# Search for Leptophilic Dark Matter at the LHeC

Guo-yuan Huang<EMAIL_ADDRESS>Max-Planck-Institut für
Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Sudip Jana
<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik, Saupfercheckweg
1, 69117 Heidelberg, Germany  Álvaro S. de Jesus<EMAIL_ADDRESS>International Institute of Physics, Universidade Federal do Rio Grande do
Norte, Campus Universitario, Lagoa Nova, Natal-RN 59078-970, Brazil
Departamento de Fisica, Universidade Federal do Rio Grande do Norte,
59078-970, Natal, RN, Brasil  Farinaldo S. Queiroz<EMAIL_ADDRESS>International Institute of Physics, Universidade Federal do Rio Grande do
Norte, Campus Universitario, Lagoa Nova, Natal-RN 59078-970, Brazil
Departamento de Fisica, Universidade Federal do Rio Grande do Norte,
59078-970, Natal, RN, Brasil  Millennium Institute for SubAtomic Physics at
the High-energy frontIeR, SAPHIR, Chile  Werner Rodejohann
<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany

###### Abstract

The Large Hadron electron Collider (LHeC) has been designed to push the field
of deep inelastic scattering to the high energy and intensity frontier using
an intense electron beam with a proton beam from the High Luminosity–Large
Hadron Collider. However, LHeC is also a great laboratory for new physics. In
this work, we propose a search for dark matter that couples with leptons. This
may yield $ej+\not{E}$ and $\mu j+\not{E}$ signals that can be potentially
observed through simple missing-energy cuts that suppress the Standard Model
background. Considering direct dark matter detection and LHC constraints, we
show that LHeC can indeed discover a weak scale dark matter fermion for masses
up to $350$ GeV, which reproduces the correct relic density, and has
interesting implications for lepton flavor violation.

##  I Introduction

There is an extensive program dedicated to dark matter searches worldwide.
Dark matter particles might leave imprints at direct and indirect detection
experiments. However, due to the nature of these searches which are subject to
nuclear and astrophysical uncertainties, respectively, it is desirable that
one could also observe a dark matter signal at colliders or accelerators. The
properties of the dark matter particle dictate the type of signals one may
observe. Several collider searches have been conducted at the Large Hadron
Collider (LHC) and other accelerators, but thus far no positive result signal
has been found.

The LHC is an excellent probe when dark matter particles interact with quarks.
Vector boson fusion production of dark matter yields interesting but less
restrictive constraints Abercrombie  _et al._ (2020). If dark matter features
leptonic interactions, obviously an $e^{+}e^{-}$ collider Dreiner  _et al._
(2013) or an LHeC that features an electron beam allowing synchronous
operation of $ep$ with $pp$ collisions at the LHC Abelleira Fernandez  _et
al._ (2012) would be helpful. Future colliders such as the Future Circular
Collider Abada  _et al._ (2019) and the International Linear Collider Baer
_et al._ (2013) are very long-term proposals, and LHeC can be seen as a very
important step towards this major new facilities in terms of physics as well
as technology. That said, could LHeC probe weak scale dark matter? As usual,
dark matter particles are missing energy at colliders as they simply leave no
imprint at the detectors, and a necessary visible counterpart is required as a
trigger.

If dark matter can be produced via $s$-channel diagrams, one can take
advantage of resonance production of the mediators and quickly conclude that
LHeC cannot be regarded as the best probe for such dark matter particles.
However, if dark matter is produced via leptonic processes, LHeC stands out.
We will present here the LHeC sensitivity to dark matter particles that have
couplings to leptons. Without loss of generality, we assume the following
Yukawa interaction

$\mathcal{L}_{Y}\supset y_{N\ell}\eta\overline{N}L_{\ell}+{\rm h.c.}$ (1)

Here $N$ is a Majorana singlet fermion and our dark matter particle, $\eta$ is
a scalar SM doublet, $L$ is a lepton doublet, and $\ell=e,\mu$. This Yukawa
interaction is often present in neutrino mass models Ma (2006), but in our
work it is treated more generally. A $Z_{2}$ symmetry has been imposed to the
new particles to forbid the Majorana fermion to couple with other SM states,
warranting its stability, as long as its mass is lighter than that of the
inert doublet. The doublet is inert either by symmetry requirements or by
theoretical constructions. The dark matter abundance of the Majorana fermion
occurs solely through $t$-channel processes into SM leptons in the standard
freeze-out regime. There are no dark matter direct detection signals at tree
level. At one-loop level, one can potentially induce dark matter-nucleus
scattering with charge-charge, dipole-charge, and dipole-dipole interactions
Schmidt  _et al._ (2012). After investigating the collider phenomenology, we
put our findings into perspective with lepton flavor violation, namely
$\mu\rightarrow e\gamma$ Toma and Vicente (2014); Lindner  _et al._ (2018),
and discuss the importance of direct dark matter detection.

In summary, Eq. (1) is commonly presented in a multitude of models, and has
implications for dark matter, lepton flavor violation as well as neutrino mass
models. In what follows, we will assess the LHeC potential to discover such
dark matter particle to solidly conclude that LHeC offers a unique opportunity
to discover a weak scale dark matter particle.

##  II Signature of Weak Scale Dark Matter at the LHeC

As explained above, there are benefits of the electron-proton collider: a
relatively clean environment for new physics searches, and the possibility to
probe weak scale dark matter that couples to leptons.

Figure 1: Feynman diagrams contributing to the typical events $ej+\not{E}$ and
$\mu j+\not{E}$.

###  II.1 Relevant Event Topologies

The relevant Feynman diagrams are shown in Fig. 1. We consider each channel at
a time. We find that the effective Lagrangian can induce the following event
topologies from electron-proton scattering at LHeC:

  * •

$ej+\not{E}$: The contributed signal process reads $ep\to ejNN$, where the
dark matter $N$ generates missing energy. The SM backgrounds for this channel
include the following: (i) $ep\to ejX$ with $X$ ($\gamma$, $G$, etc.); (ii)
$ep\to ej\nu\nu$; (iii) $ep\to ej\nu X$ ($X=\ell$); (iv) $ep\to ejXX$
($X=\gamma,G,q,\ell$). Here and in the following, $X$ denotes an in principle
visible particle that however escapes the angular acceptance of the detector.
Note that we use $\nu$ in the final state to denote both neutrino and
antineutrino for all possible flavors. In this analysis, we do not distinguish
light quarks and gluons in the final state. Processes with even higher orders,
e.g., $ep\to ej\nu\nu X$, are neglected.

  * •

$\mu j+\not{E}$: The final state with muons has fewer SM backgrounds because
of the suppression of lepton flavor violation. In comparison to the
$ej+\not{E}$ final state, we identify three types of SM backgrounds: (i)
$ep\to\mu j\nu\nu$; (ii) $ep\to\mu j\nu X$ ($X=e^{-},\mu^{+}$); (iii)
$ep\to\mu jXX$ ($X=e^{-},\mu^{+}$).

  * •

Mono $j+\not{E}$: The mono-jet topology corresponds to the process $eq\to j\nu
NN$. There are the following SM backgrounds: (i) $eq\to j\nu(X)$; (ii) $eq\to
j\nu X$; (iii) $eq\to j\nu(X)\nu(X)\nu(X)$. The leading-order process $eq\to
j\nu$ will be an overwhelmingly large background and cannot be reduced by
imposing missing energy cuts. Thus, we will focus on the previous two
topologies in our analysis.

Some of those backgrounds have a similar diagrammatic pattern as the signal
channel, hence irreducible. However, some of them can be reduced by choosing
proper cuts which we will describe later.

Event Topology | Signal $\sigma$ (fb) |  Background $\sigma$ (fb)
---|---|---
$ej+\not{E}$ | 24.4 (16.6) | $ejX$ | $ej\nu\nu$ | $ej\nu X$ | $ejXX$
$1.83\times 10^{5}$ ($0$) |  $92.9$ ($33.1$) |  $73.1$ ($2.20$) |  $5.20\times 10^{5}$ ($0$)
$\mu j+\not{E}$ | 24.5 | $\mu jX$ | $\mu j\nu\nu$ | $\mu j\nu X$ | $\mu jXX$
0 | 63.8 | 3.01 | 0.140
Table I: The cross sections for two event topologies $ej+\not{E}$ and $\mu
j+\not{E}$, assuming 60 GeV electrons colliding with 7 TeV protons. For the
signal cross section, the model parameters have been taken as
$M_{\eta}=300\leavevmode\nobreak\ {\rm GeV}$, $M_{N}=100\leavevmode\nobreak\
{\rm GeV}$ and $y_{Ne}=y_{N\mu}=1$. For $ej+\not{E}$, the cross section is
given with the cut $\not{p}_{\rm T}\gtrsim 10\leavevmode\nobreak\ {\rm GeV}$
(or $\not{p}_{\rm T}\gtrsim 100\leavevmode\nobreak\ {\rm GeV}$ for the value
in parentheses). The label ‘$X$’ in the table denotes visible particles which
have however escaped the detector angular acceptance, e.g., $\gamma$, $G$, $q$
and $\ell$.

###  II.2 Sensitivity to the Parameter Space

The sensitivity is obtained by a simple count analysis of total events without
involving detailed event distributions. Hence, the present analysis represents
a conservative result. We adopt the following chi-square to estimate the
sensitivity:

$\displaystyle\chi^{2}(M_{N},M_{\eta},y_{N})=\frac{N^{2}_{\rm sig}}{N_{\rm
sig}+N_{\rm bkg}}\;,$ (2)

where $N_{\rm sig}$ and $N_{\rm bkg}$ are the total signal and background
event numbers, respectively. The following cut will be implemented at the
parton level: $p_{\rm T}(j/b)>20\leavevmode\nobreak\ {\rm GeV}$, $p_{\rm
T}(\ell)>10\leavevmode\nobreak\ {\rm GeV}$, and the pseudorapidity
$|\eta(j/b/\ell)|<5$ Cottin  _et al._ (2021) (see also Lindner  _et al._
(2016)).

Figure 2: The efficiency of the missing-energy cut for the $ej+\not{E}$
topology. The cut is placed as $\not{p}_{\rm T}=|\bm{p}_{\rm T}(e)+\bm{p}_{\rm
T}(j)|>\not{p}_{\rm T,min}$.

Figure 3: The $3\sigma$ sensitivity region for the LHeC setup based on 60 GeV
electrons colliding with 7 TeV protons with an integrated luminosity of
$L=1\leavevmode\nobreak\ {\rm ab}^{-1}$. In the left panel, we show the
sensitivity to the inert scalar mass $M_{\eta}$ and Majorana fermion mass
$M_{N}$ by fixing the Yukawa coupling $y_{Ne},y_{N\mu}=1$. In the right panel,
we fix the inert scalar mass $M_{\eta}=400\leavevmode\nobreak\ {\rm GeV}$
instead, and scan over $y_{Ne}$ and $M_{N}$. The sensitivities for the
$ej+\not{E}$ and $\mu j+\not{E}$ topologies are displayed with orange and blue
regions, respectively. See text for details.

For the topology $ej+\not{E}$, we further impose a cut on $\not{p}_{\rm
T}=|\bm{p}_{\rm T}(e)+\bm{p}_{\rm T}(j)|$ to reject the $t$-channel
backgrounds from direct two-body scattering $eq\to ej$ (fully), $eq\to
ej\gamma$ (partly) and $ep\to ejXX$ (partly). The process $eq\to ej\gamma$
contains the initial state radiation of collinear photons and can be
efficiently reduced by imposing the missing energy cut. The effectiveness of
the cut is shown in Fig. 2 in terms of $\chi^{2}$ defined in Eq. (2), where
the model parameters have been taken as $M_{N}=100\leavevmode\nobreak\ {\rm
GeV}$, $M_{\eta}=300\leavevmode\nobreak\ {\rm GeV}$ and $y_{Ne}=1$. We
highlight that $y_{Ne}$ is the Yukawa couplings between the electron and the
fermion $N$, whereas $y_{N\mu}$ the Yukawa coupling between the muon and the
fermion $N$. Notice, there is only one Majorana fermion $N$ in our collider
study, and that for the $\mu j+\not{E}$ signal to exist, $y_{Ne}$ ought to be
non-zero (see Fig. 1). The $ej+\not{E}$ signal is insensitive to the presence
of couplings to muons, however.

That said, we find that a cut $\not{p}_{\rm T}\gtrsim 100\leavevmode\nobreak\
{\rm GeV}$ can nearly optimize the signal-to-background ratio of $ej+\not{E}$,
for which the backgrounds enhanced by collinear processes are almost
vanishing. After the cut, the dominant background stems from the process
$ep\to ej\nu\nu$. The cut adopted on $\not{p}_{T}$ is based on the $\chi^{2}$
study (see Fig. 2). For $\mu j+\not{E}$, we find that the leading background
is from the irreducible process $eq\to\mu q\nu_{e}\overline{\nu}_{\mu}$. The
obtained background event numbers for two major signal channels are collected
in Table I. Before the missing energy cut, the cross sections for $ej+\not{E}$
and $\mu j+\not{E}$ are almost identical, as their contributing diagrams are
the same. The backgrounds from $ejX$ and $ejXX$ are overwhelmingly large due
to the collinear enhancement and can be almost completely removed by imposing
the cut $\not{p}_{\rm T}\gtrsim 100\leavevmode\nobreak\ {\rm GeV}$. LHeC’s
sensitivity to the model is given in Fig. 3. The sensitivity is obtained for
the $3\sigma$ confidence level with $\chi^{2}=11.83$ for two degrees of
freedom. In the left panel, we fix the Yukawa coupling to one, and then vary
the masses of the inert scalar, $M_{\eta}$, and Majorana fermion $M_{N}$. In
the right panel, we fix the inert scalar mass to be
$M_{\eta}=400\leavevmode\nobreak\ {\rm GeV}$, and scan over the Yukawa
coupling $y_{N}$ and $M_{N}$. The orange and blue regions correspond to
$ej+\not{E}$ and $\mu j+\not{E}$ events, respectively. For the $ej+\not{E}$
event topology, the cut $\not{p}_{\rm T}\gtrsim 100\leavevmode\nobreak\ {\rm
GeV}$ has been imposed, while for $\mu j+\not{E}$ there is no necessity for
such a cut, as explained above. We observe that comparable sensitivities can
be achieved for $ej+\not{E}$ (with $\not{p}_{\rm T}$ cut) and $\mu j+\not{E}$
events, whereas for some parameter space, one may have a better sensitivity
than the other. We have investigated each scenario at a time. In other words,
for the $ej+\not{E}$ study $N$ couples only to the first generation, with
$y_{Ne}=1$. Later we probed the $\mu j+\not{E}$ signal which requires
couplings to both first and second generation of leptons, and we assumed
$y_{Ne}=y_{N\mu}=1$.

##  III Dark Matter Abundance

We emphasize that our entire study is based on three quantities, the mass of
the dark matter particle, $M_{N}$, inert doublet mass, $M_{\eta}$, and the
corresponding Yukawa coupling in Eq. (1). We remind the reader that we
consider each lepton generation at a time. Nevertheless, as far as the dark
matter relic density is concerned, they yield the same result. The dark matter
relic density is computed under the standard freeze-out regime, where the dark
matter decoupling occurs in an epoch in which the Universe is radiation-
dominated, via the Boltzmann equation

$\frac{dn_{N}}{dt}+3Hn_{N}=-\langle\sigma v\rangle(n^{2}_{N}-n^{2}_{N,\rm
eq})\,.$ (3)

Here $n_{N}$ is the number density, $H$ the Hubble rate and $\langle\sigma
v\rangle$ the annihilation cross section. The abundance is driven by
$t$-channel annihilation processes into leptons mediated by the inert doublet,
$\eta$, as dictated by Eq. (1). We have solved the Boltzmann equation
numerically with the MicrOMEGAs package Belanger  _et al._ (2007), and used it
to plot the region of parameter space that reproduces the right relic density
(green curve) in Fig. 3. In the left panel, we take $y_{Ne}=1$ and
$y_{N\mu}=1$ at a time, and perform a scan in the masses. In the right panel,
we take $M_{\eta}=400$ GeV, and scan over $y_{Ne}$. We assumed $y_{N\mu}$ to
be sufficiently small such that the process involving muon pairs is irrelevant
for the relic density. The reason for taking a suppressed value for $y_{N\mu}$
has to do with lepton flavor violation probes, but we will address this later.
Anyway, looking at Fig. 3, one can clearly conclude that LHeC can detect a
weak scale dark matter candidate that reproduces the right relic density with
masses up to $350$ GeV. The relic density curve is subject to shifts if one
embeds Eq. (1) in a non-standard cosmology scenario Allahverdi  _et al._
(2020), allowing one to get the right relic density for larger dark matter and
inert scalar masses.

Figure 4: Feynman diagram that leads to $\mu\rightarrow e\gamma$ in our model,
when both $y_{N\mu}$ and $y_{Ne}$ are different from zero. See text for
details.

##  IV Lepton Flavor Violation

Thus far, our study was based on the presence of one Majorana fermion $N$ that
couples either to first or to second generation leptons. It is clear from Fig.
3 that LHeC stands an excellent probe for leptophilic dark matter. In the
first setup, where only couplings to the first generation is present, there is
no lepton flavor violation signal, but in the second case where both couplings
to first and second generations are included, lepton flavor violation
naturally occurs Adulpravitchai  _et al._ (2009); Kubo  _et al._ (2006), as we
will explain below. When $y_{N\mu}$ and $y_{Ne}$ are non-zero as assumed in
the second setup, the lepton flavor violation muon decay occurs are displayed
in Fig. 4. The full analytical result for the branching ratio is lengthy, and
it relies on the product $(y_{N\mu}\times y_{Ne})^{2}$. This product cannot be
of order one, otherwise $BR(\mu\to e\gamma)\sim 10^{-7}$, which is orders of
magnitude above the experimental limit $BR(\mu\rightarrow e\gamma)<4.2\times
10^{-13}$ Baldini  _et al._ (2016). In the left-panel of Fig. 3 we have shown
that one can observe the $\mu j+\not{E}$ at the LHeC for larger dark matter
masses compared with the $ej+\not{E}$ channel. Now comes the importance of
complementary probes. Putting our collider findings into perspective with
lepton flavor violation, we can fiercely exclude the entire $\mu j+\not{E}$
region in Fig. 3 because there we had assumed $y_{N\mu}=y_{Ne}=1$. Lowering
$y_{N\mu}$ to values consistent with $\mu\rightarrow e\gamma$ would mean peter
out the $\mu j\not{E}$ signal. Focusing on the $ej+\not{E}$ signal, we can
estimate what should be the values of $y_{N\mu}$ could yield a
$BR(\mu\rightarrow e\gamma)$ consistent with the current limit, while
reproducing the correct dark matter relic density. With this in mind, we
display in Fig. 3 the $3\sigma$ signal region in $M_{N}$–$y_{Ne}$ plane, and
overlaid contours that yield a $BR(\mu\rightarrow e\gamma)$ in agreement with
current bounds for three different Yukawa couplings, namely
$y_{N\mu}=10^{-3}$, $y_{N\mu}=7.5\times 10^{-4}$ and $y_{N\mu}=5\times
10^{-4}$. The overlapping region between the green curve (dark matter relic
density) and orange curve (LHeC signal) which is below the contours represents
a region where LHeC can detect a weak scale dark matter candidate that
reproduces the correct relic density, and could leave some imprints on
$\mu\rightarrow e\gamma$ probes in the next generation of experiments, for
different coupling strengths to muons.

##  V Direct Detection

We have checked that direct detection bounds in this simplified case arise at
one loop via $Z$-exchange, but at with very suppressed strength. Therefore, we
do not expect any direct detection signal from Eq. (1). Considering an
extended scenario where an additional neutral fermion $N_{2}$ is present, and
a potentially sizeable direct detection signal stems from dark matter-nucleus
scattering111Electron recoils are more relevant for light dark matter Kopp
_et al._ (2009); Altmannshofer  _et al._ (2016).. At one-loop level, it leads
to the effective Lagrangian,

$\mathcal{L}_{eff}\supset\alpha\bar{N}_{2}\gamma^{\mu}N\partial^{\nu}F_{\mu\nu}+\beta\bar{N}_{2}\sigma^{\mu\nu}NF_{\mu\nu}+\gamma\bar{N}_{2}\gamma^{\mu}NA_{\mu},\\\
$

where $\alpha$, $\beta$, and $\gamma$ are coefficients that depend on the
product $(y_{N\ell}\times y_{N_{2}\ell})$, which are the Yukawa couplings of
the Majorana fermions with leptons. The terms proportional to coefficients
$\alpha$ and $\gamma$ give rise to a charge-charge operator, $\beta$ to a
dipole-charge and dipole-dipole operators Schmidt  _et al._ (2012). Anyway,
any direct detection signal is limited by $v_{\rm min}$, which is the minimum
velocity at which a dark matter particle can produce a measurable energy
recoil,

$\displaystyle v_{\rm
min}=\left(\frac{M_{A}E_{R}}{\mu_{AN}}+\delta\right)\frac{1}{\sqrt{2M_{A}E_{R}}},$
(4)

where $\delta=M_{N_{2}}-M_{N}$, and $\mu$ is the dark matter-nucleus reduced
mass. As we increase the mass difference between these two fermions, we can
only induce dark matter-nucleus scattering for large values of $v_{\rm min}$,
i.e. at the tail of the Maxwell-Boltzmann distribution Butsky  _et al._
(2016), thus weakening the direct detection bounds. For concreteness, as long
as $\delta>80$ keV, the bounds from direct detection are relatively weak
Schmidt  _et al._ (2012), thus not shown in Fig. 3. Anyway, in our collider
reach study, we have not made any assumption regarding the fermion $N_{2}$.
Therefore, we take it to be sufficiently heavier than $N$, with no prejudice.
In summary, direct detection yields less restrictive bounds compared to lepton
flavor violation, even in this extended scenario with an additional SM
singlet.

##  VI LHC Bounds

There are constraints on this scenario from LHC, which arise from slepton
searches. Since the inert scalar, $\eta$ is a $SU(2)_{L}$ doublet, one can use
the limits from left-handed selectron and smuon searches; $\eta$ will be pair-
produced and then decays back to dark matter. This analysis has been carried
out by the ATLAS collaboration Aad  _et al._ (2020), and they could probe
inert doublet masses up to $170$ GeV only. That is why we have not included
those limits in Fig. 3.

##  VII Conclusions

We have shown that LHeC, a collider that features a trade-off in the high
energy and intensity frontier, can probe a weak scale fermion dark matter
particle, $N$, for masses up to $350$ GeV, through a generic Lagrangian
containing an inert doublet. We found that the $ej+\not{E}$ and $\mu
j+\not{E}$ signal topologies are promising and derived $3\sigma$ signal
contours for a LHeC setup consisting of $60$ GeV electrons colliding with $7$
TeV protons and an integrated luminosity of $\mathcal{L}=1$ ab-1. We computed
the dark matter relic density in the standard freeze-out regime, found that
direct detection and LHC bounds are subdominant, and put our findings into
perspective with the lepton flavor violation to conclude that the $ej+\not{E}$
channels offers a unique opportunity to detect a weak scale dark matter
candidate that reproduces the correct relic density and yield a positive
signal at $\mu\rightarrow e\gamma$ probes in the next generation of
experiments.

## Acknowledgements

The authors thank Avelino Vicente, Carlos Yaguna, and Diego Restrepo for
discussions. We thank FAPESP grant 2021/01089-1, ICTP-SAIFR FAPESP grant
2016/01343-7, CNPq grant 408295/2021-0, Serrapilheira Foundation grant
Serra-1912–31613, FONDECYT Grant 1191103 and ANID-Programa Milenio-code
$ICN2019\\_044$. GYH is supported by the Alexander von Humboldt Foundation.

## References

  * Abercrombie  _et al._ (2020) D. Abercrombie _et al._ , Phys. Dark Univ. 27, 100371 (2020), arXiv:1507.00966 [hep-ex] .
  * Dreiner  _et al._ (2013) H. Dreiner, M. Huck, M. Krämer, D. Schmeier, and J. Tattersall, Phys. Rev. D 87, 075015 (2013), arXiv:1211.2254 [hep-ph] .
  * Abelleira Fernandez  _et al._ (2012) J. L. Abelleira Fernandez _et al._ (LHeC Study Group), J. Phys. G 39, 075001 (2012), arXiv:1206.2913 [physics.acc-ph] .
  * Abada  _et al._ (2019) A. Abada _et al._ (FCC), Eur. Phys. J. ST 228, 261 (2019).
  * Baer  _et al._ (2013) H. Baer _et al._ , (2013), arXiv:1306.6352 [hep-ph] .
  * Ma (2006) E. Ma, Phys. Rev. D 73, 077301 (2006), arXiv:hep-ph/0601225 .
  * Schmidt  _et al._ (2012) D. Schmidt, T. Schwetz, and T. Toma, Phys. Rev. D 85, 073009 (2012), arXiv:1201.0906 [hep-ph] .
  * Toma and Vicente (2014) T. Toma and A. Vicente, JHEP 01, 160 (2014), arXiv:1312.2840 [hep-ph] .
  * Lindner  _et al._ (2018) M. Lindner, M. Platscher, and F. S. Queiroz, Phys. Rept. 731, 1 (2018), arXiv:1610.06587 [hep-ph] .
  * Cottin  _et al._ (2021) G. Cottin, O. Fischer, S. Mandal, M. Mitra, and R. Padhan, (2021), arXiv:2104.13578 [hep-ph] .
  * Lindner  _et al._ (2016) M. Lindner, F. S. Queiroz, W. Rodejohann, and C. E. Yaguna, JHEP 06, 140 (2016), arXiv:1604.08596 [hep-ph] .
  * Belanger  _et al._ (2007) G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov, Comput. Phys. Commun. 176, 367 (2007), arXiv:hep-ph/0607059 .
  * Allahverdi  _et al._ (2020) R. Allahverdi _et al._ ,  (2020), 10.21105/astro.2006.16182, arXiv:2006.16182 [astro-ph.CO] .
  * Adulpravitchai  _et al._ (2009) A. Adulpravitchai, M. Lindner, and A. Merle, Phys. Rev. D 80, 055031 (2009), arXiv:0907.2147 [hep-ph] .
  * Kubo  _et al._ (2006) J. Kubo, E. Ma, and D. Suematsu, Phys. Lett. B 642, 18 (2006), arXiv:hep-ph/0604114 .
  * Baldini  _et al._ (2016) A. M. Baldini _et al._ (MEG), Eur. Phys. J. C 76, 434 (2016), arXiv:1605.05081 [hep-ex] .
  * Note (1) Electron recoils are more relevant for light dark matter Kopp  _et al._ (2009); Altmannshofer  _et al._ (2016).
  * Butsky  _et al._ (2016) I. Butsky, A. V. Macciò, A. A. Dutton, L. Wang, A. Obreja, G. S. Stinson, C. Penzo, X. Kang, B. W. Keller, and J. Wadsley, Mon. Not. Roy. Astron. Soc. 462, 663 (2016), arXiv:1503.04814 [astro-ph.GA] .
  * Aad  _et al._ (2020) G. Aad _et al._ (ATLAS), Phys. Rev. D 101, 052005 (2020), arXiv:1911.12606 [hep-ex] .
  * Kopp  _et al._ (2009) J. Kopp, V. Niro, T. Schwetz, and J. Zupan, Phys. Rev. D 80, 083502 (2009), arXiv:0907.3159 [hep-ph] .
  * Altmannshofer  _et al._ (2016) W. Altmannshofer, S. Gori, S. Profumo, and F. S. Queiroz, JHEP 12, 106 (2016), arXiv:1609.04026 [hep-ph] .