#  Exploring dark matter-gauge boson effective interactions at the current and
future colliders

Yu Zhang  Institutes of Physical Science and Information Technology, Anhui
University, Hefei 230601, China  School of Physics, Hefei University of
Technology, Hefei 230601,China  Yi-Wei Huang  Institutes of Physical Science
and Information Technology, Anhui University, Hefei 230601, China  Wei-Tao
Zhang  School of Physics and Optoelectronics Engineering, Anhui University,
Hefei 230601,China  Mao Song  School of Physics and Optoelectronics
Engineering, Anhui University, Hefei 230601,China  Ran Ding  Corresponding
author<EMAIL_ADDRESS>School of Physics and Optoelectronics
Engineering, Anhui University, Hefei 230601,China

###### Abstract

In this work, we investigate the collider constraints on effective
interactions between Dark Matter (DM) particles and electroweak gauge bosons
in a systematic way. We consider the simplified models in which scalar or
Dirac fermion DM candidates only couple to electroweak gauge bosons through
high dimensional effective operators. Taking into account the induced DM-
quarks and DM-gluons operators from the Renormalization Group Evolution (RGE)
running effect, we present comprehensive constraints on effective energy scale
$\Lambda$ and Wilson coefficients $C_{B}(\Lambda),\,C_{W}(\Lambda)$ from
direct detection, indirect detection, and collider searches. In particular, we
present the corresponding sensitivity from the Large Hadron Electron Collider
(LHeC) and the Future Circular Collider in electron-proton mode (FCC-ep) for
the first time, update the mono-$j$ and mono-$\gamma$ search limits at the
Large Hadron Collider (LHC), and derive the new limits at the Circular
Electron Positron Collider (CEPC).

##  I Introduction

Observations from astrophysics and cosmology have provided overwhelming
evidence for the existence of DM Aghanim:2018eyx . However, the microscopic
properties of DM are still poorly known since all of such observations are
based on its gravitational effects. The canonical DM candidate is the Weakly
Interacting Massive Particles (WIMPs). Such a scenario is easy to fulfill
relic abundance through thermal freeze-out mechanism (the “WIMP miracle”)
Lee:1977ua , and can be naturally embedded in many popular theoretical
frameworks Jungman:1995df ; Bertone:2004pz . Moreover, due to the property of
weak interaction with the SM particles, it can be probed through three
experimental prongs, i.e., collider, direct and indirect detections.

Typically, the interactions between DM and SM particles are model-dependent
and it is a great challenge to explore the complete model landscape. Instead
of exhausting all possible theoretical model parameter space, the Effective
Field Theory (EFT) framework is extensively used, which allows us to capture
some key features of the high-scale physics effects on the electroweak scale
while significantly simplifying the analysis procedures. For EFT collider
searches, various signal channels based on different SM portal effective
operators have been extensively studied at the hadron and electron colliders
Goodman:2010ku ; Fox:2011pm ; Rajaraman:2011wf ; Ding:2013nvx ; Cao:2009uw ;
Nelson:2013pqa ; Crivellin:2015wva ; Bell:2012rg ; Carpenter:2012rg ;
Alves:2015dya ; Mao:2014rga ; Lopez:2014qja ; Carpenter:2013xra ;
Berlin:2014cfa ; Bartels:2012ex ; Chae:2012bq ; Dreiner:2012xm ; Yu:2013aca ;
Richard:2014vfa ; Rossi-Torres:2015eua ; Habermehl:2017dxh ; Gao:2017tgx ;
Liu:2019ogn ; Ghosh:2019rtj ; Habermehl:2020njb ; Neng:2014mga ; Yu:2014ula ;
Liu:2017zdh ; Kadota:2018lrt ; dEnterria:2017dac .

Among them, the parameter space of DM-quark interaction which fulfills the
correct relic density has a tension with the exclusion limits from direct
detection, indirect detection, and collider searches. In this work, we
consider the dimension-6 (7) DM-diboson effective operators for scalar (Dirac
fermion) DM and examine their sensitivity at current LHC, future electron and
electron-proton colliders. The current limits of DM-diboson operators at the
LHC are mainly constrained by mono-$j$ and mono-$\gamma$ signal channels, and
mono-$\gamma$ signature is a sensitive channel at the CEPC. For electron-
proton collider, LHeC and FCC-ep are the two most promising proposals at
current. LHeC is based on an economic LHC upgrade, plan to add a 60-140 GeV
electron beam to collide with a 7 TeV proton beam in the LHC ring during the
period of High Luminosity LHC (HL-LHC) run. FCC-ep is designed to collide a 60
GeV electron with the 50 TeV proton beam. DM pairs can be produced through the
VBF process with a cleaner background, which corresponds to
${\not{E}}_{T}+e^{-}j$ signal.

The rest paper is organized as follows. In Sec.II, we firstly give a brief
description of the DM-diboson effective operators. Then relevant collider
phenomenology of ${\not{E}}_{T}+e^{-}j$ signal channel at the LHeC and FCC-ep
is discussed in Sec.III, including event simulations, kinematical
distributions of final states and cut selections related to signals and
backgrounds. In Sec.IV, we present the sensitivity at the CEPC with
mono-$\gamma$ signature. The limits from the LHC with current mono-$j$ and
mono-$\gamma$ searches are updated in Sec. V. The constraints from current
direct and indirect DM search experiments are also discussed in Sec.VI. Then
we show the results in Sec. VII. Finally, we give a summary and conclusion in
Sec.VIII.

##  II Framework of DM-gauge boson effective interactions

We consider the DM particle as a Dirac fermion ($\chi$) or scalar ($\phi$),
which connects with SM sector through high-dimensional DM-gauge boson
effective operators. At the lowest order, such operators allow DM couple to
single gauge boson, which induce electromagnetic dipole and anapole
interactions Cotta:2012nj ; Rajaraman:2012fu . However, these interactions
lead to monochromatic gamma-rays due to the unsuppressed cross-section of DM
annihilating into the $\gamma\gamma$ and $\gamma Z$ modes, which are severely
constrained by current indirect detection experiments Chen:2013gya . We thus
start from DM-diboson operators

$\displaystyle{\cal O}_{B}^{\phi}$ $\displaystyle=$ $\displaystyle\phi^{*}\phi
B^{\mu\nu}B_{\mu\nu},\qquad{\cal O}_{W}^{\phi}=\phi^{*}\phi
W^{a,\mu\nu}W^{a}_{\mu\nu},$ (1) $\displaystyle{\cal O}_{B}^{\chi}$
$\displaystyle=$ $\displaystyle\overline{\chi}\chi
B^{\mu\nu}B_{\mu\nu},\qquad{\cal O}_{W}^{\chi}=\overline{\chi}\chi
W^{a,\mu\nu}W^{a}_{\mu\nu}.$ (2)

These operators are assumed to be generated at the energy scale $\Lambda$ with
other high energy resonances much heavier than $\Lambda$ are all decoupled.
Notice that in this scenario, the coupling between DM and the SM quarks can
also be induced through the renormalization group evolution (RGE) from the
$\Lambda$ scale down to the electroweak scale $\mu_{\rm EW}$. As a
consequence, two extra dimension-6 (7) operators

$\displaystyle{\cal O}_{y}^{\phi}$ $\displaystyle=$ $\displaystyle
y_{q}\phi^{*}\phi\bar{q}Hq,\qquad{\cal
O}_{H}^{\phi}=\phi^{*}\phi(H^{\dagger}H)^{2},$ (3) $\displaystyle{\cal
O}_{y}^{\chi}$ $\displaystyle=$ $\displaystyle
y_{q}\bar{\chi}\chi\bar{q}Hq,\qquad{\cal
O}_{H}^{\phi}=\bar{\chi}\chi(H^{\dagger}H)^{2}.$ (4)

need to be taken into account. In above equation, $y_{q}=\sqrt{2}m_{q}/v$ are
the Yukawa couplings of SM quarks, and $v$ is the vacuum expectation value
(VEV) of the SM Higgs doublet $H$. Then the total effective Lagrangian at
scale $\mu$ ($\mu_{\rm EW}<\mu<\Lambda$) is given by

$\displaystyle\mathcal{L}_{\rm
eff}^{\phi,\chi}=\sum_{k=B,W,y,H}\frac{C^{\phi,\chi}_{k}(\mu)}{\Lambda^{2}}{\cal
O}_{k}\,.$ (5)

To the leading logarithmic (LL) order, the Wilson coefficients of the
effective operators ${\cal O}_{y}$ and ${\cal O}_{H}$ at the scale $\mu_{\rm
EW}$ are Crivellin:2014gpa

$\displaystyle C_{y}(\mu_{\rm EW})$ $\displaystyle\simeq$
$\displaystyle\frac{6Y_{qL}Y_{qR}\alpha_{1}}{\pi}\ln\left(\frac{\mu_{\rm
EW}}{\Lambda}\right),$ (6) $\displaystyle C_{H}(\mu_{\rm EW})$
$\displaystyle\simeq$ $\displaystyle-9\alpha^{2}_{2}\ln\left(\frac{\mu_{\rm
EW}}{\Lambda}\right)C_{W}(\Lambda),$ (7)

where $Y_{qL}\,(Y_{qR})$ denotes the hypercharges of the left-handed (right-
handed) quarks with $Y_{uL}=Y_{dL}=1/6,Y_{uR}=2/3$ and $Y_{dR}=1/3$,
$\alpha_{1}$ and $\alpha_{2}$ are the gauge coupling constants of the
$U(1)_{Y}$ and $SU(2)_{Y}$ gauge group, respectively. In our calculation, we
set $\mu_{\rm EW}\equiv m_{Z}$ with $\alpha_{1}\simeq 1/98$ and
$\alpha_{2}\simeq 1/29$. After Electroweak symmetry breaking (EWSB), the gauge
eigenstate fields $B_{\mu}$ and $W_{\mu}^{3}$ mix into the physical massless
photon field $A_{\mu}$ and massive gauge field $Z_{\mu}$. Then the effective
DM-diboson and DM-quark operators are recast as

$\displaystyle{\cal O}_{FF}^{\phi}$ $\displaystyle=\phi^{*}\phi
F^{\mu\nu}F_{\mu\nu},\quad{\cal O}_{FF}^{\chi}=\overline{\chi}\chi
F^{\mu\nu}F_{\mu\nu},$ $\displaystyle{\cal O}_{q}^{\phi}$
$\displaystyle=m_{q}\phi^{*}\phi\bar{q}q,\quad{\cal
O}_{q}^{\chi}=m_{q}\overline{\chi}\chi\bar{q}q.$ (8)

where $FF=AA,AZ,ZZ,WW$. The relevant matching conditions are

$\displaystyle{C_{AA}(\mu_{\rm EW})}$ $\displaystyle=$ $\displaystyle
C_{B}(\mu_{\rm EW})\cos^{2}\theta_{W}+C_{W}(\mu_{\rm EW})\sin^{2}\theta_{W},$
$\displaystyle{C_{ZZ}(\mu_{\rm EW})}$ $\displaystyle=$ $\displaystyle
C_{W}(\mu_{\rm EW})\cos^{2}\theta_{W}+C_{B}(\mu_{\rm EW})\sin^{2}\theta_{W},$
$\displaystyle{C_{AZ}(\mu_{\rm EW})}$ $\displaystyle=$
$\displaystyle\left(C_{W}(\mu_{\rm EW})-C_{B}(\mu_{\rm EW})\right)\sin
2\theta_{W},$ $\displaystyle{C_{WW}(\mu_{\rm EW})}$ $\displaystyle=$
$\displaystyle C_{W}(\mu_{\rm EW}),$ $\displaystyle{C_{q}(\mu_{\rm EW})}$
$\displaystyle=$ $\displaystyle C_{y}(\mu_{\rm
EW})-\frac{v^{2}}{m_{h}^{2}}C_{H}(\mu_{\rm EW}),$ (9)

with $\theta_{W}$ is the Weinberg angle.

Integrating out the heavy top quarks, the effective DM-gluon operators

$\displaystyle{\cal O}_{q}^{\phi}=\alpha_{s}\phi^{*}\phi
G^{a,\mu\nu}G_{\mu\nu}^{a},\quad{\cal
O}_{q}^{\chi}=\alpha_{s}\overline{\chi}\chi G^{a,\mu\nu}G_{\mu\nu}^{a}$ (10)

can be generated, where $G_{\mu\nu}^{a}$ denotes the gluon field strength. The
matching condition at leading order is given by Shifman:1978zn

$\displaystyle C_{G}(m_{t})=-\frac{1}{12\pi}C_{t}(m_{t}).$ (11)

When RGEs are evolved from the EW scale down to the hadronic scale $\mu_{\rm
had}\simeq 1$ GeV, ${\cal O}_{AA}$ will contribute to ${\cal O}_{q}$ through
the exchanging of virtual photons. For scale $\mu$ with $m_{q}<\mu<\mu_{\rm
EW}$, one has Frandsen:2012db

${C_{q}(\mu)}\simeq C_{q}(\mu_{\rm EW})+\frac{6Q_{q}^{2}\alpha_{\rm
em}}{\pi}\ln\left(\frac{\mu}{\mu_{\rm EW}}\right)C_{AA}(\mu_{\rm EW})$ (12)

at LL order, where $Q_{q}$ is the electric charge of quark and the
electromagnetic coupling constant $\alpha_{\rm em}\simeq 1/137$.

At $m_{b}$ and $m_{c}$ threshold, one has to integrate out the corresponding
heavy bottom and charm quarks by taking into account Eq. (11) again. Now at
scale $\mu_{\rm had}\simeq 1$ GeV, we get

$\displaystyle C_{q}\left(\mu_{\rm had}\right)$ $\displaystyle\simeq$
$\displaystyle\left(\frac{6Y_{q_{L}}Y_{q_{R}}\alpha_{1}}{\pi}C_{B}(\Lambda)+9\alpha_{2}^{2}\frac{v^{2}}{m_{h}^{2}}C_{W}(\Lambda)\right)\ln\left(\frac{\mu_{\rm
EW}}{\Lambda}\right)$ (13) $\displaystyle+$
$\displaystyle\frac{6Q_{q}^{2}\alpha_{\rm em}}{\pi}C_{AA}(\mu_{\rm
EW})\ln\left(\frac{\mu_{\rm had}}{\mu_{\rm EW}}\right),$ $\displaystyle
C_{G}\left(\mu_{\rm had}\right)$ $\displaystyle\simeq$
$\displaystyle-\frac{1}{12\pi}\Bigg{\\{}\left(\frac{\alpha_{1}}{\pi}C_{B}(\Lambda)+27\alpha_{2}^{2}\frac{v^{2}}{m_{h}^{2}}C_{W}(\Lambda)\right)\ln\left(\frac{\mu_{\rm
EW}}{\Lambda}\right)$ (14) $\displaystyle+$ $\displaystyle\frac{2\alpha_{\rm
em}}{3\pi}C_{AA}(\mu_{\rm EW})\left[\ln\left(\frac{m_{b}}{\mu_{\rm
EW}}\right)+4\ln\left(\frac{m_{c}}{\mu_{\rm EW}}\right)\right]\Bigg{\\}},$

where the Wilson coefficient $C_{AA}(\mu_{\rm EW})$ is determined by the first
matching condition in Eq. (9).

##  III Sensitivities at the LHeC and FCC-ep

In this section, we will investigate the sensitivity of the DM-gauge boson
effective operators at future electron-proton colliders.

###  III.1 Signal channels and SM backgrounds

From the effective interactions in Eqs. (1) and (2), DM pairs can be produced
at the electron-proton collider through both the Charged Current (CC) and the
Neutral Current (NC), which respectively correspond to $W$ boson mode and
$\gamma/Z$ mode of VBF processes. In CC production, DM can be generated
through the process of $e^{-}p\to\nu_{e}j\chi\bar{\chi}(\phi\phi^{*})$ via
$WW$ fusion, resulting in a mono-jet plus missing energy signature. It is
important to note that the mono-jet signature coincidentally aligns with the
background of CC deeply inelastic scattering. Due to the absence of kinematic
handles in the final state, distinguishing this signal channel from its
primary background poses a significant challenge. Consequently, our study is
primarily focused on NC production - specifically the process $e^{-}p\to
e^{-}j\chi\bar{\chi}~{}(\phi\phi^{*})$, as depicted in Fig. 1. The
corresponding signature is thereby characterized as the ${\not{E}}_{T}+e^{-}j$
channel.

Figure 1: Feynman diagrams for the processes $e^{-}p\to
e^{-}j\chi\bar{\chi}(\phi\phi^{*})$

For SM background estimation, we consider both reducible background (RB) and
irreducible background (IB). The IB mainly is due to the processes $e^{-}p\to
e^{-}j\nu_{\ell}\bar{\nu}_{\ell}~{}(\ell=e,~{}\mu,~{}\tau)$. Among them, both
$W/Z$-boson bremsstrahlung processes $e^{-}p\to
e^{-}jZ/\nu_{e}jW^{-}~{}(Z\to\nu_{e}\bar{\nu}_{e}/W^{-}\to
e^{-}\bar{\nu}_{e})$ contribute to $\nu_{e}$ final state, while for
$\nu_{\mu,\tau}$, only bremsstrahlung process $e^{-}p\to
e^{-}jZ~{}(Z\to\nu_{\mu,\tau}\bar{\nu}_{\mu,\tau})$ is relevant. For RB, a
major contribution comes from $e^{-}p\to e^{-}jW^{\pm}$ with $W$’s leptonic
decay $W^{\pm}\to\ell^{\pm}\nu_{\ell}$. There then two possibilities that need
to be taken into account. One is that electron and muon final states run out
of the detector acceptance and $\tau$ final state fakes a hard jet in the
detector. Another possibility is that the products of hadronic $\tau$ decays
are too soft to be tagged Tang:2015uha ; Han:2018rkz . As a consequence, The
SM background can be summarized as follows:

  * •

IB: $e^{-}p\to e^{-}j\nu_{\ell}\bar{\nu}_{\ell}$,

  * •

RB: $e^{-}p\to e^{-}j\ell^{\pm}\nu_{\ell}~{}(\bar{\nu}_{\ell})$,

where $\ell=e,\mu,\tau$.

###  III.2 Event simulation and kinematical cuts

For the purpose of event simulation, we generate Universal FeynRules Output
(UFO) Degrande:2011ua  model file by using FeynRules Alloul:2013bka . Which is
fed into MadGraph@NLO Alwall:2014hca  to generate parton-level events. For
both signal and backgrounds at the parton level, we apply the following pre-
selection cuts:

$\displaystyle p^{j,\ell}_{T}>5\,{\rm GeV},\quad\ |\eta_{j,\ell}|\ <5,$
$\displaystyle\Delta R_{j\ell}>0.4,\quad\Delta R_{\ell\ell}>0.4,$ (15)

where $p_{T}$ and $\eta$ are the transverse momentum and pseudorapidity for
the corresponding particles, and $\Delta R=\sqrt{\Delta\phi^{2}+\Delta y^{2}}$
is the separation in the azimuthal angle-rapidity ($\phi-y$) plane. Notice
that all of the cuts in Eq. (15) are defined in the lab frame. We then use
Pythia6.4 Sjostrand:2006za  to implement parton shower and hadronization, and
Delphes3.4 Ovyn:2009tx  to fast detector simulation according to the LHeC
designed parameters AbelleiraFernandez:2012cc .

In order to suppress RB, we apply the following veto criteria Han:2018rkz  as
basic cuts:

  * •

events are required to contain exactly one hard electron, one hard jet, and
${\not{E}}_{T}$,

  * •

events containing any extra jets with $p_{T}^{j}>3$ GeV and $|\eta_{j}|<2$ or
leptons with $p_{T}^{e,\mu}>5$ GeV or tagged $\tau$ jets are voted.

In Fig. 2 and Fig. 3, we respectively display the normalized distributions of
missing transverse energy ${\not{E}}_{T}$, the invariant mass of tagged
electron and jet system $M(e^{-},j)$ and inelasticity variable $y$ for both
signal and backgrounds after the basic cuts, at the 140 GeV $\otimes$ 7 TeV
LHeC and 60 GeV $\otimes$ 50 TeV FCC-ep. For signal, we assumed the benchmark
values $m_{\chi,~{}\phi}=1$ GeV, $C_{B,~{}W}=1$ and $\Lambda=500$ GeV. For
inelasticity variable $y$, we follow the definition in Ref. Tang:2015uha
which is given as

$\displaystyle y\equiv\frac{k_{P}\cdot(k_{e}-p_{e})}{k_{P}\cdot k_{e}},$ (16)

where $k_{P,~{}e}$ are the 4-momenta of the initial proton and electron, and
$p_{e}$ is the 4-momenta of the final electron. One can observe the following
features:

  * •

for ${\not{E}}_{T}$ distribution, SM RB tends to possess smaller value
${\not{E}}_{T}<100$ GeV compared with signals, while SM IB has similar with
signals,

  * •

for $y$ distribution, SM RB and IB respectively tend to distribute at small
and large $y$ values, while signals present relatively flat distribution.

Such distinct behaviors of signals and SM backgrounds indicate that we can
impose the following kinematical cuts for LHeC (FCC-ep) to further extract
signals:

$\displaystyle{\not{E}}_{T}>100\quad(220)\,{\rm GeV},$ $\displaystyle
M(e^{-},j)>100\quad(200)\,{\rm GeV},$ $\displaystyle
0.3<y<0.8\quad(0.1<y<0.9).$ (17)

Figure 2: The normalized distribution for the missing transverse energy
${\not{E}}_{T}$(left), the invariant mass of electron-jet system $M(e^{-},j)$
(middel) and the inelasticity variable $y$ (right) with $m_{\chi}=m_{\phi}=1$
GeV, $\Lambda=500$ GeV and $C_{B}=C_{W}=1$ at the 140 GeV $\otimes$ 7 TeV
LHeC. The red and black curves respectively correspond to scalar and fermion
DM signals, and SM RB and IB are shown by green and blue lines.

Figure 3: Same as Fig. 2, but at the 60 GeV $\otimes$ 50 TeV FCC-ep.  Process | Signal ($\phi$) | Signal ($\chi$) | RB | IB | ${\cal S}_{\phi}$ | ${\cal S}_{\chi}$ | ${\cal R}_{\phi}$ | ${\cal R}_{\chi}$
---|---|---|---|---|---|---|---|---
pre-selection cuts | 7.57 | 10.28 | 1421.87 | 260.39 | 8.26 | 11.20 | 0.0045 | 0.0061
basic cuts | 4.91 | 6.84 | 316.83 | 155.21 | 10.11 | 14.08 | 0.010 | 0.014
${\not{E}}_{T}>100$ GeV | 1.71 | 2.62 | 18.28 | 38.33 | 10.16 | 15.58 | 0.030 | 0.046
$M(e^{-},j)>100$ GeV | 1.56 | 2.33 | 15.46 | 29.28 | 10.46 | 15.56 | 0.035 | 0.052
$0.3<y<0.8$ | 1.01 | 1.54 | 5.40 | 9.16 | 11.80 | 18.05 | 0.070 | 0.106
Table 1: The remaining cross-section (in a unit of fb) of signal and background after corresponding cuts with $m_{\chi,~{}\phi}=1$ GeV, $C_{B,~{}W}=1$ and $\Lambda=500$ GeV at 140 GeV $\otimes$ 7 TeV LHeC. The corresponding significance ${\cal S}=N_{S}/\sqrt{N_{B}}$ and the signal-to-background ratio ${\cal R}=N_{S}/N_{B}$ are obtained for $L=2~{}{\rm ab}^{-1}$, where $N_{S}$ and $N_{B}$ are the number of signal and background events. Process | Signal ($\phi$) | Signal ($\chi$) | RB | IB | ${\cal S}_{\phi}$ | ${\cal S}_{\chi}$ | ${\cal R}_{\phi}$ | ${\cal R}_{\chi}$
---|---|---|---|---|---|---|---|---
pre-selection cuts | 18.17 | 64.85 | 1232.28 | 222.80 | 21.31 | 76.03 | 0.012 | 0.045
basic cuts | 11.56 | 42.58 | 244.95 | 125.09 | 26.87 | 98.98 | 0.031 | 0.12
${\not{E}}_{T}>220$ GeV | 3.10 | 13.44 | 0.86 | 4.32 | 60.88 | 264.17 | 0.060 | 2.60
$M(e^{-},j)>200$ GeV | 2.59 | 10.71 | 0.61 | 2.13 | 70.21 | 289.95 | 0.95 | 3.92
$0.1<y<0.9$ | 2.41 | 9.48 | 0.57 | 1.44 | 75.94 | 298.87 | 1.19 | 4.71
Table 2: Same as Table 1, but at the 60 GeV $\otimes$ 50 TeV FCC-ep.

In Table. 1 and Table. 2, we show the cut-flows for the fermion and scalar DM
signals and backgrounds at LHeC and FCC-ep, respectively. The corresponding
significance ${\cal S}=N_{S}/\sqrt{N_{B}}$ and signal-to-background ratio
${\cal R}=N_{S}/N_{B}$ with the total integrated luminosity $L=2~{}{\rm
ab}^{-1}$ are also shown, where $N_{S,~{}B}=\sigma_{S,~{}B}\cdot L$, are
respectively the survived event number of signals and backgrounds. One can see
that both ${\cal S}$ and ${\cal R}$ get increasing with each cut imposed.
After the pre-selection cuts, RB contribution is still huge, while the basic
cuts suppress RB by a factor of 4.5 (5.0) at the LHeC (FCC-ep). Moreover, the
RB is dramatically reduced by the ${\not{E}}_{T}$ cut, which only survived
about $1.26\%$ (0.07%) at the LHeC (FCC-ep).

At the LHeC, after all cuts, the fermion (scalar) DM signal can survive
$13.2\%$ ($14.9\%$), while RB (IB) only survived $0.37\%$ ($3.5\%$). At the
FCC-ep, the signal for fermion (scalar) DM survived $13.3\%$ ($14.6\%$), and
RB (IB) survived $0.047\%$ ($0.65\%$) at the end. It is worth noting that the
remaining cross-section of background at the FCC-ep are only $14\%$ of ones
that at the LHeC, while signal for fermion (scalar) DM production at the FCC-
ep could reach $6$ ($1.5$) times than that at the LHeC, which implies FCC-ep
has considerable improvement on significance compared with LHeC.

##  IV Sensitivity at the CEPC in three different modes

Due to the effective DM-diboson operators in Eqs. (1) and (2), the WIMP can be
produced at $e^{+}e^{-}$ colliders via the processes $e^{+}e^{-}\to
V^{\prime*}\to\phi\phi^{*}(\bar{\chi}\chi)+V\ (V=\gamma,Z)$, which give rise
to mono-photon or mono-$Z$ signatures. The corresponding Feynman diagrams are
displayed in Fig. 4. In this section, we will focus on the mono-photon
signature at future projected CEPC CEPCStudyGroup:2018rmc ;
CEPCStudyGroup:2018ghi  with three different running modes, including the
Higgs factory mode for seven years running at $\sqrt{s}=240\ \mathrm{GeV}$
with a total luminosity of $\sim 5.6\ \mathrm{ab}^{-1}$, the $Z$ factory mode
for two years running at $\sqrt{s}=91.2\ \mathrm{GeV}$ with a total luminosity
of $\sim 16\ \mathrm{ab}^{-1}$, and the $W^{+}W^{-}$ threshold scan mode for
one year running at $\sqrt{s}\sim$ $158-172\ \mathrm{GeV}$ with a total
luminosity of $\sim 2.6\ \mathrm{ab}^{-1}$. 111We take $\sqrt{s}=160$ GeV for
the $W^{+}W^{-}$ threshold scan mode throughout our analysis.

Figure 4: Feynman diagrams for the processes
$e^{+}e^{-}\to\phi\phi^{*}(\bar{\chi}\chi)+Z$ or
$e^{+}e^{-}\to\phi\phi^{*}(\bar{\chi}\chi)+\gamma$.

For the monophoton signature at CEPC, an irreducible background arises from
the neutrino production $e^{+}e^{-}\to\nu_{l}\bar{\nu}_{l}\gamma$ processes,
where $\nu_{l}=\nu_{e},\nu_{\mu},\nu_{\tau}$. Due to the Breit-Wigner
distribution of $Z$ boson in the irreducible BG, there is a resonance in the
final photon energy spectrum, which is located at

$E_{\gamma}^{Z}=\frac{s-M_{Z}^{2}}{2\sqrt{s}}$ (18)

with a full-width-at-half-maximum as
$\Gamma_{\gamma}^{Z}=M_{Z}\Gamma_{Z}/\sqrt{s}$. In order to suppress the
irreducible background contribution, we will veto the events within
$5\Gamma_{\gamma}^{Z}$ at the $Z$ resonance in the monophoton energy spectrum
Liu:2019ogn . The vetoing cut can be written as

$|E_{\gamma}-E_{\gamma}^{Z}|<5\Gamma_{\gamma}^{Z}.$ (19)

The major reducible SM backgrounds come from the
$e^{+}e^{-}\to\gamma+{\not{X}}$ processes, where only one photon is visible in
the final state, and ${\not{X}}$ denotes the other undetectable particle(s)
due to the limited detection capability of the detectors. In our analysis, the
parameters for the EMC coverage: $|\cos\theta|<0.99$ and $E>0.1$ GeV, are
adopted following CEPC CDR CEPCStudyGroup:2018ghi . The processes
$e^{+}e^{-}\to f\bar{f}\gamma$ and $e^{+}e^{-}\to\gamma\gamma\gamma$ provide
dominate contributions to the reducible background when final $f\bar{f}$ and
$\gamma\gamma$ are emitted with $|\cos\theta|>0.99$. Due to the momentum
conservation in the transverse direction and energy conservation, the maximum
photon energy as a function of its polar angle can be obtained as Liu:2019ogn

$E_{\gamma}^{m}(\theta_{\gamma})=\sqrt{s}\left(\frac{\sin\theta_{\gamma}}{\sin\theta_{b}}\right)^{-1},$
(20)

where the polar angle $\theta_{b}$ corresponds to the boundary of the EMC with
$|\cos\theta|<0.99$. In order to suppress the monophoton events arising from
the reducible background, we adopt the detector cut
$E_{\gamma}>E_{\gamma}^{m}(\theta_{\gamma})$ on the final state photon in our
analysis.

##  V Update limits from the LHC mono-$j$ and mono-$\gamma$ searches

Due to the DM-diboson operators, the WIMP can be produced via the processes
$qq^{\prime}\to V^{\prime*}\to\phi\phi^{*}(\bar{\chi}\chi)+V\ (V=\gamma,Z,W)$,
which can exhibit a mono-photon (${\not{E}_{T}+\gamma)}$ or a mono-jet
(${\not{E}_{T}+W/Z\ (\to{\rm hardons})}$) signature, respectively. The
representative Feynman diagrams are shown in Fig. 5. Besides, the loop-induced
effective DM-quarks (${\cal O}_{q}$) and DM-gluons (${\cal O}_{G}$) operators,
could also result in the mono-jet signature via $q\bar{q}\to\bar{\chi}\chi+g$,
$gg\to\bar{\chi}\chi+g$, and $qg\to\bar{\chi}\chi+q$ processes. The
representative Feynman diagrams are shown in Fig. 6.

Figure 5: Representative Feynman diagrams for a mono-photon (Left) and a mono-
jet signature (Right) arising from DM-diboson operators.

Figure 6: Representative Feynman diagrams for a mono-jet signature arising
from loop-induced effective DM-quarks and DM-gluons operators.

In this section, we consider the sensitivities from LHC mono-$j$ and
mono-$\gamma$ searches. To simulate the signal from dark matter, the parton-
level events are generated by MadGraph@NLO Alwall:2014hca , and the following
parton shower and hadronization are dealt with Pythia6.4 Sjostrand:2006za . We
use Delphes3.4 Ovyn:2009tx  to carry out a fast detector simulation for the
ATLAS or CMS detector with the corresponding parameter setup. We follow the
procedure of the mono-photon Aad:2020arf  and mono-jet Aad:2021egl  latest
analysis by the ATLAS Collaboration with an integrated luminosity of 139 fb-1
at a center-of-mass energy of 13 TeV.

In the case of ${\not{E}_{T}+\gamma}$ search channel, events in our analysis
are required to have a leading $\gamma$ with
$E_{{T}}^{\gamma}>150\mathrm{GeV},\ |\eta^{\gamma}|<1.37\text{ or
}1.52<|\eta^{\gamma}|<2.37,\ \Delta\phi\left(\gamma,\ \bm{E}_{{T}}^{\text{miss
}}\right)>0.4,$ 0 or 1 jet with $p_{{T}}>30\mathrm{GeV},\ |\eta|<4.5\text{ and
}\Delta\phi\left(\mathrm{jet},\ \bm{E}_{{T}}^{\mathrm{miss}}\right)>0.4$, and
none leptons. Besides, the ATLAS Collaboration performs the measurement in
seven different signal regions with a varying cut on the missing transverse
momentum ${E}_{{T}}^{\mathrm{miss}}$. We find that the strongest bounds on the
parameter space in our case are given with the most severe cut of
${E}_{{T}}^{\mathrm{miss}}>375$ GeV. The corresponding 95% C.L. limit on the
fiducial cross-section reads $\sigma_{\rm fid}<$ 0.53 fb.

For mono-jet search, events having identified muons, electrons, photons or
$\tau$-leptons in the final state are vetoed. Selected events have a leading
jet with $p_{T}>150$ GeV and $|\eta|<2.4$, and up to three additional jets
with $p_{T}>30$ GeV and $|\eta|<2.8$. Separation in the azimuthal angle
between the missing transverse momentum direction and each selected jet
$\Delta\phi\left(\mathrm{jet},\ \bm{E}_{{T}}^{\mathrm{miss}}\right)>0.4(0.6)$
is required for events with ${E}_{\mathrm{T}}^{\mathrm{miss}}>250$ GeV
($200\mathrm{GeV}<{E}_{{T}}^{\mathrm{miss}}<250$ GeV). It turns out that the
stronger restriction of ${E}_{{T}}^{\mathrm{miss}}>1200$ GeV provides the best
limits. At 95% C.L. the bound on the corresponding fiducial cross section is
given by $\sigma_{\rm fid}<$ 0.3 fb.

In Tab. 3, we present the ratio of the number of events for mono-jet signature
at LHC from the process ${pp\to\chi\bar{\chi}+W/Z\ (\to{\rm hardons})}$ via
dimension-7 DM-diboson operators with the Feynman diagrams shown in Fig. 5 to
one from the process ${pp\to\chi\bar{\chi}+q/g}$ via loop-induced DM-quarks
and DM-gluons operators with the Feynman diagrams shown in Fig. 6 with
$\Lambda$=1000 GeV. There we consider three cases, i.e., $C_{B}=C_{W}=500$,
$C_{B}=0,C_{W}=500$, and $C_{B}=500,~{}C_{W}=0$. It can be found that the
ratios in the three cases all increase with the increment of the DM mass.
Besides, to the mono-jet signature at LHC the contributions from tree-level
DM-diboson operators are about two orders of magnitudes over from loop-induced
DM-quarks and DM-gluons operators, especially about three orders in the case
of $C_{B}=1,~{}C_{W}=0$, thus we ignore the contribution from DM-quarks and
DM-gluons operators in our following analysis.

$m_{\chi}$ | 1 | 300 | 800 | 2000 | 3000
---|---|---|---|---|---
$C_{B}=500,C_{W}=0$ | 638.22 | 699.35 | 752.11 | 1435.32 | 2733.95
$C_{B}=0,C_{W}=500$ | 95.00 | 99.55 | 104.29 | 181.55 | 302.49
$C_{B}=500,C_{W}=500$ | 79.20 | 80.33 | 91.64 | 158.94 | 271.41
Table 3: The ratio of the number of events for the mono-jet signature at LHC
from the process ${pp\to\chi\bar{\chi}+W/Z\ (\to{\rm hardons})}$ via DM-
diboson operators with the Feynman diagrams shown in Fig. 5 to
${pp\to\chi\bar{\chi}+q/g}$ via loop-induced DM-quarks and DM-gluons operators
with the Feynman diagrams shown in Fig. 6 with $\Lambda$=1000 GeV.

##  VI Relic abundance, Direct and Indirect Constraints

In this section, we discuss the bounds on $\Lambda$ from current direct and
indirect search experiments. The effective operators ${\cal O}_{B}$ and ${\cal
O}_{W}$ lead to the non-relativistic cross-sections of DM annihilation
$\chi\bar{\chi},\ \phi\phi^{*}\to\gamma\gamma/\gamma Z/ZZ/WW$ as

$\langle\sigma_{k}v\rangle=a_{k}+b_{k}v^{2}+{\cal O}(v^{4})$ (21)

with $k=B,W$ annihilation channels. The coefficients of $a$ and $b$ are
Chen:2013gya

$\displaystyle\chi\left\\{\begin{aligned} a_{B}&=a_{W}=0,\\\
b_{B}&=\frac{\left|C_{B}(\Lambda)\right|^{2}}{\pi}\frac{m_{\chi}^{4}}{\Lambda^{6}}\Big{[}c_{w}^{4}+\frac{c_{w}^{2}s_{w}^{2}}{8}\beta_{\gamma
Z}^{2}\left(x_{Z}-4\right)^{2}\Theta(M_{Z}<2m_{\chi})+\frac{s_{w}^{4}}{8}\beta_{ZZ}\left(3x_{Z}^{2}-8x_{Z}+8\right)\Theta(M_{Z}<m_{\chi})\Big{]},\\\
b_{W}&=\frac{\left|C_{W}(\Lambda)\right|^{2}}{\pi}\frac{m_{\chi}^{4}}{\Lambda^{6}}\Big{[}s_{w}^{4}+\frac{c_{w}^{2}s_{w}^{2}}{8}\beta_{\gamma
Z}^{2}\left(x_{Z}-4\right)^{2}\Theta(M_{Z}<2m_{\chi})+\frac{c_{w}^{4}}{8}\beta_{ZZ}\left(3x_{Z}^{2}-8x_{Z}+8\right)\Theta(M_{Z}<m_{\chi})\\\
&+\frac{1}{4}\beta_{WW}\left(3x_{W}^{2}-8x_{W}+8\right)\Theta(M_{W}<m_{\chi})\Big{]},\end{aligned}\right.$
(22) $\displaystyle\phi\left\\{\begin{aligned}
a_{B}&=\frac{\left|C_{B}(\Lambda)\right|^{2}}{\pi}\frac{2m_{\phi}^{2}}{\Lambda^{4}}\Big{[}c_{w}^{4}+\frac{c_{w}^{2}s_{w}^{2}}{8}\beta_{\gamma
Z}^{2}\left(x_{Z}-4\right)^{2}\Theta(M_{Z}<2m_{\phi})+\frac{s_{w}^{4}}{8}\beta_{ZZ}\left(3x_{Z}^{2}-8x_{Z}+8\right)\Theta(M_{Z}<m_{\phi})\Big{]},\\\
a_{W}&=\frac{\left|C_{W}(\Lambda)\right|^{2}}{\pi}\frac{2m_{\phi}^{2}}{\Lambda^{4}}\Big{[}s_{w}^{4}+\frac{c_{w}^{2}s_{w}^{2}}{8}\beta_{\gamma
Z}^{2}\left(x_{Z}-4\right)^{2}\Theta(M_{Z}<2m_{\phi})+\frac{c_{w}^{4}}{8}\beta_{ZZ}\left(3x_{Z}^{2}-8x_{Z}+8\right)\Theta(M_{Z}<m_{\phi})\\\
&+\frac{1}{4}\beta_{WW}\left(3x_{W}^{2}-8x_{W}+8\right)\Theta(M_{W}<m_{\phi})\Big{]},\\\
b_{B}&=\frac{\left|C_{B}(\Lambda)\right|^{2}}{\pi}\frac{2m_{\phi}^{2}}{\Lambda^{4}}\Big{[}c_{w}^{4}/2+\frac{c_{w}^{2}s_{w}^{2}}{4}\beta_{\gamma
Z}^{2}\left(x_{Z}-4\right)\Theta(M_{Z}<2m_{\phi})+s_{w}^{4}\beta_{ZZ}\left(1/2-x_{Z}\right)\Theta(M_{Z}<m_{\phi})\Big{]},\\\
b_{W}&=\frac{\left|C_{W}(\Lambda)\right|^{2}}{\pi}\frac{2m_{\phi}^{2}}{\Lambda^{4}}\Big{[}s_{w}^{4}/2+\frac{c_{w}^{2}s_{w}^{2}}{4}\beta_{\gamma
Z}^{2}\left(x_{Z}-4\right)\Theta(M_{Z}<2m_{\phi})+c_{w}^{4}\beta_{ZZ}\left(1/2-x_{Z}\right)\Theta(M_{Z}<m_{\phi})\\\
&+\beta_{WW}\left(1-2x_{Z}\right)\Theta(M_{W}<m_{\phi})\Big{]}.\end{aligned}\right.$
(23)

In above equations, we have defined the phase space factor
$\beta_{ij}=\sqrt{1-\left(m_{i}+m_{j}\right)^{2}/\left(4m_{\rm
DM}^{2}\right)}$ and $x_{i}=m_{i}^{2}/m_{\rm DM}^{2}$ with ${\rm
DM}=\chi,\phi$. Using the results for $a_{B,W}$ and $b_{B,W}$, the relic
density can then be calculated with

$\Omega_{\rm DM}h^{2}=\frac{1.09\times
10^{9}x_{F}\mathrm{GeV}^{-1}}{M_{Pl}\sqrt{g_{*}\left(x_{F}\right)}\left(a+3b/x_{F}\right)},$
(24)

where $a=a_{B}+a_{W}$ and $b=b_{B}+b_{W}$. $x_{F}=m_{\rm DM}/T_{F}$, being the
ratio of the mass of DM $m_{\rm DM}$ and the early-universe freeze-out
temperature $T_{F}$, can be obtained by solving

$x_{F}=\ln\left[c(c+2)\sqrt{\frac{45}{8}}\frac{m_{\rm
DM}M_{Pl}(a+6b/x_{F})}{2\pi^{3}\sqrt{g_{*}(x_{F})}}\right].$ (25)

The loop-induced effective DM-quarks (${\cal O}_{q}$) and DM-gluons (${\cal
O}_{G}$) operators, could result in the interaction between DM and nucleon.
The spin-independent (SI) cross section for elastic scalar and Dirac WIMP
scattering on a nucleon has the form

$\displaystyle\sigma_{\rm SI}^{\mathrm{\phi}}$ $\displaystyle\simeq$
$\displaystyle\frac{\mu_{N}^{2}m_{N}^{2}}{\pi
m_{\phi}^{2}\Lambda^{4}}\left|\frac{\alpha
Z^{2}}{A}f_{A}^{N}C_{AA}\left(\mu_{\rm
had}\right)+\frac{Z}{A}f_{p}\left(\mu_{\rm
had}\right)+\frac{A-Z}{A}f_{n}\left(\mu_{\rm had}\right)\right|^{2}\,,$ (26)
$\displaystyle\sigma_{\rm SI}^{\mathrm{\chi}}$ $\displaystyle\simeq$
$\displaystyle\frac{\mu_{N}^{2}m_{N}^{2}}{\pi\Lambda^{6}}\left|\frac{\alpha
Z^{2}}{A}f_{A}^{N}C_{AA}\left(\mu_{\rm
had}\right)+\frac{Z}{A}f_{p}\left(\mu_{\rm
had}\right)+\frac{A-Z}{A}f_{n}\left(\mu_{\rm had}\right)\right|^{2}\,.$ (27)

In above equation, $m_{N}\simeq 0.939$ GeV is the average nucleon mass, and
$\mu_{N}=m_{\phi,\chi}m_{N}/(m_{\phi,\chi}+m_{N})$ is the reduced mass of the
DM-nucleon system. The form factor $f_{A}^{N}$ describes the Rayleigh
scattering of two photons on the entire nucleus. At zero-momentum transfer
limit, $f_{A}^{N}\simeq 0.08$ for xenon target. The form factor $f_{N=p,n}$
describing the couplings between the DM and nucleon reads Weiner:2012cb ;
Frandsen:2012db

$f_{N}\left(\mu_{\rm had}\right)=\sum_{q=u,d,s}f_{q}^{N}C_{q}\left(\mu_{\rm
had}\right)-\frac{8\pi}{9}f_{G}^{N}C_{G}\left(\mu_{\rm had}\right),$ (28)

where from factors $f_{q}^{N}$ describing the scalar couplings between quarks
and nucleon are given by Belanger:2013oya

$f_{d}^{p}=0.0191,\ f_{u}^{p}=0.0153,\ f_{d}^{n}=0.0273,\ f_{u}^{n}=0.0110,\
f_{s}^{p}=f_{s}^{n}=0.0447$ (29)

and

$f_{G}^{N}=1-\sum_{q=u,d,s}f_{q}^{N}.$ (30)

##  VII Results

We apply simple $\chi^{2}$ analysis to derive the lower bounds on $\Lambda$
for effective operators in Eq. (5) at future LHeC, FCC-ep, and CEPC at $95\%$
C.L., by requiring $\chi^{2}=N_{S}^{2}/N_{B}=3.84$ Zyla:2020zbs  with the
specified luminosities. For LHeC and FCC-ep, the total integrated luminosities
are assumed as $L=2~{}{\rm ab}^{-1}$. The limits of $\Lambda$ as a function of
the mass of Dirac fermion and scalar DM are displayed in Figs. 7 and 8,
respectively. We also present the exclusion limits derived from the mono-
photon Aad:2020arf  search and mono-jet Aad:2021egl  search at the LHC and
from the direct DM experiments XENON1T XENON:2017vdw ; XENON:2018voc , XENONnT
XENON:2023sxq  and PandaX-4T PandaX-4T:2021bab . For illustration, we plot the
contours of the relic abundance $\Omega_{\rm DM}h^{2}=0.1186\pm 0.0020$
Planck:2018vyg . The neutrino floor is also shown, which represents the WIMP-
discovery limit obtained with an assuming exposure of 1000 ${}^{8}B$ neutrinos
detected on a Xenon target Ruppin:2014bra . For the scalar DM, we also show
the constraints from the DM indirect searches through the gamma-ray and $WW$
observations by Fermi-LAT collaboration Fermi-LAT:2015att . Here we consider
three typical cases, i.e., $C_{B}=0,~{}C_{W}=500$, $C_{B}=500,~{}C_{W}=0$, and
$C_{B}=C_{W}=500$.

For fermion DM, the bounds from CEPC with all the three running modes can
touch the neutrino floor with the DM mass less than a few GeV, while the
$e^{+}e^{-}$ collider CEPC is not competitive with other $ep$ and $pp$
colliders. At LHC, the sensitivity from mono-jet search is better with
$C_{B}=0,C_{W}=500$, and $C_{B}=C_{W}=500$, but is slightly worse with
$C_{B}=500,~{}C_{W}=0$ than mono-photon search. Besides, the LHC mono-jet
search can give the best constraint with $C_{B}=0,C_{W}=500$ compared with
other collider experiments. From the figures, it is further validated that
FCC-ep has great advantages to constrain DM-gauge boson effective interactions
compared with LHeC as mentioned before. In the cases of $C_{B}=500,~{}C_{W}=0$
and $C_{B}=C_{W}=500$, FCC-ep can touch the region unconstrained by other
experiments when the DM mass in less than several hundred GeV, and push the
limits up to about 8000 GeV and 9000 GeV respectively.

For scalar DM, in the case of $C_{B}=0,C_{W}=500$ all the collider searches
cannot escape the neutrino floor, while the indirect searches of DM by the
Fermi-LAT collaboration provide the leading constraints and can touch the
neutrino floor in the mass region of $5\ {\rm GeV}\lesssim m_{\phi}\lesssim 7$
GeV and $m_{\phi}\gtrsim 130$ GeV with gamma-ray observation, and
$m_{\phi}\gtrsim 280$ GeV with $WW$ observation. In the case of
$C_{B}=500,~{}C_{W}=500$, except FCC-ep can reach the neutrino floor in the
mass region of $5\ {\rm GeV}\lesssim m_{\phi}\lesssim 6$ GeV, other collider
searches cannot escape the neutrino floor. Similar to the case of
$C_{B}=0,C_{W}=500$, DM searches by the Fermi-LAT collaboration can touch the
neutrino floor for $3\ {\rm GeV}\lesssim m_{\phi}\lesssim 7$ GeV and
$m_{\phi}\gtrsim 80$ GeV with cosmic gamma-ray observation, and $m_{\phi}>250$
GeV with $WW$ observation. In the case of $C_{B}=500,~{}C_{W}=0$, the cosmic
gamma-ray observation by the Fermi-LAT collaboration almost lies above the
neutrino floor in the plotted region, and $WW$ observation cannot provide any
constraint since $\phi\bar{\phi}W^{+}W^{-}$ coupling vanishes. For the
collider searches, the future FCC-ep provides the most stringent restrictions
relative to the other colliders for all the considered three cases with
$m_{\phi}$ less than a few hundred GeV. Besides, in the case of
$C_{B}=500,~{}C_{W}=0$, FCC-ep and CEPC in the $Z$ factory mode can probe a
light DM ( $m_{\phi}\lesssim 7$ GeV ) touching the neutrino floor.

Figure 7:  Constraints in the $m_{\chi}-\Lambda$ plane for the fermion DM with
$C_{B}^{6(7)}=0,C_{W}^{6(7)}=500$ (left panel) ,
$C_{B}^{6(7)}=500,C_{W}^{6(7)}=0$ (middle panel) and
$C_{B}^{6(7)}=500,C_{W}^{6(7)}=500$ (right panel). purple and orange solid
lines denote the exclusion limits from the mono-photon Aad:2020arf  and mono-
jet Aad:2021egl  searches at the 95% C.L. at the 13 TeV LHC. For the direct
searches, recent bounds from XENON1T XENON:2017vdw ; XENON:2018voc , XENONnT
XENON:2023sxq  and PandaX-4T PandaX-4T:2021bab  are shown as cyan, dark cyan
and steel blue solid lines, respectively. For illustration, the contours of
the relic abundance $\Omega_{\rm DM}h^{2}=0.1186$ are also plotted with gray
dash-dotted curves. The neutrino floor is shown in green-shaded regions. The
expected 95% C.L. bounds at the future LHeC (blue dotted lines), FCC-ep (dark
blue dashed lines), and CEPC (red dotted lines for $Z$ factory mode, red dash-
dotted lines for $W^{+}W^{-}$ threshold scan mode, red dashed lines for Higgs
factory mode) are shown.  Figure 8: Same as Fig. 7, but in the
$m_{\phi}-\Lambda$ plane for the fermion DM. The constraints from the indirect
search of DM through the cosmic gamma-ray (black solid line) and $WW$ (black
dashed line) observations by the Fermi-LAT collaboration Fermi-LAT:2015att
are also shown.

##  VIII Summary and conclusion

In this work, we focus on the constraints on the dimension-6 (7) effective
operators between scalar (Dirac fermion) DM and SM gauge bosons at future $ep$
colliders LHeC and FCC-ep via ${\not{E}}_{T}+e^{-}j$ signature for the first
time. We find that SM irreducible background and reducible background can be
effectively suppressed by imposing appropriate kinematic cuts. We also
consider the sensitivity at the future $e^{+}e^{-}$ collider CEPC in three
different modes with mono-photon signature and update the limits from the LHC
with current mono-$j$ and mono-$\gamma$ searches. Besides the contribution
from ${\not{E}_{T}+W/Z\ (\to{\rm hardons})}$ channel due to the DM-diboson
operators for mono-$j$ at LHC, we also investigate the contribution from the
loop induced effective DM-quarks and DM-glouns operators. We find that the
contribution from DM-quarks and DM-gluons operators can be ignored. We present
the constraints on effective energy scale $\Lambda$ as a function of DM mass
with three typical cases of Wilson coefficients:
$C_{B}^{6(7)}=C_{W}^{6(7)}=500$, $C_{B}^{6(7)}=0,C_{W}^{6(7)}=500$, and
$C_{B}^{6(7)}=500,C_{W}^{6(7)}=0$. The FCC-ep presents better sensitivity than
LHeC in all cases for scalar and Dirac fermion DM. It can be found that the
collider searches can probe light DM touching the neutrino floor.

###### Acknowledgements.

RD is supported in part by the National Key R&D Programme of China
(2021YFC2203100). YZ is supported in part by the National Natural Science
Foundation of China (Grant no. 11805001) and the Fundamental Research Funds
for the Central Universities (Grant no.JZ2023HGTB0222).

## References

  * (1) N. Aghanim et al. [Planck], Astron. Astrophys. 641 (2020), A6 [erratum: Astron. Astrophys. 652 (2021), C4] doi:10.1051/0004-6361/201833910 [arXiv:1807.06209 [astro-ph.CO]].
  * (2) B. W. Lee and S. Weinberg, Phys. Rev. Lett. 39 (1977), 165-168 doi:10.1103/PhysRevLett.39.165
  * (3) G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. 267 (1996), 195-373 doi:10.1016/0370-1573(95)00058-5 [arXiv:hep-ph/9506380 [hep-ph]].
  * (4) G. Bertone, D. Hooper and J. Silk, Phys. Rept. 405 (2005), 279-390 doi:10.1016/j.physrep.2004.08.031 [arXiv:hep-ph/0404175 [hep-ph]].
  * (5) J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T. M. P. Tait and H. B. Yu, Phys. Rev. D 82 (2010), 116010 doi:10.1103/PhysRevD.82.116010 [arXiv:1008.1783 [hep-ph]].
  * (6) P. J. Fox, R. Harnik, J. Kopp and Y. Tsai, Phys. Rev. D 85 (2012), 056011 doi:10.1103/PhysRevD.85.056011 [arXiv:1109.4398 [hep-ph]].
  * (7) A. Rajaraman, W. Shepherd, T. M. P. Tait and A. M. Wijangco, Phys. Rev. D 84 (2011), 095013 doi:10.1103/PhysRevD.84.095013 [arXiv:1108.1196 [hep-ph]].
  * (8) R. Ding, Y. Liao, J. Y. Liu and K. Wang, JCAP 05 (2013), 028 doi:10.1088/1475-7516/2013/05/028 [arXiv:1302.4034 [hep-ph]].
  * (9) Q. H. Cao, C. R. Chen, C. S. Li and H. Zhang, JHEP 08 (2011), 018 doi:10.1007/JHEP08(2011)018 [arXiv:0912.4511 [hep-ph]].
  * (10) A. Nelson, L. M. Carpenter, R. Cotta, A. Johnstone and D. Whiteson, Phys. Rev. D 89 (2014) no.5, 056011 doi:10.1103/PhysRevD.89.056011 [arXiv:1307.5064 [hep-ph]].
  * (11) A. Crivellin, U. Haisch and A. Hibbs, Phys. Rev. D 91 (2015), 074028 doi:10.1103/PhysRevD.91.074028 [arXiv:1501.00907 [hep-ph]].
  * (12) N. F. Bell, J. B. Dent, A. J. Galea, T. D. Jacques, L. M. Krauss and T. J. Weiler, Phys. Rev. D 86 (2012), 096011 doi:10.1103/PhysRevD.86.096011 [arXiv:1209.0231 [hep-ph]].
  * (13) L. M. Carpenter, A. Nelson, C. Shimmin, T. M. P. Tait and D. Whiteson, Phys. Rev. D 87 (2013) no.7, 074005 doi:10.1103/PhysRevD.87.074005 [arXiv:1212.3352 [hep-ex]].
  * (14) A. Alves and K. Sinha, Phys. Rev. D 92 (2015) no.11, 115013 doi:10.1103/PhysRevD.92.115013 [arXiv:1507.08294 [hep-ph]].
  * (15) M. Song, G. Li, W. G. Ma, R. Y. Zhang and J. Y. Guo, JHEP 09 (2014), 069 doi:10.1007/JHEP09(2014)069 [arXiv:1403.2142 [hep-ph]].
  * (16) N. Lopez, L. M. Carpenter, R. Cotta, M. Frate, N. Zhou and D. Whiteson, Phys. Rev. D 89 (2014) no.11, 115013 doi:10.1103/PhysRevD.89.115013 [arXiv:1403.6734 [hep-ph]].
  * (17) L. Carpenter, A. DiFranzo, M. Mulhearn, C. Shimmin, S. Tulin and D. Whiteson, Phys. Rev. D 89 (2014) no.7, 075017 doi:10.1103/PhysRevD.89.075017 [arXiv:1312.2592 [hep-ph]].
  * (18) A. Berlin, T. Lin and L. T. Wang, JHEP 06 (2014), 078 doi:10.1007/JHEP06(2014)078 [arXiv:1402.7074 [hep-ph]].
  * (19) C. Bartels, M. Berggren and J. List, Eur. Phys. J. C 72 (2012), 2213 doi:10.1140/epjc/s10052-012-2213-9 [arXiv:1206.6639 [hep-ex]].
  * (20) Y. J. Chae and M. Perelstein, JHEP 05 (2013), 138 doi:10.1007/JHEP05(2013)138 [arXiv:1211.4008 [hep-ph]].
  * (21) H. Dreiner, M. Huck, M. Krämer, D. Schmeier and J. Tattersall, Phys. Rev. D 87 (2013) no.7, 075015 doi:10.1103/PhysRevD.87.075015 [arXiv:1211.2254 [hep-ph]].
  * (22) Z. H. Yu, Q. S. Yan and P. F. Yin, Phys. Rev. D 88 (2013) no.7, 075015 doi:10.1103/PhysRevD.88.075015 [arXiv:1307.5740 [hep-ph]].
  * (23) F. Richard, G. Arcadi and Y. Mambrini, Eur. Phys. J. C 75 (2015), 171 doi:10.1140/epjc/s10052-015-3379-8 [arXiv:1411.0088 [hep-ex]].
  * (24) F. Rossi-Torres and C. A. Moura, Phys. Rev. D 92 (2015) no.11, 115022 doi:10.1103/PhysRevD.92.115022 [arXiv:1503.06475 [hep-ph]].
  * (25) M. Habermehl, K. Fujii, J. List, S. Matsumoto and T. Tanabe, PoS ICHEP2016 (2016), 155 doi:10.22323/1.282.0155 [arXiv:1702.05377 [hep-ex]].
  * (26) M. Jin and Y. Gao, Eur. Phys. J. C 78 (2018) no.8, 622 doi:10.1140/epjc/s10052-018-6093-5 [arXiv:1712.02140 [hep-ph]].
  * (27) Z. Liu, Y. H. Xu and Y. Zhang, JHEP 06 (2019), 009 doi:10.1007/JHEP06(2019)009 [arXiv:1903.12114 [hep-ph]].
  * (28) D. K. Ghosh, T. Katayose, S. Matsumoto, I. Saha, S. Shirai and T. Tanabe, Phys. Rev. D 101 (2020) no.1, 015007 doi:10.1103/PhysRevD.101.015007 [arXiv:1906.06864 [hep-ph]].
  * (29) M. Habermehl, M. Berggren and J. List, Phys. Rev. D 101 (2020) no.7, 075053 doi:10.1103/PhysRevD.101.075053 [arXiv:2001.03011 [hep-ex]].
  * (30) N. Wan, M. Song, G. Li, W. G. Ma, R. Y. Zhang and J. Y. Guo, Eur. Phys. J. C 74 (2014) no.12, 3219 doi:10.1140/epjc/s10052-014-3219-2 [arXiv:1403.7921 [hep-ph]].
  * (31) Z. H. Yu, X. J. Bi, Q. S. Yan and P. F. Yin, Phys. Rev. D 90 (2014) no.5, 055010 doi:10.1103/PhysRevD.90.055010 [arXiv:1404.6990 [hep-ph]].
  * (32) J. Liu, L. T. Wang, X. P. Wang and W. Xue, Phys. Rev. D 97 (2018) no.9, 095044 doi:10.1103/PhysRevD.97.095044 [arXiv:1712.07237 [hep-ph]].
  * (33) K. Kadota and A. Spray, JHEP 02 (2019), 017 doi:10.1007/JHEP02(2019)017 [arXiv:1811.00560 [hep-ph]].
  * (34) D. d’Enterria, PoS ICHEP2016 (2017), 434 doi:10.22323/1.282.0434 [arXiv:1701.02663 [hep-ex]].
  * (35) R. C. Cotta, J. L. Hewett, M. P. Le and T. G. Rizzo, Phys. Rev. D 88 (2013), 116009 doi:10.1103/PhysRevD.88.116009 [arXiv:1210.0525 [hep-ph]].
  * (36) A. Rajaraman, T. M. P. Tait and A. M. Wijangco, Phys. Dark Univ. 2 (2013), 17-21 doi:10.1016/j.dark.2012.12.001 [arXiv:1211.7061 [hep-ph]].
  * (37) J. Y. Chen, E. W. Kolb and L. T. Wang, Phys. Dark Univ. 2 (2013), 200-218 doi:10.1016/j.dark.2013.11.002 [arXiv:1305.0021 [hep-ph]].
  * (38) A. Crivellin and U. Haisch, Phys. Rev. D 90 (2014), 115011 doi:10.1103/PhysRevD.90.115011 [arXiv:1408.5046 [hep-ph]].
  * (39) M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Phys. Lett. B 78 (1978), 443-446 doi:10.1016/0370-2693(78)90481-1
  * (40) M. T. Frandsen, U. Haisch, F. Kahlhoefer, P. Mertsch and K. Schmidt-Hoberg, JCAP 10 (2012), 033 doi:10.1088/1475-7516/2012/10/033 [arXiv:1207.3971 [hep-ph]].
  * (41) Y. L. Tang, C. Zhang and S. h. Zhu, Phys. Rev. D 94 (2016) no.1, 011702 doi:10.1103/PhysRevD.94.011702 [arXiv:1508.01095 [hep-ph]].
  * (42) C. Han, R. Li, R. Q. Pan and K. Wang, Phys. Rev. D 98 (2018) no.11, 115003 doi:10.1103/PhysRevD.98.115003 [arXiv:1802.03679 [hep-ph]].
  * (43) C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer and T. Reiter, Comput. Phys. Commun. 183 (2012), 1201-1214 doi:10.1016/j.cpc.2012.01.022 [arXiv:1108.2040 [hep-ph]].
  * (44) A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, Comput. Phys. Commun. 185 (2014), 2250-2300 doi:10.1016/j.cpc.2014.04.012 [arXiv:1310.1921 [hep-ph]].
  * (45) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli and M. Zaro, JHEP 07 (2014), 079 doi:10.1007/JHEP07(2014)079 [arXiv:1405.0301 [hep-ph]].
  * (46) T. Sjostrand, S. Mrenna and P. Z. Skands, JHEP 05 (2006), 026 doi:10.1088/1126-6708/2006/05/026 [arXiv:hep-ph/0603175 [hep-ph]].
  * (47) S. Ovyn, X. Rouby and V. Lemaitre, [arXiv:0903.2225 [hep-ph]].
  * (48) J. L. Abelleira Fernandez et al. [LHeC Study Group], J. Phys. G 39 (2012), 075001 doi:10.1088/0954-3899/39/7/075001 [arXiv:1206.2913 [physics.acc-ph]].
  * (49) [CEPC Study Group], [arXiv:1809.00285 [physics.acc-ph]].
  * (50) J. B. Guimarães da Costa et al. [CEPC Study Group], [arXiv:1811.10545 [hep-ex]].
  * (51) G. Aad et al. [ATLAS], JHEP 02 (2021), 226 doi:10.1007/JHEP02(2021)226 [arXiv:2011.05259 [hep-ex]].
  * (52) G. Aad et al. [ATLAS], Phys. Rev. D 103 (2021) no.11, 112006 doi:10.1103/PhysRevD.103.112006 [arXiv:2102.10874 [hep-ex]].
  * (53) N. Weiner and I. Yavin, Phys. Rev. D 86 (2012), 075021 doi:10.1103/PhysRevD.86.075021 [arXiv:1206.2910 [hep-ph]].
  * (54) G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 185 (2014), 960-985 doi:10.1016/j.cpc.2013.10.016 [arXiv:1305.0237 [hep-ph]].
  * (55) P. A. Zyla et al. [Particle Data Group], PTEP 2020 (2020) no.8, 083C01 doi:10.1093/ptep/ptaa104
  * (56) E. Aprile et al. [XENON], Phys. Rev. Lett. 119 (2017) no.18, 181301 doi:10.1103/PhysRevLett.119.181301 [arXiv:1705.06655 [astro-ph.CO]].
  * (57) E. Aprile et al. [XENON], Phys. Rev. Lett. 121 (2018) no.11, 111302 doi:10.1103/PhysRevLett.121.111302 [arXiv:1805.12562 [astro-ph.CO]].
  * (58) E. Aprile et al. [XENON], Phys. Rev. Lett. 131 (2023) no.4, 041003 doi:10.1103/PhysRevLett.131.041003 [arXiv:2303.14729 [hep-ex]].
  * (59) Y. Meng et al. [PandaX-4T], Phys. Rev. Lett. 127 (2021) no.26, 261802 doi:10.1103/PhysRevLett.127.261802 [arXiv:2107.13438 [hep-ex]].
  * (60) N. Aghanim et al. [Planck], Astron. Astrophys. 641 (2020), A6 [erratum: Astron. Astrophys. 652 (2021), C4] doi:10.1051/0004-6361/201833910 [arXiv:1807.06209 [astro-ph.CO]].
  * (61) F. Ruppin, J. Billard, E. Figueroa-Feliciano and L. Strigari, Phys. Rev. D 90 (2014) no.8, 083510 doi:10.1103/PhysRevD.90.083510 [arXiv:1408.3581 [hep-ph]].
  * (62) M. Ackermann et al. [Fermi-LAT], Phys. Rev. Lett. 115 (2015) no.23, 231301 doi:10.1103/PhysRevLett.115.231301 [arXiv:1503.02641 [astro-ph.HE]].