e^{-},e^{+})|=(80,0)$, (80,20) and (80,30). In each case, we can also have
four different polarization configurations, namely, ${\rm
sign}(P(e^{-}),P(e^{+}))=(+,+)$, $(+,-)$, $(-,+)$, and $(-,-)$, where $+$ and
$-$ denote the right- and left-handed helicities respectively. According to
the H20 running scenario, the standard sharing between the different beam
helicity configurations is 40% for $(+,-)$ and $(-,+)$ each and 10% for
$(+,+)$ and $(-,-)$ each [125]. With the luminosity upgrade, the target
integrated luminosities per beam helicity configuration at $1$ TeV machine
energy is 3200 fb-1 for $(+,-)$ and $(-,+)$ each and 800 fb-1 for $(+,+)$ and
$(-,-)$ each, with the proposed total luminosity of 8000 fb-1 [125]. In our
analysis, we have considered the H20 scenario as well as the optimal beam
polarization case for better comparison at $8000$ fb-1.

In Table 1, we show the effect of different schemes of polarizations and
helicity orientations on the monophoton signal and background cross sections.
It is evident from the table that the radiative Bhabha background remains
almost unchanged, as mentioned earlier. On the other hand, electron-beam
polarization is very effective in reducing the neutrino background, as an 80%
_right_ -handed electron beam can reduce the neutrino background to $23\%$ of
the unpolarized case, even without any polarization on the positron beam. The
effect is further enhanced by a _left_ -handed positron beam. We see that for
$20\%$ and $30\%$ left-handed positron beam polarization, the neutrino
background is reduced to $26\%$ and $18\%$ of its unpolarized value,
respectively.

The signals are also affected to some extent by beam polarization and the
optimal helicity configuration depends on the operator type. For SP- and TAT-
type operators we see no effect of electron-beam polarization, but a 20% (30%)
right-handed positron beam can enhance the signal by 16% (24%). The VA-type
signal, on the other hand, prefers the $(+,-)$ helicity configuration — the
same choice for which the neutrino background is minimized. With the
$(+80\%,-30\%)$ configuration, the VA-type signal is enhanced by a factor of
2.3, whereas the $(+80\%,+30\%)$ configuration enhances it by a modest 26%. In
Table 1, it can be noted that the cross section of SP- and TAT-types are same
for $(+,+)$ and $(-,-)$ helicity orientations, but it can be seen that the
neutrino background enhances with $(-,-)$ helicity, whereas the Bhabha
background remains same. This effectively renders the $(-80,-30)$ polarization
choice to be nonoptimal for these two operator types. In Table 1, the optimal
polarization choices that give the largest signal-to-background ratio are
shown in boldfaced fonts.

###  III.3 Cut-based analysis

Now we analyze various kinematic distributions and perform a cut-based
analysis to optimize the signal-to-background ratio. This of course depends on
the DM mass, so in Table 2, we list three benchmark points (BPs) with
$m_{\chi}=100$, 250, and 350 GeV, respectively, and present the corresponding
selection cuts optimized for each case. Here we fix $\Lambda=3$ TeV for
illustration, but in the next subsection, we will vary both $m_{\chi}$ and
$\Lambda$ to obtain the $3\sigma$ sensitivity limits. As for the choice of the
DM mass values, since it was seen from Fig. 3 that the signal cross sections
are barely sensitive to the DM mass up to around $100$ GeV, our BP1
essentially captures the light DM scenario. Similarly, our BP3 is chosen
moderately close to the kinematic limit of $\sqrt{s}/2$ (going too close to
$\sqrt{s}/2$ will result in cross section values too low to give sizable event
counts after all the selection cuts). BP2 is chosen for an intermediate DM
mass in between BP1 and BP3.

Table 2: Monophoton selection cuts for different BPs across all operator types. Because of the dynamic nature of these cuts, the backgrounds will have to be separately analyzed for each BP. | BP1 | BP2 | BP3
---|---|---|---
Definition | $m_{\chi}=100\text{ GeV, }$ | $m_{\chi}=250\text{ GeV, }$ | $m_{\chi}=350\text{ GeV, }$
$\Lambda=3\text{ TeV}$ | $\Lambda=3\text{ TeV}$ | $\Lambda=3\text{ TeV}$
SP-type / VA-type / TAT-type
Baseline selection | $E_{\gamma}>5\text{ GeV},\;\;|\eta_{\gamma}|<2.8,$
Cut 1 |  $E_{\gamma}<450\text{ GeV}$00 |  0$E_{\gamma}<350\text{ GeV}$00 |  0$E_{\gamma}<250\text{ GeV}$
Cut 2 | $|\cos\theta_{\gamma}|<0.93$
Cut 3 | Charged particle veto with $p_{T}>5$ GeV
Cut 4 | BeamCal veto

Figure 4: Normalized differential distributions after baseline selection cuts for the set of five kinematic observables considered in Table 2, namely, $E_{\gamma}$, $\cos\theta_{\gamma}$ for the unpolarized case (the polarized case histograms are in the Appendix, see Fig. 15). In each panel, the shaded green histograms are for the neutrino background and the cyan hatched histograms are for the radiative Bhabha background. The unshaded red, green and blue histograms are for BP1, BP2, and BP3, respectively. The left, middle, and right panels correspond to the SP-, VA-, and TAT-type operators, respectively. Table 3: Cut-flow chart for different operators and BPs for the monophoton signal and backgrounds. The cut efficiencies are calculated with respect to the baseline selection (BS) cuts given in Eq. (11), with the corresponding event numbers shown in the first row. Note that depending on the operator type we are interested in for the signal, the selection cuts are different (cf. Table 2). Therefore, the corresponding backgrounds will also vary as shown here. Selection cuts | Event Numbers (Cut efficiencies)
---|---
Neutrino pair | Radiative Bhabha | Signal
BP1 | BP2 | BP3 | BP1 | BP2 | BP3 | BP1 | BP2 | BP3
SP-type:
BS | $3.09\times 10^{7}$ | $6.70\times 10^{7}$ | $2.01\times 10^{5}$ | $1.45\times 10^{5}$ | $8.74\times 10^{4}$
$(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $1.76\times 10^{7}$ | $1.73\times 10^{7}$ | $1.70\times 10^{7}$ | $6.08\times 10^{5}$ | $6.08\times 10^{5}$ | $6.08\times 10^{5}$ | $1.24\times 10^{5}$ | $8.97\times 10^{4}$ | $5.50\times 10^{4}$
$(56.75\%)$ | $(56.04\%)$ | $(54.95\%)$ | $(0.91\%)$ | $(0.91\%)$ | $(0.91\%)$ | $(61.68\%)$ | $(61.97\%)$ | $(62.97\%)$
VA-type:
BS | $3.09\times 10^{7}$ | $6.70\times 10^{7}$ | $2.71\times 10^{5}$ | $2.10\times 10^{5}$ | $1.41\times 10^{5}$
$(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $1.76\times 10^{7}$ | $1.73\times 10^{7}$ | $1.70\times 10^{7}$ | $6.08\times 10^{5}$ | $6.08\times 10^{5}$ | $6.08\times 10^{5}$ | $1.66\times 10^{5}$ | $1.30\times 10^{5}$ | $8.84\times 10^{4}$
$(56.75\%)$ | $(56.04\%)$ | $(54.95\%)$ | $(0.91\%)$ | $(0.91\%)$ | $(0.91\%)$ | $(61.07\%)$ | $(61.97\%)$ | $(62.78\%)$
TAT-type:
BS | $3.09\times 10^{7}$ | $6.70\times 10^{7}$ | $2.88\times 10^{5}$ | $2.67\times 10^{5}$ | $2.13\times 10^{5}$
$(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $1.76\times 10^{7}$ | $1.73\times 10^{7}$ | $1.70\times 10^{7}$ | $6.08\times 10^{5}$ | $6.08\times 10^{5}$ | $6.08\times 10^{5}$ | $1.78\times 10^{5}$ | $1.64\times 10^{5}$ | $1.34\times 10^{5}$
$(56.75\%)$ | $(56.04\%)$ | $(54.95\%)$ | $(0.91\%)$ | $(0.91\%)$ | $(0.91\%)$ | $(61.77\%)$ | $(61.67\%)$ | $(62.64\%)$

Baseline selection cuts: We define our monophoton signals by those events that
pass through the baseline selection criteria as defined below, in addition to
the cuts given in Eq. (7),

$E_{\gamma}>5\text{ GeV},\;\;|\eta_{\gamma}|<2.8$ (11)

where the photon with highest transverse momentum in an event is considered as
the signal photon.

After implementing these baseline selection criteria, the signal and
background events (selection efficiencies) are given in the first rows of
Table 3. We find that the signal and the neutrino background are reduced to
about $80\%$ and $76\%$ of their original values, respectively as in Table 1.
Similarly, the actual Bhabha-induced background relevant for our signal is
found to be only about 34% of its original value quoted in Table 1 after the
baseline selection cuts. To further enhance our signal-to-background ratio, we
then examine the signal versus background distributions of some relevant
kinematic variables and devise further cuts. The normalized distributions are
given in Fig. 4 for the unpolarized case. The corresponding distributions for
the optimally polarized case are shown in Fig. 15 in the Appendix. From Fig.
4, we see some variations of the signal distributions between the three BPs.
Therefore, the specialized selection criteria discussed below are dynamic with
respect to different BPs, as also summarized in Table 2.

Cut 1: From the $E_{\gamma}$ distribution in Fig. 4 (top row), we see that
although the neutrino background is mostly signallike, it has a large number
of events around $E_{\gamma}\approx 496$ GeV, due to the radiative return of
the $Z$ resonance which is determined by $(s-M_{Z}^{2})/2\sqrt{s}$. On the
other hand, the $E_{\gamma}$ distribution for the signal drops at a lower
value, depending on the DM mass, due to phase-space suppression. Therefore, we
impose a dynamic maximum $E_{\gamma}$ cut (cf. Table 2) to account for both
the effects. Cut 2: Detection of collinear photons is not possible. The
limitation of the detector reflects in the baseline pseudorapidity cut
$|\eta_{\gamma}|<2.8$ or $7^{\circ}<\theta<173^{\circ}$. However, for the
purpose of background suppression, a stricter cut in the form of
$\cos{\theta_{\gamma}}$ was applied, based on the event distribution in Fig.
4. This cut is particularly effective in reducing the Bhabha background
significantly to about 16%-17% of its yield after baseline selection, whereas
the neutrino-pair background and signal drop to about 56%-58% and 65%-66%
respectively.

Cut 3: Next we apply a veto condition on the charged particles that are
reconstructed as electrons as well as muon and jets. Here, we selected the
jets and muons to have $p_{T}>5$ GeV and $|\eta|<2.8$, whereas for the
electrons we required $p_{T}>2$ GeV. This kind of selection criteria was also
used in Ref. [138] which substantially reduces the Bhabha background. We note
that after this cut the Bhabha event yield drops from around $17\%$ (after cut
2) to approximately $7\%$.

Cut 4: Finally, we apply a veto condition to reject all the events that have
energy deposits in the electromagnetic calorimeter in the very forward
direction of the beamline (BeamCal) [139]. This calorimeter is useful in
detecting and tagging the highly energetic electrons from the Bhabha events
that otherwise escape the central tracker, thereby minimizing their
contribution to the background. As can be seen in Table 3, the Bhabha
background is now reduced to $\sim 2\%$, whereas neutrino-pair background and
signal events are left at $\sim 55\%$ and $\sim 60\%$ of their respective
numbers after baseline selection.

The above-mentioned cuts are optimized for the unpolarized beam case, and so
in the cut-flow table we only show the numbers for the unpolarized case. A
similar table for polarized case can be found in the Appendix in Table 14,
where we see that the final selection efficiencies are similar. However, as
shown in the Table 1, due to enhanced cross sections of the signals and
reduced production of the neutrino-pair background (while Bhabha remains the
same) in the respective optimal polarization scenarios of the operators, it
translates into significantly better signal significance than the unpolarized
case as discussed in the next section.

###  III.4 Signal significance

Table 4: Signal significance in the monophoton channel at $\sqrt{s}=1$ TeV with $\mathcal{L}_{\rm int}=8~{}{\rm ab}^{-1}$ integrated luminosity. These are the optimized numbers after implementing all the selection cuts. The values in the parentheses denote the significances with background systematic uncertainty. | Signal significance for ${\cal L}_{\rm int}=8\,{\rm ab}^{-1}$
---|---
Unpolarized beams | H20 scenario | Optimally polarized beams
Operator type | BP1 | BP2 | BP3 | BP1 | BP2 | BP3 | BP1 | BP2 | BP3
SP | $29.0\;(3.5)$ | $21.1\;(2.5)$ | $13.1\;(1.6)$ | $40.1\;(15.5)$ | $29.7\;(11.5)$ | $18.5\;(7.1)$ | $64.6\;(12.6)$ | $47.8\;(9.3)$ | $29.8\;(5.8)$
VA | $38.7\;(4.6)$ | $30.6\;(3.7)$ | $21.0\;(2.5)$ | $129.3\;(45.6)$ | $104.3\;(36.5)$ | $72.6\;(25.2)$ | $199.3\;(44.7)$ | $161.1\;(35.8)$ | $112.3\;(24.6)$
TAT | $41.5\;(5.0)$ | $38.6\;(4.6)$ | $31.7\;(3.8)$ | $57.6\;(22.4)$ | $54.2\;(21.0)$ | $44.5\;(17.3)$ | $92.6\;(18.2)$ | $86.8\;(17.1)$ | $71.5\;(14.1)$

After implementing all the cuts mentioned above, we calculate the final signal
significance for our benchmark scenarios using the definition

${\rm
Sig}=\frac{S}{\sqrt{S+\sum_{i}\left(B_{i}+\epsilon_{i}^{2}B_{i}^{2}\right)}}\,,$
(12)

where $S$ and $B_{i}$ are the number of signal and ($i$th channel) background
events, respectively, for a given integrated luminosity, and $\epsilon_{i}$ is
the corresponding systematic uncertainty in the background estimation. Our
results are given in Table 4 for the three BPs. We show the numbers for an
ideal case with zero systematics and also for a more conservative case with
0.2% and 1% systematics for the dominant and subdominant backgrounds,
respectively (in parentheses). The results are significantly weakened in the
latter case because of the relatively large background compared to the signal.
For the mixed polarization case in H20 scenario, we calculated the
significances for individual polarization cases and assumed them to be
uncorrelated, calculated the final signal significance as

${\rm Sig_{\,H20}}=\sqrt{{\rm Sig}_{(+,+)}^{2}+{\rm Sig}_{(+,-)}^{2}+{\rm
Sig}_{(-,+)}^{2}+{\rm Sig}_{(-,-)}^{2}}$ (13)

The possible sources of systematic uncertainties at the ILC are [138, 114,
115] (i) event estimation, (ii) integrated luminosity, (iii) beam
polarization, and (iv) shape of the luminosity spectrum. Here we did a
simplified analysis and considered only uncertainties in background estimation
in a cut-and-count method. We considered here an uncertainty of $0.2\%$ for
the neutrino-pair background and $1\%$ for the radiative Bhabha background
[138, 140, 136]. We admit that for a more realistic uncertainty analysis one
should consider all the sources of uncertainty based on the event
distributions and their possible correlations. Also it is possible to
constrain systematic uncertainties with the data alone, i.e., by looking at
the $Z$-return peak [138]. Since, in practice, uncertainty in background
normalization is $100\%$ correlated between different polarizations, its
impact on the signal sensitivity will be reduced compared to our predicted
sensitivities. However, such a detailed analysis of systematic uncertainties
is beyond the scope of our work. When the actual experiment is performed, we
expect the realistic sensitivities to lie somewhere between the two cases
presented here, i.e., the ideal situation with no systematics and the
conservative case with our choice of systematics.

From Table 4, we see that the significance enhances as we go to lower DM mass
regions, as expected because of kinematic reasons. Operatorwise we see that
TAT- and VA-type operators perform better than the SP-type. We also find
substantial increase in significance on application of optimal beam
polarization. We note that the optimal beam polarization gives the best
significance when there are no systematics, but the mixed polarization case of
the H20 running scenario gives better significance when uncertainty in
background estimation is taken into account.

Using a machine-learning-based multivariate (MV) analysis might improve the
signal significance over that obtained here using a simple cut-based analysis,
but a detailed MV analysis is beyond the scope of the current work and will be
reported in a follow-up study.

Going beyond the three BPs, we now vary the DM mass and calculate the signal
significance following the same cut-based analysis procedure outlined above.
Our results for the $3\sigma$ sensitivity contours in the $(m_{\chi},\Lambda)$
plane are shown in Fig. 5 for all the operator types. The green, blue and red
contours are for the unpolarized, optimal and mixed polarization cases,
respectively, and the solid (dashed) contours are assuming zero (1%)
background systematics. The shaded regions are excluded by various
constraints, as follows: first of all, for $\Lambda<{\rm
max}\\{\sqrt{s},3m_{\chi}\\}$, our EFT framework is not valid (cf. Sec. II).
This is shown by the navy-blue-shaded regions in Fig. 5. For $\sqrt{s}=1$ TeV
as considered here, this EFT validity limit supersedes the previous limit from
the Large Electron-Positron collider [95].

Figure 5: $3\sigma$ sensitivity contours in the monophoton channel for the SP-
type (top left panel), VA-type (top right panel) and TAT-type (bottom panel)
operators with unpolarized (Unpol, green lines), mixed polarized (Mix-Pol, red
lines) and optimally polarized (Opt-pol, blue lines) $e^{+}e^{-}$ beams at
$\sqrt{s}=1$ TeV center-of-mass energy and with ${\cal L}_{\rm int}=8$ ab-1
integrated luminosity. The solid (dashed) contours are assuming zero (1%)
background systematics. The various shaded regions are excluded by direct
detection (XENON1T, PANDAX-4T), indirect detection (Fermi-LAT, AMS),
astrophysics (SN1987A) and cosmology (CMB) constraints. In the shaded region
below $\Lambda={\rm max}\\{\sqrt{s},3m_{\chi}\\}$, our EFT framework is not
valid. Along the dot-dashed line, the observed DM relic density is reproduced
for a thermal WIMP assuming only DM-electron effective coupling.

The same effective operator given in Eq. (1) also gives rise to DM scattering
with electrons $\chi e^{-}\to\chi e^{-}$. The exact analytic expressions for
these cross sections in our EFT framework can be found in Appendix C of Ref.
[85] for all the operator types. Up to velocity or electron mass suppression,
the typical size of the cross section is given by

$\displaystyle\sigma^{0}_{\chi e}\simeq\frac{m_{e}^{2}}{\pi\Lambda^{4}}\approx
3.2\times 10^{-47}{\rm cm}^{2}\left(\frac{1~{}{\rm
TeV}}{\Lambda}\right)^{4}\,.$ (14)

Comparing this with the experimental upper limits on $\sigma_{\chi e}$ from
dedicated direct detection experiments [77, 78], we can derive a lower limit
on the cutoff scale $\Lambda$ as a function of the DM mass $m_{\chi}$.
However, the current best limit on $\sigma_{\chi e}$ from XENON1T is at the
level of $\mathcal{O}(10^{-39})$ cm2 [78], which translates into a very weak
bound on $\Lambda$ and is not relevant for our study. Even the future
ambitious proposals like DARKSPHERE can only reach up to ${\cal
O}(10^{-42})\>{\rm cm}^{2}$ [141], still 5 orders of magnitude weaker than
that needed to probe a TeV-scale $\Lambda$ value [cf. Eq. (14)].

Figure 6: $3\sigma$ sensitivity reach of different operators in the monophoton
channel with DM mass of $1$ GeV at $\sqrt{s}=1$ TeV and $\mathcal{L}_{\rm
int}=8~{}{\rm ab}^{-1}$. The green, red and blue bars show the corresponding
sensitivities with unpolarized, mixed polarized and optimally polarized beams
respectively, and the lighter (darker) shade corresponds to zero (1%)
background systematics.

However, more stringent limits can be derived from DM-nucleon scattering
searches. Even for a leptophilic DM as in our case, DM-nucleon couplings are
necessarily induced at loop level from photon exchange between virtual leptons
and the quarks. In fact, as shown in Ref. [29], the loop-induced DM-nucleon
scattering almost always dominates over the DM-electron scattering. The
analytic expressions for the one- and two-loop DM-nucleon scattering cross
sections can be found in Ref. [29]. The typical size of the one-loop cross
section is [116]

$\displaystyle\sigma^{1}_{\chi N}$
$\displaystyle\simeq\frac{\mu_{N}^{2}c_{e}^{2}}{9\pi
A^{2}}\left(\frac{\alpha_{\rm
em}Z}{\pi\Lambda^{2}}\right)^{2}\left[\log\left(\frac{\Lambda^{2}}{\Lambda_{\rm
NR}^{2}}\right)\right]^{2}$ $\displaystyle\approx\,1.5\times 10^{-45}{\rm
cm}^{2}\left(\frac{\mu_{N}}{10~{}{\rm GeV}}\right)^{2}\left(\frac{1~{}{\rm
TeV}}{\Lambda}\right)^{4}$ (15)

where $\mu_{N}=m_{N}m_{\chi}/(m_{N}+m_{\chi})$ is the reduced mass of the DM-
nucleon system, $Z$ and $A$ are, respectively, the atomic number and atomic
mass of the given nucleus, $\alpha_{\rm em}$ is the fine-structure constant,
$\Lambda_{\rm NR}$ is the direct detection scale, and $c_{e}$ is the operator
coefficient value at the scale $\Lambda_{\rm NR}$. In the last step of Eq.
(15), we have used $\Lambda_{\rm NR}=2$ GeV and $c_{e}=0.965$ following Ref.
[116]. Since the most stringent DM-nucleon spin-independent cross section
limits come from XENON1T [11] and PANDAX-4T [12] experiments, we have also set
$Z=54$ and $A=131$ for the xenon nucleus and have translated the experimental
upper limits onto the $(m_{\chi},\Lambda)$ plane, as shown by the yellow- and
grey-shaded regions, respectively, in Fig. 5. Note that these limits are only
applicable for the vector and tensor lepton currents, i.e.
$\Gamma_{\ell}=\gamma_{\mu},\ \sigma_{\mu\nu}$ in Eq. (1). For the scalar
lepton current, $\Gamma_{\ell}=1$, the one-loop DM-nucleon coupling vanishes,
and one has to go to two loops, which is suppressed by $\alpha_{\rm em}^{2}$
for the scalar-scalar-type coupling and $\alpha_{\rm em}^{2}v^{2}$ (where
$v\sim 10^{-3}$ is the DM velocity) for the pseudoscalar-scalar-type coupling.
In contrast, for pseudoscalar and axial-vector lepton currents, i.e.,
$\Gamma_{\ell}=\gamma_{5},\ \gamma_{\mu}\gamma_{5}$, the DM-nucleon coupling
vanishes to all orders. Therefore, we have not shown the XENON1T and PANDAX-4T
limits for the SP-type operator on the top left panel of Fig. 5. The same
effective operator given in Eq. (1) also enables DM annihilation into
electrons $\chi\overline{\chi}\to e^{+}e^{-}$. The exact analytic expressions
for these cross sections in our EFT framework can be found in Appendix C of
Ref. [85] for all the operator types. Using these, we calculate the thermal-
averaged cross section times relative velocity $\langle\sigma v\rangle$, which
goes as $m_{\chi}^{2}/\Lambda^{4}$, and compare it with the existing indirect
detection upper limits on $\langle\sigma v\rangle$ in the $e^{+}e^{-}$ channel
to put a lower bound on $\Lambda$ as a function of the DM mass. This is shown
in Fig. 5 by the red- and brown-shaded regions, respectively, for the Fermi-
LAT [142] and AMS-02 [106] constraints on $\langle\sigma v\rangle$. Similar
constraints on $\langle\sigma v\rangle$ can be derived using CMB anisotropies
[142], which is shown by the cyan-shaded region in Fig. 5, assuming an
$s$-wave annihilation (for $p$-wave annihilation, the CMB bound will be much
weaker).

Along the dot-dashed line in Fig. 5, the observed relic density can be
reproduced for a WIMP DM. In principle, the region to the left and above of
this line is disfavored for a thermal WIMP, because in this region
$\langle\sigma v\rangle$ is smaller than the observed value of
$\sim(2-5)\times 10^{-26}$ cm3 sec-1 (depending on the DM mass [143]), which
leads to an overabundance of DM, since $\Omega_{\chi}h^{2}\propto
1/\langle\sigma v\rangle$. However, this problem can be circumvented by either
opening up additional leptonic annihilation channels (like $\mu^{+}\mu^{-}$,
$\tau^{+}\tau^{-}$, and $\nu\bar{\nu}$) or even going beyond the WIMP paradigm
and invoking, e.g., the freeze-in mechanism [144]. This will not affect the
main results of our paper, since the collider phenomenology discussed here
only depends on the DM coupling to electrons. Also shown in Fig. 5 is the
supernova constraint, which excludes the magenta-shaded region from
consideration of energy loss and optical depth criteria from the observation
of SN1987A [85]. Here we have used an average supernova core temperature of 30
MeV. Note that the supernova bound is only applicable for DM mass below $\sim$
200 MeV or so, and for a certain range of $\Lambda$ values, above which the DM
particles cannot be efficiently produced in the supernova core, and below
which they will no longer free stream.

From Fig. 5, we find that, in spite of a large irreducible background, the
accessible range of the cutoff scale $\Lambda$ at $\sqrt{s}=1$ TeV ILC looks
quite promising in the monophoton channel, especially for low-mass DM, where
the collider sensitivity is almost flat, whereas the existing direct and
indirect detection constraints are much weaker. This complementarity makes the
collider searches for DM very promising. With unpolarized beams, the 3$\sigma$
reach for the SP-type operator can be up to 5.4 TeV, while for the VA- and
TAT-type operators, it can be up to 5.8 TeV. With optimally polarized beams,
i.e., with $(+80\%,+30\%)$ for the SP- and TAT-types and $(+80\%,-30\%)$ for
the VA type, the sensitivity reach can be extended to 6.6 (SP), 8.8 (VA), and
7.1 TeV (TAT), as shown in Fig. 5. This increase in sensitivity results mainly
from the background suppression due to the beam polarization. But the
systematic uncertainties in background estimation play a crucial role here. It
is clearly visible from Fig. 5 that the unpolarized case suffers the most due
to the systematics, whereas the mixed polarization case is affected the least.
As a result, we see that although the optimal polarization choice gives the
best sensitivities without systematics, with systematic uncertainties included
the mixed polarization case of the H20 scenario performs better. The latter
gives $3\sigma$ sensitivities up to 4.6, 6.0, and 5.0 TeV for SP-, VA-, TAT-
type operator, respectively, as compared to 3.4, 6.0, and 4.7 TeV in the
optimal polarization case.111In Refs. [138, 114], it was shown that with a
better treatment of the systematic uncertainties, their effect on the final
sensitivity is much less ($\sim 10\%$) in the case of H20 mixed polarization
scenario unlike in our case where it reduces the $\Lambda$-scale sensitivity
by up to $\sim 40\%\text{-}50\%$. This is mainly due to our simplified
approach [cf. Eq. (13) and subsequent discussion] which results in a very
conservative estimate. Even then, we find the collider sensitivities to be
comparable to or better than the existing limits from direct and indirect
detection searches, which makes ILC a promising tool for leptophilic DM.

Now that we have the results for the SP-, VA-, and TAT-type assuming all the
relevant coefficients in Eq. (2) to be unity, it is instructive to see how the
sensitivity reach differs for other choices of the coefficients. This is shown
in Fig. 6 for a fixed DM mass of 1 GeV. The green, blue and red bars show the
$3\sigma$ ILC sensitivity with unpolarized, optimally polarized beams and
mixed polarization case, respectively, whereas the lighter (darker) shade
corresponds to without (with) background systematics. The optimal choice of
polarization is in general the $(+80\%,+30\%)$ with exceptions of V+A-, V-,
A-type operators with $(+80\%,-30\%)$ and V$-$A with $(-80\%,+30\%)$. Here
S+P, V+A, and T+AT refer to our default choice with all relevant coefficients
equal to 1, whereas S$-$P refers to the case with $c_{S}^{\chi}=c_{S}^{e}=1$
and $c_{P}^{\chi}=c_{P}^{e}=-1$ (similarly for V$-$A and T$-$AT), whereas pure
S type (P type) means the pseudoscalar (scalar) coefficients are zero (and
similarly for pure vector, axial-vector, tensor, and axial-tensor cases). We
see that with the unpolarized beams, the best sensitivity is obtained for the
T$\pm$AT operators, while with the optimally polarized beams as well as mixed
polarization case, the V+A operator gives the best sensitivity.

In any case, all the operator types can be probed up to multi-TeV cutoff
scales at $\sqrt{s}=1$ TeV with ${\cal L}_{\rm int}=8$ ab-1 integrated
luminosity.

##  IV Mono-$Z$ channel

In addition to the monophoton channel discussed in the previous section,
another useful channel for leptophilic DM search at lepton colliders is the
mono-$Z$ channel, where the $Z$ boson is emitted from one of the initial
states as shown in the Feynman diagram in Fig. 7.

In Fig. 8, we show the variations of the mono-$Z$ production cross section for
the signal as a function of the DM mass and the cutoff scale in the left and
right panels, respectively, for all three operator types, namely, SP (solid),
VA (dashed) and TAT (dotted) types. We find that the unpolarized cross section
is the smallest (largest) for the VA (TAT)-type operator at any given DM mass.
In the left panel, the cross section drops rapidly as $m_{\chi}$ approaches
$\sqrt{s}/2-m_{Z}$ due to phase-space suppression. Otherwise, for smaller DM
masses, the cross section for a given operator type and a given cutoff scale
is almost independent of the DM mass, like in the monophoton case (cf. Fig.
3). In the right panel, we see that for a given DM mass the cross section
drops as $\Lambda^{-4}$, as expected. For subsequent decay of the $Z$ boson
the nature of the graphs remains the same, and only the numbers are shifted as
per the branching fractions to different channels. In the following, we
discuss and compare how the leptonic and hadronic decay channels perform as a
probe of the fermionic DM scenario, in comparison to the monophoton case
discussed above.

Figure 7:  Feynman diagram for the mono-$Z$ channel. The $Z$ boson can also be
emitted from the positron leg. Depending on the decay mode the particle $f$
can be leptons ($\ell^{\pm}\equiv e^{\pm},\mu^{\pm}$) as well as jets
($j\equiv u,c,d,s,b$).

Figure 8:  Variation of the mono-$Z$ production cross section with the DM mass
(left) and cutoff scale (right) at $\sqrt{s}=1$ TeV ILC. The solid, dashed and
dotted lines are for the SP-, VA- and TAT-type operators, respectively. In the
left panel, the red, green and blue curves correspond to different values of
the cutoff scale $\Lambda=1$, 3, and 5 TeV respectively, while in the right
panel, they correspond to different values of the DM mass $m_{\chi}=$100, 250,
and 450 GeV, respectively.

###  IV.1 Leptonic mode

For the leptonic decay of the $Z$ boson, we examine the process
$e^{+}e^{-}\to\chi\overline{\chi}Z(\to\ell^{-}\ell^{+})$, as shown in Fig. 7.
We will only consider $\ell=e,\mu$ for simplicity and use the lepton pair as
the visible particles for tagging. The main SM background for this channel is
$e^{+}e^{-}\to\nu\overline{\nu}\;\ell^{+}\ell^{-}$, and it is polarization
dependent.

####  IV.1.1 Unpolarized and polarized cross sections

For the signal and background simulation, we generated the UFO library for our
EFT framework using FeynRules [134] and then generated events for both signal
and background using MadGraph5 [145], with the following basic baseline cuts
to the parameter space:

$p_{T}(\ell)>2\text{ GeV},\;\;\;\;|\eta_{\ell}|\leq 3.5,\;\;\;\;\Delta
R_{\ell\ell}\geq 0.2\,.$ (16)

For the signal, the $Z$ bosons are decayed into the charged lepton pairs via
the MadSpin [146, 147] package, which is implemented in MadGraph5, to take
care of the spin-correlation effects of the lepton pairs. A fast detector
simulation to these events is done using Delphes3 [137] with the same
configuration card (ILCgen) as discussed in subsection III.1.

Similar to the monophoton case, we also examine the effect of polarization on
the signal and background cross sections, as shown in Table 5. The neutrino
background can be reduced to 28% of its original value by making the electron
beam $+80\%$ polarized and further reduced to 21% of its original value by
additionally making the positron beam $-30\%$ polarized. The $(+80\%,-30\%)$
polarization configuration also enhances the VA-type signal by a factor of
2.4. However, the $(+80\%,+30\%)$ configuration is better for the SP- and TAT-
type signals. For ease of comparison between different operator types, we
choose to work with the $(+80\%,+30\%)$ configuration democratically for all
operator types, as well as for the background, unless otherwise specified.

Table 5: Comparison of the leptonic mono-$Z$ background and signal cross sections for different choices of beam polarization for $m_{\chi}=100\text{ GeV}$ and $\Lambda=3\text{ TeV}$ at $\sqrt{s}=1\text{ TeV}$ ILC. The numbers in bold highlight the optimal polarization choice for a given operator type. |  |  | Polarized cross section (fb)
---|---|---|---
Process type | Unpolarized cross section (fb) | Polarization $P(e^{-},e^{+})$ | $(+,+)$ | $(+,-)$ | $(-,+)$ | $(-,-)$
$\nu\overline{\nu}\ell^{-}\ell^{+}$ |  | $(80,0)$ |  $2.60\times 10^{2}$1 |  1$2.60\times 10^{2}$1 |  1$1.26\times 10^{3}$1 |  1$1.26\times 10^{3}$
$7.66\times 10^{2}$ | $(80,20)$ |  $3.01\times 10^{2}$1 |  1$2.21\times 10^{2}$1 |  1$1.48\times 10^{3}$1 |  1$1.06\times 10^{3}$
| $(80,30)$ |  $3.22\times 10^{2}$1 |  1$2.01\times 10^{2}$1 |  1$1.59\times 10^{3}$1 |  1$9.53\times 10^{2}$
SP |  | (80, 0) | $0.28$ | $0.28$ | $0.28$ | $0.28$
$0.28$ | (80,20) | $0.32$ | $0.23$ | $0.23$ | $0.32$
| $(80,30)$ | $0.35$ | $0.21$ | $0.21$ | $0.34$
VA |  | (80, 0) | $0.15$ | $0.15$ | $0.02$ | $0.02$
$0.08$ | (80,20) | $0.12$ | $0.18$ | $0.01$ | $0.02$
| $(80,30)$ | $0.10$ | $0.19$ | $0.01$ | $0.02$
TAT |  | (80, 0) | $0.67$ | $0.67$ | $0.68$ | $0.68$
$0.68$ | (80,20) | $0.79$ | $0.57$ | $0.57$ | $0.79$
| $(80,30)$ | $0.84$ | $0.51$ | $0.51$ | $0.84$

####  IV.1.2 Cut-based analysis

Now we proceed with our cut-based analysis to enhance the signal-to-background
ratio.

##### Baseline selection cuts:

We define our signals by those events that pass through the baseline selection
criteria as defined below where the $Z$ boson is reconstructed by the
condition that all final state lepton-pairs are oppositely charged and of same
flavor (OSSF).

$p_{T,\ell}>5\text{ GeV, }\;\;\;|\eta_{\ell}|<2.8\,$ (17)

Other selection criteria are dynamic with respect to different BPs, as defined
in Table 6. We have taken the same three BPs as in the monophoton case to
probe different regions of the parameter space, namely, BP1 essentially
represents all light DM region, BP3 represents the region close to the
kinematic limit of $\sqrt{s}/2-m_{Z}$, whereas BP2 captures the intermediate
DM mass region.

After implementing the baseline selection criteria given in Eq. (17), the
signal and background events are given in the first row of Table 7. We find
that the background is reduced to about $31\%$ of its original value in Table
5 for the unpolarized case, whereas the signals are reduced to about 60%-80%
of their original values across VA-type to TAT-type operator.

We now consider various kinematic distributions for the signal and background,
as shown in Fig. 9 for the unpolarized case. Based on these histograms, some
specialized selection cuts are chosen as follows (also summarized in Table 6)
which enhance the signal significance:

Table 6: Leptonic mono-$Z$ selection cuts for different BPs across operator types. | BP1 | BP2 | BP3
---|---|---|---
Definition |  $m_{\chi}=100\text{ GeV, }$ $\Lambda=3\text{ TeV}$ |  $m_{\chi}=250\text{ GeV, }$ $\Lambda=3\text{ TeV}$ |  $m_{\chi}=350\text{ GeV, }$ $\Lambda=3\text{ TeV}$
Baseline selection | OSSF lepton-pairs with $\;\;p_{T,\ell}>5\text{ GeV, }\;\;\;|\eta_{\ell}|<2.8$
SP type
Cut 1 | $80\text{ GeV}\leq M_{\rm inv}(\ell^{-}\ell^{+})\leq 100\text{ GeV}$
Cut 2 |  $540\text{ GeV}<\not{E}<780\text{ GeV}$00 |  0$620\text{ GeV}<\not{E}<850\text{ GeV}$00 |  0$740\text{ GeV}<\not{E}<900\text{ GeV}$
Cut 3 | $\cos\theta_{\rm miss}<0.79$
Cut 4 | $\Delta{R}_{\ell\ell}<1.1\text{ rad}$ | $\Delta{R}_{\ell\ell}<1.3\text{ rad}$ | $\Delta{R}_{\ell\ell}<1.6\text{ rad}$
VA type
Cut 1 | $80\text{ GeV}\leq M_{\rm inv}(\ell^{-}\ell^{+})\leq 100\text{ GeV}$
Cut 2 | $610\text{ GeV}<\not{E}$ | $650\text{ GeV}<\not{E}$ | $750\text{ GeV}<\not{E}$
Cut 3 | $\cos{\theta_{\rm miss}}<0.95$
TAT type
Cut 1 | $80\text{ GeV}\leq M_{\rm inv}(\ell^{-}\ell^{+})\leq 100\text{ GeV}$
Cut 2) | $520\text{ GeV}<\not{E}<800\text{ GeV}$ | $620\text{ GeV}<\not{E}<780\text{ GeV}$ | $740\text{ GeV}<\not{E}<930\text{ GeV}$
Cut 3 | $\cos\theta_{\rm miss}<0.8$
Cut 4 | $\Delta{R}_{\ell\ell}<1.3\text{ rad}$ | $\Delta{R}_{\ell\ell}<1.3\text{ rad}$ | $\Delta{R}_{\ell\ell}<1.5\text{ rad}$

Figure 9: Normalized differential distributions for leptonic mono-$Z$ background (shaded) and signals after baseline selection cuts for the kinematic variables shown in Table 6 with unpolarized beams. The corresponding distributions with polarized beams are given in Fig. 16 in the Appendix. Table 7: Cut-flow table for different operators and BPs for the leptonic mono-$Z$ signal and background. The cut efficiencies are calculated with respect to the baseline selection cuts given in Eq. (17), with the corresponding event numbers shown in the first row. The background event yields vary across operator types because of the dynamic nature of the cuts chosen (cf. Table 6).  | Event Numbers (Cut efficiencies)
---|---
| $\nu\bar{\nu}\ell^{+}\ell^{-}$ | Signal
Selection cuts | BP1 | BP2 | BP3 | BP1 | BP2 | BP3
SP-type:
Baseline selection | $1.92\times 10^{6}$ | $1.60\times 10^{3}$ | $7.14\times 10^{2}$ | $1.71\times 10^{2}$
$(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $2.72\times 10^{4}$ | $4.37\times 10^{4}$ | $4.77\times 10^{4}$ | $7.79\times 10^{2}$ | $4.05\times 10^{2}$ | $9.7\times 10^{1}$
$(1.42\%)$ | $(2.28\%)$ | $(2.49\%)$ | $(48.72\%)$ | $(56.72\%)$ | $(56.73\%)$
VA-type:
Baseline selection | $1.92\times 10^{6}$ | $5.14\times 10^{2}$ | $3.43\times 10^{2}$ | $1.64\times 10^{2}$
$(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $2.75\times 10^{5}$ | $2.64\times 10^{5}$ | $2.15\times 10^{5}$ | $3.98\times 10^{2}$ | $2.76\times 10^{2}$ | $1.38\times 10^{2}$
$(14.36\%)$ | $(13.79\%)$ | $(11.23\%)$ | $(77.43\%)$ | $(80.47\%)$ | $(84.15\%)$
TAT-type:
Baseline selection | $1.92\times 10^{6}$ | $3.54\times 10^{3}$ | $2.07\times 10^{3}$ | $5.99\times 10^{2}$
$(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $4.10\times 10^{4}$ | $3.00\times 10^{4}$ | $4.06\times 10^{4}$ | $2.30\times 10^{3}$ | $1.10\times 10^{3}$ | $3.29\times 10^{2}$
$(2.14\%)$ | $(1.56\%)$ | $(2.12\%)$ | $(65.03\%)$ | $(53.11\%)$ | $(54.92\%)$

Cut 1: For the $\nu\overline{\nu}\ell^{-}\ell^{+}$ background, the lepton
pairs could come from different channels (such as from the $t$-channel $W$ and
$Z$ boson exchange, in addition to the on/off shell $WW$ and $ZZ$ decays), the
invariant mass of the lepton pairs are distributed over a wide range. But for
the signals, the lepton pairs come as decay products of the on shell $Z$ boson
giving a sharp peak at the $Z$ boson mass, as is evident in Fig. 9 (top row).
Thus, selecting events that fall within a narrow window of dilepton invariant
mass around the $Z$ mass, i.e. $M_{\rm
inv}(\ell^{+}\ell^{-})\in[80,\,100]\text{ GeV}$, suppresses the background
significantly (to 22% of the baseline value with the unpolarized beams) while
keeping most of the signals ($\sim 94\%$).

Cut 2: The next variable is the missing energy which is reconstructed using
the initial and detected visible 4-momentum of the events. It is basically the
energy of the 4-momentum, $\mathbf{p}_{\rm miss}=\mathbf{p}_{\rm
initial}-\mathbf{p}_{\ell^{+}\ell^{-}}$, where $\mathbf{p}_{\rm
initial}=(1000,0,0,0)$ GeV (neglecting the 14 mrad crossing angle since
Delphes does not consider it) and $\mathbf{p}_{\ell^{+}\ell^{-}}$ is the
momentum of the dielectron system. We see that as the DM mass increases the
curve gets narrower with the peak shifting toward the total CM energy, while
for the lighter DM it is more spread. So we use a dynamic cut across the
benchmark points.

Cut 3: The third variable is the cosine of the missing polar angle,
$\theta_{\rm miss}$. This variable appears to be very effective since the
heavier candidates of the missing momentum tend to scatter at a larger angle
than the SM ones. We see that this cut reduces the backgrounds to $\sim 2.5\%$
for SP- and TAT-type operators to $\sim 11\%\text{-}14\%$ for the VA-type
operators while keeping most of the signal intact.

Cut 4: Our final selection cut is on the distance of the two leptons in the
$(\eta,\phi)$, defined as $R=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$, where
$\Delta\eta\;(\Delta\phi)$ is the difference in pseudorapidity (azimuthal
angle) of the two leptons. The lepton pairs from the $Z$ boson decay are more
collinear than the ones in the SM case. So a strict cut on this variable
enhances the signal significance. We do not use this cut for the VA-type
operator since (as evident from the distribution) it is not sensitive for this
particular case. The cut efficiencies and the selected events after this cut
are reported in Table 7. The corresponding number for the case of optimally
polarised beams are given in Table 15 in the Appendix.

In Table 7, we tabulate the number of events and efficiencies after baseline
selection and final selection cut for both background as well as signal for
different operators. We find that, after applying cuts 1–4, we can still
retain about $48\%-65\%$ of the signal (77%-84% for VA-type), whereas the
background is reduced to below percent level (3.5% in the case of VA-type
operator) of the original values given in Table 5.

Table 8: Signal significance in the mono-$Z$ leptonic channel at $\sqrt{s}=1$ TeV and $\mathcal{L}_{\rm int}=8\,{\rm ab}^{-1}$ after implementing all the cuts mentioned in the text. The values in the parentheses denote the significances with a 1% background systematic uncertainty.  | Signal significance for ${\cal L}_{\rm int}=8\,{\rm ab}^{-1}$
---|---
| Unpolarized beams | H20 scenario | Optimally polarized beams
Operator Type | BP1 | BP2 | BP3 | BP1 | BP2 | BP3 | BP1 | BP2 | BP3
SP | $4.7\;(2.4)$ | $1.9\;(0.8)$ | $0.4\;(0.2)$ | $6.7\;(6.2)$ | $2.9\;(2.6)$ | $0.7\;(0.6)$ | $10.6\;(8.2)$ | $4.5\;(3.1)$ | $1.1\;(0.7)$
VA | $0.8\;(0.14)$ | $0.5\;(0.10)$ | $0.3\;(0.06)$ | $3.0\;(1.9)$ | $2.1\;(1.4)$ | $1.9\;(0.8)$ | $4.6\;(2.1)$ | $3.3\;(1.5)$ | $1.8\;(0.9)$
TAT | $11.1\;(5.0)$ | $6.2\;(3.2)$ | $1.6\;(0.7)$ | $15.3\;(13.8)$ | $8.9\;(8.2)$ | $2.4\;(2.2)$ | $24.1\;(17.6)$ | $14.0\;(10.7)$ | $3.9\;(2.7)$

Figure 10: $3\sigma$ sensitivity contours in the mono-$Z$ leptonic channel for
the SP-type (top left panel), VA-type (top right panel) and TAT-type (bottom
panel) operators with unpolarized (green lines), mixed polarized (red lines)
and optimally polarized (blue lines) $e^{+}e^{-}$ beams at $\sqrt{s}=1$ TeV
center-of-mass energy and with ${\cal L}_{\rm int}=8$ ab-1 integrated
luminosity. The solid (dashed) contours are assuming zero (1%) background
systematics. The various shaded regions are excluded by direct detection
(XENON1T, PANDAX-4T), indirect detection (Fermi-LAT, AMS), astrophysics
(SN1987A) and cosmology (CMB) constraints. In the shaded region below
$\Lambda={\rm max}\\{\sqrt{s},3m_{\chi}\\}$, our EFT framework is not valid.
Along the dot-dashed line, the observed DM relic density is reproduced for a
thermal WIMP assuming only DM-electron effective coupling.

####  IV.1.3 Results

After implementing all these cuts, we calculate the final signal significance
for the three BPs using Eq. (12). Our results are given in Table 8 for an
integrated luminosity of ${\cal L}_{\rm int}=8$ ab-1. We see that as we go
higher up in the DM mass the signal significance drops. We also find that the
best-performing operator type is the TAT-type, for which more than $97\%$ of
the background events are removed after all the selection cuts. For the signal
we retain $54\%$–$65\%$ of the events. We also analyzed the effect of beam
polarization, in particular, the mixed polarization as per the H20 operating
scenario. We note, unlike the monophoton case, the optimal polarization case
gives the best significance both with and without systematic uncertainty. The
systematic uncertainty affects more if the background is large. Compared to
monophoton case, the background here is not large enough and hence its effect
on the signal-to-background ratio is relatively small, resulting in less
suppression of the significance for the optimally polarized case.
Nevertheless, we observe positive effect of beam polarization with excess of 2
times increase in significance across the two cases discussed.

Figure 11: $3\sigma$ sensitivity reach of different operators in the leptonic
mono-$Z$ channel with DM mass of $1$ GeV at $\sqrt{s}=1$ TeV and
$\mathcal{L}_{\rm int}=8\,{\rm ab}^{-1}$. The green, red, and blue bars show
the sensitivities with unpolarized, mixed and optimally polarized beams
respectively and the lighter (darker) shade corresponds to zero (1%)
background systematics.

Going beyond the three BPs, we now vary the DM mass and present the $3\sigma$
sensitivity reach for this channel in Fig. 10 for all the operators. The
labels and shaded regions are the same as in the monophoton case (cf. Fig. 5).
We see that the accessible range of the cutoff scale $\Lambda$ for the
unpolarized beams can reach up to 4.2 TeV for the TAT-type operator, whereas
for the SP and VA-type, it can reach up to 3.5 and 1.7 TeV, respectively. But
with the application of optimally polarized beams as discussed earlier, we see
an increase by about $20\%\text{--}25\%\;(55\%)$ of the $3\sigma$ reach on the
$\Lambda$ scale of SP and TAT-type (VA-type), up to 4.3, 2.7, and 5.2 TeV for
the for SP, VA, and TAT-type operators, respectively. With the mixed
polarization case we obtain 3.8, 2.4, and 4.7 TeV for the three operators,
respectively.

We also compare the leptonic mono-$Z$ sensitivities for different operator
types in Fig. 11 for a fixed DM mass of 1 GeV. As in Fig. 6 for the monophoton
case, the green, blue and red bars show the $3\sigma$ sensitivity with
unpolarized, optimally polarized and mixed polarization cases and the lighter
(darker) shade corresponds to zero ($1\%$) background systematics. We again
find the best sensitivity for T$\pm$ AT operators, and the S$\pm$ P operators
also perform well, while, unlike the monophoton case, the vector- and axial-
vector-type operators have the weakest sensitivity in this channel mainly
because of its similarity to the background in the event distributions. Also,
for the reasons discussed above the optimal polarization case does best even
with background systematics.

###  IV.2 Hadronic mode

Now we look at the process $e^{+}e^{-}\to\chi\overline{\chi}Z(\to jj)$, where
$j\equiv u,d,c,s,b$ quarks (as shown in Fig. 7). The relevant SM background
processes for this channel are $e^{+}e^{-}\to\nu\overline{\nu}jj$ and
$e^{+}e^{-}\to jj\ell\nu$ (with one charged lepton escaping the detector)
where the jets and leptons in the final state can come from any possible
source (not necessarily from an on shell $Z$).

####  IV.2.1 Unpolarized and polarized cross sections

We use the same UFO library as before which is implemented using FeynRules
[134] and simulate the events for the signal and backgrounds via MadGraph5
[145] with the following basic cuts to the parameter space:

$\displaystyle p_{T}(j)>5\text{ GeV},\;\;\;\;p_{T}(\ell)>2\text{ GeV},$
$\displaystyle|\eta_{j}|\leq 3.2,\;\;\;|\eta_{\ell}|\leq 3.5,\;\;\;\Delta
R_{jj,\ell j}\geq 0.2\,.$ (18)

For the signals, as in the leptonic case, the on shell $Z$ bosons are decayed
into the pairs of jets using the MadSpin package [146, 147], implemented in
MadGraph5. Both the signal and background samples are hadronized using
Pythia8.3 [148] and then the fast detector simulation is done using Delphes3.5
[137] with the same configuration card as discussed in subsection III.1. The
particle flow objects built with the reconstructed tracks and calorimeter
towers and rejected to be isolated leptons and photons are clustered with the
Valencia algorithm for linear colliders (usually referred to as the VLC
algorithm) with $R=2,\;\beta=1,\;\gamma=0$ which mimics the Durham
(ee_kt_algorithm) algorithm. It is important to mention here that a beam-
induced background from $\gamma\gamma\to{\rm hadrons}$ [149], where the
photons are radiated from incoming beams (bremsstrahlung and beamstrahlung),
are relevant in linear colliders and have been discussed in the full
simulation studies. To consider that, one has to externally generate the
events and overlay with detector simulation (as possibly pileup events) [150].
At the ILC, with larger bunch spacing ($\sim 500~{}{\rm ns}$), individual
bunch crossings are better distinguishable, thus minimizing this background
[150]; hence it will not change our results much.

We then examine different choices of beam polarization on both the event
samples for this channel, as shown in Table 9. We find that both the
backgrounds are polarization dependent and fall off significantly for a right-
handed electron beam and with increasing degree of polarization for a left-
handed positron beam. We observe as earlier that the optimal polarization
choice is $P(e^{-},e^{+})=(+80\%,+30\%)$ for the SP- and TAT-type operators
and $(+80\%,-30\%)$ for the VA-type operators.

####  IV.2.2 Cut-based analysis

After obtaining the signal and background cross sections as reported in Table
9, we proceed with our cut-based analysis to optimize the signal significance,
as follows.

Table 9: Comparison of the hadronic mono-$Z$ signal and background cross sections for different choices of beam polarization with $m_{\chi}=100~{}\rm{GeV}$ and $\Lambda=3~{}\rm{TeV}$ at $\sqrt{s}=1~{}\rm{TeV}$ ILC. The numbers in bold highlight the optimal polarization choice for a given operator type. |  |  | Polarized cross section (fb)
---|---|---|---
Process type | Unpolarized cross section (fb) | Polarized $P(e^{-},e^{+})$ | $(+,+)$ | $(+,-)$ | $(-,+)$ | $(-,-)$
|  | $(80,0)$ |  $1.73\times 10^{2}$1 |  1$1.73\times 10^{2}$1 |  1$1.41\times 10^{3}$1 |  1$1.41\times 10^{3}$
$\nu\overline{\nu}jj$ | $8.39\times 10^{2}$ | $(80,20)$ |  $1.99\times 10^{2}$1 |  1$1.47\times 10^{2}$1 |  1$1.68\times 10^{3}$1 |  1$1.08\times 10^{3}$
|  | $(80,30)$ |  $2.31\times 10^{2}$1 |  1$1.47\times 10^{2}$1 |  1$1.93\times 10^{3}$1 |  1$1.04\times 10^{3}$
|  | $(80,0)$ |  $3.26\times 10^{2}$1 |  1$3.26\times 10^{2}$1 |  1$2.22\times 10^{3}$1 |  1$2.22\times 10^{3}$
$\ell\nu jj$ | $2.14\times 10^{3}$ | $(80,20)$ |  $3.85\times 10^{2}$1 |  1$2.66\times 10^{2}$1 |  1$2.65\times 10^{3}$1 |  1$1.83\times 10^{3}$
|  | $(80,30)$ |  $8.39\times 10^{2}$1 |  1$4.83\times 10^{2}$1 |  1$4.56\times 10^{3}$1 |  1$2.69\times 10^{3}$
|  | $(80,0)$ | 2.77 | 2.77 | 2.78 | 2.78
SP | $2.78$ | $(80,20)$ | 3.23 | 2.34 | 2.34 | 3.22
|  | $(80,30)$ | 3.45 | 2.11 | 2.11 | 3.44
|  | $(80,0)$ | 1.50 | 1.50 | 0.17 | 0.17
VA | $0.83$ | $(80,20)$ | 1.20 | 1.79 | 0.13 | 0.20
|  | $(80,30)$ | 1.05 | 1.95 | 0.12 | 0.22
|  | $(80,0)$ | 6.74 | 6.74 | 6.78 | 6.21
TAT | $6.77$ | $(80,20)$ | 7.86 | 5.70 | 5.70 | 7.88
|  | $(80,30)$ | 8.40 | 5.14 | 5.14 | 8.38

Baseline selection cuts: We select the events that contain at least two jets
with the following transverse momentum and pseudorapidity requirements:

$p_{T,j}>10\text{ GeV, }\;\;\;\;|\eta_{j}|<2.6\,,$ (19)

and do not have any isolated lepton and photon, i.e., electromagnetic (EM)
particles, with $p_{T}>5$ GeV and $|\eta|<2.8$. The two highest $p_{T}$ jets
are required to reconstruct the $Z$ boson. Further selection cuts are applied,
some of which depend on the DM mass. So, as in the leptonic channel, we have
taken the same three BPs with varying DM mass and imposed dynamic cuts. The
benchmark parameters and the different selection cuts are defined in Table 10.
The number of signal and background events after implementing the baseline
selection criteria are given in the first row of Table 11.

Table 10: Different BPs and selection cuts across operator types for mono-$Z$ hadronic channel. | BP1 | BP2 | BP3
---|---|---|---
Definition |  $m_{\chi}=100\text{ GeV, }$ $\Lambda=3\text{ TeV}$ |  $m_{\chi}=250\text{ GeV, }$ $\Lambda=3\text{ TeV}$ |  $m_{\chi}=350\text{ GeV, }$ $\Lambda=3\text{ TeV}$
Baseline selection | $\;\;p_{T,j}>10\text{ GeV, }\;\;\;|\eta_{j}|<2.6,\;\text{no EM particles}\;$
SP type
Cut 1 | $85\text{ GeV}\leq M_{\rm inv}(jj)\leq 100\text{ GeV}$
Cut 2 | $510\text{ GeV}<\not{E}<760\text{ GeV}\phantom{0}$ |  $\phantom{0}610\text{ GeV}<\not{E}<840\text{ GeV}$0 |  0$710\text{ GeV}<\not{E}<860\text{ GeV}$
Cut 3 | $|\cos\theta_{\rm miss}|<0.79$
Cut 4 | $\Delta R_{jj}<1.3\text{ rad}$ | $\Delta R_{jj}<1.4\text{ rad}$ | $\Delta R_{jj}<1.5\text{ rad}$
VA type
Cut 1 | $85\text{ GeV}\leq M_{\rm inv}(jj)\leq 100\text{ GeV}$
Cut 2 | $590\text{ GeV}<\not{E}$ | $690\text{ GeV}<\not{E}$ | $750\text{ GeV}<\not{E}$
Cut 3 | $|\cos\theta_{\rm miss}|<0.9$
TAT type
Cut 1 | $85\text{ GeV}\leq M_{\rm inv}(jj)\leq 100\text{ GeV}$
Cut 2 |  $510\text{ GeV}<\not{E}<770\text{ GeV}$00 |  0$610\text{ GeV}<\not{E}<780\text{ GeV}$00 |  0$720\text{ GeV}<\not{E}<850\text{ GeV}$
Cut 3 | $|\cos\theta_{\rm miss}|<0.79$
Cut 4 | $\Delta R_{jj}<1.3\text{ rad}$ | $\Delta R_{jj}<1.4\text{ rad}$ | $\Delta R_{jj}<1.6\text{ rad}$

Figure 12: Normalized differential distributions for hadronic mono-$Z$ backgrounds (shaded) and signals after baseline selection cuts for the kinematic variables shown in Table 10 with unpolarized beams. The corresponding distributions for polarized beams are given in Fig. 17 in the Appendix. Table 11: Cut-flow chart for different operators and BPs for the mono-$Z$ hadronic signal and backgrounds at $\sqrt{s}=1$ TeV and $\mathcal{L}_{\rm int}=8\,{\rm ab}^{-1}$. The cut efficiencies are calculated with respect to Eq. (19), with the corresponding event numbers shown in the first row. The background cut efficiencies vary across operator types because of the dynamic nature of the cuts chosen (cf. Table 10). | Event Numbers (Cut efficiencies)
---|---
| $\nu_{\ell}\overline{\nu_{\ell}}jj$ | $\ell\nu_{\ell}jj$ | Signal
Selection cuts | BP1 | BP2 | BP3 | BP1 | BP2 | BP3 | BP1 | BP2 | BP3
SP type:
BS | $6.05\times 10^{6}$ | $4.07\times 10^{6}$ | $2.16\times 10^{4}$ | $1.32\times 10^{4}$ | $2.13\times 10^{3}$
| $(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $3.16\times 10^{5}$ | $6.09\times 10^{5}$ | $5.86\times 10^{5}$ | $1.59\times 10^{4}$ | $4.13\times 10^{4}$ | $4.95\times 10^{4}$ | $8.62\times 10^{3}$ | $4.57\times 10^{3}$ | $0.97\times 10^{3}$
$(5.23\%)$ | $(10.08\%)$ | $(9.70\%)$ | $(0.39\%)$ | $(1.01\%)$ | $(1.22\%)$ | $(39.87\%)$ | $(34.60\%)$ | $(45.64\%)$
VA type:
BS | $6.05\times 10^{6}$ | $4.07\times 10^{6}$ | $6.12\times 10^{3}$ | $4.10\times 10^{3}$ | $1.97\times 10^{3}$
| $(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $2.30\times 10^{6}$ | $2.12\times 10^{6}$ | $1.88\times 10^{6}$ | $2.44\times 10^{5}$ | $2.36\times 10^{5}$ | $2.2\times 10^{5}$ | $3.38\times 10^{3}$ | $2.29\times 10^{3}$ | $1.20\times 10^{3}$
$(38.01\%)$ | $(35.04\%)$ | $(31.05\%)$ | $(6.00\%)$ | $(5.81\%)$ | $(5.48\%)$ | $(55.28\%)$ | $(55.93\%)$ | $(61.03\%)$
TAT type:
BS | $6.05\times 10^{6}$ | $4.07\times 10^{6}$ | $5.30\times 10^{4}$ | $2.67\times 10^{4}$ | $7.46\times 10^{3}$
| $(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $3.51\times 10^{5}$ | $3.59\times 10^{5}$ | $5.90\times 10^{5}$ | $1.8\times 10^{4}$ | $1.90\times 10^{4}$ | $4.72\times 10^{4}$ | $2.62\times 10^{4}$ | $1.15\times 10^{4}$ | $3.66\times 10^{3}$
$(5.81\%)$ | $(5.94\%)$ | $(9.76\%)$ | $(0.45\%)$ | $(0.47\%)$ | $(1.16\%)$ | $(49.37\%)$ | $(43.04\%)$ | $(49.12\%)$
Table 12: Signal significances of the mono-$Z$ hadronic channel at $\sqrt{s}=1~{}{\rm TeV}$ and $\mathcal{L}_{\rm int}=8~{}{\rm ab}^{-1}$ after implementing all analysis cuts mentioned in the text. The values in the parentheses denote the significance with a 1% background systematic uncertainty. | Signal significance for ${\cal L}_{\rm int}=8\,{\rm ab}^{-1}$
---|---
| Unpolarized beams | H20 scenario | Optimally polarized beams
Operator types | BP1 | BP2 | BP3 | BP1 | BP2 | BP3 | BP1 | BP2 | BP3
SP | $14.8\;(2.7)$ | $5.6\;(0.7)$ |  $1.2\;(0.2)$0 |  0$20.2\;(12.7)$ | $8.2\;(4.3)$ |  $1.8\;(1.0)$0 |  0$32.8\;(11.6)$ | $12.9\;(3.4)$ | $2.8\;(0.8)$
VA | $2.1\;(0.1)$ | $1.5\;(0.1)$ |  $0.8\;(0.1)$0 |  0$7.8\;(2.3)$ | $5.5\;(1.7)$ |  $3.1\;(1.0)$0 |  0$12.1\;(2.3)$ | $8.5\;(1.7)$ | $4.8\;(1.0)$
TAT | $41.6\;(7.3)$ | $18.4\;(3.1)$ |  $4.6\;(0.6)$0 |  0$54.7\;(34.7)$ | $26.3\;(16.5)$ |  $6.7\;(3.6)$0 |  0$87.0\;(31.6)$ | $41.4\;(14.2)$ | $10.5\;(2.8)$

Cut 1: The contribution to the $\nu\overline{\nu}jj$ background comes mainly
from the process $e^{+}e^{-}\to Z+\gamma/Z$, where the dijet comes from the
$\gamma/Z$ boson, whereas the $\ell\nu jj$ background comes largely from
$e^{+}e^{-}\to W^{+}W^{-}$, where the $W$ boson pair decays semileptonically.
So, by selecting a dijet-invariant mass around the $Z$ mass, i.e., $85\leq
M_{\rm inv}(jj)\leq 100\text{ GeV}$, the backgrounds are reduced considerably.
In particular, the $\ell\nu jj$ background with a peak around the $W$ boson
mass (see Fig. 12 top row) is reduced by $90\%$, whereas $42\%$ of the
$\nu\overline{\nu}jj$ background after baseline selection is rejected (cf.
Table 11). Cut 2: As was the case for the leptonic channel, the missing energy
($\not{E}$) distribution for the signal depends on the DM mass. As the mass of
the DM increases, the distribution of missing energy ($\not{E}$) becomes
narrower and shift toward the value of the c.m. energy. So we choose a dynamic
cut (with respect to the benchmark points) to efficiently select the signal
events and reject as many background events as possible (as shown in Table
10). We see that this reduces the $\nu\overline{\nu}jj$ background up to
$16\%$ and the $\ell\nu jj$ background up to $1.7\%$ of their respective
values after BS, varying with different operators, while keeping more than
$50\%$ of the signal events.

Cut 3: The cosine of the polar angle of the missing momentum is again very
efficient as a variable selection cut for pretty much the same reason as in
the leptonic channel, that heavier invisible particles tend to scatter at
larger angles with beam direction.

Cut 4: We put a cut on $\Delta R_{jj}$ to remove a significant amount of
events for both the backgrounds that are less collinear than the signals where
the jets come from on shell resonant decay of the $Z$ boson, as discussed for
the leptonic channel, and as shown in Fig. 13 (final row). We avoid using this
cut for the VA-type operator because, similar to the leptonic case, here also
it appears not to be sensitive (also evident from the distribution) after the
previous three cuts. In Table 11, we reported the number of selected events
and the selection efficiencies after the application of all the cuts for
unpolarised beams. The corresponding numbers for the polarised beams are in
Table 16 in the Appendix.

Figure 13: $3\sigma$ sensitivity contours in the mono-$Z$ hadronic channel.
The labels are the same as in Fig. 10.  Figure 14: $3\sigma$ sensitivity reach
of different operators in the mono-$Z$ hadronic channel with DM mass of $1$
GeV at $\sqrt{s}=1$ TeV and $\mathcal{L}_{\rm int}=8~{}{\rm ab}^{-1}$. The
green, red and blue bars show the sensitivities with unpolarized, mixed and
optimally polarized beams respectively and the lighter (darker) shade
corresponds to zero (1%) background systematics.

####  IV.2.3 Results

The signal significances calculated using Eq. (12) are tabulated in Table 12.
We see similar behavior for the different BPs as in the previously discussed
channels, i.e., enhanced signal significance with decreasing mass of the DM.
The selection cuts are most efficient for SP- and TAT-type operators. For
BP$1$ we remove more than $94\%$–$97\%$ of the background events, while
keeping more than $43\%$ of the signal events for the two operator types,
yielding a large signal significance, especially for the TAT-type operator
with polarized beams.

Varying the DM mass, we display the $3\sigma$ sensitivity contours for all
three operators in Fig. 13. It is clear that the TAT-type operator has the
best sensitivity, which reaches up to $5.5$ TeV with unpolarized beams and
$6.7$ TeV with optimally polarized beams. The SP-type operator also has a
sensitivity comparable to the monophoton channel and can reach up to
$4.4\;(5.5)$ TeV with unpolarized (optimally polarized) beams. The VA-type
operator has a modest sensitivity in this channel, only up to $2.7\;(4.2)$ TeV
with unpolarized (optimally polarized) beams. The mixed polarization case of
the H20 scenario also produces good numbers particularly with systematic
uncertainty outperforming the optimal polarization case with the $3\sigma$
reach of $\Lambda$ reaching up to $4.9$ ($4.0$), $3.8$ ($2.8$), and $6.0$
($5.0$), respectively, for SP-, VA-, and TAT-type operators without (with)
uncertainty in background estimation.

We also compare the hadronic mono-$Z$ sensitivities for different operator
types in Fig. 14 for a fixed DM mass of 1 GeV. The green, blue and red bars
show the $3\sigma$ sensitivity with unpolarized, optimal and mixed
polarization case, respectively, and the lighter (darker) shade corresponds to
zero (1%) background (both) systematics. We again find the best sensitivity
for T$\pm$AT operators, followed by the S$\pm$P operators, while the vector-
and axial-vector-type operators have the weakest sensitivity in this channel,
mainly because of its similarity to the background in the event distributions,
just as in the leptonic mono-$Z$ case. In general, the hadronic mono-$Z$
channel is more effective than the leptonic one, primarily because of a larger
signal cross section due to higher hadronic branching ratio (69.9%) of the $Z$
boson, as compared to its leptonic branching ratio to electrons and muons
(only 6.7%).

Table 13: Summary of our results for the $3\sigma$ sensitivity reach of the cutoff scale $\Lambda$ in the three different channels discussed in the text. Here we have fixed the DM mass at 1 GeV. The numbers in parentheses are with 1% background systematics. The numbers in bold show the highest $\Lambda$ value that can be probed for a given operator. |  |  $3\sigma$ sensitivity reach of $\Lambda$ (TeV)
---|---|---
Process type | Beam configuration0 | SP | VA | TAT
Mono-$\gamma$ | Unpolarized |  05.37 (3.15)0 |  05.78 (3.39)0 |  05.80 (3.41)
Optimal pol |  6.60 (3.38) |  8.76 (5.95) |  7.14 (4.73)
H20 scenario | 5.84 (4.60) | 7.86 (5.99) | 6.32 (4.97)
Mono-$Z$ leptonic | Unpolarized | 3.47 (2.90) | 1.73 (1.45) | 4.22 (3.53)
Optimal pol | 4.30 (3.98) | 2.69 (2.56) | 5.24 (4.85)
H20 scenario | 3.84 (3.75) | 2.41 (2.36) | 4.68 (4.57)
Mono-$Z$ hadronic | Unpolarized | 4.41 (2.58) | 2.74 (1.40) | 5.46 (3.18)
Optimal pol | 5.45 (3.77) | 4.24 (2.78) | 6.74 (4.65)
H20 scenario | 4.85 (4.02) | 3.80 (2.79) | 6.01 (4.96)

##  V Conclusion

Very little is known about the nature of the DM and its interaction with the
SM particles. It is possible to think of a scenario where DM only couples to
the SM leptons, but not to quarks at tree level. We have explored the physics
potential of the future $e^{+}e^{-}$ colliders in probing such leptophilic DM
in a model-independent way. As a case study, we have taken the $\sqrt{s}=1$
TeV ILC with an integrated luminosity of $8$ ab-1 and have analyzed the pair-
production of fermionic DM using leptophilic dimension-six operators of all
possible bilinear structures, namely, scalar-pseudoscalar, vector-axial
vector, and tensor-axial tensor. We have performed a detailed cut-based
analysis for each of these operators in three different channels based on the
tagged particle, namely, monophoton, mono-$Z$ leptonic, and hadronic.

We have taken into account one of the most important and powerful features of
lepton colliders, i.e., the possibility of beam polarization with different
degrees of polarization and helicity orientations. We find that the ${\rm
sign}(P(e^{-}),P(e^{+}))=(+,+)$ beam configuration is optimal for the SP- and
TAT-type operators, while the $(+,-)$ configuration is better for probing the
VA-type operators. The maximum value of the cutoff scale $\Lambda$ that can be
probed in each channel at $3\sigma$ is given in Table 13. We find that the
monophoton channel provides the best sensitivity across all operator types
both with and without background systematics. The mono-$Z$ hadronic channel
also performs well particularly for the TAT-type operator with systematics. A
comparison between the sensitivity reaches of different operators can also be
seen from Figs. 6, 11, and 14. We also demonstrate the complementarity of our
lepton collider study with other existing direct and indirect detection
searches for leptophilic DM (cf. Figs. 5, 10, and 13). In particular, we show
that lepton colliders will be able to provide the best-ever sensitivity in the
still unexplored light DM regime.

## Acknowledgments

S.K. would like to thank Kajari Mazumdar for useful discussions. He would also
like to thank Tanumoy Mandal and Suman Kumar Kundu for the help with using
ROOT software package. P.K.D. thanks K. C. Kong and Partha Konar for the
discussions. B.D. would like to thank Juri Smirnov for a discussion on the
feebly interacting massive particle DM possibility. A.G. would like to thank
the Department of Physics, IISER Pune and Chungnam National University for
their support. The work of P.K.D. is partly supported by the SERB Grant No.
EMR/2016/002651. The work of B.D. is supported in part by the U.S. Department
of Energy under Award No. DE-SC0017987 and by a Fermilab Intensity Frontier
Fellowship. This work was partly performed at the Aspen Center for Physics,
which is supported by National Science Foundation Grant No. PHY-1607611. Work
of A.G. is supported by the National Research Foundation of Korea
(NRF-2019R1C1C1005073).

##  Appendix A KINEMATIC DISTRIBUTIONS AND CUT-FLOW TABLES FOR POLARIZED BEAMS

For completeness, we present in Figs. 15–17 the kinematic distributions with
optimally polarized beams for the same set of variables as shown in Figs. 4,
9, and 12 for the monophoton, mono-$Z$ leptonic, and hadronic channels,
respectively.

Figure 15: Same as in Fig. 4, but for optimally polarised beams.  Table 14: Same as Table 3, but for optimally polarised beams. | Event Numbers (Cut efficiencies)
---|---
| Neutrino pair | Radiative Bhabha | Signal
Selection cuts | BP1 | BP2 | BP3 | BP1 | BP2 | BP3 | BP1 | BP2 | BP3
SP type:
BS | $8.55\times 10^{6}$ | $6.87\times 10^{7}$ | $2.49\times 10^{5}$ | $1.79\times 10^{5}$ | $1.09\times 10^{5}$
$(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $4.70\times 10^{6}$ | $4.61\times 10^{6}$ | $4.50\times 10^{6}$ | $7.19\times 10^{5}$ | $7.19\times 10^{5}$ | $7.19\times 10^{5}$ | $1.52\times 10^{5}$ | $1.11\times 10^{5}$ | $6.84\times 10^{4}$
$(54.97\%)$ | $(53.88\%)$ |  $(52.60\%)$0 |  0$(1.05\%)$ | $(1.05\%)$ |  $(1.05\%)$0 |  0$(61.14\%)$ | $(62.10\%)$ | $(62.79\%)$
VA type:
BS | $5.41\times 10^{6}$ | $6.85\times 10^{7}$ | $6.32\times 10^{5}$ | $4.92\times 10^{5}$ | $3.30\times 10^{5}$
$(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $2.74\times 10^{6}$ | $2.63\times 10^{6}$ | $2.55\times 10^{6}$ | $6.42\times 10^{5}$ | $6.42\times 10^{5}$ | $6.42\times 10^{5}$ | $3.87\times 10^{5}$ | $3.05\times 10^{5}$ | $2.07\times 10^{5}$
$(50.72\%)$ | $(48.66\%)$ | $(47.18\%)$ | $(0.94\%)$ | $(0.94\%)$ | $(0.94\%)$ | $(61.22\%)$ | $(61.93\%)$ | $(62.78\%)$
TAT type:
BS | $8.55\times 10^{6}$ | $6.87\times 10^{7}$ | $3.56\times 10^{5}$ | $3.30\times 10^{5}$ | $2.65\times 10^{5}$
$(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $4.70\times 10^{6}$ | $4.61\times 10^{6}$ | $4.50\times 10^{6}$ | $7.19\times 10^{5}$ | $7.19\times 10^{5}$ | $7.19\times 10^{5}$ | $2.20\times 10^{5}$ | $2.04\times 10^{5}$ | $1.66\times 10^{5}$
$(54.97\%)$ | $(53.88\%)$ | $(52.60\%)$ | $(1.05\%)$ | $(1.05\%)$ | $(1.05\%)$ | $(61.73\%)$ | $(61.77\%)$ | $(62.52\%)$

Figure 16: Same as in Fig. 9, but for optimally polarised beams. Table 15: Same as Table 7, but for optimally polarised beams.  | Event Numbers (Cut efficiencies)
---|---
| $\nu\bar{\nu}\ell^{+}\ell^{-}$ | Signal
Selection cuts | BP1 | BP2 | BP3 | BP1 | BP2 | BP3
SP:
Baseline selection | $7.08\times 10^{5}$ | $1.99\times 10^{4}$ | $8.84\times 10^{2}$ | $2.13\times 10^{2}$
$(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $7.41\times 10^{3}$ | $1.16\times 10^{4}$ | $1.24\times 10^{4}$ | $9.70\times 10^{2}$ | $5.00\times 10^{2}$ | $1.21\times 10^{2}$
$(1.04\%)$ | $(1.64\%)$ |  $(1.75\%)$00 |  00$(48.87\%)$ | $(56.56\%)$ | $(56.81\%)$
VA:
Baseline selection | $4.63\times 10^{5}$ | $1.20\times 10^{3}$ | $8.05\times 10^{2}$ | $3.85\times 10^{2}$
$(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $3.97\times 10^{4}$ | $3.79\times 10^{4}$ | $3.08\times 10^{4}$ | $9.34\times 10^{2}$ | $6.49\times 10^{2}$ | $3.24\times 10^{2}$
$(8.59\%)$ | $(8.20\%)$ | $(6.66\%)$ | $(77.77\%)$ | $(80.62\%)$ | $(84.16\%)$
TAT:
Baseline selection | $7.08\times 10^{5}$ | $4.40\times 10^{3}$ | $2.57\times 10^{3}$ | $7.44\times 10^{2}$
$(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $1.12\times 10^{4}$ | $8.17\times 10^{3}$ | $1.06\times 10^{4}$ | $2.86\times 10^{3}$ | $1.36\times 10^{3}$ | $4.08\times 10^{2}$
$(1.58\%)$ | $(1.15\%)$ | $(1.49\%)$ | $(65.03\%)$ | $(52.98\%)$ | $(54.84\%)$

Figure 17: Same as in Fig. 12, but for optimally polarised beams. Table 16: Same as Table 11, but for optimally polarised beams. | Event Numbers (Cut efficiencies)
---|---
| $\nu_{\ell}\overline{\nu_{\ell}}jj$ | $\ell\nu_{\ell}jj$ | Signal
Selection cuts | BP1 | BP2 | BP3 | BP1 | BP2 | BP3 | BP1 | BP2 | BP3
SP type:
BS | $1.65\times 10^{6}$ | $2.07\times 10^{6}$ | $2.68\times 10^{4}$ | $1.13\times 10^{4}$ | $2.65\times 10^{3}$
$(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $8.54\times 10^{4}$ | $1.59\times 10^{5}$ | $1.53\times 10^{5}$ | $9.80\times 10^{3}$ | $2.54\times 10^{4}$ | $2.94\times 10^{4}$ | $1.07\times 10^{4}$ | $5.63\times 10^{3}$ | $1.19\times 10^{3}$
$(5.18\%)$ | $(9.65\%)$ |  $(9.27\%)$0 |  0$(0.47\%)$ | $(1.23\%)$ |  $(1.42\%)$0 |  0$(39.86\%)$ | $(49.90\%)$ | $(44.96\%)$
VA type:
BS | $1.01\times 10^{6}$ | $1.22\times 10^{6}$ | $1.43\times 10^{4}$ | $9.58\times 10^{3}$ | $4.60\times 10^{3}$
$(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $3.29\times 10^{5}$ | $3.01\times 10^{5}$ | $2.66\times 10^{5}$ | $8.60\times 10^{4}$ | $8.32\times 10^{4}$ | $7.81\times 10^{4}$ | $7.88\times 10^{3}$ | $5.33\times 10^{3}$ | $2.81\times 10^{3}$
$(32.44\%)$ | $(29.69\%)$ | $(26.23\%)$ | $(7.05\%)$ | $(6.82\%)$ | $(6.40\%)$ | $(55.10\%)$ | $(55.64\%)$ | $(61.17\%)$
TAT:
BS | $1.65\times 10^{6}$ | $2.07\times 10^{6}$ | $6.57\times 10^{4}$ | $3.31\times 10^{4}$ | $9.27\times 10^{3}$
$(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$ | $(100\%)$
After Final cut | $9.45\times 10^{4}$ | $9.44\times 10^{4}$ | $1.53\times 10^{5}$ | $1.12\times 10^{4}$ | $1.15\times 10^{4}$ | $2.86\times 10^{4}$ | $3.23\times 10^{4}$ | $1.44\times 10^{4}$ | $4.55\times 10^{3}$
$(5.73\%)$ | $(5.72\%)$ | $(9.30\%)$ | $(0.54\%)$ | $(0.55\%)$ | $(1.38\%)$ | $(49.21\%)$ | $(43.43\%)$ | $(48.99\%)$

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