aainstitutetext: Centro de Investigaciones, Universidad Antonio Nariño
Carrera 3 este # 47A-15, Bogotá, Colombiabbinstitutetext: Department of
Physics, Indian Institute of Technology Guwahati, Assam 781039,
Indiaccinstitutetext:  C. N. Yang Institute for Theoretical Physics and
Department of Physics and Astronomy,
Stony Brook University, Stony Brook, New York 11794, USA

# Catch ’em all: Effective Leptophilic WIMPs at the $e^{+}\,e^{-}$ Collider

Basabendu Barman b Subhaditya Bhattacharya c Sudhakantha Girmohanta b Sahabub
Jahedi<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider higher-dimensional effective operators consisting of fermion dark
matter (DM) connecting to Standard Model (SM) leptons upto dimension six.
Considering all operators together and assuming the DM to undergo thermal
freeze-out, we find out relic density allowed parameter space in terms of DM
mass and New Physics (NP) scale with one loop direct search constraints from
XENON1T experiment. Allowed parameter space of the model is probed at the
proposed International Linear Collider (ILC) via monophoton signal. We also
find out the precision with which the Wilson coefficients of the effective
operators can be extracted via optimal observable technique (OOT) and comment
on the separability of one model from another, assuming the presence of few
selected operators for each case.

###### Keywords:

Beyond the Standard Model, Dark Matter, $e^{+}e^{-}$ Experiments

PI/UAN-2021-700FT

##  1 Introduction

The existence of dark matter (DM) is motivated from several astrophysical
Zwicky:1933gu ; Zwicky:1937zza ; Rubin:1970zza ; Clowe:2006eq  and
cosmological Hu:2001bc ; Aghanim:2018eyx  evidences (for a review, see, e.g.
Refs. Jungman:1995df ; Bertone:2004pz ; Feng:2010gw ), although a laboratory
discovery is still awaited. Excepting some broad characteristics like
electromagnetic charge neutrality, stability over the Universe’s life time
etc., other properties like mass, spin, interactions (other than
gravitational) are still unknown. Anisotropy in the cosmic microwave
background (CMB) radiation provides the most precise measurement of the DM
relic density, usually expressed as $\Omega_{\tt DM}h^{2}\simeq 0.12$
Aghanim:2018eyx , where $\Omega$ refers to cosmological density and $h$ is the
reduced Hubble constant in the unit of 100 km/sec/Mpc. Since the Standard
Model (SM) of particle physics fails to offer a viable particle DM, one has to
explore beyond the realms of the SM.

The weakly interacting massive particle (WIMP) Jungman:1995df ; Kolb:1990vq
by far is the most popular DM candidate. WIMPs are assumed to be in thermal
and chemical equilibrium in the early universe due to sizeable DM-SM coupling
and undergoes a thermal freeze-out once the interaction rate falls below the
Hubble expansion rate ($\mathcal{H}$) of the universe. The interaction cross-
section that gives rise to the observed relic abundance of DM turns out to be
of the order of the weak interaction ($\sim 10^{-10}~{}\rm{GeV}^{-2}$),
suggesting the name WIMP. This very fact also opens up the possibility for
WIMPs to be probed in a plethora of experimental frontiers like direct DM
search, collider search and indirect searches (for a review, see
Arcadi:2017jqd ; Roszkowski:2017nbc ). However, other possibilities like
Feebly Interacting Massive Particle, FIMP Hall:2009bx ; Bernal:2017kxu  or
Strongly Interacting Massive Particle, SIMP Hochberg:2014dra  etc. are getting
more attention due to null experimental observation so far, but the quest for
hunting WIMP-like DM is still on.

Theoretical and phenomenological studies of particle DM have been done mainly
in two ways: constructing a UV complete model as an extension of the SM or by
constructing DM-SM effective (EFT) operators, where the later is the focus of
the present paper. DM EFT operators can be cooked up in a standard
prescription with the Lorentz invariant contact interactions of the form
$\sim\mathcal{O}_{\text{DM}}\,\mathcal{O}_{\text{SM}}$ (suppressed by
appropriate powers of the new physics (NP) scale), where
$\mathcal{O}_{\text{DM}}$ consists of dark sector fields and obey some dark
symmetry, while $\mathcal{O}_{\text{SM}}$ contains SM fields and follow SM
gauge symmetry (for a review see Bhattacharya:2021edh ). Such EFT
constructions can describe DM relic density, direct search Fitzpatrick:2012ib
; Fitzpatrick:2012ix ; Cirelli:2013ufw ; SuperCDMS:2015lcz ; Bishara:2016hek ;
DeSimone:2016fbz ; DEramo:2016gos ; Brod:2017bsw ; Bishara:2017pfq ;
Li:2018orw ; Criado:2021trs , indirect search Beltran:2008xg ; Cao:2009uw ;
Cheung:2010gj ; Goodman:2010qn ; Cheung:2010ua ; Blumenthal:2014cwa ;
Klasen:2015uma ; Bell:2016uhg ; Bell:2019pyc , and production at collider in
terms of just three parameters of the theory, namely, the NP scale $\Lambda$,
the DM mass $m_{\chi}$ and the coupling $c_{i}$ in a model-independent
way111However, these effective operators can be associated to an UV complete
description involving all the relevant fields and symmetry at an energy scale
larger than the NP scale of interaction..

Considering collider signal, the DM is missed in the detector, but its
production in association with any initial state radiation (like a photon or a
jet) gives rise to mono-$X$ plus missing energy signal, a typical one in the
context of DM EFT frameworks222Similar searches are also found in ‘simplified
model’ with $s/t$-channel mediators (see, for example, Abdallah:2015ter ;
Abercrombie:2015wmb ).. Mono-$X$ signal is searched extensively at LHC (for a
review, see Kahlhoefer:2017dnp ; Penning:2017tmb ; Boveia:2018yeb ) and the
absence of an excess has provided bounds on the DM parameter space333Higgs and
Z-invisible decays 10.1093/ptep/ptaa104  also provide a collider probe for DM
having mass less than $m_{h/Z}/2$.. Similar analysis have also been done in
context of lepton colliders Dreiner:2012xm ; Chae:2012bq ; deBlas:2018mhx ;
Habermehl:2020njb . A crucial point is to associate the collider signal with
the DM bounds viz., relic density, direct detection and indirect searches,
which has not been strictly followed in several existing analyses.

The study of the complementarity between (in)direct and collider searches
adopting the EFT approach have also been done extensively in context of LHC
Goodman:2010yf ; Goodman:2010ku ; Bai:2010hh ; Rajaraman:2011wf ;
Buckley:2011kk ; Fox:2011pm ; Dreiner:2013vla ; Buchmueller:2013dya ;
Petrov:2013nia ; Chang:2013oia ; Altmannshofer:2014cla ; Bell:2015sza ;
Belyaev:2016pxe ; Belyaev:2018pqr  as well as in context of lepton collider
($e^{+}e^{-}$) or proposed International Linear Collider (ILC) Fox:2011fx ;
Yu:2013aca ; Essig:2013vha ; Kadota:2014mea ; Yu:2014ula ; Freitas:2014jla ;
Dutta:2017ljq ; Liu:2019ogn ; Choudhury:2019sxt ; Bharadwaj:2020aal  with
possible UV completions Dutta:2017ljq ; Liu:2019ogn ; Bharadwaj:2020aal . More
exhaustive connections are also thought of, for example in Bertuzzo:2017lwt ,
where the bounds on MeV-scale DM are concocted with CMB, BBN, LHC, LEP, direct
detection experiments and meson decays.

Two more observations related to DM EFT are as follows: first, EFT approach
primarily dictates to consider all such operators having same mass dimension
on equal footing 444With an exception for potential tree generated (PTG) or
loop generated (LG) operators GonzalezMacias:2015rxl ., although many of the
analyses have been projected by taking one operator at a time; secondly, in
collider study of DM EFT one needs to ensure that the center-of-mass (CM)
energy of the reaction lies below the mass of the NP scale ($\Lambda$). In a
hadron collider like the LHC, the partonic CM energy ($\sqrt{\hat{s}}$) is
unknown, hence it is not possible to guarantee that $\sqrt{\hat{s}}<\Lambda$
holds, particularly for DM pair production whose invariant mass can not be
constructed555In Busoni:2013lha ; Busoni:2014haa ; Busoni:2014sya  the
validity of EFT in context of DM collider search has been studied in detail in
presence of $s$ or $t$ channel mediators.. This is in contrast to DM direct
searches, where the momentum transfer involved in the scattering of DM
particles with heavy nuclei are of the order of tens of keV, way below the NP
scale, making EFT description more reliable. As a consequence, even though
hadron colliders have a larger reach, applicability of DM EFT is questionable.
On the other hand, in leptonic colliders like ILC, knowledge of CM energy of
the reaction, together with symmetric beams and the possibility of polarizing
the beams to reduce SM background contribution, make DM EFT studies much more
concrete. Motivated from these, in this work, we explore the DM EFT in
$e^{+}\,e^{-}$ collider, assuming the DM to be a Dirac or Majorana fermion,
where it interacts preferentially to the SM leptons. By considering all the
operators of dimension six together, we explore the resulting parameter space
for the DM abiding bounds from Planck observed relic abundance and also limits
from spin-independent direct search experiments.

We finally take up an wishful exercise in probing the Wilson coefficients of
these leptophilic EFT operators at the ILC to one sigma $(1\sigma)$ deviation
from their seed values, and find the separation between different possible
models having a few operators together in each case using the optimal
observable technique (OOT), which ensures the error to be minimal Diehl:1993br
; Gunion:1996vv . This technique has been employed in probing Higgs Physics
Hagiwara:2000tk ; Dutta:2008bh , top-quark interactions Grzadkowski:1996pc ;
Grzadkowski:1997cj ; Grzadkowski:1998bh ; Grzadkowski:1999kx  and heavy
charged fermions Bhattacharya:2021ltd  in context of ILC.

The paper is organized as follows: DM-SM effective operators are noted in
Section 2, while the DM constraints are described in Section 3; collider
prospects of leptophilic DM at ILC is elaborated in Section 4, followed by
optimal statistical analysis in Section 5. We remark on the UV completion of
our effective description in Section 6 and finally summarize in Section 7.
Appendices A contain all the relevant annihilation cross-section formulae; B
contain DM constraints and OOT analysis for some special choices of the Wilson
coefficients.

##  2 DM-SM Effective Operators

We take up EFT approach to study DM physics, which is also motivated from the
absence of a specific hint of dark sector particles from ongoing experiments
and assume effective DM-SM interactions of the following form :

${\cal L}_{\tt
int}=\frac{c}{\Lambda^{d+d^{\prime}-4}}\mathcal{O}^{(d)}_{\text{SM}}\mathcal{O}^{(d^{\prime})}_{\text{DM}}\,,$
(1)

where $\Lambda$ denotes NP scale, $\mathcal{O}^{(d)}_{\text{SM}}$ consists of
SM fields (having mass dimension $d$) and are invariant under ${\cal G}_{\tt
SM}$ (SM gauge symmetry), while $\mathcal{O}^{(d^{\prime})}_{\text{DM}}$
consists of DM fields (having mass dimension $d^{\prime}$) and is invariant
under dark symmetry ${\cal G}_{\tt DM}$; necessitated by the stability of DM
upto a scale of universe life time and $c$ denotes dimensionless couplings,
also called the Wilson coefficients. We also assume $\mathcal{O}_{\text{DM}}$
is singlet under ${\cal G}_{\tt SM}$ and $\mathcal{O}_{\text{SM}}$ is singlet
under ${\cal G}_{\tt DM}$666Exceptions to this simplification have been
studied in context of many UV complete models, where dark sector particles
transform nontrivially under ${\cal G}_{\tt SM}$, see for example, Diaz-
Cruz:2010czr .. Also note that $\mathcal{O}_{\text{DM}}$ must have at least
two dark-sector fields, since all dark fields transform non-trivially under
${\cal G}_{\tt DM}$. For simplicity, we assume here ${\cal G}_{\tt DM}=Z_{2}$.
Eq. (1) dictates the freeze-out when DM is assumed to be WIMP or freeze-in
when it is assumed to be FIMP, as well as its interaction in direct, indirect
and collider search experiments.

DM EFT operators involving a scalar ($\Phi$), fermion (Dirac or Majorana)
($\chi$) or vector bosons ($X$) as DM, have been constructed in several works
(see, for example, Matsumoto:2014rxa ; Duch:2014yma ; Duch:2014xda ;
Macias:2015cna ). Let us consider the DM to be a fermion $\chi$ which
transforms under a dark symmetry ${\cal G}_{\tt DM}=Z_{2}$, and write all
possible EFT operators in Eq. (2) (dimension five) and Eq. (3) (dimension
six), suppressed by appropriate powers of NP scale $\Lambda$.

$\begin{array}[]{ll}\mathcal{O}_{D1}^{5}=\frac{g}{8\pi^{2}\Lambda}\left(\overline{\chi}\sigma_{\mu\nu}\chi\right)B^{\mu\nu}\,,&~{}\mathcal{O}_{D2}^{5}=\frac{g}{8\pi^{2}\Lambda}\left(\overline{\chi}i\sigma_{\mu\nu}\gamma^{5}\chi\right)B^{\mu\nu}\,,\cr\mathcal{O}_{3}^{5}=\frac{c_{H_{1}}}{\Lambda}\left(\overline{\chi}\chi\right)\left(H^{\dagger}H\right)\,,&~{}\mathcal{O}_{4}^{5}=\frac{c_{H_{2}}}{\Lambda}\left(\overline{\chi}i\gamma^{5}\chi\right)\left(H^{\dagger}H\right).\end{array}$
(2)

$\begin{array}[]{ll}\mathcal{O}_{DQ}^{6}=\frac{c_{q_{1}}}{\Lambda^{2}}\left(\overline{\chi}\gamma_{\mu}\chi\right)\left(\overline{{\mathfrak{q}}}\gamma^{\mu}{\mathfrak{q}}\right)\,,&\mathcal{O}_{DL}^{6}=\frac{c_{\ell_{1}}}{\Lambda^{2}}\left(\overline{\chi}\gamma_{\mu}\chi\right)\left(\overline{\ell}\gamma^{\mu}\ell\right)\,,\cr\mathcal{O}_{Q}^{6}=\frac{c_{q_{2}}}{\Lambda^{2}}\left(\overline{\chi}\gamma_{\mu}\gamma^{5}\chi\right)\left(\overline{{\mathfrak{q}}}\gamma^{\mu}{\mathfrak{q}}\right)\,,&\mathcal{O}_{L}^{6}=\frac{c_{\ell_{2}}}{\Lambda^{2}}\left(\overline{\chi}\gamma_{\mu}\gamma^{5}\chi\right)\left(\overline{\ell}\gamma^{\mu}\ell\right)\,,\cr\mathcal{O}_{Q1}^{6}=\frac{c_{q_{3}}}{\Lambda^{2}}\left(\overline{\chi}\gamma_{\mu}\gamma^{5}\chi\right)\left(\overline{{\mathfrak{q}}}\gamma^{\mu}\gamma_{5}{\mathfrak{q}}\right)\,,&\mathcal{O}_{L1}^{6}=\frac{c_{\ell_{3}}}{\Lambda^{2}}\left(\overline{\chi}\gamma_{\mu}\gamma^{5}\chi\right)\left(\overline{\ell}\gamma^{\mu}\gamma_{5}\ell\right)\,,\cr\mathcal{O}_{DQ1}^{6}=\frac{c_{q_{4}}}{\Lambda^{2}}\left(\overline{\chi}\gamma_{\mu}\chi\right)\left(\overline{{\mathfrak{q}}}\gamma^{\mu}\gamma_{5}{\mathfrak{q}}\right)\,,&\mathcal{O}_{DL1}^{6}=\frac{c_{\ell_{4}}}{\Lambda^{2}}\left(\overline{\chi}\gamma_{\mu}\chi\right)\left(\overline{\ell}\gamma^{\mu}\gamma_{5}\ell\right)\,,\cr\mathcal{O}_{DH}^{6}=\frac{c_{H_{3}}}{\Lambda^{2}}\left(\overline{\chi}\gamma^{\mu}\chi\right)(H^{\dagger}i\overset{\leftrightarrow}{D}_{\mu}H)\,,&\mathcal{O}_{H}^{6}=\frac{c_{H_{4}}}{\Lambda^{2}}\left(\overline{\chi}\gamma^{\mu}\gamma^{5}\chi\right)(H^{\dagger}i\overset{\leftrightarrow}{D}_{\mu}H).\end{array}$
(3)

Here $H$ stands for the SM Higgs isodoublet, ${\mathfrak{q}}$ stands for
either left handed (LH) doublet or right handed (RH) singlet SM quarks (of all
flavors), and $\ell$ stands for SM lepton, LH doublet or RH singlet. SM gauge
invariance ensures only LH doublets or RH singlets to appear in the SM fermion
current. For brevity we omit flavour indices and for simplicity assume the
interactions to be flavour diagonal777Flavour non-diagonal DM interactions
have been studied in context of lepton flavour universality violation, as in
DAmbrosio:2002vsn .. The covariant derivative $D_{\mu}$ is defined as

$D_{\mu}=\partial_{\mu}-ig\frac{\sigma^{i}}{2}W_{\mu}^{i}-i\frac{g^{\prime}}{2}B_{\mu}~{},$

where $g,g^{\prime}$ stands for $SU\left(2\right)_{L}$ and
$U\left(1\right)_{Y}$ gauge coupling strengths, $\sigma^{i}$ are the Pauli
spin matrices with $i\in 1,2,3$, and $W_{\mu}^{i}$ and $B_{\mu}$ are the
$SU\left(2\right)_{L}$ and $U\left(1\right)_{Y}$ gauge bosons.
$B^{\mu\nu}=\partial^{\mu}B^{\nu}-\partial^{\nu}B^{\mu}$ represents the
$U(1)_{Y}$ gauge field strength tensor, which is itself SM gauge invariant.
The hermitian conjugate of covariant derivatives is defined as
$\overset{\leftrightarrow}{D}_{\mu}\equiv
D_{\mu}-\overset{\leftarrow}{D}_{\mu}^{\dagger}$. We also assume the operators
to have different couplings to SM leptons ($c_{\ell_{i}}$), quarks
($c_{q_{i}}$) and SM Higgs ($c_{H_{i}}$). Note further, that the operators
with tag $D$ (for example, $\mathcal{O}_{DQ}^{6},\mathcal{O}_{DH}^{6}$ etc.)
are present only when DM $\chi$ is a Dirac fermion, which naturally indicates
all the $D$-tagged operators to be dropped for Majorana $\chi$. In the
following, we consider both these possibilities and indicate the distinction
they provide in subsequent phenomenology. We further point out that any
operator having an interaction with gauge field strength tensor (for example,
$B^{\mu\nu}$) are generated at least in a one-loop level via NP in a
perturbative UV theory and hence classified as loop generated operators (LG)
having additional suppression factor $\sim 1/(16\pi^{2})$ (see, for example,
Macias:2015cna ). Therefore, $\mathcal{O}_{D{1,2}}^{5}$ will have smaller
contribution and their presence can be ignored compared to those other
operators, which are potential tree-generated (PTG).

A priori for a fermion DM, dimension five operators (Eq. (2)) naturally
dominate over others and subsequent phenomenology is that of Higgs portal
interaction, which have been studied extensively Fedderke:2014wda ;
Matsumoto:2014rxa ; Bishara:2015cha . However, one may think of NP scenarios
which gives rise to predominantly DM-SM fermion interactions, where the Higgs
portal interactions can be neglected and the four fermion operators (as in Eq.
(3)) having the following form start playing a crucial role:

$\frac{c_{f}}{\Lambda^{2}}\Bigl{(}\overline{\chi}\Gamma^{\mu}_{\chi}\chi\Bigr{)}\Bigl{(}\overline{f}\Gamma_{\mu,f}f\Bigr{)}\,;~{}~{}\Gamma^{\mu}_{\chi/f}:\\{\gamma^{\mu},\gamma^{\mu}\gamma^{5}\\}\,;~{}~{}f=\\{{\mathfrak{q}},\ell\\}\,.$
(4)

Further classification may also emerge if the DM couple preferentially to SM
leptons or quarks (due to some specific NP scenarios as we will highlight
later) as:

  * •

Leptophilic: $c_{\ell_{i}}=1,c_{H_{i}}=c_{q_{i}}=0$ ;

  * •

Hadrophilic: $c_{q_{i}}=1,c_{H_{i}}=c_{\ell_{i}}=0$ .

It is easy to see that Leptophilic or Hadrophilic operators provide completely
different phenomenology; for example, Hadrophilic DM operators are more prone
to direct search constraints and can be produced at the LHC. On the other
hand, Leptophilic DM is interesting for at least two reasons; they can hide
from direct search constraints to a great extent and can be probed at ILC in
its effective limit. We note that the non-zero values of the Wilson
coefficients chosen here is representative and can be compensated by a new
choice of $\Lambda$888However, when several operators are considered together,
as we do here, different relative coupling strengths can’t be reproduced by
naive scaling of $\Lambda$.. By Leptophilic DM ($c_{\ell_{i}}=1$), we usually
refer to the situation where all the operators ($i=\\{1,..4\\}$) are assumed
to have equal coupling strength, unless explicitly specified to the cases
where some of them are considered absent. We will also see that the sign of
the couplings will play an important part, particularly in collider
phenomenology. We comment on some possible UV completion of leptophilic models
considered here in section 6.

##  3 DM constraints on Leptophilic Operators

The first exercise is to find constraints on the operators from existing
experimental limits from DM searches. They mainly include relic density,
direct search and indirect searches, which we are going to discuss in the
following subsections, highlighting the processes which contribute to these
observables and finally the allowed parameter space available to such
operators after these constraints.

###  3.1 Relic Density

DM Relic density provides the most important constraints on the parameter
space of the EFT model. Here we assume $\chi$ to be a WIMP-like fermion DM in
thermal and chemical equilibrium with SM particles in the early universe,
which decouples at some later epoch as the universe expands and cools down.
The the relic density within the WIMP scenario (see Kolb:1990vq  for details)
is obtained from the solution to the Boltzmann equation:

$\frac{dY_{\chi}}{dx}=-1.32\,\sqrt{g_{*}}~{}M_{\tt
pl}~{}\frac{m_{\chi}}{x^{2}}~{}\left\langle\sigma v\right\rangle_{2_{\rm
DM}\rightarrow 2_{\rm SM}}~{}\left({Y_{\chi}}^{2}-Y_{\tt eq}^{2}\right)\,,$
(5)

where $M_{\tt pl}$ denotes the reduced Planck mass, $Y_{\chi}=n/s$ refers to
DM yield ($n$ is the DM density, $s$ is the total entropy density) and
$x=m_{\chi}/T$ ($T$ is the temperature). $Y_{\tt eq}$ denotes the value of
$Y_{\chi}$ in thermal equilibrium given by Maxwell Boltzmann distribution for
non-relativistic species:

$Y_{\tt eq}(x)=0.145~{}\frac{g_{\tt DM}}{g_{*}}x^{3/2}e^{-x}\,.$ (6)

In the above, $g_{\tt DM}$ refers to the number of DM internal states; $g_{*}$
denotes the effective relativistic degrees of freedom

$g_{*}=\sum_{i=\rm bosons}g_{i}\theta(T-m_{i})+\frac{7}{8}\sum_{i=\rm
fermions}g_{i}\theta(T-m_{i})\,,$ (7)

where $g_{i}$ are the internal degrees of freedom of particle $i$ with mass
$m_{i}$. The key in Eq. (5) is to assume that $Y_{\chi}=Y_{\tt eq}$ at
$x\rightarrow 0$, ensuring DM to be in thermal bath. Finally, in Eq. (5),
$\left\langle\sigma v\right\rangle_{2_{\rm DM}\rightarrow 2_{\rm SM}}$ denotes
the thermal average of the DM annihilation cross-section$\times$ velocity for
the process $\chi\bar{\chi}\rightarrow\ell\bar{\ell}$ mediated by operators
with $c_{\ell_{i}}=1$. It is straightforward to compute the corresponding
annihilation cross-sections for all these operators. For example, the operator
$\mathcal{O}_{DL}^{6}$ provides the following annihilation cross-section for a
Dirac DM $\chi$

$\displaystyle\langle{\sigma v}\rangle_{2_{\rm DM}\rightarrow 2_{\rm SM}}$
$\displaystyle=\frac{m_{\chi}^{2}}{2\pi\Lambda^{4}}\sum_{\ell}\left(c_{\ell_{1}}\right)^{2}\left\\{2+\frac{m_{\ell}^{2}}{m_{\chi}^{2}}+\left[\frac{8m_{\chi}^{4}-4m_{\ell}^{2}m_{\chi}^{2}+5m_{\ell}^{4}}{24m_{\chi}^{2}(m_{\chi}^{2}-m_{\ell}^{2})}\right]v^{2}\right\\}\sqrt{1-\frac{m_{\ell}^{2}}{m_{\chi}^{2}}}\,,$
$\displaystyle=a+bv^{2}\,,$ (8)

where $m_{\ell}$ is the lepton mass, $v$ is the Möller velocity
$v=\sqrt{(p_{\chi}.p_{\bar{\chi}})^{2}-m_{\chi}^{4}}/(E_{\chi}E_{\bar{\chi}})$
Gondolo:1990dk ; Edsjo:1997bg  and we have ignored higher powers of $v$. This
interaction then gives rise to s-wave ($\propto a$) and p-wave ($\propto b$)
contributions Kolb:1990vq ; Bauer:2017qwy . Annihilation cross-section for
other operators are furnished in Appendix A.

Figure 1: Left: Evolution of DM (Dirac) yield $Y_{\chi}$ with $x=m_{\chi}/T$
for operator $\mathcal{O}_{DL}^{6}$ (see Eq. (3)), for three different choices
of the EFT scale $\Lambda=\\{4.0,2.3,1.5\\}$ TeV shown respectively in red,
blue and green keeping $c_{\ell_{1}}=1$ and $m_{\chi}=$ 300 GeV fixed. Right:
Same as left, but for a fixed $\Lambda=3$ TeV with three choices of
$c_{\ell_{1}}=\\{0.7,1.7,3.5\\}$ shown in red, blue and green respectively. In
both the plots the thick black curve shows the equilibrium DM yield ($Y_{\tt
eq}$) and the grey dashed straight line indicates the yield that produces
correct relic abundance (see Eq. (11)).

The relic abundance of DM after thermal freeze-out has an approximate
analytical form

$\Omega_{\tt DM}\,{\sf h}^{2}\simeq\frac{(1.04\times
10^{9}\,\text{GeV})x_{f}}{M_{\tt pl}\sqrt{g_{*}}(a+3b/x_{f})}\,,$ (9)

with $\Omega_{\tt DM}=\rho_{\tt DM}/\rho_{c}$, where $\rho_{c},\rho_{\tt DM}$
denote the critical and DM densities respectively and
$x_{f}=m_{\chi}/{T_{f}}$, with $T_{f}$ the freeze-out temperature when the DM
decouples from thermal bath. In presence of all the operators with
$c_{\ell_{i}}=1$, all the individual amplitudes add together to the total
annihilation cross-section, with

$\langle{\sigma v}\rangle_{\rm
tot}\sim\Big{|}\sum_{\mathcal{O}_{i}}\mathcal{M}_{\mathcal{O}_{i}}\Big{|}^{2}\,,$
(10)

where
$\mathcal{M}_{\mathcal{O}_{i}}=\frac{c_{\ell_{i}}}{\Lambda^{2}}(\bar{\chi}\Gamma\chi)(\bar{\ell}\Gamma\ell);~{}\rm{with}~{}\Gamma=\\{\gamma^{\mu},\gamma^{\mu}\gamma^{5}\\}$
denote the matrix elements for DM pair annihilation to SM via respective
operators as in Eq. (3). The relic density obtained from the model is then
compared to the current Planck data Aghanim:2018eyx

$\Omega_{\tt DM}{\sf h}^{2}=0.11933\pm 0.00091\,,$ (11)

that finds relic density allowed parameter space of the model. In Fig. 1, we
show the thermal freeze-out of the DM following Eq. (5) in $Y_{\chi}-x$ plane,
for operator $\mathcal{O}_{DL}^{6}$ (see Eq. (3)) assuming a Dirac DM $\chi$.
The equilibrium distribution $Y_{\tt eq}$ is shown by black thick line, where
we choose $m_{\chi}=300$ GeV. Horizontal grey dashed line corresponds to
$Y_{\chi}$ value for correct DM relic density following Eq. (11), which is
approximately given by the condition $m_{\chi}Y_{\chi}\approx 0.44$ eV. In the
left panel, we vary the EFT scale $\Lambda=\\{4.0,2.3,1.5\\}$ TeV, shown
respectively by red, green and blue curves assuming $c_{\ell_{1}}=1$ and we
see that the green curve with $\Lambda=2.3$ TeV provides the correct relic
after freeze-out, whereas the case with larger (smaller) $\Lambda$,
corresponding to the red (blue) curve in Fig. 1 provides over (under)
abundance following the inverse dependence of relic density to annihilation
cross-section as in Eq. (9). In the right panel of Fig. 1, we show the thermal
freeze-out for three different choices of $c_{\ell_{1}}=\\{0.7,1.7,3.5\\}$ in
red, green and blue respectively for a fixed $\Lambda=3$ TeV. Again, we see
that the case with $c_{\ell_{1}}=1.7$ satisfy correct relic while the smaller
(larger) $c_{\ell_{1}}$ provides over (under) abundance attributed to Eq. (9).
The relic density allowed parameter space of the model can then be found in
terms of $\\{\Lambda,m_{\chi}\\}$ assuming $c_{\ell_{i}}=1$. We use the
numerical tool MicrOmegas Belanger:2010pz  to compute relic density for the
model for both Dirac and Majorana cases. The results, together with direct
search constraints, will be discussed in the next section.

###  3.2 Direct detection

Direct DM search relies on the DM scattering with earthly detectors and
observation of nuclear recoil to confirm the presence of DM interaction.
Unfortunately no DM signal has been confirmed yet, resulting a strong bound on
the DM-SM interaction cross-section, the latest from XENON1T XENON:2018voc .
Leptophilic DM as defined, do not have a direct coupling to detector nucleus.
But, it is possible to induce couplings to quarks via photon exchange at one
loop level as in Fig. 2, where in the loop one may have any charged lepton to
which the DM couples to999Similar diagrams with a Z, or Higgs-boson propagator
are suppressed in comparison with the virtual photon mediation by a factor
$\simeq(k-k^{\prime})^{2}/m_{Z,H}^{2}$ PhysRevD.80.083502 .. Instead of a
nuclear recoil, it is also possible to knock off an electron from the atomic
orbital of the detector material directly via the effective DM interaction
with the lepton. However, as it has been shown in PhysRevD.80.083502 , the
loop-induced DM-nucleon scattering always dominates over the DM-electron
scattering cross-section, where the latter is suppressed by the momentum wave
function. Therefore, we ignore the DM-electron scattering here and focus on
the one loop DM-nucleus interaction via one loop. Now, following the most
general 4-fermion interaction as in Eq. (4) Kopp:2009et , the one loop
contribution to DM-nucleon interaction involves the following integration:

$\displaystyle\int\frac{d^{4}q}{\left(4\pi\right)^{4}}\text{Tr}\Biggl{[}\Gamma_{\ell}\frac{\not{q}^{\prime}+m_{\ell}}{q^{{}^{\prime}2}-m_{\ell}^{2}}\gamma^{\mu}\frac{\not{q}+m_{\ell}}{q^{2}-m_{\ell}^{2}}\Biggr{]}\,,$
(12)

where $q$ and $q^{\prime}=k-k^{\prime}+q$ denote the loop momentum, with
$k,k^{\prime}$ as the momenta carried by the incoming and outgoing DM
particles respectively. Note that, the loop contribution is non-zero only for
vector and tensor lepton currents
$\Gamma^{\mu}_{\ell}=\gamma^{\mu},\sigma^{\mu\nu}$. For scalar, pseudoscalar
and axial-vector currents
$\Gamma^{\mu}_{\ell}=1,\gamma_{5},\gamma^{\mu}\gamma_{5}$ this contribution in
Eq. (12) identically vanish. Therefore, non-zero direct search contribution
for leptophilic DM arises only for operators
$\mathcal{O}_{DL}^{6},\mathcal{O}_{L}^{6}$. However, the contribution from
$\mathcal{O}_{L}^{6}$ to the direct detection cross-section vanishes in the
non-relativistic limit and $\mathcal{O}_{DL}^{6}$ doesn’t apply to Majorana
DM. Therefore, direct detection constraints do not yield any bound on the
parameter space for leptophilic (or hadrophilic) Majorana DM. Subsequently, we
focus on the vector type interaction, namely
$\mathcal{O}_{DL}^{6}=\frac{c_{\ell_{1}}}{\Lambda^{2}}(\bar{\chi}\gamma^{\mu}\chi)(\bar{\ell}\gamma_{\mu}\ell)$,
and calculate the loop-induced matrix element relevant for nuclear recoil
experiments.

Figure 2: Loop-induced DM-nucleon direct detection mediated by a virtual
photon. Here $q$ is the loop momentum, and $q^{\prime}=k-k^{\prime}+q$. The
blob represents the effective nature of the vertex.

The matrix element for $\chi N\rightarrow\chi N$ scattering (where $N$ denotes
detector nucleus) via one-loop interaction, modulo implicit sum over the light
quark generations with charge $Q_{j}$ is thus given by:

$\displaystyle\begin{split}&\mathcal{M}=L^{(1)}\Bigl{(}\overline{u}_{\chi}\,\gamma^{\mu}\,u_{\chi}\Bigr{)}\,\Bigl{\langle}N\Bigl{|}Q_{j}\,\Bigl{(}\overline{{\mathfrak{q}}}_{j}\gamma_{\mu}{\mathfrak{q}}_{j}\Bigr{)}\Bigr{|}N\Bigr{\rangle}=L^{(1)}\Bigl{(}\overline{u}_{\chi}\,\gamma^{\mu}\,u_{\chi}\Bigr{)}Z\,F\left(q\right)\,\Bigl{(}\overline{u}_{N}\,\gamma_{\mu}\,u_{N}\Bigr{)}\,,\end{split}$
(13)

where $F\left(q\right)$ denotes the nuclear form factor, $Z$ denotes the
Atomic number of the nucleus and the loop factor $L^{(1)}$ is given by

$\displaystyle
L^{(1)}=\frac{2\alpha_{\text{em}}}{\pi}\frac{c_{\ell_{1}}}{\Lambda^{2}}\int_{0}^{1}\,dx\,x\left(1-x\right)\ln\Biggl{[}\frac{-x\left(1-x\right)\mathcal{Q}^{2}+m_{\ell}^{2}}{\mu^{2}}\Biggr{]}\,.$
(14)

$\mathcal{Q}$ in the above expression momentum transfer
($\mathcal{Q}=k-k^{\prime}$) in the process and $\mu$ is the renormalization
scale. For $m_{\ell}\gg\left|\bf{\mathcal{Q}}\right|$ one may neglect the
momentum transfer and obtain

$\displaystyle
L_{LL}^{(1)}\approx\frac{2\alpha_{\text{em}}}{\pi}\frac{c_{\ell_{1}}}{\Lambda^{2}}\underbrace{\int_{0}^{1}\,dx\,x\left(1-x\right)\ln\Biggl{[}\frac{m_{\ell}^{2}}{\mu^{2}}\Biggr{]}}_{=\frac{1}{6}\ln\Biggl{[}\frac{m_{\ell}^{2}}{\mu^{2}}\Biggr{]}}=\frac{\alpha_{\text{em}}}{3\pi}\frac{c_{\ell_{1}}}{\Lambda^{2}}\ln\Biggl{[}\frac{m_{\ell}^{2}}{\mu^{2}}\Biggr{]}+\mathcal{O}\Bigl{(}\left|\bf{\mathcal{Q}}\right|/m_{\ell}\Bigr{)}\,,$
(15)

where $LL$ stands for the “leading log” contribution, neglecting the remaining
logarithmic dependence on momentum transfer. With the leading log
approximation the matrix element takes the form

$\displaystyle\begin{split}&\mathcal{M}=\Biggl{(}\frac{\alpha_{\text{em}}}{3\pi}\frac{c_{\ell_{1}}}{\Lambda^{2}}\log\Biggl{[}\frac{m_{\ell}^{2}}{\mu^{2}}\Biggr{]}ZF\left(q\right)\Biggr{)}\Bigl{(}\overline{u}_{\chi}\,\gamma^{\mu}\,u_{\chi}\Bigr{)}\Bigl{(}\overline{u}_{N}\,\gamma_{\mu}\,u_{N}\Bigr{)}.\end{split}$
(16)

For a single operator it is possible to compute the direct search cross-
section analytically as illustrated Kopp:2009et . However, since we are
considering all operators together, it is difficult to obtain an analytical
expression for the loop-induced direct detection cross-section. More
importantly, the Wilson coefficients ($c_{\ell_{i}}$) can vary substantially
from the typical energies for obtaining DM relic density (where they are
defined) to the typical energies of DM direct search experiments ($\sim$ GeV).
We use the runDM DEramo:2016gos  package to obtain couplings at the energy
scale of direct detection $\sim 2~{}\rm GeV$ starting from energy scale of
$\sim$ TeV where the Wilson coefficients are assumed to be $c_{\ell_{i}}=1$.
The renormalization group evolution (RGE) typically introduces mixing between
different DM-SM interactions, affecting the size of couplings, and even
inducing new couplings which do not appear in a naive comparison. The relic
density calculation however remains unaffected with or without the running.

Note that the matrix element in Eq. (16) depend on the renormalization scale
$\mu$. To extract physical quantities that does not depend on $\mu$, we need
to define the renormalization condition. In our case this condition is that at
scale $\Lambda\sim 1$ TeV, the coefficient
($c_{q_{1}}^{({\mathfrak{q}}_{j})}$) of the effective DM-quark operator
$(\bar{\chi}\gamma^{\mu}\chi)(\bar{{\mathfrak{q}}}_{j}\gamma_{\mu}{\mathfrak{q}}_{j})$
is zero. As an illustrative example with only one generation of lepton, in the
leading-log approximation (following Eq. (16)), at the $\overline{\text{MS}}$
scale $\mu$, the coefficient $c_{q_{1}}^{({\mathfrak{q}}_{j})}$ reads

$c_{q_{1}}^{({\mathfrak{q}}_{j})}(\mu)\approx Q_{j}\frac{\alpha_{\rm
em}}{3\pi}\frac{c_{\ell_{1}}}{\Lambda^{2}}\ln\bigg{(}\frac{m_{\ell}^{2}}{\mu^{2}}\bigg{)}\,.$
(17)

Therefore, our renormalization condition reads

$\displaystyle c_{q_{1}}^{({\mathfrak{q}}_{j})}(\Lambda)\approx
Q_{j}\frac{\alpha_{\rm
em}}{3\pi}\frac{c_{\ell_{1}}}{\Lambda^{2}}\ln\bigg{(}\frac{m_{\ell}^{2}}{\Lambda^{2}}\bigg{)}=0\,,$
(18)

where we set $\mu\equiv\Lambda$ Kopp:2009et . Using Eq. (18), it is easy to
infer the coefficient at any energy scale $\Lambda_{f}$ ($\leq\Lambda$) as

$c_{q_{1}}^{({\mathfrak{q}}_{j})}(\Lambda_{f})\approx Q_{j}\frac{\alpha_{\rm
em}}{3\pi}\frac{c_{\ell_{1}}}{\Lambda^{2}}\ln\bigg{(}\frac{\Lambda^{2}}{\Lambda_{f}^{2}}\bigg{)}\
.$ (19)

Relevant Scales | $c_{q_{1}}^{(u)}$ | $c_{q_{1}}^{(d)}$ | $c_{q_{1}}^{(c)}$ | $c_{q_{1}}^{(s)}$ | $c_{q_{1}}^{(b)}$ | $c_{\ell_{1}}^{(e)}$ | $c_{\ell_{1}}^{(\mu)}$ | $c_{\ell_{1}}^{(\tau)}$
---|---|---|---|---|---|---|---|---
$\Lambda=3$ TeV | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1
$\Lambda_{\rm NR}=2$ GeV | 0.024 | -0.012 | 0.024 | -0.012 | -0.012 | 0.965 | 0.965 | 0.965

Table 1: Running of the coupling coefficients $c_{q_{1}}^{({\mathfrak{q}})}$,
and $c_{\ell_{1}}^{(\ell)}$ of the operators ${\cal O}_{DQ}^{6}$ and ${\cal
O}_{DL}^{6}$ respectively (see Eq. (3)) using runDM package, from an initial
scale $\Lambda=3$ TeV to the non-relativistic scale $\Lambda_{\rm NR}=2$ GeV
relevant for DM-nucleon scattering. For Leptophilic case, only non-zero
coefficient of the operator ${\cal O}_{DL}^{6}$ is assumed with
$c_{\ell_{1}}^{(\ell)}=1$, at the scale $\Lambda$. Note that this is only
relevant for Dirac DM, as the vector DM current is zero for the Majorana case.

Notice in Eq. (19), that $c_{q_{1}}^{({\mathfrak{q}}_{j})}$ is zero at the
scale $\Lambda_{f}=\Lambda$, as per renormalization condition. The DM-nucleon
cross section at direct detection scale $\Lambda_{\text{NR}}$, is then given
by

$\displaystyle\sigma_{\rm SI}\approx\frac{\mu_{p}^{2}c_{\ell_{1}}^{2}}{9\pi
A^{2}}\Biggl{(}\frac{\alpha_{\text{em}}Z}{\pi\Lambda^{2}}\Biggr{)}^{2}\Biggl{[}\ln\Biggl{(}\frac{\Lambda^{2}}{\Lambda_{\text{NR}}^{2}}\Biggr{)}\Biggr{]}^{2}\,,$
(20)

where we have ignored the term proportional to DM velocity $v\sim 10^{-6}$,
used $F(q)=1$ for brevity and
$\mu_{p}=m_{p}\,m_{\chi}/\left(m_{p}+m_{\chi}\right)$ as the reduced mass of
the DM-nucleon system. Its worth reminding that in deriving Eq. (20), we have
assumed the presence of only the vector operator
$\mathcal{O}_{DL}^{6}=\frac{c_{\ell_{1}}}{\Lambda^{2}}(\bar{\chi}\gamma^{\mu}\chi)(\bar{\ell}\gamma_{\mu}\ell)$
at the effective scale $\Lambda$. The running of the couplings to a lower
scale ($\Lambda_{\rm{NR}}$) generates the DM-quark interaction via operator
mixing, thereby producing the spin-independent direct detection cross-section
as in Eq. (20). In Table 1, we show the values of the running coupling
coefficients $c_{q_{1}}^{({\mathfrak{q}})}$, and $c_{\ell_{1}}^{(\ell)}$ at
direct search experiment scale $\Lambda_{\text{NR}}=2$ GeV evaluated using
runDM package, while the input values are assumed at NP scale $\Lambda=3$ TeV.
Using them, together with $m_{\chi}=500$ GeV, we find $\sigma_{\rm SI}\approx
3\times 10^{-47}~{}\text{cm}^{2}$, close to the sensitivity of existing direct
search cross-section. It is clear, that the loop-induced WIMP-nucleon spin-
independent scattering cross-section can provide a very strong bound on the
leptophilic DM parameter space, which will be shown together with the relic
density allowed parameter space in Subsection. 3.4.

###  3.3 Indirect detection

Although DM occasionally annihilate after thermal-freeze-out, it can still
occur in regions with very large DM density, for example, at the centre of
galaxies. The annihilations produce a bunch of SM particles in pair of various
kinds, such as:

$\displaystyle\chi+\bar{\chi}\rightarrow
f\bar{f},~{}\gamma\gamma,...~{}~{}f:~{}\text{SM ~{}fermions};$

where the presence of anti-particle ($\bar{f}$) and photons in particular,
play crucial role in elucidating the presence of DM, if an excess is found.
Various satellite and ground based telescopes (like the Fermi Large Area
Telescope (FermiLAT) Fermi-LAT:2016afa  or Cherenkov Telescope Array (CTA)
CTAConsortium:2012fwj ) have searched for excess in anti-particle or photons
beyond astrophysical processes which may arise from the DM annihilation. For
example, the production rate ($\Gamma$) of SM particles from DM annihilation
is crucially dependent on parameters like DM density ($\rho_{\tt DM}$) and
strength of annihilation cross-section $(\sigma v)$ Cirelli:2010xx ;
Slatyer:2017sev ; Hooper:2018kfv

$\displaystyle\Gamma\sim\int_{V}\left(\frac{\rho^{2}_{\tt
DM}}{m_{\chi}^{2}}\right)dV\times(\sigma v)(N_{\text{SM}});$ (21)

where $N_{\text{SM}}$ represents number of SM particles per annihilation
event. Note that annihilation cross-section present in the equation above is
also crucial to produce thermal freeze-out for WIMP like DM to produce correct
relic density. In context of the effective leptophilic DM operators that we
are concerned, we can therefore calculate annihilation cross-section to
produce current indirect search observation and relate to the annihilation
cross-section to produce correct relic density and thus produce a bound on the
model. We further note that significant constraint can only be obtained for
leptophilic Dirac DM as Majorana DM would produce velocity dependent
annihilation cross-section, which results in a suppressed annihilation rate,
and hence does not provide any reasonable bound from indirect detection. A
detailed analysis of indirect detection bound on the DM parameter space is
beyond the scope of this paper. We simply calculate the thermal averaged DM
pair annihilation cross-section into different SM leptonic final states as
shown in table 2, and project the bound from the non-observation of gamma-ray
signals from DM annihilation in dwarf satellite galaxies MAGIC:2016xys . In
the absence of any boost factor viz., Sommerfeld enhancement Feng:2010zp , the
bounds from indirect search turns out to be much milder than that due to
direct detection, and constraints the relic density allowed parameter space
for the DM only at very low $m_{\chi}$.

$\langle\sigma v\rangle_{\chi\bar{\chi}\rightarrow e^{+}e^{-}}$ $({\rm cm}^{3}/{\rm s})$ |  $\langle\sigma v\rangle_{\chi\bar{\chi}\rightarrow\mu^{+}\mu^{-}}$ $({\rm cm}^{3}/{\rm s})$ |  $\langle\sigma v\rangle_{\chi\bar{\chi}\rightarrow\tau^{+}\tau^{-}}$ $({\rm cm}^{3}/{\rm s})$
---|---|---
$7.5\times 10^{-27}$ | $7.5\times 10^{-27}$ | $7.5\times 10^{-27}$
Table 2: Velocity averaged annihilation cross-section $\langle\sigma v\rangle$
(s-wave) of dominating channels for the relic density satisfying points. We
have chosen illustrative values $c_{\ell_{i}}=1$ (Dirac DM). The annihilation
to quark final states are negligibly small in comparison to the leptonic final
states. Note that the annihilation to neutrino final states are identically
zero due to interference effects (e.g., see Eq. (37)).

###  3.4 Viable parameter space

Figure 3: Relic density and direct search (XENON1T) allowed parameter space
for hadrophilic ($c_{q_{i}}=1,c_{\ell_{i}}=0$), and leptophilic
($c_{q_{i}}=0,c_{\ell_{i}}=1$) cases in the top panel and Higgs portal case
($c_{H_{i}}=1,c_{\ell_{i}}=c_{q_{i}}=0$) in the bottom panel. Top Left (right)
figure describes Dirac (Majorana) DM. The coupling values
$(c_{\ell_{1}},c_{\ell_{2}},c_{\ell_{3}},c_{\ell_{4}})$ is denoted as an
ordered set for Dirac DM and $(c_{\ell_{2}},c_{\ell_{3}})$ for Majorana DM.
XENON1T allowed region and disallowed regions are marked in different colours
(see figure inset and text for details); for example, in hadrophilic Dirac DM
case (top left), XENON1T allowed region is shown in cyan, while the disallowed
region is shown in black. Note, there is no spin-independent direct detection
bound on Majorana DM.

After discussing the major constraints on DM model coming from relic density,
direct and indirect search experiments, we summarise here the available DM
parameter space for leptophilic DM operators from all of these constraints, in
terms of $m_{\chi}-\Lambda$. In Fig. 3 we show the Planck observed relic
density allowed parameter space for Dirac (top left) and Majorana (top right)
DM in $\Lambda-m_{\chi}$ plane by considering all relevant operators together
for both hadrophilic ($c_{\ell_{i}}=0,c_{q_{i}}=1$) and leptophilic
($c_{\ell_{i}}=1,c_{q_{i}}=0$) cases. We vary the two free parameters of the
theory, namely the cut-off scale $\Lambda$ and the DM mass $m_{\chi}$, in the
following range

$\Lambda:\\{1-10^{4}\\}~{}\text{GeV};~{}m_{\chi}:\\{1-10^{4}\\}~{}\text{GeV}$

while considering $\Lambda\gtrsim m_{\chi}/2\pi$ in order to make sure that
the effective theory description remains valid. First of all, notice that in
all plots, with the increase in DM mass, $\Lambda$ also rises to satisfy the
relic abundance, following that the relic density of thermal WIMP goes as
$\Omega_{\chi}h^{2}\sim 1/\langle\sigma v\rangle\sim\Lambda^{4}/m_{\chi}^{2}$.
XENON1T allowed region and disallowed regions are marked in different colours.

In the top left plot of Fig. 3, for hadrophilic case, the relic density
allowed region is significantly ruled out from spin-independent direct
detection bound upto $m_{\chi}\lesssim 6.4~{}\rm TeV$ for $\Lambda\lesssim
18~{}\rm TeV$ (shown by black points) making it difficult for collider probe.
In the same plot, we also show the leptophilic cases where coupling values
$(c_{\ell_{1}},c_{\ell_{2}},c_{\ell_{3}},c_{\ell_{4}})$ is denoted as an
ordered set for Dirac DM. Amongst different possibilities considered, we see
that vector coupling plays a crucial role as those cases with $c_{\ell_{1}}=1$
almost overlap and segregates from those with $c_{\ell_{1}}=0$. As explained
earlier in Sec. 3.2, the loop-induced direct search plays an important role as
this rules out leptophilic DM mass $m_{\chi}$ and $\Lambda$ significantly. For
example, (1,1,1,1) case is ruled out upto $(m_{\chi},\Lambda)\sim$(345, 2780)
GeV, and (1,0,0,0) case is ruled out upto $(m_{\chi},\Lambda)\sim$(380, 2700)
GeV. Other choices of model parameters, such as (1,1,0,0), (1,1,1,0) etc. are
very close to the (1,0,0,0) case and hence we did not show them here. Note
that for the cases with $c_{\ell_{1}}=0$, the spin-independent direct search
cross-section vanishes, and there is no bound on $\Lambda-m_{\chi}$ parameter
space from direct detection. Also notice that for the same $m_{\chi}$, the
leptophilic cases requires a lower $\Lambda$ than the hadrophilic case to
satisfy relic abundance. This stems from larger annihilation cross-section due
to the color factor to quark final states, which therefore requires a
comparatively larger $\Lambda$ to tame down the annihilation cross-section to
satisfy the DM relic.

The same exercise is done for Majorana DM in the top right panel of Fig. 3 for
both leptophilic (red, blue, and magenta) and hadrophilic (cyan) case.
Couplings $(c_{\ell_{2}},c_{\ell_{3}})$ are used as an ordered pair for
defining Majorana DM models. We again see here, that the hadrophilic case
requires larger $\Lambda$ than the leptophilic cases following the same line
of arguments as before. Here one should notice the presence of two prominent
bumps corresponding to the opening of $b\bar{b}$ and $t\bar{t}$ final states
for the hadrophilic case. For Majorana DM, there is no significant direct
detection bound because of the absence of DM vector current, and due to the
fact that the contribution of ${\cal O}_{L}^{6}$ (see Eq. (3)) to the direct
detection cross-section vanishes in the non-relativistic limit. This makes all
of the relic density allowed parameter space available starting from DM mass
as low as 1 GeV with $\Lambda\sim 10~{}\rm GeV$.

We have also shown the allowed parameter space corresponding to the Higgs
portal operators with $(c_{H_{i}}=1,c_{q_{i}}=c_{\ell_{i}}=0)$ case for
comparison in the bottom panel of Fig. 3. The vertical black dashed lines
correspond to (from left to right) resonances due to $Z$ and Higgs boson,
opening of $Z$, Higgs and top quark final states respectively. Here we see
both relic density and direct detection bounds are satisfied for
$m_{\chi}\gtrsim 6.2~{}\rm TeV$ and $\Lambda\gtrsim 7~{}\rm TeV$, again making
this model difficult to probe in ongoing collider experiment. We carry out the
analysis of viable parameter space with negative Wilson coefficients in
Appendix B.

Figure 4: Indirect detection bound on Dirac DM annihilation cross-section. The
gray region is excluded from the combined data analysis of MAGIC cherenkov
telescopes and Fermi large area telescope (FermiLAT) that searched for the
gamma-ray signals from DM annihilation MAGIC:2016xys . The blue points
represent thermally averaged annihilation cross-section ($\langle\sigma
v\rangle)$ for DM annihilating into $\tau^{+}\tau^{-}$ pair. As can be seen
from the figure, this bound rules out DM having mass below $30$ GeV.  Figure
5:  Benchmark points (BP4 in red, BP5 in purple and BP6 in magenta: see table
3) for Dirac DM confronted with XENON1T bound shown by gray excluded region in
$m_{\chi}$ (in GeV) versus the spin-independent direct detection cross-section
$\sigma_{\rm SI}$ (in cm2) plane. The Yellow region at the bottom corresponds
to neutrino floor. Note that Majorana DM do not face any direct search
constraints in leptophilic case as considered.

Let us now turn to indirect search bounds. The constraint on thermally-
averaged cross-section for $\tau^{+}\tau^{-}$ final state in our model yields
a bound on Dirac DM mass, namely $m_{\chi}\lesssim 30$ GeV (Fig. 4). This
bound is much less stringent than the bound from direct detection (as in Fig.
3). In further analysis for collider search, we therefore address benchmark
points where DM constraints are coming mainly from relic density and direct
search101010A recent analysis concerning bound on leptophilic DM from AMS-02
data can be found in John:2021ugy .. A few such benchmark points are tabulated
in table 3 where we specify the DM mass ($m_{\chi}$), corresponding NP scale
($\Lambda$) and Wilson coefficients as input model parameter in an ordered set
$(c_{\ell_{1}},c_{\ell_{2}},c_{\ell_{3}},c_{\ell_{4}})$ for Dirac case and
$(c_{\ell_{2}},c_{\ell_{3}})$ for Majorana DM cases. The benchmark points for
Dirac DM are also represented in the spin-independent direct detection cross-
section ($\sigma_{\rm SI}$ (in cm2) plane) versus DM mass plane in Fig. 5 with
XENON1T bound. As Majorana DM do not face any bound from direct search,
corresponding benchmark points can’t be shown in this plane.

DM scenario | Benchmark Points | Model |  $m_{\chi}$ (GeV) |  $\Lambda$ (GeV)
---|---|---|---|---
| BP1 | $(1,1)$ | 70 | 816
Majorana | BP2 | $(1,1)$ | 112 | 1025
| BP3 | $(1,-1)$ | 70 | 990
| BP4 | $(1,1,1,1)$ | 350 | 2800
Dirac | BP5 | $(1,1,1,1)$ | 400 | 3000
| BP6 | $(1,1,-1,-1)$ | 350 | 3300
Table 3: Benchmarks points satisfying DM relic density and direct detection
constraints for Majorana (BP1, BP2, BP3) and Dirac (BP4, BP5, BP6) DM scenario
considering all leptophilic operators together. Different choices for the
Wilson coefficients are denoted as different models and represented as an
ordered set of allowed $\\{c_{\ell_{i}}\\}$. For example, $(1,1,-1,-1)$ in BP6
refers to the choice
$(c_{\ell_{1}}=1,c_{\ell_{2}}=1,c_{\ell_{3}}=-1,c_{\ell_{4}}=-1)$, whereas
$(1,-1)$ in BP3 denotes $(c_{\ell_{2}}=1,c_{\ell_{3}}=-1)$.

Before closing this section, we would like to mention that the recent result
from PandaX-4T PandaX-4T:2021bab  puts a stringent bound on the spin-
independent DM-nucleon cross-section, for example, $3.3\times
10^{-47}~{}\text{cm}^{2}(90\%$ C.L.) for $\hbox{m}_{\chi}\sim$ 30 GeV.
Satisfying this exclusion limit follows a similar exercise as above and
requires an even larger mass ($m_{\chi}$) and effective scale ($\Lambda$) for
Dirac DM (when $c_{\ell_{1}}\neq 0$), like
$\\{m_{\chi},\Lambda\\}:\\{430,3150\\}~{}\rm GeV$. Note that this can still be
probed at a 1 TeV lepton collider. However, for further analysis we content
ourselves with the already published limit from XENON1T data XENON:2018voc .

##  4 Collider search prospects of Leptophilic DM

It is obvious that DM operators that connects to SM leptons have very
suppressed production (via loop) at the currently running LHC and given the SM
background contribution, it is difficult to find them at LHC.

Figure 6: Mono-photon with missing energy $\gamma+\not{E_{T}}$ signature at
the collider where the photon is irradiated from the initial state charged
leptons. The blob represents the effective nature of the vertex.

On the other hand, such operators can be probed at $e^{+}e^{-}$ machine. The
main production of DM occurs via the contact interaction dictated by the EFT
operators. However, DM being invisible at the detector a pair production of DM
would not produce a signal at the detector, so one needs to rely upon one or
more initial state radiation (ISR) photon. Therefore, the signal in such a
case is predominantly given by

$\displaystyle
e^{+}e^{-}\rightarrow\chi\bar{\chi}+\gamma\rightarrow\text{ME}+\gamma\,;$ (22)

where ME refers to missing energy with monophoton. The Feynman graph is shown
in Fig. 6. Note that higher multiplicity of radiated photon will diminish the
cross-section of the process further. Therefore, mono-photon events with
missing energy is a vanilla EFT signature of DM production at colliders that
have been extensively studied in the literature in context with both hadron
and lepton colliders.

As has been emphasized already, in a hadron collider, since the partonic CM
frame is not the same as the hadronic CM frame, it is rather inconsistent to
apply effective theory. The leptonic collider, on the other hand, provides a
simple kinematics where a higher NP scale ($\Lambda$) compared to the known CM
energy ($\sqrt{s}$), with $\Lambda>\sqrt{s}$ makes the EFT limit validated. As
we have seen from the previous section, for a hadrophilic Dirac DM, the spin-
independent direct search limit on the relic density allowed parameter space
becomes so severe that it only allows DM as heavy as about $m_{\chi}\sim$ 6
TeV with the NP scale as large as $\Lambda\sim$ 18 TeV, that lies almost out
of LHC sensitivities. For a leptophilic Dirac DM, on the other hand, it is
possible to bring down the allowed DM mass to about 345 GeV with $\Lambda\sim
2.8~{}\rm TeV$. Such a DM can not certainly be pair-produced at a lepton
collider with $\sqrt{s}=500~{}\rm GeV$, but can be probed with CM energy of 1
TeV. For leptophilic Majorana DM since all of its relic density satisfying
parameter space is also allowed by direct search limits, this scenario is
testable at a lepton collider with CM energy of 500 GeV or even below. In the
following we perform event-level collider simulation for both Dirac and
Majorana DM at those select benchmark points mentioned in Table 3.

It is worth mentioning here that constraints on leptophilic DM from mono-
photon searches at LEP have been analyzed in Fox:2011fx  using effective
interactions and investigated with experimental data from DELPHI
DELPHI:2003dlq ; DELPHI:2008uka . This results in a bound on the effective DM-
electron coupling scale, $\Lambda:250-500$ GeV for DM mass $m_{\chi}\lesssim
80$ GeV, depending on the operator structure. The most stringent bound on
$\Lambda$ arises from vector current (i.e., $O^{6}_{DL}$ in our case) which
disallows $\Lambda<500$ GeV at 90% C.L. for $m_{\chi}\lesssim 60$ GeV. For DM
mass $m_{\chi}\gtrsim 100$ GeV, LEP is completely insensitive for kinematic
reasons. On the other hand, LHC bound on leptophilic DM has been discussed in
Bell:2014tta , where the DM communicates with the SM via leptophilic
$Z^{\prime}$ and the interaction with quarks (partons) take place via 1-loop.
However, the bounds are model-dependent, and analyzed for $4\ell+\not{E_{T}}$
final states, taking each lepton flavour in turn. Hence, this is not directly
applicable for our framework. In Table 3, all the benchmark points are well
above the LEP exclusion limit.

###  4.1 DM Production cross-section and beam polarization

The central point of the collider signal and statistical analysis depend on
the DM production cross-section at $e^{+}e^{-}$ collider. Here we elaborate on
that. Since the electroweak part of the SM is chiral, appropriate beam
polarization may be helpful to reduce SM backgrounds and increase NP signal,
so we focus on polarized beams. The general expression for differential cross-
section of the process $e^{+}e^{-}\rightarrow\chi\bar{\chi}$ with partial beam
polarization ($-1<P_{e^{\pm}}<1$)111111Note that
$P_{e^{\pm}}=\frac{n_{R}-n_{L}}{n_{R}+n_{L}}$ refers to longitudinal beam
polarisation containing a mixture of $n_{L}$ left handed and $n_{R}$ right
handed electrons or positrons. is given by Fujii:2018mli ; Bambade:2019fyw

$\begin{split}\frac{d\sigma(P_{e^{-}},P_{e^{+}})}{d\Omega}&=\frac{1}{4}\Biggl{\\{}\Bigl{(}1+P_{e^{-}}\Bigr{)}\Bigl{(}1+P_{e^{+}}\Bigr{)}\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm
RR}+\Bigl{(}1-P_{e^{-}}\Bigr{)}\Bigl{(}1-P_{e^{+}}\Bigr{)}\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm
LL}\\\
&+\Bigl{(}1+P_{e^{-}}\Bigr{)}\Bigl{(}1-P_{e^{+}}\Bigr{)}\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm
RL}+\Bigl{(}1-P_{e^{-}}\Bigr{)}\Bigl{(}1+P_{e^{+}}\Bigr{)}\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm
LR}\Biggr{\\}}\,,\end{split}$ (23)

where $\sigma_{ij}$ with $i,j\in L,R$ is the cross section for a given process
with completely polarized beams of the four possible orientations. The final
state of our interest are fermion pairs and the operators that mediate the DM-
SM interactions are either vector or axial-vector operators. Hence, spin
conservation indicates that the total spin of initial state should either be
$\pm 1$, indicating that $\sigma_{\rm RR}=\sigma_{\rm LL}=0$ always, reducing
the cross-section to the form:

$\displaystyle\frac{d\sigma(P_{e^{-}},P_{e^{+}})}{d\Omega}$
$\displaystyle=\frac{(1+P_{e^{-}})(1-P_{e^{+}})}{4}\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm
RL}$
$\displaystyle+\frac{(1-P_{e^{-}})(1+P_{e^{+}})}{4}\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm
LR}\,,$ $\displaystyle=\sum_{i=1}^{3}g_{i}f_{i};$ (24)

where

$\displaystyle\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}^{\tt dirac}_{\rm RL}$
$\displaystyle=\frac{s\sqrt{1-\beta_{\chi}^{2}}}{32\pi^{2}\Lambda^{4}}\bigg{[}\Big{\\{}\
(1+\beta^{2}_{\chi})(c_{\ell_{1}}+c_{\ell_{4}})^{2}+(1-\beta^{2}_{\chi})(c_{\ell_{2}}+c_{\ell_{3}})^{2}\Big{\\}}+4\sqrt{1-\beta^{2}_{\chi}}$
$\displaystyle\quad\Big{\\{}(c_{\ell_{1}}+c_{\ell_{4}})(c_{\ell_{2}}+c_{\ell_{3}})\Big{\\}}\cos\theta+(1-\beta^{2}_{\chi})\Big{\\{}(c_{\ell_{1}}+c_{\ell_{4}})^{2}+(c_{\ell_{2}}+c_{\ell_{3}})^{2}\Big{\\}}\cos^{2}\theta\bigg{]}\
,$ (25)

$\displaystyle\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}^{\tt dirac}_{\rm LR}$
$\displaystyle=\frac{s\sqrt{1-\beta_{\chi}^{2}}}{32\pi^{2}\Lambda^{4}}\bigg{[}\Big{\\{}\
(1+\beta^{2}_{\chi})(c_{\ell_{1}}-c_{\ell_{4}})^{2}+(1-\beta^{2}_{\chi})(c_{\ell_{2}}-c_{\ell_{3}})^{2}\Big{\\}}+4\sqrt{1-\beta^{2}_{\chi}}$
$\displaystyle\quad\Big{\\{}(c_{\ell_{4}}-c_{\ell_{1}})(c_{\ell_{2}}-c_{\ell_{3}})\Big{\\}}\cos\theta+(1-\beta^{2}_{\chi})\Big{\\{}(c_{\ell_{1}}-c_{\ell_{4}})^{2}+(c_{\ell_{2}}-c_{\ell_{3}})^{2}\Big{\\}}\cos^{2}\theta\bigg{]}\
.$ (26)

Now, we decompose the differential cross-section for Dirac DM in terms of
$g_{i}$ and $f_{i}$, where $g_{i}$ represents dimensionless coupling
coefficients, and $f_{i}$ contains functions involving phase space variable
($\cos\theta$ here, where $\theta$ is the angle of emergence of $\chi$ from
beam axis), then a linearly independent set of $f_{i}$ can be achieved as
follows:

$\\{f_{1},\,f_{2},\,f_{3}\\}=\frac{s\sqrt{1-\beta_{\chi}^{2}}}{32\pi^{2}\Lambda^{4}}\left\\{1,\,4\sqrt{1-\beta_{\chi}^{2}}\cos\theta,\,(1-\beta_{\chi}^{2})\cos^{2}\theta\right\\}\,,$
(27)

with

$\displaystyle g_{1}=$
$\displaystyle\frac{1-P_{e^{-}}P_{e^{+}}}{2}\bigg{[}\bigg{\\{}(c_{\ell_{1}}^{2}+c_{\ell_{4}}^{2})(1+\beta_{\chi}^{2})+(c_{\ell_{2}}^{2}+c_{\ell_{3}}^{2})(1-\beta_{\chi}^{2})\bigg{\\}}+$
$\displaystyle{\hskip
56.9055pt}+2\,P_{\text{eff}}\Bigl{\\{}c_{\ell_{1}}\,c_{\ell_{4}}\,\left(1+\beta_{\chi}^{2}\right)+c_{\ell_{2}}\,c_{\ell_{3}}\,\left(1-\beta_{\chi}^{2}\right)\Bigr{\\}}\bigg{]};$
$\displaystyle g_{2}=$
$\displaystyle\frac{1-P_{e^{-}}P_{e^{+}}}{2}\bigg{[}\bigg{\\{}c_{\ell_{1}}c_{\ell_{2}}+c_{\ell_{3}}c_{\ell_{4}}\bigg{\\}}+P_{\rm
eff}\bigg{\\{}c_{\ell_{1}}c_{\ell_{3}}+c_{\ell_{2}}c_{\ell_{4}}\bigg{\\}}\bigg{]};$
$\displaystyle g_{3}=$
$\displaystyle\frac{1-P_{e^{-}}P_{e^{+}}}{2}\bigg{[}\bigg{\\{}c_{\ell_{1}}^{2}+c_{\ell_{2}}^{2}+c_{\ell_{3}}^{2}+c_{\ell_{4}}^{2}\bigg{\\}}+2P_{\rm
eff}\bigg{\\{}c_{\ell_{1}}c_{\ell_{4}}+c_{\ell_{2}}c_{\ell_{3}}\bigg{\\}}\bigg{]}.$
(28)

Here, we define

$P_{\rm eff}=\frac{P_{e^{-}}-P_{e^{+}}}{1-P_{e^{-}}P_{e^{+}}},\hskip
28.45274pt\beta_{\chi}=\frac{2m_{\chi}}{\sqrt{s}}.$ (29)

This decomposition as $\frac{d\sigma}{d\Omega}=\sum g_{i}f_{i}$ turns crucial
for statistical analysis taken up in the next section. For Majorana DM, we can
use Eq. (24) for generic differential cross-section along with:

$\displaystyle\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}^{\tt majorana}_{\rm RL}$
$\displaystyle=\frac{s}{16\pi^{2}\Lambda^{4}}\Big{[}\Big{(}c_{\ell_{2}}+c_{\ell_{3}}\Big{)}^{2}\Big{(}1-\beta_{\chi}^{2}\Big{)}^{3/2}(1+\cos^{2}\theta)\Big{]}\
,$ (30)

and

$\displaystyle\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}^{\tt majorana}_{\rm LR}$
$\displaystyle=\frac{s}{16\pi^{2}\Lambda^{4}}\Big{[}\Big{(}c_{\ell_{2}}-c_{\ell_{3}}\Big{)}^{2}\Big{(}1-\beta_{\chi}^{2}\Big{)}^{3/2}(1+\cos^{2}\theta)\Big{]}\
.$ (31)

Note here that for Majorana DM scenario, no suitable choice of linearly
independent set of $f_{i}$’s can be found as the phase space dependence to all
combinations of $c_{i}$ remains the same, which prohibits a possible optimal
statistical analysis to extract the corresponding Wilson coefficients for
Majorana DM EFT operators.

Furthermore, it is evident from Eq. (25) and Eq. (26) that the combination
$\\{c_{\ell_{2}},c_{\ell_{3}}\\}$, and $\\{c_{\ell_{1}},c_{\ell_{4}}\\}$
breaks the Parity symmetry, as these specific combination of coefficients
gives rise to $V\pm A$ type interactions. It is then straightforward to find
the analytical expression of unpolarized cross-sections ($\sigma_{0}$) for
leptophilic Majorana and Dirac DM production via designated operators as given
by,

$\displaystyle\sigma^{\tt dirac}_{0}$ $\displaystyle=$
$\displaystyle\frac{s}{12\pi\Lambda^{4}}\sqrt{1-\beta_{\chi}^{2}}\bigg{[}\Big{(}c_{\ell_{1}}^{2}+c_{\ell_{4}}^{2}\Big{)}\Big{(}1+\frac{\beta_{\chi}^{2}}{2}\Big{)}+\Big{(}c_{\ell_{2}}^{2}+c_{\ell_{3}}^{2}\Big{)}\Big{(}1-\beta_{\chi}^{2}\Big{)}\bigg{]}\,,$
(32) $\displaystyle=$
$\displaystyle\frac{s}{3\pi\Lambda^{4}}\left(1-\frac{m_{\chi}^{2}}{s}\right)\sqrt{1-\frac{4m_{\chi}^{2}}{s}}\,;~{}{\rm
where}~{}c_{\ell_{i}}=1~{}(i=1{\text{-}}4)\,,$

and

$\displaystyle\sigma^{\tt majorana}_{0}$ $\displaystyle=$
$\displaystyle\frac{s}{6\pi\Lambda^{4}}\Big{(}c_{\ell_{2}}^{2}+c_{\ell_{3}}^{2}\Big{)}\left(1-\beta_{\chi}^{2}\right)^{3/2}\,,$
(33) $\displaystyle=$
$\displaystyle\frac{s}{3\pi\Lambda^{4}}\left(1-\frac{4m_{\chi}^{2}}{s}\right)^{3/2}\,;~{}~{}~{}{\rm
where}~{}c_{\ell_{i}}=1~{}(i=2,3)\,.$

It is worthy mentioning that polarised cross-section can also be written as
Bambade:2019fyw

$\displaystyle\sigma\Bigl{(}P_{e^{-}},P_{e^{+}}\Bigr{)}=2\sigma_{0}\,\Bigl{(}\mathcal{L}_{\text{eff}}/\mathcal{L}\Bigr{)}\,\Bigl{[}1-\mathcal{A}_{LR}\,P_{\text{eff}}\Bigr{]}\,;$
(34)
$\displaystyle\sigma_{0}=\frac{1}{4}\Biggl{[}\sigma_{\text{RR}}+\sigma_{\text{LL}}+\sigma_{\text{RL}}+\sigma_{\text{LR}}\Biggr{]}\,;~{}~{}\mathcal{L}_{\text{eff}}=\frac{1}{2}\Bigl{(}1-P_{e^{-}}\,P_{e^{+}}\Bigr{)}\mathcal{L}\,;$
(35)

where $\mathcal{L}_{\text{eff}}$ refers to effective Luminosity of ILC.

We further note from the Eqs. (25), (26), (31) and (30) that in the limit of
all $c_{\ell_{i}}=1$:

$\sigma_{LR}=0;~{}~{}\mathcal{A}_{LR}=\frac{\sigma_{\text{LR}}-\sigma_{\text{RL}}}{\sigma_{\text{LR}}+\sigma_{\text{RL}}}=-1.$
(36)

The above feature can also be understood from the spinor structure for
$e^{+}e^{-}$. Consider for example, the electron current
$\bar{v}(p^{\prime})\gamma^{\mu}\gamma_{5}u(p)$ corresponding to Majorana DM
operator $\mathcal{O}_{L1}^{6}$. Let us now consider right-polarised electron
and replace the current by
$\bar{v}(p^{\prime})\gamma^{\mu}\gamma_{5}\mathcal{P}_{R}u(p)$, where
$\mathcal{P}_{R}=\frac{1+\gamma_{5}}{2}$ represents right chiral projection
operator. Clearly this is same as
$(\mathcal{P}_{R}v(p^{\prime}))^{\dagger}\gamma^{0}\gamma^{\mu}\gamma_{5}u(p)$,
which is non-zero only for left-handed positrons, i.e. with $RL$ combination.
Trivially, this can be seen to be true for the opposite helicity combination
by replacing $\mathcal{P}_{R}\rightarrow\mathcal{P}_{L}$, i.e. with $LR$
combination. When we take all operators together (for both Dirac and Majorana
cases), we practically sum the corresponding amplitudes. In the limit of
$c_{\ell_{i}}=1$, this results in

$\displaystyle\mathcal{L}_{\rm
eff}=\sum_{i}\mathcal{O}_{i}\sim\begin{cases}\left(\bar{\chi}\gamma^{\mu}\gamma^{5}\chi\right)\bar{e}\gamma_{\mu}\left(1+\gamma^{5}\right)e&\text{Majorana
DM}\\\
\left(\bar{\chi}\gamma^{\mu}\gamma^{5}\chi+\bar{\chi}\gamma^{\mu}\chi\right)\bar{e}\gamma_{\mu}\left(1+\gamma^{5}\right)e&\text{Dirac
DM}\end{cases}.$ (37)

This expression is non-vanishing only when ${\mathcal{P}_{R}\,e}\neq 0$, i.e.,
the electron is right-handed. This automatically implies that the positron is
left-handed, resulting in a non-vanishing $\sigma_{RL}$ combination. In other
words, we infer that the signal cross-section is maximum for fully right
handed electron and left handed positron ($\\{+,-\\}$ combination) of the
$e^{-}e^{+}$ beam. This is evident from Table 4, where we show the signal
production cross-section for the benchmark points for different choices of the
beam polarization. Note here following ILC design, the maximum right polarised
electron and left polarised positron is possible upto
$\\{P_{e^{-}}:P_{e^{+}}\\}=\\{+80\%:-30\%\\}$, which we abide by.

A different choice for the signs of the Wilson coefficients will however alter
this behaviour. Let us consider a distinct set of values for the Wilson
coefficients, namely, $c_{\ell_{i}}\equiv\\{1,1,-1,-1\\}$ for Dirac DM and
$c_{\ell_{i}}\equiv\\{1,-1\\}$ for Majorana DM, which yields effective
Lagrangian of the form

$\displaystyle\mathcal{L}_{\rm
eff}=\begin{cases}\left(\bar{\chi}\gamma^{\mu}\gamma^{5}\chi+\bar{\chi}\gamma^{\mu}\chi\right)\,\bar{e}\gamma_{\mu}\left(1-\gamma^{5}\right)e&\text{Dirac
DM}\\\
\left(\bar{\chi}\gamma^{\mu}\gamma^{5}\chi\right)\,\bar{e}\gamma_{\mu}\left(1-\gamma^{5}\right)e&\text{Majorana
DM}\end{cases}$ (38)

which clearly shows that the interaction vanishes for $\mathcal{P}_{L}\,e=0$
and results in $\sigma_{\text{RL}}=0$ as can also be inferred from Eq. (25).
As a result, for this case, ${\mathcal{A}_{LR}}=+1$, and one has to choose
left-polarized electron and right-polarized positron beam to maximize the
signal production cross-section. This choice of the beam polarization, in
context with the ILC, is maximally possible with
$\\{P_{e^{-}}:P_{e^{+}}\\}=\\{-80\%:+30\%\\}$. However, this choice will
result in a suppressed signal cross-section compared to the SM background, as
seen from Table 4 and Table 5. Consequently, the signal significance will
deteriorate, making this Dirac DM scenario more difficult to probe at the ILC.
Therefore, we do not consider $\\{1,1,-1,-1\\}$ combination of the Wilson
coefficients for event simulation. Worthy to mention here that, in analyzing
the signal and the background we have adopted the ILC specifications regarding
beam bunch length, bunch population, horizontal and vertical beam size etc
following Behnke:2013xla .

Beam | Production cross-section ($\sigma_{e^{+}e^{-}\rightarrow\chi\bar{\chi}}$ ) (pb)
---|---
polarization | Majorana DM ($\sqrt{s}=$ 250 GeV) | Dirac DM ($\sqrt{s}=$ 1 TeV)
$P_{e^{-}}$ | $P_{e^{+}}$ | BP1 | BP2 | BP3 | BP4 | BP5 | BP6
-0.8 | +0.3 |  0.40 |  0.02 |  0.78 |  0.05 |  0.03 |  0.42
+0.8 | -0.3 |  6.73 |  0.34 |  0.05 |  0.81 |  0.47 |  0.03
0.0 | 0.0 |  2.87 |  0.14 |  0.33 |  0.35 |  0.20 |  0.18
Table 4: Production cross-section for the benchmark points in table 3 for
different choices of the beam polarization.

Figure 7: $\sigma_{e^{+}e^{-}\rightarrow\chi\bar{\chi}}$ as function of
$\sqrt{s}$ (top panel), $\Lambda$ (middle panel) and $m_{\chi}$ (bottom panel)
for Majorana DM (BP1) in the left column and Dirac DM (BP4) in the right
column for three choices of polarisation:
$\\{P_{e^{-}}:P_{e^{+}}\\}=\\{0,0\\};\\{+0.8,-0.3\\};\\{-0.8,+0.3\\}$ as
mentioned in the figure inset. The other parameters kept fixed are mentioned
in Figure heading.

We plot the variation of DM production cross-section
($\sigma_{e^{+}e^{-}\rightarrow\chi\bar{\chi}}$) for Majorana DM (BP1) in the
left column and Dirac DM (BP4) in the right column of Fig. 7. In the top,
middle and bottom panel we show the variation with respect to $\sqrt{s}$,
$\Lambda$ and $m_{\chi}$ respectively each for three different choices of the
beam polarization:
$\\{P_{e^{-}}:P_{e^{+}}\\}=\\{0,0\\};\\{+0.8,-0.3\\};\\{-0.8,\\\ +0.3\\}$
shown by solid, dashed-dot and dashed black (red) curves for Majorana (Dirac)
DM cases with all accessible operators put together. The other parameters kept
fixed are mentioned in the Figure headings. We find the signal cross-section
to be maximum for the $\\{+80\%,-30\%\\}$ case as elaborated above, with all
Wilson coefficients chosen positive. The shape of the plot in the left panel
(for Majorana DM) and in the right panel (for Dirac DM) can be verified from
the Eq. (33) and Eq. (32) respectively where we find that total cross-section
to increase with the CM energy ($\sqrt{s}$) upto effective limit
$\Lambda>\sqrt{s}$ (top panel). In the middle panel, the cross-section falls
with larger $\Lambda$ as $\sigma_{e\bar{e}\rightarrow\chi\bar{\chi}}\sim
1/\Lambda^{4}$ for a fixed DM mass. The cross-section slowly falls with
$m_{\chi}$ and vanishes for $\sqrt{s}\lesssim 2m_{\chi}$, as expected from the
phase-space dependence shown in the bottom panel of Fig. 7.

Figure 8: Top: SM background due to $e^{+}e^{-}\rightarrow\nu\bar{\nu}$ process. Bottom: Bhabha scattering contributing to the background for mono-photon with missing energy process where both $e^{+},e^{-}$ in the final state are undetected in the collider. In all cases radiative photons may arise either from the initial or final state radiation or both. $\sqrt{s}$ | $P_{e^{-}}$ | $P_{e^{+}}$ | $\sigma_{ee\rightarrow\nu\bar{\nu}}$ | $\sigma_{ee\rightarrow ee}$
---|---|---|---|---
(GeV) |  |  | (pb) | (pb)
| -0.8 | +0.3 | 102.0 | 618.8
250 | +0.8 | -0.3 | 14.72 | 628.3
| 0.0 | 0.0 | 52.71 | 610.3
| -0.8 | +0.3 | 130.93 | 622.06
1000 | +0.8 | -0.3 | 7.88 | 603.33
| 0.0 | 0.0 | 56.03 | 596.29
Table 5: Production cross-section for dominant SM backgrounds at $\sqrt{s}=$
250 and 1000 GeV.

Since we are typically interested in mono-photon signal with missing energy at
the $e^{+}e^{-}$ collider, there are two main SM backgrounds which can mimic
such a signal final state

  * •

$e^{+}e^{-}\rightarrow\nu\bar{\nu}\gamma$ is an irreducible background
Liu:2019ogn  since neutrinos will also be missed inside the detector like the
DM. Such a process is contributed by two Feynman diagrams: one involves the
$t$-channel $W$ boson exchange and the other involves the $s$-channel $Z$
boson exchange as shown in the top panel of Fig. 8.

  * •

Another possible background is the radiative Bhabha scattering
$e^{+}e^{-}\rightarrow e^{+}e^{-}\gamma$ as shown in the bottom panel of Fig.
8, where both the final state leptons go undetected. Note here, that
$e^{+}e^{-}\rightarrow f\bar{f}\gamma$ where $f$ is any SM fermion, lepton or
jet forms a subset for this class of backgrounds, where $f$ has to be missed
in the detector.

We tabulate the production cross-section dominant SM background in table 5 for
different choices of the beam polarizations which can be realized in ILC set-
up. The SM background contribution varies quite differently with polarization.
This can be understood easily. For $e^{+}e^{-}\rightarrow\nu\bar{\nu}$, the
Feynman diagrams involving $t$-channel $W$ boson exchange only contributes to
a non-zero $\sigma_{\text{LR}}$ since only left-handed fermions and right-
handed anti-fermions take part in the weak charge current interaction. This
indicates that the asymmetry parameter $\mathcal{A}_{\text{LR}}=1$. For the
$s$-channel diagram for $e^{+}e^{-}\rightarrow\nu\bar{\nu}$, exchange of a
vector boson ($Z$-boson) ensures non-zero $\sigma_{\text{LR,RL}}$ due to spin
conservation. Since the coupling of $Z$-boson to $e_{L}^{-}e_{R}^{+}$ is
stronger than $e_{R}^{-}e_{L}^{+}$, the diagrams involving $s$-channel
$Z$-exchange has more contributions to $\sigma_{\text{LR}}$ than to
$\sigma_{\text{RL}}$. Therefore, for
$\sigma_{e^{+}e^{-}\rightarrow\nu\bar{\nu}}$ diminishes for polarization
configuration $\\{P_{e^{-}}:P_{e^{+}}\\}=\\{+0.8,-0.3\\}$, contrary to the
signal for operators with all positive Wilson coefficients, which provides the
best way to probe leptophilic DM at ILC. Bhaba scattering
($e^{+}e^{-}\rightarrow e^{+}e^{-}\gamma$) cross-section although still looks
quite substantial, but we will be able to tame it down significantly, which we
discuss in context of the event selection criteria.

###  4.2 Event distribution and analysis

We generate the parton-level signal events for the benchmark points in Table 3
using the batch mode of CalcHEP-3.8.10 Belyaev:2012qa . The events are then
fed to Pythia-6.4.28 Sjostrand:2006za  for showering utilising in-built
switches for initial and final state radiation (ISR/FSR). The SM background
events are generated using MadGraph Alwall:2014hca  and then the event files
are again analysed through Pythia. In collider environment, we reconstruct the
following objects and define a few observables as

  * •

Lepton ($l=e,\mu$): Leptons are required to have a minimum transverse momentum
$p_{T}>20$ GeV and pseudorapidity $|\eta|<2.5$. Two leptons are isolated
objects if their mutual distance in the $\eta-\phi$ plane is $\Delta
R=\sqrt{\left(\Delta\eta\right)^{2}+\left(\Delta\phi\right)^{2}}\geq 0.2$,
while the separation between a lepton and a jet has to satisfy $\Delta R\geq
0.4$.

  * •

Jets ($j$): All the partons within $\Delta R=0.4$ from the jet initiator cell
are included to form the jets using the cone jet algorithm PYCELL built in
Pythia. We require $p_{T}>10$ GeV for a clustered object to be considered as
jet. Jets are isolated from unclustered objects if $\Delta R>0.4$.

  * •

Photons ($\gamma$): Photons are identified to register in the detector with
minimum energy $E_{\gamma}>0.1$ GeV.

  * •

Unclustered Objects: All the final state objects which are neither clustered
to form jets, nor identified as leptons, belong to this category. All
particles with $0.5<p_{T}<20$ GeV and $|\eta|<5$, are considered as
unclustered.

  * •

Missing Transverse Energy or MET ($\not{E}_{T}$): The transverse momentum of
all the missing particles (those are not registered in the detector) can be
estimated from the momentum imbalance in the transverse direction associated
to the visible particles. Thus MET is defined as:

$\displaystyle\not{E}_{T}=-\sqrt{(\sum_{\ell,j}p_{x})^{2}+(\sum_{\ell,j}p_{y})^{2}},$
(39)

where the sum runs over all visible objects that include the leptons, jets and
the unclustered components.

  * •

Missing Energy or ME ($\not{E}$): The energy which is carried away by the
missing particles can be identified at lepton collider given the knowledge of
com energy of the reaction as

$\displaystyle\not{E}=\sqrt{s}-\sum_{\ell,j,\gamma}E;$ (40)

We eventually put zero lepton and jet veto on the final state events of our
interest. Following detector cuts are further used on the photons identified
in Pythia

  * •

In order to ensure most of the events are localized around the central region
of the detector we choose the pseduorapidity $\left|\eta_{\gamma}\right|\leq
0.24$.

  * •

We also ensure the final state events contain at least one “hard” photon by
choosing a cut on the transverse momentum of the photons: $p_{T}^{\gamma}>5$
GeV.

Figure 9: Top left: Normalized inclusive mono-photon event distribution for
the BP1 and BP2 in table 3 and the SM backgrounds as a function of the photon
energy $E_{\gamma}$. Top right: Normalized distribution of events for the
signal and the SM backgrounds with missing energy (MET). Bottom: Normalized
signal and background distribution with $\theta_{\gamma}$. In all cases the
signals are shown in red and blue curves, while the SM background is shown by
the dashed black curve for $e^{+}e^{-}\rightarrow\nu\bar{\nu}$. We simulate
the events at $\sqrt{s}=250$ GeV with
$\\{P_{e^{-}}:P_{e^{+}}\\}=\\{+0.8:-0.3\\}$ beam polarization.

Figure 10: Same as Fig. 9 with BP4 and BP5, but with $\sqrt{s}=1$ TeV.

Due to collinear singularity, the radiative Bhabha scattering process has
large cross section when both final state electron and positron go along the
beam directions. However, as shown in Liu:2019ogn , such backgrounds can be
efficiently removed by adopting $E_{\gamma}>E_{\text{B}}^{m}(\theta_{\gamma})$
on the final state mono photon with

$\displaystyle
E_{\text{B}}^{m}(\theta_{\gamma})=\sqrt{s}\Biggl{[}1+\frac{\sin\theta_{\gamma}}{\sin\theta_{\text{bnd}}}\Biggr{]}^{-1}$
(41)

where $|\cos\theta_{\text{bnd}}|=0.99$ corresponds to the boundary of the
electromagnetic calorimeter (EMC). Thus, here onwards, we will omit SM
background due to Bhabha scattering process and only consider the irreducible
neutrino background. We would like to further mention that this particular
choice of cut on the final state mono-photon energy (Eq. (41)) also
efficiently eliminates reducible backgrounds due to processes like
$e^{+}\,e^{-}\rightarrow f\,\bar{f}\,\gamma$ and
$e^{+}\,e^{-}\rightarrow\gamma\,\gamma\,\gamma$ as shown in Liu:2019ogn .

The distribution of normalized number of inclusive mono-photon events
corresponding to the signal for Majorana DM as well as to the backgrounds, as
a function of different observables, are illustrated in Fig. 9. As seen from
the top left panel, the mono-photon energy $E_{\gamma}$ spectrum does not
exhibit any peak structure, whereas $e^{+}e^{-}\rightarrow\nu\bar{\nu}\gamma$
shows a peak around $E_{\gamma}=\frac{s-m_{Z}^{2}}{2\sqrt{s}}$. It is easy to
show from the 2-body kinematics, the maximum photon energy due to the signal
is given by Liu:2019ogn

$E_{\gamma}<E_{\chi}^{\text{max}}=\frac{1}{2\sqrt{s}}\left(s-4m_{\chi}^{2}\right)\,,$
(42)

which exactly determines the end-point of the signal distribution. As a
result, we see, for a heavier DM the radiated photons are less energetic. Note
that the distribution for $E_{\gamma}$ is identical to ME ($\not{E}$)
distribution given the presence of only one photon in the event. The
normalized event distribution for MET (top right) are also similar to photon
transverse momenta $p_{T}^{\gamma}$ for the same reason. Also a more massive
DM results in a less-energetic radiated photon since most of the energy is
carried by the DM itself. In the present scenario, since the photons come from
the ISR, they tend to be soft and collinear. This makes their
$\theta_{\gamma}$ distribution rather flat as shown in the bottom panel. A
similar set of distributions is shown for Dirac DM in Fig. 10, with a higher
CM energy and the inferences broadly remain the same. In passing we note that
the distributions as shown in Figs. 9, 10 primarily depend on the kinematics,
therefore conclusions for other benchmark points as in Table 3 can be easily
gauged.

###  4.3 Cut-flow and signal significance

Only for a small fraction of neutrino pair production events the radiated
photon can be measured in the detector Kalinowski:2020lhp . Therefore, as
mentioned earlier, we demand the final event selection to have photons with
transverse momentum $p_{T}^{\gamma}>p_{T}^{\text{min}}=5$ GeV. On top of this
we apply the following cuts in order to reduce the SM background as
efficiently as possible, without harming the signal events.

  * •

Cut1 ($\mathcal{C}_{1}$) : Events with zero lepton and jet-veto with exactly
one photon in the final state.

  * •

Cut2 ($\mathcal{C}_{2}$): We choose photons with energy lying within the
window $0.1\leq E_{\gamma}<60(100)$ GeV or Majorana (Dirac) DM scenario. This
helps to avoid the background events around the $Z$-mass window by retaining
majority of the signal events.

  * •

Cut3 ($\mathcal{C}_{3}$): We then employ an upper cut on photon $p_{T}$ by
demanding $p_{T}^{\gamma}\leq 20$ GeV on top of $\mathcal{C}_{1,2}$.

  * •

Cut4 ($\mathcal{C}_{4}$): Finally we apply a cut on the missing energy
$\not{E_{T}}\leq 30(50)$ GeV for for Majorana (Dirac) DM scenario.

Cuts | BP1 | BP2 | $\nu\bar{\nu}\gamma$ | $\epsilon^{\tt BP1}$ | $\epsilon^{\tt BP2}$
---|---|---|---|---|---
| (fb) | (fb) | (fb) |  |
$\mathcal{C}_{1}$ | 72.53 | 1.32 | 303.62 | 0.011 | 0.004
$\mathcal{C}_{2}$ | 71.99 | 1.31 | 208.91 | 0.011 | 0.004
$\mathcal{C}_{3}$ | 71.08 | 1.31 | 204.69 | 0.011 | 0.004
$\mathcal{C}_{4}$ | 64.26 | 1.29 | 178.08 | 0.010 | 0.004
Table 6: Cut-efficiency for the benchmark points for Majorana DM along with the irreducible SM background.  Cuts | BP4 | BP5 | $\nu\bar{\nu}\gamma$ | $\epsilon^{\tt BP4}$ | $\epsilon^{\tt BP5}$
---|---|---|---|---|---
| (fb) | (fb) | (fb) |  |
$\mathcal{C}_{1}$ | 32.43 | 18.21 | 421.45 | 0.040 | 0.040
$\mathcal{C}_{2}$ | 29.12 | 16.39 | 301.05 | 0.035 | 0.036
$\mathcal{C}_{3}$ | 28.32 | 15.94 | 294.32 | 0.035 | 0.035
$\mathcal{C}_{4}$ | 25.22 | 14.12 | 272.26 | 0.031 | 0.031
Table 7: Cut-efficiency for the benchmark points (BP4, BP5) for Dirac DM
scenario along with the irreducible SM background.
Figure 11: Signal significance for exclusive mono-photon with missing energy
final state events, as a function of integrated luminosity for the chosen
benchmark points in table 3. The black solid, dashed and dotdashed horizontal
straight lines denote respectively 3$\sigma$, 5$\sigma$ and 10$\sigma$
discovery limit. The red (blue) lines correspond to Majorana (Dirac) DM.

In Table 6 we have tabulated the signal for Majorana DM and background event
cross-sections with the cuts employed following the sequence mentioned above.
We find, after applying $\mathcal{C}_{4}$ we are left with considerable final
state signal event cross-section. The background cross-section albeit larger
than any the signal cross-section, but still gives rise to a large
significance typically for the lighter DM with large production cross-section.
A similar observation can be made for Dirac DM from table 7, although we
require higher CM energy to produce them as dictated by relic density and
direct search constraints. We also quote the efficiency factor
$\epsilon=\frac{\sigma^{\tt sig}}{\sigma^{\tt prod}}$, which depicts the loss
of events in the process of employing selection cuts on the final state
events, where $\sigma^{\tt prod(sig)}$ denotes the production (signal)-level
cross-section. The significance of signal events with respect to SM
backgrounds $\sigma=\frac{S}{\sqrt{S+B}}$ is plotted in terms of luminosity in
Fig. 11. Here $S(=\sigma^{\tt sig}\times\mathcal{L})$ denotes the number of
signal (final state) events for a given luminosity, while $B$ corresponds to
the number of background events at the same luminosity. We see that mass of
the DM plays a crucial role; while for a luminosity of 1 $\text{fb}^{-1}$, BP1
(red line) with $m_{\chi}=70$ GeV can reach 5$\sigma$ significance, BP2 (red
dashed) with $m_{\chi}=112$ GeV requires at least
$\mathcal{L}>10^{3}~{}\text{fb}^{-1}$ to be probed with 5$\sigma$ confidence.
The signal cross-section for BP4 and BP5 (Dirac case) is smaller than BP1 and
larger than BP2 (although with a different CM energy), so are the discovery
reaches at those points, as shown by blue thick and dashed lines.

Figure 12: Parameter space in $\Lambda-m_{\chi}$ plane considering the vector
operator $(\overline{\chi}\gamma^{\mu}\chi)(\overline{\ell}\gamma_{\mu}\ell)$,
where the solid lines represent 3$\sigma$ significance contour for ILC at a CM
energy of $\sqrt{s}=1$ TeV with integrated luminosity of
$\\{10,100,300,500\\}\text{fb}^{-1}$ shown in black, red, green, and orange
respectively; the dashed diagonal line corresponds to the Planck observed
relic density for DM and the solid blue curve shows the exclusion limit from
spin-independent XENON1T data.

It is useful to study the interplay of collider vs non-collider bounds in the
context of effective operator as this provides information about the reach for
different experiments in probing the DM parameter space. For hadrophilic DM
such a study has been performed in many instances (for example, in
Belyaev:2018pqr ). Here we would like to illustrate such an example by
considering a single operator in action in context of leptophlic DM scenario.
In Fig. 12 we show the complementarity of direct search sensitivity to that of
the collider search in mono-photon final state for the operator
$\mathcal{O}^{6}_{DL}$ in DM mass ($m_{\chi}$) versus NP scale ($\Lambda$)
plane. The gray shaded region is discarded from the XENON1T exclusion limit,
while along the black dashed straight diagonal line the Planck observed relic
density is satisfied. The black, red, green and orange solid curves indicate
3$\sigma\equiv\frac{S}{\sqrt{S+B}}$ confidence limit at the ILC for a CM
energy of $\sqrt{s}=1$ TeV, corresponding to luminosities
$\mathcal{L}=\\{10,100,300,500\\}\,\text{fb}^{-1}$ respectively. Note that,
all the collider confidence contours converge at $m_{\chi}\simeq\sqrt{s}/2\sim
500$ GeV for a CM energy of 1 TeV. It is easy to understand that with larger
luminosity the collider sensitivity overpowers the direct detection.

##  5 Optimal observable analysis

In this section, we discuss the possibility of extracting Wilson coefficients
($c_{\ell_{i}}$) concerning leptophilic DM operators at $e^{+}e^{-}$ collider
using optimal observable technique (OOT). We will first describe the construct
of an optimal covariance matrix and $\chi^{2}$, followed by 1$\sigma$ contours
in the $c_{\ell_{i}}$ coupling planes. We then comment on the separability of
two such models having different EFT operators.

###  5.1 Optimal uncertainties

The first step is to identify an observable ($\mathcal{O}$), which in context
of a collider search is the differential DM pair production cross-section

$\mathcal{O}=\frac{d\sigma^{e^{+}e^{-}\rightarrow\chi\bar{\chi}}}{d\Omega}=\sum_{i}g_{i}f_{i}(\cos\theta)\,,$
(43)

where the coefficients $g_{i}$ are functions of NP coupling parameters and
numerical constants and $f_{i}$’s are linearly independent set of functions of
the phase space variable $\cos\theta$. A decomposition to this end has already
been made in context of Dirac DM in Eqs. (25), (26), (28), and (27). We will
use the decomposition to construct optimal covariance matrix. We need to
remind at this moment that for Majorana case, such decomposition isn’t
possible given the sensitivity of each coefficient proportional to same phase
space variable. Therefore, we will focus only on extracting the Dirac DM
coefficients in this section.

Following three important caveats: (i) the SM contribution to $\mathcal{O}$ is
negligible, (ii) the experiment runs over a finite time so that event numbers
follow a Poisson distribution, (iii) an optimal weighting function $w_{i}$ can
be defined as Bhattacharya:2021ltd

$w_{i}=\frac{1}{\mathcal{L}}\frac{f_{i}}{\mathcal{O}}\,,$ (44)

the optimal covariance matrix $V$ turns out to be:

$V=\frac{1}{\mathcal{L}}M^{-1}\,;\quad
M_{ij}=\left<{\frac{f_{i}f_{j}}{\mathcal{O}}}\right>\,,$ (45)

where $\bigg{<}..\bigg{>}=\int d\cos\theta$ and $\mathcal{L}$ denotes
integrated luminosity.

If we assume that $g_{i}^{0}$ define the ‘seed’ values of the Wilson
coefficients of the EFT operators, and $g_{i}$ is the measured value that can
be extracted, subject to corresponding statistical uncertainties
$g_{i}-g_{i}^{0}$, then the $\chi^{2}$ function

$\chi^{2}=\epsilon\sum_{\\{i,j\\}=1}^{n}(g_{i}-g_{i}^{0})\,(g_{j}-g_{j}^{0})\left(V_{0}^{-1}\right)_{ij},\qquad
V_{0}=\left.V\right|_{g=g^{0}}\,;$ (46)

describes the statistical uncertainties in the measurement of the coefficients
$g_{i}$; the region $\chi^{2}\leq n^{2}$ therefore corresponds $n\sigma$
standard deviation in the most optimal way as the covariance matrix ($V_{0}$)
constructed is minimal with respect to $g_{i}$ coefficients. This procedure is
thus referred as the optimal observable technique (OOT).

Note here that the decomposition of $g_{i}$ and $f_{i}$ is not unique; any
combination that respects Eq. (43) and keeps $\chi^{2}$ in Eq. (46) invariant
is possible. The seed coefficients $g_{i}^{0}$ take different values depending
on the types of NP considered. Given that a model has parameters
$c_{\ell_{i}}$, then $g_{i}=g_{i}(c_{\ell_{i}})$ and $\delta
g_{i}=g_{i}(c_{\ell_{i}}^{0}+\delta c_{\ell_{i}})-g_{i}(c_{\ell_{i}}^{0})$;
from which the statistical uncertainties and correlations of the
$c_{\ell_{i}}$ can be extracted; as we do in the next section.

###  5.2 1$\sigma$ regions in 2D plane

Figure 13: $1\sigma$ regions for different combinations of model parameters
$\\{c_{\ell_{i}}\\}$ (see plot headings) for both unpolarized (light red) and
polarized ($P_{e^{\pm}}=^{-30\%}_{+80\%}$) beams (in light blue). The
efficiency factor has been taken as $\epsilon=0.001$. The other parameters
kept fixed are $m_{\chi}=350$ GeV, $\sqrt{s}=1$ TeV,
$\mathcal{L}=567\,\text{fb}^{-1}$ and $\Lambda=2800$ GeV.

The 1$\sigma$ regions for different combinations of $c_{\ell_{i}}$ are shown
in Fig. 13 for both unpolarized (light red shaded region) and polarized
($P_{e^{\pm}}=^{-30\%}_{+80\%}$) beams (shown by blue shaded regions). We
consider here $\epsilon=0.001$121212Although we have found $\epsilon\sim 0.03$
previously, here we use a more conservative $\epsilon=0.001$ because in
realistic experimental scenario, we may have less number of final state events
than what we have predicted with more stringent cuts and efficient background
reduction. as reasonable estimate of the efficiency of signal detection. The
combinations of $c_{\ell_{i}}$ used for each plot is mentioned in the figure
heading. We must clarify here, that the covariance matrix $V$ is evaluated
assuming each coefficient $c_{\ell_{i}}$ to be one, while evaluating the
surface, we consider the variation of any two of them, keeping the other two
fixed at the seed values. We see that the correlations for different operators
are different in each of these combinations yielding different shapes and
sizes. Clearly as expected, the uncertainties for unpolarised beam is larger
than the polarized beam, thanks to larger signal cross-section for the
polarised beam as already argued. Note here that the $1\sigma$ regions for
negative Wilson coefficients are shown in Appendix B.

The features are more clear from the 1-$\sigma$ uncertainties tabulated in
Table 8. For example, we can see that when we try to find the correlation
between $\mathcal{O}^{6}_{DL}$ and $\mathcal{O}^{6}_{L}$ (top left panel of
Fig. 13), the error in extracting $c_{\ell_{2}}$ is larger compared to
$c_{\ell_{1}}$, so that $\mathcal{O}^{6}_{L}$ is harder to probe. Similarly,
when $\mathcal{O}^{6}_{DL}$ is correlated to $\mathcal{O}^{6}_{L1}$ (top right
of Fig. 13) then $\mathcal{O}^{6}_{L1}$ is harder to probe, i.e. error in
extracting $c_{\ell_{3}}$ is larger. Now, when $\mathcal{O}^{6}_{DL}$ is
correlated to $\mathcal{O}^{6}_{DL1}$ then both $c_{\ell_{1}}$ and
$c_{\ell_{4}}$ has similar uncertainties, so is the case for
$\mathcal{O}^{6}_{L}$ and $\mathcal{O}^{6}_{L1}$ (middle panel of Fig. 13).
When we see the correlation of $\mathcal{O}^{6}_{L}$ or $\mathcal{O}^{6}_{L1}$
with $\mathcal{O}^{6}_{DL1}$ (bottom panel of Fig. 13) then
$\mathcal{O}^{6}_{DL1}$ is always easier to probe, i.e. the error in
extracting $c_{\ell_{4}}$ is less.

We would like to mention that for two parameter space, as we have done here,
one often uses $\chi^{2}=2.3$, which provides 68% confidence level of
estimating the errors in the NP parameters which is equivalent to 1$\sigma$
error for one parameter case. It is easy to appreciate that this exercise will
just enlarge the $\chi^{2}$ regions by an appropriate factor keeping the major
conclusions unchanged.

Couplings ($c_{i}$’s) | $P_{e^{\pm}}=0$ | $P_{e^{\pm}}=^{-30\%}_{+80\%}$
---|---|---
a | b | $\pm\Delta a$ | $\pm\Delta b$ | $\pm\Delta a$ | $\pm\Delta b$
$c_{1}=1$ | $c_{2}=1$ | $+0.09$ | $+0.23$ | $+0.06$ | $+0.15$
$-0.12$ | $-0.18$ | $-0.08$ | $-0.12$
$c_{1}=1$ | $c_{3}=1$ | $+0.09$ | $+0.61$ | $+0.06$ | $+0.15$
$-0.40$ | $-0.18$ | $-0.08$ | $-0.12$
$c_{1}=1$ | $c_{4}=1$ | $+0.26$ | $+0.26$ | $+0.24$ | $+0.24$
$-0.38$ | $-0.38$ | $-0.33$ | $-0.33$
$c_{2}=1$ | $c_{3}=1$ | $+0.38$ | $+0.38$ | $+0.36$ | $+0.36$
$-0.37$ | $-0.37$ | $-0.36$ | $-0.36$
$c_{2}=1$ | $c_{4}=1$ | $+0.61$ | $+0.09$ | $+0.15$ | $+0.06$
$-0.17$ | $-0.39$ | $-0.12$ | $-0.07$
$c_{3}=1$ | $c_{4}=1$ | $+0.23$ | $+0.09$ | $+0.15$ | $+0.06$
$-0.18$ | $-0.12$ | $-0.12$ | $-0.07$
Table 8: Optimal 1$\sigma$ statistical uncertainty in the $c_{\ell_{i}}$
couplings for both unpolarized and polarized ($P_{e^{\pm}}=^{-30\%}_{+80\%}$)
beams. The efficiency factor has been taken as $\epsilon=0.001$ in agreement
to the signal estimate. The illustrative choices of $m_{\chi}=350$ GeV and
$\Lambda=2800$ GeV are consistent with DM constraints. The DM mass requires
the EFT framework to be probed with $\sqrt{s}=1$ TeV. We have chosen
$\mathcal{L}=567\,\text{fb}^{-1}$.

###  5.3 Distinction of models

| Significance ($\Delta\sigma$)
---|---
Base Model | New hypothesis | $P_{e^{\pm}}=0$ | $P_{e^{\pm}}=^{-30\%}_{+80\%}$
$c_{1}=0,c_{2}=1$ | $c_{1}=1,c_{2}=0$ | 4.72 | 7.41
$c_{1}=1,c_{2}=1$ | 7.71 | 12.24
$c_{1}=1,c_{2}=1$ | $c_{1}=1,c_{2}=0$ | 7.53 | 9.18
$c_{1}=0,c_{2}=1$ | 6.48 | 11.18
$c_{1}=1,c_{2}=0$ | $c_{1}=1,c_{2}=1$ | 5.70 | 8.89
$c_{1}=0,c_{2}=1$ | 3.81 | 6.00
Table 9: Separation of any hypothesis form a base model for both unpolarized
and polarized ($P_{e^{\pm}}=^{-30\%}_{+80\%}$) beams. The efficiency factor
has been taken as $\epsilon=0.001$. The illustrative choices made for the
analysis are $m_{\chi}=350$ GeV, $\sqrt{s}=1$ TeV,
$\mathcal{L}=567\,\text{fb}^{-1}$, and $\Lambda=2800$ GeV.

Figure 14: Separation of two benchmark models. 2$\sigma$, 3$\sigma$ and
5$\sigma$ regions are shown in red, blue and green colors respectively. Yellow
color indicates the separation of a specific model from the base model. Top:
Separation of $\\{c_{\ell_{1}}=1,c_{\ell_{2}}=0\\}$ (left) and
$\\{c_{\ell_{1}}=1,c_{\ell_{2}}=1\\}$ (right) from
$\\{c_{\ell_{1}}=0,c_{\ell_{2}}=1\\}$. Middle: Separation of
$\\{c_{\ell_{1}}=1,c_{\ell_{2}}=0\\}$ (left) and
$\\{c_{\ell_{1}}=0,c_{\ell_{2}}=1\\}$ (right) from
$\\{c_{\ell_{1}}=c_{\ell_{2}}=1\\}$. Bottom: Separation of
$\\{c_{\ell_{1}}=c_{\ell_{2}}=1\\}$ (left) and
$\\{c_{\ell_{1}}=0,c_{\ell_{2}}=1\\}$ (right) from
$\\{c_{\ell_{1}}=1,c_{\ell_{2}}=0\\}$. The reference model is marked by star
($\star$) and the model to be separated is marked by “x”. We consider
unpolarized beams here. The efficiency factor has been taken as
$\epsilon=0.001$. The parameters kept fixed are $m_{\chi}=350$ GeV,
$\sqrt{s}=1$ TeV, $\mathcal{L}=567\,\text{fb}^{-1}$, and $\Lambda=2800$ GeV.

Figure 15: Same as Fig. 14 with polarized beams
($P_{e^{\pm}}=^{-30\%}_{+80\%}$).

One of the key features of statistical analysis is to find out the separation
between two hypotheses, the one which is separated (with coefficients
$c_{\ell_{i}}$) is called the test hypothesis and the reference model is
termed to be base hypothesis denoted as $c_{\ell_{i}}^{0}$. Using Eq. (45), we
define the separation or statistical significance with respect to base
hypothesis as:

$\left[\Delta\sigma(g_{i}^{0};\,\bar{g}_{i})\right]^{2}=\epsilon\sum_{i,j}\left(g^{0}_{i}-\bar{g}_{i}\right)\left(g^{0}_{j}-\bar{g}_{j}\right)\left(V^{-1}_{0}\right)_{ij}\,,\quad
V_{0}=V(g=g^{0})\,;$ (47)

(where $g^{0}_{i}=g_{i}(c_{l_{i}}^{0}),\,\bar{g}_{i}=g_{i}(\bar{c}_{l_{i}})$)
which we take as a measure of the degree to which the
$c_{l_{i}}=\bar{c}_{l_{i}},\,c_{l_{j}}=\bar{c}_{l_{j}}$ hypothesis can be
distinguished from the base hypothesis
$c_{l_{i}}=c_{l_{i}}^{0},\,c_{l_{j}}=c_{l_{j}}^{0}$. Similar to $\chi^{2}$,
the expression in Eq. (47) depend on the efficiency ($\epsilon$) and the
luminosity (${\cal L}$). Again, we use $\epsilon=0.001$ and ${\cal
L}=567\,\text{fb}^{-1}$. If we consider $\\{c_{\ell_{1}}=0,c_{\ell_{2}}=1\\}$
as our base model then $\\{c_{\ell_{1}}=1,c_{\ell_{2}}=0\\}$ and
$\\{c_{\ell_{1}}=1,c_{\ell_{2}}=1\\}$ hypotheses are 4.72$\sigma$ and
7.72$\sigma$ sigma away respectively. If we chose
$\\{c_{\ell_{1}}=1,c_{\ell_{2}}=1\\}$ as base model, then both
$\\{c_{\ell_{1}}=1,c_{\ell_{2}}=0\\}$ and
$\\{c_{\ell_{1}}=0,c_{\ell_{2}}=1\\}$ are 5$\sigma$ away from the base model.
So, both hypotheses are well distinguishable. Now, if we take
$\\{c_{\ell_{1}}=1,c_{\ell_{2}}=0\\}$ as base model then
$\\{c_{\ell_{1}}=0,c_{\ell_{2}}=1\\}$ is 3.81$\sigma$ away from the base model
so that this model is indistinguishable 131313We consider $5\sigma$ to be the
reference to distinguish between two models. from the base model; but
$\\{c_{\ell_{1}}=1,c_{\ell_{2}}=1\\}$ model is 5.70$\sigma$ away. A specific
choice of beam polarization ($P_{e^{\pm}}=^{-30\%}_{+80\%}$) helps us to
provide better distinction for any hypothesis from a base model. Different
cases of such separation is shown in the Fig. 14 and Fig. 15 for polarized and
unpolarized beams. The numerical values of separation between different models
for unpolarized beams and polarized ($P_{e^{\pm}}=^{-30\%}_{+80\%}$) beams are
tabulated in the table 9.

##  6 Possible UV completion

In this section we sketch a few possible UV complete frameworks that are
capable of generating leptophilic DM operators discussed so far in the draft.
One of the simplest such possibilities can be found in Fox:2008kb , where the
SM gauge sector is extended by a dark abelian gauge symmetry
$U(1)_{\text{DS}}$. There is also a Dirac fermion $\chi$, odd under a dark
sector parity, that can be a potential DM candidate (all SM fields are even
under the same parity). In order to ensure the dark gauge boson $Z^{\prime}$
to be leptophilic, one has to ensure that it couples with equal and opposite
charge to two generations of leptons for anomaly cancellation (for example, in
a gauged $L_{i}-L_{j}$ model He:1990pn ; Ma:2001md ; Baek:2001kca ;
Baek:2008nz ; Heeck:2011wj ; Das:2013jca ; Biswas:2016yan ; Duan:2017qwj ;
Foldenauer:2018zrz  with $i,j\in e,\mu,\tau$)141414Leptophilic Majorana DM,
discussed in the context of the anomaly-free theories can be found in Refs.
Duerr:2013dza ; Schwaller:2013hqa ; FileviezPerez:2019cyn .. The qurak
interaction is prohibited since they do not carry any $L_{i}-L_{j}$ charge. At
a scale $\mu\ll m_{Z^{\prime}}$, the heavy dark gauge boson can be integrated
out resulting in an operator of the form
$\bar{\chi}\,\Gamma\,\chi\bar{\ell}\,\Gamma^{\prime}\,\ell/m_{Z^{\prime}}^{2}$,
where $\Gamma^{(^{\prime})}\equiv\\{\gamma^{\mu},\gamma^{\mu}\,\gamma^{5}\\}$.

Extra-dimensional model where the fermions have strong localizations at
various points in the extra dimensions as2000 ; ms2000 ; Nussinov:Shrock02
furnishes another interesting possibility where the DM can couple to the
leptons, while the coupling with the quarks is exponentially suppressed.
Coupling fermions with appropriate kink (for one-extra dimension) or vortex
(for two-extra dimensions) can localize the (zero-mode) fermions in the extra
dimensions Rubakov:1983bb ; Kaplan:1995pe ; Dvali:2000ha . Assuming that the
fermions have support in the interval $[0,L]$ in the extra dimensions,
constraints from precision electroweak searches, flavor-changing neutral
current, and collider searches can be accommodated by choosing
$L^{-1}\equiv\Lambda_{L}\gtrsim 100$ TeV Delgado:1999sv ; Dobrescu01 . One
appealing feature of this kind of extra-dimensional model is that the large
hierarchy in SM fermion masses can be obtained by separating out the chiral
parts in the extra dimension(s), without the requirement of large range of
dimensionless Yukawa couplings. Furthermore, proton decay can be exponentially
suppressed to safety by separating out quark and lepton wavefunctions in the
extra dimensions as2000 ; bvd , although this does not hinder $n-\bar{n}$
oscillations nnb02 ; Nussinov:Shrock02 ; Wise13 ; nnblrs . Set of solutions of
SM fermion wavefunction centers exists that reproduce neutrino masses and
mixing, and is consistent with experimental data nuled ; barenboim01 .
Therefore, if the DM candidate $\chi$ is localized near the leptons in the
extra dimensions, then the experimental bound from proton decay implies that
$\chi$ is separated widely from the quark wavefunctions. Hence, the Wilson
coefficients of
$(\bar{\chi}\Gamma_{\chi}^{\mu}\chi)(\bar{\ell}\Gamma_{\ell,\mu}\ell)$ will be
unsuppressed, while the Wilson coefficients for
$(\bar{\chi}\Gamma_{\chi}^{\mu}\chi)(\bar{\mathfrak{q}}\Gamma_{{\mathfrak{q}},\mu}{\mathfrak{q}})$
will be exponentially suppressed by the separation distance between the
corresponding fermion wavefunctions, thereby generating an effective
leptophilic interaction as we study here dmled .

##  7 Summary and Conclusions

Dark matter (DM) in effective theory (EFT) formalism has been studied
extensively due to the unknown nature of dark sector and economy of free
parameters ($m_{\chi},\Lambda$) without loss of predictability of the theory
itself. However, DM operators coupling to SM quarks face severe constraints
from direct DM searches as well as from LHC, caveat to appropriate validation
of EFT limit at hadron collider. Leptophilic DM, on the other hand, is
motivated from the fact that it not only provides a model independent way to
probe DM physics in EFT formalism, but presents an opportunity to study DM
production at $e^{+}e^{-}$ collider abiding the EFT limit after addressing
relic density. Dirac DM operators can additionally be probed in the future
sensitivities of direct search experiment via one loop interaction with the SM
quarks, while Majorana DM operators remain absent in direct detection due to
the Lorentz structure of the current.

We study the contribution from all possible DM operators of dimension six
simultaneously, assuming the Wilson coefficients to contribute with equal
strengths. Direct search for the Dirac DM is performed by taking into account
the RG evolution of the Wilson coefficients from a high NP scale $\Lambda$ to
low energy non-relativistic scale. Mono-photon arising from the initial state
radiation (ISR), together with missing energy, turns out to be a potential
signal to probe such operators at the ILC. For Dirac DM one requires higher CM
energy ($\sqrt{s}=1$ TeV) satisfying the relic density and spin-independent
direct search exclusion, while a comparatively lower CM energy ($\sqrt{s}=500$
GeV) is suffice to probe Majorana DM due to the absence of any direct search
limit. The SM backgrounds can be tamed down with judicious choice of cuts on
observables like missing energy, photon energy, photon transverse momentum
etc. The key is to choose a maximally right polarised electron beam and left
polarized positron beam ($P_{e^{\pm}}=^{-30\%}_{+80\%}$), which helps to
enhance the signal with positive Wilson coefficients and reduce the SM
background significantly. A discovery reach of 5$\sigma$ can be achieved with
luminosity $\mathcal{L}\simeq 1\,\text{fb}^{-1}$ for low mass Majorana DM
($m_{\chi}\sim 70$ GeV), while that for Dirac DM with masses above 300 GeV
requires larger luminosity $\mathcal{L}\gtrsim 50\,\text{fb}^{-1}$.

We finally perform a statistical analysis using optimal observable technique
(OOT) to extract the 1$\sigma$ uncertainties of the Wilson coefficients of
Dirac DM operators to show that the operators
$\frac{c_{\ell_{1}}}{\Lambda^{2}}\left(\overline{\chi}\gamma_{\mu}\chi\right)\left(\overline{\ell}\gamma^{\mu}\ell\right)$
and
$\frac{c_{\ell_{4}}}{\Lambda^{2}}\left(\overline{\chi}\gamma_{\mu}\chi\right)\left(\overline{\ell}\gamma^{\mu}\gamma_{5}\ell\right)$
are easier to probe than
$\frac{c_{\ell_{2}}}{\Lambda^{2}}\left(\overline{\chi}\gamma_{\mu}\gamma^{5}\chi\right)\left(\overline{\ell}\gamma^{\mu}\ell\right)$
and
$\frac{c_{\ell_{3}}}{\Lambda^{2}}\left(\overline{\chi}\gamma_{\mu}\gamma^{5}\chi\right)\left(\overline{\ell}\gamma^{\mu}\gamma_{5}\ell\right)$,
which never has been studied before. We also find that the separation between
models having some selective operators can go beyond 5$\sigma$ particularly in
presence of polarised beams.

###### Acknowledgements.

BB received funding from the Patrimonio Autónomo - Fondo Nacional de
Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José
de Caldas (MinCiencias - Colombia) grant 80740-465-2020. SG acknowledges
support from the U.S. National Science Foundation Grant NSF-PHY-1915093. This
project has received funding /support from the European Union’s Horizon 2020
research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 860881-HIDDeN. SB would like to acknowledge DST-SERB grant
CRG/2019/004078 from Govt. of India and Prof. Jose Wudka for useful
discussions. SG would like to thank Prof. Robert Shrock for helpful
discussions. The authors would like to thank Prof. Alexander Pukhov and Prof.
M. C. Kumar for helping with the model implementation.

##  Appendix A Annihilation cross-sections for leptophilic operators

In this appendix we list the annihilation cross-section times the relative
velocity $v$, expanded in powers of $v$ for different leptophilic operators.

$\displaystyle\big{(}\sigma v\big{)}_{{\cal O}_{DL}^{6}}$
$\displaystyle=\frac{c_{\ell_{1}}^{2}}{2\pi\Lambda^{4}}m_{\chi}^{2}\sum_{\ell}\sqrt{1-\delta_{\chi\ell}^{2}}\bigg{[}\Big{(}2+\delta_{\chi\ell}^{2}\Big{)}+\frac{v^{2}}{24\Big{(}1-\delta_{\chi\ell}^{2}\Big{)}}\bigg{(}8-4\delta_{\chi\ell}^{2}+5\delta_{\chi\ell}^{4}\bigg{)}\bigg{]}+{\cal
O}(v^{4}),$ (48) $\displaystyle\big{(}\sigma v\big{)}_{{\cal O}_{L}^{6}}$
$\displaystyle=\kappa^{2}\frac{c_{\ell_{2}}^{2}}{12\pi\Lambda^{4}}m_{\chi}^{2}\sum_{\ell}\sqrt{1-\delta_{\chi\ell}^{2}}\Big{(}2+\delta_{\chi\ell}^{2}\Big{)}v^{2}+{\cal
O}(v^{4}),$ (49) $\displaystyle\big{(}\sigma v\big{)}_{{\cal O}_{L1}^{6}}$
$\displaystyle=\kappa^{2}\frac{c_{\ell_{3}}^{2}}{2\pi\Lambda^{4}}m_{\chi}^{2}\sum_{\ell}\sqrt{1-\delta_{\chi\ell}^{2}}\bigg{[}\delta_{\chi\ell}^{2}+\frac{v^{2}}{24\Big{(}1-\delta_{\chi\ell}^{2}\Big{)}}\bigg{(}8-22\delta_{\chi\ell}^{2}+17\delta_{\chi\ell}^{4}\bigg{)}\bigg{]}+{\cal
O}(v^{4}),$ (50) $\displaystyle\big{(}\sigma v\big{)}_{{\cal O}_{DL1}^{6}}$
$\displaystyle=\frac{c_{\ell_{4}}^{2}}{\pi\Lambda^{4}}m_{\chi}^{2}\sum_{\ell}\bigg{[}\Big{(}1-\delta_{\chi\ell}^{2}\Big{)}^{3/2}+\frac{v^{2}}{24}\sqrt{1-\delta_{\chi\ell}^{2}}\Big{(}4+5\delta_{\chi\ell}^{2}\Big{)}\bigg{]}+{\cal
O}(v^{4}),$ (51)

where $\kappa=1(2)$ for a Dirac (Majorana) DM $\chi$, and
$\delta_{\chi\ell}\equiv\frac{m_{\ell}}{m_{\chi}}$.

##  Appendix B Models with negative Wilson coefficients

The cases where Wilson coefficients are assumed negative (particularly for
$c_{\ell_{3}}$ and $c_{\ell_{4}}$), offer different phenomenology. In this
appendix, we elaborate such cases. First, we show the relic density satisfying
parameter space constrained from direct detection bound from XENON1T for
($1,1,1,-1$), ($1,1,-1,1$), and $(1,1,-1,-1)$ choices of the $c_{\ell_{i}}$
for Dirac DM. To compare this with the BPs of the collider analyses in the
text, we choose a few benchmark points as shown in Table 10.

$c_{\ell_{i}}$ | ($m_{\chi},\Lambda$) (GeV)
---|---
$(1,1,1,-1)$ | ($350,3300$)
$(1,1,-1,1)$ | ($350,2800$)
$(1,1,-1,-1)$ | ($350,3300$)
Table 10: Benchmark points for Dirac DM for negative choices of the Wilson
coefficients $c_{\ell_{3}}$ and/or $c_{\ell_{4}}$.

The DM relic density and direct search allowed parameter space is shown in
Fig. 16. The blue, and green points satisfy XENON1T bound on the spin-
independent direct detection cross-section, while the red and purple points
are ruled out from the same. The cross-over point from red to blue is
$(275,2950)$ GeV, and similar cross-over point for purple to green is
$(345,2800)$ GeV.

Figure 16: Relic density satisfying parameter space for different choices of
the Wilson coefficients. The coupling values
$(c_{\ell_{1}},c_{\ell_{2}},c_{\ell_{3}},c_{\ell_{4}})$ is denoted as an
ordered set. XENON1T disallowed parameter space is marked in separate colour
(see figure inset). The case $(1,1,-1,-1)$ is not shown here as it is
approximately degenerate with the $(1,1,1,-1)$ case.

We plot next the variation of DM production cross-section
($\sigma_{e^{+}e^{-}\rightarrow\chi\bar{\chi}}$) for Majorana DM in the left
column and Dirac DM in the right column of Fig. 17. In the top, middle and
bottom panel we show the variation with respect to $\sqrt{s}$, $\Lambda$ and
$m_{\chi}$ respectively for three different choices of the beam polarization:
$\\{P_{e^{-}}:P_{e^{+}}\\}=\\{0,0\\};\\{+0.8,-0.3\\};\\{-0.8,\\\ +0.3\\}$
shown by solid, dashed-dot and dashed black (red) curves for Majorana (Dirac)
DM cases with operators having negative Wilson coefficients. See Figure
caption for details. The features remain almost the same, excepting for the
fact that cross-section for this case enhances for other polarization
configuration, namely $\\{P_{e^{-}}:P_{e^{+}}\\}=\\{-0.8,+0.3\\}$, unlike the
cases with positive Wilson coefficients. Recall, that we could reduce the
neutrino background significantly using the other polarisation
$\\{P_{e^{-}}:P_{e^{+}}\\}=\\{+0.8,-0.3\\}$, which reduces the signal with
this particular choices of Wilson coefficients. That is why, such a
possibility is harder to probe at ILC.

Figure 17: $\sigma_{e^{+}e^{-}\rightarrow\chi\bar{\chi}}$ as function of
$\sqrt{s}$ (top panel), $\Lambda$ (middle panel) and $m_{\chi}$ (bottom panel)
for Majorana DM (BP3) in the left column and Dirac DM (BP6) in the right
column for three choices of polarisation:
$\\{P_{e^{-}}:P_{e^{+}}\\}=\\{0,0\\};\\{+0.8,-0.3\\};\\{-0.8,+0.3\\}$ as
mentioned in the figure inset. The other parameters kept fixed are mentioned
in Figure heading.

Now we turn to the optimal analysis of Wilson coefficients in this scenario.
1$\sigma$ regions for benchmark point ($1,1,-1,-1$) is shown in different
$c_{\ell_{i}}$ plane in the Fig. 18. Here, we can see that the uncertainties
are reduced when electron beam is left polarized ($P_{e^{-}}=-80\%$) and
positron beam is right polarized ($P_{e^{+}}=+30\%$). The corresponding
uncertainties are tabulated in Table 11. We clearly see that the error in
extracting the Wilson coefficients is much larger here than those with
positive input values.

Figure 18: 1$\sigma$ regions for $\\{m_{\chi},\Lambda\\}=\\{350,3300\\}$ GeV. Light red (blue) color indicate unpolarized (polarized: $P_{e^{\pm}}=^{+30\%}_{-80\%}$) beam. We consider CM energy ($\sqrt{s})=1$ TeV, $\epsilon=0.001$ and $\mathcal{L}=567\,\text{fb}^{-1}$.  Couplings ($c_{i}$’s) | $P_{e^{\pm}}=0$ | $P_{e^{\pm}}=^{-30\%}_{+80\%}$
---|---|---
a | b | $\pm\Delta a$ | $\pm\Delta b$ | $\pm\Delta a$ | $\pm\Delta b$
$c_{1}=1$ | $c_{2}=1$ | $+0.12$ | $+0.67$ | $+0.08$ | $+0.20$
$-0.44$ | $-0.23$ | $-0.10$ | $-0.17$
$c_{1}=1$ | $c_{3}=-1$ | $+0.12$ | $+0.25$ | $+0.08$ | $+0.17$
$-0.20$ | $-0.35$ | $-0.10$ | $-0.20$
$c_{1}=1$ | $c_{4}=-1$ | $+0.30$ | $+0.46$ | $+0.29$ | $+0.45$
$-0.46$ | $-0.30$ | $-0.45$ | $-0.29$
$c_{2}=1$ | $c_{3}=-1$ | $+0.45$ | $+0.43$ | $+0.44$ | $+0.42$
$-0.43$ | $-0.45$ | $-0.42$ | $-0.44$
$c_{2}=1$ | $c_{4}=-1$ | $+0.35$ | $+0.20$ | $+0.20$ | $+0.10$
$-0.25$ | $-0.12$ | $-0.10$ | $-0.08$
$c_{3}=-1$ | $c_{4}=-1$ | $+0.23$ | $+0.12$ | $+0.17$ | $+0.10$
$-0.67$ | $-0.44$ | $-0.20$ | $-0.08$
Table 11: Optimal 1$\sigma$ statistical uncertainty in the $c_{\ell_{i}}$
couplings for both unpolarized and polarized ($P_{e^{\pm}}=^{+30\%}_{-80\%}$)
beams. The efficiency factor has been taken as $\epsilon=0.001$ in agreement
to the signal estimate. The illustrative choices of $m_{\chi}=350$ GeV and
$\Lambda=3300$ GeV are consistent with DM constraints. The DM mass requires
the EFT framework to be probed with $\sqrt{s}=1$ TeV. We have chosen
$\mathcal{L}=567\,\text{fb}^{-1}$.

## References

  * (1) F. Zwicky, _Die Rotverschiebung von extragalaktischen Nebeln_ , _Helv. Phys. Acta_ 6 (1933) 110.
  * (2) F. Zwicky, _On the Masses of Nebulae and of Clusters of Nebulae_ , _Astrophys. J._ 86 (1937) 217.
  * (3) V. C. Rubin and W. K. Ford, Jr., _Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions_ , _Astrophys. J._ 159 (1970) 379.
  * (4) D. Clowe, M. Bradac, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones et al., _A direct empirical proof of the existence of dark matter_ , _Astrophys. J. Lett._ 648 (2006) L109 [astro-ph/0608407].
  * (5) W. Hu and S. Dodelson, _Cosmic microwave background anisotropies_ , _Ann. Rev. Astron. Astrophys._ 40 (2002) 171 [astro-ph/0110414].
  * (6) Planck collaboration, _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6 [1807.06209].
  * (7) G. Jungman, M. Kamionkowski and K. Griest, _Supersymmetric dark matter_ , _Phys. Rept._ 267 (1996) 195 [hep-ph/9506380].
  * (8) G. Bertone, D. Hooper and J. Silk, _Particle dark matter: Evidence, candidates and constraints_ , _Phys. Rept._ 405 (2005) 279 [hep-ph/0404175].
  * (9) J. L. Feng, _Dark Matter Candidates from Particle Physics and Methods of Detection_ , _Ann. Rev. Astron. Astrophys._ 48 (2010) 495 [1003.0904].
  * (10) E. W. Kolb and M. S. Turner, _The Early Universe_ , _Front. Phys._ 69 (1990) 1.
  * (11) G. Arcadi, P. Ghosh, Y. Mambrini, M. Pierre and F. S. Queiroz, _$Z^{\prime}$ portal to Chern-Simons Dark Matter_, _JCAP_ 1711 (2017) 020 [1706.04198].
  * (12) L. Roszkowski, E. M. Sessolo and S. Trojanowski, _WIMP dark matter candidates and searches—current status and future prospects_ , _Rept. Prog. Phys._ 81 (2018) 066201 [1707.06277].
  * (13) L. J. Hall, K. Jedamzik, J. March-Russell and S. M. West, _Freeze-In Production of FIMP Dark Matter_ , _JHEP_ 03 (2010) 080 [0911.1120].
  * (14) N. Bernal, M. Heikinheimo, T. Tenkanen, K. Tuominen and V. Vaskonen, _The Dawn of FIMP Dark Matter: A Review of Models and Constraints_ , _Int. J. Mod. Phys._ A32 (2017) 1730023 [1706.07442].
  * (15) Y. Hochberg, E. Kuflik, T. Volansky and J. G. Wacker, _Mechanism for Thermal Relic Dark Matter of Strongly Interacting Massive Particles_ , _Phys. Rev. Lett._ 113 (2014) 171301 [1402.5143].
  * (16) S. Bhattacharya and J. Wudka, _Effective Theories with Dark Matter Applications_ , 2104.01788.
  * (17) A. L. Fitzpatrick, W. Haxton, E. Katz, N. Lubbers and Y. Xu, _Model Independent Direct Detection Analyses_ , 1211.2818.
  * (18) A. L. Fitzpatrick, W. Haxton, E. Katz, N. Lubbers and Y. Xu, _The Effective Field Theory of Dark Matter Direct Detection_ , _JCAP_ 02 (2013) 004 [1203.3542].
  * (19) M. Cirelli, E. Del Nobile and P. Panci, _Tools for model-independent bounds in direct dark matter searches_ , _JCAP_ 10 (2013) 019 [1307.5955].
  * (20) SuperCDMS collaboration, _Dark matter effective field theory scattering in direct detection experiments_ , _Phys. Rev. D_ 91 (2015) 092004 [1503.03379].
  * (21) F. Bishara, J. Brod, B. Grinstein and J. Zupan, _Chiral Effective Theory of Dark Matter Direct Detection_ , _JCAP_ 02 (2017) 009 [1611.00368].
  * (22) A. De Simone and T. Jacques, _Simplified models vs. effective field theory approaches in dark matter searches_ , _Eur. Phys. J. C_ 76 (2016) 367 [1603.08002].
  * (23) F. D’Eramo, B. J. Kavanagh and P. Panci, _You can hide but you have to run: direct detection with vector mediators_ , _JHEP_ 08 (2016) 111 [1605.04917].
  * (24) J. Brod, A. Gootjes-Dreesbach, M. Tammaro and J. Zupan, _Effective Field Theory for Dark Matter Direct Detection up to Dimension Seven_ , _JHEP_ 10 (2018) 065 [1710.10218].
  * (25) F. Bishara, J. Brod, B. Grinstein and J. Zupan, _From quarks to nucleons in dark matter direct detection_ , _JHEP_ 11 (2017) 059 [1707.06998].
  * (26) T. Li and Y. Liao, _Constraints on dimension-seven operators with a derivative in effective field theory for Dirac dark matter_ , _Phys. Lett. B_ 796 (2019) 6 [1811.08200].
  * (27) J. C. Criado, A. Djouadi, M. Perez-Victoria and J. Santiago, _A complete effective field theory for dark matter_ , _JHEP_ 07 (2021) 081 [2104.14443].
  * (28) M. Beltran, D. Hooper, E. W. Kolb and Z. C. Krusberg, _Deducing the nature of dark matter from direct and indirect detection experiments in the absence of collider signatures of new physics_ , _Phys. Rev. D_ 80 (2009) 043509 [0808.3384].
  * (29) Q.-H. Cao, C.-R. Chen, C. S. Li and H. Zhang, _Effective Dark Matter Model: Relic density, CDMS II, Fermi LAT and LHC_ , _JHEP_ 08 (2011) 018 [0912.4511].
  * (30) C. Cheung, G. Elor, L. J. Hall and P. Kumar, _Origins of Hidden Sector Dark Matter I: Cosmology_ , _JHEP_ 03 (2011) 042 [1010.0022].
  * (31) J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T. M. P. Tait and H.-B. Yu, _Gamma Ray Line Constraints on Effective Theories of Dark Matter_ , _Nucl. Phys. B_ 844 (2011) 55 [1009.0008].
  * (32) K. Cheung, P.-Y. Tseng and T.-C. Yuan, _Cosmic Antiproton Constraints on Effective Interactions of the Dark Matter_ , _JCAP_ 01 (2011) 004 [1011.2310].
  * (33) J. Blumenthal, P. Gretskov, M. Krämer and C. Wiebusch, _Effective field theory interpretation of searches for dark matter annihilation in the Sun with the IceCube Neutrino Observatory_ , _Phys. Rev. D_ 91 (2015) 035002 [1411.5917].
  * (34) M. Klasen, M. Pohl and G. Sigl, _Indirect and direct search for dark matter_ , _Prog. Part. Nucl. Phys._ 85 (2015) 1 [1507.03800].
  * (35) N. F. Bell, Y. Cai and R. K. Leane, _Impact of mass generation for spin-1 mediator simplified models_ , _JCAP_ 1701 (2017) 039 [1610.03063].
  * (36) N. F. Bell, G. Busoni and S. Robles, _Capture of Leptophilic Dark Matter in Neutron Stars_ , _JCAP_ 06 (2019) 054 [1904.09803].
  * (37) J. Abdallah et al., _Simplified Models for Dark Matter Searches at the LHC_ , _Phys. Dark Univ._ 9-10 (2015) 8 [1506.03116].
  * (38) D. Abercrombie et al., _Dark Matter Benchmark Models for Early LHC Run-2 Searches: Report of the ATLAS/CMS Dark Matter Forum_ , _Phys. Dark Univ._ 27 (2020) 100371 [1507.00966].
  * (39) F. Kahlhoefer, _Review of LHC Dark Matter Searches_ , _Int. J. Mod. Phys. A_ 32 (2017) 1730006 [1702.02430].
  * (40) B. Penning, _The pursuit of dark matter at colliders—an overview_ , _J. Phys. G_ 45 (2018) 063001 [1712.01391].
  * (41) A. Boveia and C. Doglioni, _Dark Matter Searches at Colliders_ , _Ann. Rev. Nucl. Part. Sci._ 68 (2018) 429 [1810.12238].
  * (42) P. D. Group, P. A. Zyla, R. M. Barnett, J. Beringer, O. Dahl, D. A. Dwyer et al., _Review of Particle Physics_ , _Progress of Theoretical and Experimental Physics_ 2020 (2020)  [https://academic.oup.com/ptep/article-pdf/2020/8/083C01/34673722/ptaa104.pdf].
  * (43) H. Dreiner, M. Huck, M. Krämer, D. Schmeier and J. Tattersall, _Illuminating Dark Matter at the ILC_ , _Phys. Rev. D_ 87 (2013) 075015 [1211.2254].
  * (44) Y. J. Chae and M. Perelstein, _Dark Matter Search at a Linear Collider: Effective Operator Approach_ , _JHEP_ 05 (2013) 138 [1211.4008].
  * (45) J. de Blas et al., _The CLIC Potential for New Physics_ , 1812.02093.
  * (46) M. Habermehl, M. Berggren and J. List, _WIMP Dark Matter at the International Linear Collider_ , _Phys. Rev. D_ 101 (2020) 075053 [2001.03011].
  * (47) J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T. M. P. Tait and H.-B. Yu, _Constraints on Light Majorana dark Matter from Colliders_ , _Phys. Lett. B_ 695 (2011) 185 [1005.1286].
  * (48) J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T. M. Tait and H.-B. Yu, _Constraints on Dark Matter from Colliders_ , _Phys. Rev. D_ 82 (2010) 116010 [1008.1783].
  * (49) Y. Bai, P. J. Fox and R. Harnik, _The Tevatron at the Frontier of Dark Matter Direct Detection_ , _JHEP_ 12 (2010) 048 [1005.3797].
  * (50) A. Rajaraman, W. Shepherd, T. M. P. Tait and A. M. Wijangco, _LHC Bounds on Interactions of Dark Matter_ , _Phys. Rev. D_ 84 (2011) 095013 [1108.1196].
  * (51) M. R. Buckley, _Asymmetric Dark Matter and Effective Operators_ , _Phys. Rev. D_ 84 (2011) 043510 [1104.1429].
  * (52) P. J. Fox, R. Harnik, J. Kopp and Y. Tsai, _Missing Energy Signatures of Dark Matter at the LHC_ , _Phys. Rev. D_ 85 (2012) 056011 [1109.4398].
  * (53) H. Dreiner, D. Schmeier and J. Tattersall, _Contact Interactions Probe Effective Dark Matter Models at the LHC_ , _EPL_ 102 (2013) 51001 [1303.3348].
  * (54) O. Buchmueller, M. J. Dolan and C. McCabe, _Beyond Effective Field Theory for Dark Matter Searches at the LHC_ , _JHEP_ 01 (2014) 025 [1308.6799].
  * (55) A. A. Petrov and W. Shepherd, _Searching for dark matter at LHC with Mono-Higgs production_ , _Phys. Lett. B_ 730 (2014) 178 [1311.1511].
  * (56) S. Chang, R. Edezhath, J. Hutchinson and M. Luty, _Effective WIMPs_ , _Phys. Rev. D_ 89 (2014) 015011 [1307.8120].
  * (57) W. Altmannshofer, P. J. Fox, R. Harnik, G. D. Kribs and N. Raj, _Dark Matter Signals in Dilepton Production at Hadron Colliders_ , _Phys. Rev. D_ 91 (2015) 115006 [1411.6743].
  * (58) N. F. Bell, Y. Cai, J. B. Dent, R. K. Leane and T. J. Weiler, _Dark matter at the LHC: Effective field theories and gauge invariance_ , _Phys. Rev. D_ 92 (2015) 053008 [1503.07874].
  * (59) A. Belyaev, L. Panizzi, A. Pukhov and M. Thomas, _Dark Matter characterization at the LHC in the Effective Field Theory approach_ , _JHEP_ 04 (2017) 110 [1610.07545].
  * (60) A. Belyaev, E. Bertuzzo, C. Caniu Barros, O. Eboli, G. Grilli Di Cortona, F. Iocco et al., _Interplay of the LHC and non-LHC Dark Matter searches in the Effective Field Theory approach_ , _Phys. Rev. D_ 99 (2019) 015006 [1807.03817].
  * (61) P. J. Fox, R. Harnik, J. Kopp and Y. Tsai, _LEP Shines Light on Dark Matter_ , _Phys. Rev. D_ 84 (2011) 014028 [1103.0240].
  * (62) Z.-H. Yu, Q.-S. Yan and P.-F. Yin, _Detecting interactions between dark matter and photons at high energy $e^{+}e^{-}$ colliders_, _Phys. Rev. D_ 88 (2013) 075015 [1307.5740].
  * (63) R. Essig, J. Mardon, M. Papucci, T. Volansky and Y.-M. Zhong, _Constraining Light Dark Matter with Low-Energy $e^{+}e^{-}$ Colliders_, _JHEP_ 11 (2013) 167 [1309.5084].
  * (64) K. Kadota and J. Silk, _Constraints on Light Magnetic Dipole Dark Matter from the ILC and SN 1987A_ , _Phys. Rev. D_ 89 (2014) 103528 [1402.7295].
  * (65) Z.-H. Yu, X.-J. Bi, Q.-S. Yan and P.-F. Yin, _Dark matter searches in the mono- $Z$ channel at high energy $e^{+}e^{-}$ colliders_, _Phys. Rev. D_ 90 (2014) 055010 [1404.6990].
  * (66) A. Freitas and S. Westhoff, _Leptophilic Dark Matter in Lepton Interactions at LEP and ILC_ , _JHEP_ 10 (2014) 116 [1408.1959].
  * (67) S. Dutta, D. Sachdeva and B. Rawat, _Signals of Leptophilic Dark Matter at the ILC_ , _Eur. Phys. J. C_ 77 (2017) 639 [1704.03994].
  * (68) Z. Liu, Y.-H. Xu and Y. Zhang, _Probing dark matter particles at CEPC_ , _JHEP_ 06 (2019) 009 [1903.12114].
  * (69) D. Choudhury and D. Sachdeva, _Model independent analysis of MeV scale dark matter. II. Implications from $e^{-}e^{+}$ colliders and direct detection_, _Phys. Rev. D_ 100 (2019) 075007 [1906.06364].
  * (70) H. Bharadwaj and A. Goyal, _Effective Field Theory approach to lepto-philic self conjugate dark matter_ , _Chin. Phys. C_ 45 (2021) 023114 [2008.13621].
  * (71) E. Bertuzzo, C. J. Caniu Barros and G. Grilli di Cortona, _MeV Dark Matter: Model Independent Bounds_ , _JHEP_ 09 (2017) 116 [1707.00725].
  * (72) V. Gonzalez Macias and J. Wudka, _Effective theories for Dark Matter interactions and the neutrino portal paradigm_ , _JHEP_ 07 (2015) 161 [1506.03825].
  * (73) G. Busoni, A. De Simone, E. Morgante and A. Riotto, _On the Validity of the Effective Field Theory for Dark Matter Searches at the LHC_ , _Phys. Lett. B_ 728 (2014) 412 [1307.2253].
  * (74) G. Busoni, A. De Simone, T. Jacques, E. Morgante and A. Riotto, _On the Validity of the Effective Field Theory for Dark Matter Searches at the LHC Part III: Analysis for the $t$-channel_, _JCAP_ 09 (2014) 022 [1405.3101].
  * (75) G. Busoni, A. De Simone, J. Gramling, E. Morgante and A. Riotto, _On the Validity of the Effective Field Theory for Dark Matter Searches at the LHC, Part II: Complete Analysis for the $s$-channel_, _JCAP_ 06 (2014) 060 [1402.1275].
  * (76) M. Diehl and O. Nachtmann, _Optimal observables for the measurement of three gauge boson couplings in $e^{+}e^{-}$ —$>$ $W^{+}W^{-}$_, _Z. Phys. C_ 62 (1994) 397.
  * (77) J. F. Gunion, B. Grzadkowski and X.-G. He, _Determining the $t\bar{t}$ and $ZZ$ couplings of a neutral Higgs boson of arbitrary CP nature at the NLC_, _Phys. Rev. Lett._ 77 (1996) 5172 [hep-ph/9605326].
  * (78) K. Hagiwara, S. Ishihara, J. Kamoshita and B. A. Kniehl, _Prospects of measuring general Higgs couplings at $e^{+}e^{-}$ linear colliders_, _Eur. Phys. J. C_ 14 (2000) 457 [hep-ph/0002043].
  * (79) S. Dutta, K. Hagiwara and Y. Matsumoto, _Measuring the Higgs-Vector boson Couplings at Linear $e^{+}e^{-}$ Collider_, _Phys. Rev. D_ 78 (2008) 115016 [0808.0477].
  * (80) B. Grzadkowski and Z. Hioki, _CP violating lepton energy correlation in $e^{-}e^{+}$ —$>$ $t\bar{t}$_, _Phys. Lett. B_ 391 (1997) 172 [hep-ph/9608306].
  * (81) B. Grzadkowski, Z. Hioki and M. Szafranski, _Four Fermi effective operators in top quark production and decay_ , _Phys. Rev. D_ 58 (1998) 035002 [hep-ph/9712357].
  * (82) B. Grzadkowski and Z. Hioki, _Probing top quark couplings at polarized NLC_ , _Phys. Rev. D_ 61 (2000) 014013 [hep-ph/9805318].
  * (83) B. Grzadkowski and J. Pliszka, _Testing top quark Yukawa interactions in $e^{+}e^{-}$ —$>$ $t\bar{t}Z$_, _Phys. Rev. D_ 60 (1999) 115018 [hep-ph/9907206].
  * (84) S. Bhattacharya, S. Jahedi and J. Wudka, _Probing heavy charged fermions at $e^{+}e^{-}$ collider using the Optimal Observable Technique_, 2106.02846.
  * (85) J. L. Diaz-Cruz and E. Ma, _Neutral SU(2) Gauge Extension of the Standard Model and a Vector-Boson Dark-Matter Candidate_ , _Phys. Lett. B_ 695 (2011) 264 [1007.2631].
  * (86) S. Matsumoto, S. Mukhopadhyay and Y.-L. S. Tsai, _Singlet Majorana fermion dark matter: a comprehensive analysis in effective field theory_ , _JHEP_ 10 (2014) 155 [1407.1859].
  * (87) M. Duch, _Effective Operators for Dark Matter Interactions_ , Ph.D. thesis, Warsaw U., 2014.  1410.4427.
  * (88) M. Duch, B. Grzadkowski and J. Wudka, _Classification of effective operators for interactions between the Standard Model and dark matter_ , _JHEP_ 05 (2015) 116 [1412.0520].
  * (89) V. Gonzalez Macias and J. Wudka, _Effective theories for Dark Matter interactions and the neutrino portal paradigm_ , _JHEP_ 07 (2015) 161 [1506.03825].
  * (90) G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, _Minimal flavor violation: An Effective field theory approach_ , _Nucl. Phys. B_ 645 (2002) 155 [hep-ph/0207036].
  * (91) M. A. Fedderke, J.-Y. Chen, E. W. Kolb and L.-T. Wang, _The Fermionic Dark Matter Higgs Portal: an effective field theory approach_ , _JHEP_ 08 (2014) 122 [1404.2283].
  * (92) F. Bishara, J. Brod, P. Uttayarat and J. Zupan, _Nonstandard Yukawa Couplings and Higgs Portal Dark Matter_ , _JHEP_ 01 (2016) 010 [1504.04022].
  * (93) P. Gondolo and G. Gelmini, _Cosmic abundances of stable particles: Improved analysis_ , _Nucl. Phys. B_ 360 (1991) 145.
  * (94) J. Edsjo and P. Gondolo, _Neutralino relic density including coannihilations_ , _Phys. Rev. D_ 56 (1997) 1879 [hep-ph/9704361].
  * (95) M. Bauer and T. Plehn, _Yet Another Introduction to Dark Matter: The Particle Physics Approach_ , vol. 959 of _Lecture Notes in Physics_. Springer, 2019, 10.1007/978-3-030-16234-4, [1705.01987].
  * (96) G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, _micrOMEGAs: A Tool for dark matter studies_ , _Nuovo Cim. C_ 033N2 (2010) 111 [1005.4133].
  * (97) XENON collaboration, _Dark Matter Search Results from a One Ton-Year Exposure of XENON1T_ , _Phys. Rev. Lett._ 121 (2018) 111302 [1805.12562].
  * (98) J. Kopp, V. Niro, T. Schwetz and J. Zupan, _Dama/libra data and leptonically interacting dark matter_ , _Phys. Rev. D_ 80 (2009) 083502.
  * (99) J. Kopp, V. Niro, T. Schwetz and J. Zupan, _DAMA/LIBRA and leptonically interacting Dark Matter_ , _Phys. Rev. D_ 80 (2009) 083502 [0907.3159].
  * (100) Fermi-LAT collaboration, _Sensitivity Projections for Dark Matter Searches with the Fermi Large Area Telescope_ , _Phys. Rept._ 636 (2016) 1 [1605.02016].
  * (101) CTA Consortium collaboration, _Dark Matter and Fundamental Physics with the Cherenkov Telescope Array_ , _Astropart. Phys._ 43 (2013) 189 [1208.5356].
  * (102) M. Cirelli, G. Corcella, A. Hektor, G. Hutsi, M. Kadastik, P. Panci et al., _PPPC 4 DM ID: A Poor Particle Physicist Cookbook for Dark Matter Indirect Detection_ , _JCAP_ 03 (2011) 051 [1012.4515].
  * (103) T. R. Slatyer, _Indirect Detection of Dark Matter_ , in _Theoretical Advanced Study Institute in Elementary Particle Physics: Anticipating the Next Discoveries in Particle Physics_ , pp. 297–353, 2018, 1710.05137, DOI.
  * (104) D. Hooper, _TASI Lectures on Indirect Searches For Dark Matter_ , _PoS_ TASI2018 (2019) 010 [1812.02029].
  * (105) MAGIC, Fermi-LAT collaboration, _Limits to Dark Matter Annihilation Cross-Section from a Combined Analysis of MAGIC and Fermi-LAT Observations of Dwarf Satellite Galaxies_ , _JCAP_ 02 (2016) 039 [1601.06590].
  * (106) J. L. Feng, M. Kaplinghat and H.-B. Yu, _Sommerfeld Enhancements for Thermal Relic Dark Matter_ , _Phys. Rev. D_ 82 (2010) 083525 [1005.4678].
  * (107) I. John and T. Linden, _Cosmic-Ray Positrons Strongly Constrain Leptophilic Dark Matter_ , 2107.10261.
  * (108) PandaX-4T collaboration, _Dark Matter Search Results from the PandaX-4T Commissioning Run_ , 2107.13438.
  * (109) DELPHI collaboration, _Photon events with missing energy in e+ e- collisions at s**(1/2) = 130-GeV to 209-GeV_ , _Eur. Phys. J. C_ 38 (2005) 395 [hep-ex/0406019].
  * (110) DELPHI collaboration, _Search for one large extra dimension with the DELPHI detector at LEP_ , _Eur. Phys. J. C_ 60 (2009) 17 [0901.4486].
  * (111) N. F. Bell, Y. Cai, R. K. Leane and A. D. Medina, _Leptophilic dark matter with $Z^{\prime}$ interactions_, _Phys. Rev. D_ 90 (2014) 035027 [1407.3001].
  * (112) K. Fujii et al., _The role of positron polarization for the inital $250$ GeV stage of the International Linear Collider_, 1801.02840.