Secluded Scalar Dark Matter and the Muon Anomalous Magnetic Moment Karim
Ghorbani
Physics Department, Faculty of Science, Arak University, Arak 38156-8-8349,
Iran

We consider a dark matter model with a singlet scalar, $\chi$, as our dark
matter (DM) candidate which is secluded from the Standard Model (SM) and
annihilates to the singlet scalar, $\phi$, via a contact interaction. The
singlet scalar, $\phi$, has a leptophilic interaction with the SM leptons and
may decay leptonically at tree level, and decays into a pair of photons at
loop level. The focus in this work is to consider DM masses below 10 GeV. It
is found a viable secluded region in the parameter space after imposing the
observed relic density. There is a one-loop interaction between scalar dark
matter and the atomic electron in this model. We then apply the available
direct detection bounds from Xenon10, Xenon1T, and DarkSide on the DM-electron
elastic scattering cross section. While the model can explain the muon
anomalous magnetic moment, we put bounds from current and future lepton
collider experiments.

##  1 Introduction

There are two well known mechanisms for dark matter (DM) production in the
early universe. One of them is called freeze-out paradigm [1] where DM
particles are in thermal equilibrium with the standard model (SM) particles in
the early time. The second type of DM genesis is called freeze-in mechanism
[2] in which DM particles and the SM particles have never been in thermal
equilibrium.

A prominent question about the nature of the dark matter (DM) is its
interaction type with the normal matter. We may classify the DM interaction
with ordinary matter into two types; interaction with atomic electrons and
interaction with nucleons. Beside the interaction type, the mass of the DM
particle is another important ingredient in direct search for the DM
particles, the so-called direct detection (DD) experiments. The viable range
of DM mass is quite broad within the weakly interacting massive particles
(WIMPs) paradigm [3, 4, 5]. In recent years direct search for dark matter with
DM-nucleon interaction has been under tense scrutiny in DD underground
experiments reaching unprecedented upper limits from LUX [6], Xenon [7], and
SuperCDMS [8]. In these experiments the most sensitivity is reached around the
DM mass $\sim 10-100$ GeV. In the DM mass range below 10 GeV, direct detection
experiments searching for enticing signal from DM-electron interaction become
relevant, for instance, in Xenon10 [9], Xeno1T [10], and DarkSide [11].

In light of the recent confirmation of the muon anomalous magnetic moment [12,
13], new types of interactions of the SM leptons in scenarios beyond the SM
are intriguing. The same type of interaction might be responsible for the DM-
electron interaction in direct detection experiments. The model we introduce
in this work contains two singlet scalars, which are coupled via a contact
interaction, one of which plays the role of the DM candidate and the other
scalar has interaction with the SM leptons. This latter interaction being
leptophilic is motivated from various UV complete models [14, 15, 16, 17, 18].
The dominant annihilation of the thermal DM is into a pair of singlet scalars,
within the so-called secluded region in the parameter space [19, 20], where
the DM mass is adequately larger than the singlet scalar. One interesting
feature of the present model is that the interaction of the scalar DM with the
atomic electron is induced via a one loop Feynman diagram. In a different
scenario the scalar DM which is able to account for ($g-2$) anomaly, can also
have direct detection DM-nucleon interaction [21].

The structure of the paper is the following. The model is described in section
2, in which the decay width of the singlet scalar is provided, and the DM-
electron elastic scattering cross section, and annihilation cross section are
obtained. We provide the relevant constraints in section 3. Discussions on the
relic density calculation are done in section 4, and our numerical results
concerning the viable parameter space after imposing all the presented bounds
are given in section 5. We then finish with a conclusion.

##  2 The Model

We consider a model which contains two scalar fields, singlet under the SM
gauges, one acting as our DM candidate is stable and the other scalar has a
contact interaction with dark matter particle. The second scalar is unstable
and can decay to the SM leptons at the classical level. The relevant
Lagrangian has the structure,

${\cal
L}_{\text{DM}}=\lambda\chi^{2}\phi^{2}+\alpha\sum_{l=e,\mu,\tau}\frac{m_{l}}{v}~{}\phi
l^{+}l^{-}\,,$ (1)

where $v=246$ GeV is the vacuum expectation value of the SM Higgs doublet, and
the leptophilic scalar $\phi$ interaction with the SM leptons has a mass-
proportional strength. This type of interaction is motivated in a UV complete
model with two Higgs-doublet and an additional scalar singlet [14, 15, 16]. As
an alternative UV completion for this model one may consider a scenario with
vector-like fermions at the weak scale [17].

The scalar $\phi$ may decay to the SM fermions via $\phi\to
e^{+}e^{-},\mu^{+}\mu^{-},\tau^{+}\tau^{-}$ with the following decay width for
the leptonic process, $\phi\to l^{+}l^{-}$,

$\Gamma_{l^{+}l^{-}}=\frac{\alpha^{2}}{8\pi}\frac{m_{l}^{2}}{v^{2}}(1-m_{l}^{2}/m_{\phi}^{2})^{3/2}\,.$
(2)

The singlet scalar with an arbitrary mass can decay to a pair of photons at
loop level with the following decay width [22],

$\Gamma_{\gamma\gamma}=\frac{\alpha^{2}\alpha_{em}^{2}}{256\pi^{3}}\frac{m_{\phi}^{3}}{v^{2}}\lvert\sum_{l=e,\mu,\tau}{\cal
M}_{1/2}(s_{l})\rvert^{2}\,,$ (3)

where $s_{l}=m_{\phi}^{2}/4m_{l}^{2}$, and ${\cal
M}_{1/2}(s_{l})=2[s_{l}+(s_{l}-1)f(s_{l})]/s^{2}_{l}$ with

$f(s_{l})=\begin{cases}Arcsin^{2}\sqrt{s_{l}}&s_{l}\leq 1\,,\\\
-\frac{1}{4}\Big{(}\log\frac{\sqrt{s_{l}}+\sqrt{s_{l}-1}}{\sqrt{s_{l}}-\sqrt{s_{l}-1}}-i\pi\Big{)}^{2}&s_{l}>1\,.\end{cases}$

Figure 1: Shown is the leading Feynman diagram for the elastic scattering of
the DM candidate off the atomic electron.

Since the scalar DM has a contact interaction with the scalar $\phi$, when
$m_{\phi}\lesssim m_{\chi}$, then the DM annihilation cross section times the
relative velocity reads

$(\sigma_{\text{anni}}v_{\text{rel}})_{\chi\chi\to\phi\phi}=\frac{\lambda^{2}}{\pi
s}\sqrt{1-4m_{\phi}^{2}/s}\,.$ (4)

This formula is needed when computing the DM relic density. The mandelstam
parameter, $s$, is the center of mass energy of the two annihilating dark
matter particles.

In the present model there is no tree level interaction between the DM scalar
and the SM leptons. The leading DM-electron interaction induced via a loop
interaction as depicted in Fig. 1. The scattering amplitude for the DM-
interaction reads

${\cal
M}=4\lambda\frac{\alpha^{2}m_{e}^{2}}{v^{2}}\int\frac{d^{4}l}{(2\pi)^{4}}\frac{\bar{e}(p_{2})(\not{l}+m_{e})e(p_{1})}{[(p_{2}-l)^{2}-m^{2}_{\phi}][(p_{1}-l)^{2}-m^{2}_{\phi}][l^{2}-m^{2}_{e}]}\,.$
(5)

Taking the form factor $F_{\text{DM}}(q)=1$ [23], the scattering cross section
for DM-electron is obtained as

$\sigma_{e}\sim\frac{\lambda^{2}g^{4}_{e}}{128\pi^{5}m^{2}_{\text{DM}}}|{\cal
F}(m_{\phi})|^{2}\,,$ (6)

where $g_{e}=\alpha m_{e}/v$, and for ${\cal F}(m_{\phi})$ we have

${\cal
F}(m_{\phi})=1+\frac{m^{2}_{\phi}-3m^{2}_{e}}{2m^{2}_{e}}~{}\text{log}(\frac{m^{2}_{e}}{m^{2}_{\phi}})+\frac{(m^{2}_{\phi}-m^{2}_{e})\sqrt{m^{4}_{\phi}-4m^{2}_{\phi}m^{2}_{e}}}{m^{2}_{e}m^{2}_{\phi}}~{}\text{log}\Big{(}\frac{m^{2}_{\phi}+\sqrt{m^{4}_{\phi}-4m^{2}_{\phi}m^{2}_{e}}}{2m_{\phi}m_{e}}\Big{)}\,.$
(7)

The singlet scalar has a loop contribution to the anomalous magnetic moment of
the muon, the so called $g-2$ anomaly [24, 25]. The new physics correction is
obtained as

$\Delta
a_{\mu}=\frac{\alpha^{2}m^{2}_{\mu}}{v^{2}}\int^{1}_{0}dy\frac{(1+y)(1-y)^{2}}{(1-y)^{2}+y(m_{\phi}/m_{\mu})^{2}}\,,$
(8)

where the correction depends on the coupling $\alpha$ and $m_{\phi}$ as free
parameters. In this model there are four independent free parameters, namely,
$m_{\chi}$, $m_{\phi}$, $\lambda$, and $\alpha$. We may use $m_{\chi}$ and
$m_{\text{DM}}$ interchangeably.

##  3 Constraints

In this section we collect all the constraints which are relevant for the
model in this work.

  * •

Relic density: The dark matter relic abundance is measured by WMAP and Planck
satellite with unprecedented accuracy, $\Omega h^{2}=0.12\pm 0.001$ [26]. This
will put constraints on the three parameters of the model, namely, the
coupling $\lambda$, and masses $m_{\text{DM}}$ and $m_{\phi}$.

  * •

$(g-2)_{\mu}$ anomaly: There has been a long-standing discrepancy on the
measurement of the muon magnetic moment and its theoretical prediction within
the SM [27]. This discrepancy is defined as $\Delta
a_{\mu}=a_{\mu}(\text{Exp})-a_{\mu}(\text{SM})$. A recent update provided by
FNAL supports the deviation with a significant of $\sim 4.2\sigma$ [12], with
$\Delta a_{\mu}=(25.1\pm 5.9)\times 10^{-10}$. The muon ($g-2$) collaboration
has also released their results based on the data collected in 2010 and 2020
[13]. The results reads $\Delta a_{\mu}=(24.9\pm 4.8)\times 10^{-10}$ at $5.1$
standard deviation, indicating an improvement in precision by a factor of two.
We will use an updated world average for this quantity in our numerical
results.

  * •

BABAR experiment: The production of a new scalar is feasible via the process
$e^{-}e^{+}\to\mu^{-}\mu^{+}\phi$ in lepton colliders, where the scalar $\phi$
will decay into a pair of muons subsequently. The BABAR experiment have found
strong constraints on the coupling $\alpha$ in the channel with four muons in
the final state [28]. In addition, for a leptophilic scalar, $\phi_{L}$, which
decays mainly to the SM leptons, the BABAR experiment provides constraints on
the muon-scalar coupling [29].

  * •

Belle II: This is a high-luminosity B-factory which collected a significant
number of $\tau^{+}\tau^{-}$ in association with scalars. Since the scalar
coupling is proportional to the lepton mass then the scalar production from
taus is the dominant process, see [14].

  * •

ILC as future international linear collider: This is a future lepton collider
as Higgs factory which runs at the center of mass $\sqrt{s}\sim 250$ GeV [30,
31]. We will apply the constraints from an ILC machine at 250 GeV with
integrated luminosity $2000fb^{-1}$. [32]

  * •

The Light Dark Matter eXperiment (LDMX): This is a multi-purpose future
experiment [33] which provides a high-luminosity measurement of missing
momentum. The primary goal of LDMX is to search for light dark matter in the
sub-GeV mass range. We will apply the projections of LDMX in Phase.1 and
Phase.2 with the measurement of the muon missing momentum (M3).

##  4 Relic Density

Figure 2: The Hubble rate and the total decay width of the singlet scalar as a
function of the parameter, $x$, are shown for three values of the singlet
scalar mass.

In the present model there is no dominant DM annihilation to the SM particles
at tree level in perturbation theory. There is rather a secluded dark matter
annihilation to a pair of singlet scalars. In the thermal bath, the DM entropy
is transferred via the secluded annihilation $\chi\chi\to\phi\phi$. The
relevant interaction is a contact one with the strength proportional to
$\lambda$. The annihilation of DM to the SM leptons is also possible via off-
shell singlet scalars, however, these interactions are negligible, leaving us
a secluded region in the parameter space to probe.

The singlet scalar in the thermal bath can decay to the SM leptons when
kinematically allowed. If the strength of the scalar-lepton interaction is
sufficiently strong then it is possible to maintain the dark sector and the SM
sector in kinetic equilibrium. As long as the DM particles have strong enough
interaction with the singlet scalar, to maintain the kinetic equilibrium, it
is sufficient that the singlet scalar and the SM particles be in kinetic
equilibrium. To this end, we constrain the total decay width of the scalar to
be larger that the Hubble rate at the freeze-out temperature,
$H(T_{f})\leq\Gamma(\phi\to\text{SM})$, where $T_{f}=m_{\text{DM}}/x_{f}$, and
according to the standard lore expectation is $x_{f}\sim 15-20$. We examine
this condition for three scalar masses, $m_{\phi}=0.1,1$ and 10 GeV, in Fig.
2, with the scalar coupling $\alpha=1$. It is evident that in the range of
interest for the scalar mass this condition is held. Taking larger value for
$\alpha$, the condition is satisfied even easier. In addition, there is
another condition for the scalar lifetime, where the scalar should decay
before the BBN epoch, $t\sim 1$ sec. This can be seen from the scalar decay
widths in Fig. 2, that the BBN condition is easily fulfilled in the region of
the parameter space of our interest.

Now, the secluded annihilation channel is open even when the singlet scalar
mass is slightly larger than the DM mass, due to the kinematical tail in the
Boltzmann distribution. To describe the time evolution of the number density
of the scalar DM via freeze-out mechanism, one needs to solve the Boltzmann
equation,

$\frac{dn_{\chi}}{dt}+3Hn_{\chi}=-\langle(\sigma_{\text{ann}}v_{\text{rel}})_{\chi\chi\to\text{SM}}\rangle[n^{2}_{\chi}-(n^{\text{eq}}_{\chi})^{2}]-\langle(\sigma_{\text{ann}}v_{\text{rel}})_{\chi\chi\to\phi\phi}\rangle[n^{2}_{\chi}-(n^{\text{eq}}_{\chi})^{2}(\frac{n_{\phi}}{n^{\text{eq}}_{\phi}})^{2}]\,,$
(9)

where, the Hubble parameter is denoted by $H$, and the DM number density at
equilibrium is
$n^{\text{eq}}_{\chi}(T)=\frac{m_{\chi}^{2}}{2\pi^{2}}TK_{2}\left(\frac{m_{\chi}}{T}\right)$.
Now, since in our model
$(\sigma_{\text{ann}}v_{\text{rel}})_{\chi\chi\to\text{SM}}\sim 0$, and the
singlet scalar has been in thermal equilibrium with the SM plasma, then
$n_{\phi}=n^{\text{eq}}_{\phi}$. The Boltzmann equation then takes on the
form,

$\frac{dn_{\chi}}{dt}+3Hn_{\chi}=-\langle(\sigma_{\text{ann}}v_{\text{rel}})_{\chi\chi\to\phi\phi}\rangle[n^{2}_{\chi}-(n^{\text{eq}}_{\chi})^{2}]\,.$
(10)

The thermal average of the DM annihilation cross section times the velocity is
given by

$\langle\sigma_{\text{ann}}v_{\text{rel}}\rangle=\frac{1}{8m_{\chi}^{4}TK^{2}_{2}(\frac{m_{\chi}}{T})}\int^{\infty}_{4m^{2}_{\chi}}ds~{}(s-4m^{2}_{\chi})\sqrt{s}~{}K_{1}(\frac{\sqrt{s}}{T})~{}\sigma_{\text{ann}}(s)\,,$
(11)

where, the modified Bessel functions of first and second rank are denoted by
$K_{1,2}$. The package Micromegas will be employed to evaluate the DM relic
density numerically [34].

Figure 3: The dark coupling, $\lambda$, as a function of the DM mass is given
after imposing observed DM relic density, for three values of the singlet
scalar mass.

The only coupling, $\lambda$, controls the size of the DM relic density. Thus,
by imposing the observed relic density, we find the viable values for the
coupling in terms of the DM mass for the singlet scalar mass
$m_{\phi}=(0.1,0.9,1.1)m_{\text{DM}}$. The results given in Fig. 3 show that
by increasing the DM mass, larger value for the coupling is picked out. This
is expected, since the annihilation cross section in Eq. 4 falls off by
increasing the DM mass, and thus the coupling $\lambda$ should grow in such a
way to compensate the reduction of the cross section. The case with
$m_{\phi}=1.1m_{\text{DM}}$, where the singlet scalar is slightly larger than
the DM mass, can also give rise to the correct relic density since the DM
velocity lies in the tail of the Boltzmann distribution.

##  5 Results

Figure 4: The region consistent with the discrepancy in the muon ($g-2$)
anomaly is shown in the plane $\alpha-m_{\phi}$. The collider bounds from the
current and future experiments are placed.

In this section we will put together all the astrophysical, collider, and
direct detection bounds which are relevant for the parameter space of the
present model. The aim is to find the viable regions which comply with
discrepancy found in the muon magnetic moment. First we display in the
$\alpha-m_{\phi}$ plane, the region consistent with the muon anomalous
magnetic moment, ($g-2$) band, see Fig. 4. Then we apply in the same figure a
bound from BABAR experiment, which excludes the ($g-2$) band in the singlet
scalar mass range $m_{\phi}\sim 1.5-5$ GeV. Along the same line Belle II,
first super B-Factory experiment, is able to reach a significantly higher
sensitivity to exclude the scalar mass range $m_{\phi}\sim 0.3-3.5$ GeV. As
depicted in Fig. 4 the expected bounds from a future experiment, LDMX, can
probe regions in the parameter space with scalar mass up to about 0.5 GeV in
its first phase and masses up to about 1.5 GeV in its second phase. Finally,
we present constraints from ILC, as a future international linear collider,
which is sensitive to scalar masses above 5 GeV.

In the next step we go on to compute the DM-electron elastic cross section as
a function of the DM mass for points in the parameter space which already
respect the bound on the relic density provided by WMAP and Planck. We scan
regions with 0.1 GeV $<m_{\text{DM}}<$ 10 GeV, 0.1 GeV $<m_{\phi}<$ 10 GeV,
$10^{-5}<\lambda<1$ and $1<\alpha<150$. In Fig. 5 we can find regions
respecting the direct detection bounds from Xenon10, Xenon1T and DarkSide. The
bound from the neutrino floor restricts the direct detection cross section
from below. As can be seen in Fig. 5, over the range 0.1 GeV $<m_{\text{DM}}<$
10 GeV, regions respecting both direct detection bounds and observed relic
density are plausible, wherein the respecting values for the scalar mass,
$m_{\phi}$, and the scalar coupling, $\alpha$, are found. Given the viable
regions in parameters space which satisfy the Xenon1T bound and observed relic
density, it can be seen that in these regions there are points with $\alpha$
and $m_{\phi}$ that can explain the muon anomalous magnetic moment shown as a
red band in Fig. 4. The future colliders are then capable to probe the allowed
parameter space we have found here after imposing all the accessible bounds.

Figure 5:  The DM-electron elastic scattering cross section is shown as a
function of the DM mass. In the left panel the scalar coupling, $\alpha$, is
indicated by the color spectrum vertical bar, while in the right panel, color
spectrum shows the values for $m_{\phi}$. In both plots bound from the
observed relic density is imposed. Direct detection bounds from Xenon10,
Xenon1T and DarkSide are placed. As well, the neutrino floor is shown in both
plots.

##  6 Conclusion

We have studied a model with two singlet scalars, one a stable particle as our
DM candidate and the other decaying into the SM leptons, to explain the muon
anomalous magnetic moment. First we found a secluded region in the parameter
space respecting the observed relic abundance. Then we consider the DM-
electron elastic scattering cross section being a one-loop process, and
constrain the viable space further by applying the direct detection bounds. On
the other hand, bounds from the present and future lepton colliders can reach
a sensitivity to probe the remaining viable space. The scalar mass range
$m_{\phi}\sim 1.5-5$ GeV, is already excluded by BABAR experiment, as can be
seen in Fig. 4. LDMX (Phase.2) and ILC are capable to reach a sensitivity in
order to exclude the rest of singlet scalar mass range respecting the muon
($g-2$) anomaly. On the other side, taking the strongest direct detection
bounds there is a large parameter space in the range $m_{\text{DM}}=0.1-10$
GeV above the neutrino floor and the subject of further probe by the future
direct detection experiments.

## References

  * [1] G. Steigman, B. Dasgupta, and J. F. Beacom, “Precise Relic WIMP Abundance and its Impact on Searches for Dark Matter Annihilation,” Phys. Rev. D 86 (2012) 023506, arXiv:1204.3622 [hep-ph].
  * [2] L. J. Hall, K. Jedamzik, J. March-Russell, and S. M. West, “Freeze-In Production of FIMP Dark Matter,” JHEP 03 (2010) 080, arXiv:0911.1120 [hep-ph].
  * [3] M. Kawasaki and K. Nakayama, “Axions: Theory and Cosmological Role,” Ann. Rev. Nucl. Part. Sci. 63 (2013) 69–95, arXiv:1301.1123 [hep-ph].
  * [4] K. Griest and M. Kamionkowski, “Unitarity limits on the mass and radius of dark-matter particles,” Phys. Rev. Lett. 64 (Feb, 1990) 615–618.
  * [5] K. Blum, Y. Cui, and M. Kamionkowski, “An Ultimate Target for Dark Matter Searches,” Phys. Rev. D 92 no. 2, (2015) 023528, arXiv:1412.3463 [hep-ph].
  * [6] LUX Collaboration, D. S. Akerib et al., “Results from a search for dark matter in the complete LUX exposure,” Phys. Rev. Lett. 118 no. 2, (2017) 021303, arXiv:1608.07648 [astro-ph.CO].
  * [7] XENON Collaboration, E. Aprile et al., “First Dark Matter Search Results from the XENON1T Experiment,” Phys. Rev. Lett. 119 no. 18, (2017) 181301, arXiv:1705.06655 [astro-ph.CO].
  * [8] SuperCDMS Collaboration, R. Agnese et al., “Results from the Super Cryogenic Dark Matter Search Experiment at Soudan,” Phys. Rev. Lett. 120 no. 6, (2018) 061802, arXiv:1708.08869 [hep-ex].
  * [9] R. Essig, A. Manalaysay, J. Mardon, P. Sorensen, and T. Volansky, “First Direct Detection Limits on sub-GeV Dark Matter from XENON10,” Phys. Rev. Lett. 109 (2012) 021301, arXiv:1206.2644 [astro-ph.CO].
  * [10] XENON Collaboration, E. Aprile et al., “Light Dark Matter Search with Ionization Signals in XENON1T,” Phys. Rev. Lett. 123 no. 25, (2019) 251801, arXiv:1907.11485 [hep-ex].
  * [11] DarkSide Collaboration, P. Agnes et al., “Constraints on Sub-GeV Dark-Matter–Electron Scattering from the DarkSide-50 Experiment,” Phys. Rev. Lett. 121 no. 11, (2018) 111303, arXiv:1802.06998 [astro-ph.CO].
  * [12] Muon g-2 Collaboration, B. Abi et al., “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm,” Phys. Rev. Lett. 126 (2021) 141801, arXiv:2104.03281 [hep-ex].
  * [13] Muon g-2 Collaboration, D. P. Aguillard et al., “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm,” arXiv:2308.06230 [hep-ex].
  * [14] B. Batell, N. Lange, D. McKeen, M. Pospelov, and A. Ritz, “Muon anomalous magnetic moment through the leptonic Higgs portal,” Phys. Rev. D 95 no. 7, (2017) 075003, arXiv:1606.04943 [hep-ph].
  * [15] K. Ghorbani, “Light vector dark matter with scalar mediator and muon g-2 anomaly,” Phys. Rev. D 104 no. 11, 115008, arXiv:2104.13810 [hep-ph].
  * [16] L.-B. Jia, “A dark leptophilic scalar with the updated muon $g-2$ anomaly,” Eur. Phys. J. C 82 no. 4, (2022) 321, arXiv:2105.13805 [hep-ph].
  * [17] C.-Y. Chen, H. Davoudiasl, W. J. Marciano, and C. Zhang, “Implications of a light “dark Higgs” solution to the $g_{\mu}$-2 discrepancy,” Phys. Rev. D 93 no. 3, (2016) 035006, arXiv:1511.04715 [hep-ph].
  * [18] P. Ghorbani, “Dark matter and muon g $-$ 2 anomaly via scale symmetry breaking,” JHEP 04 (2022) 170, arXiv:2203.03964 [hep-ph].
  * [19] M. Pospelov, A. Ritz, and M. B. Voloshin, “Secluded WIMP Dark Matter,” Phys. Lett. B 662 (2008) 53–61, arXiv:0711.4866 [hep-ph].
  * [20] M. Pospelov and A. Ritz, “Astrophysical Signatures of Secluded Dark Matter,” Phys. Lett. B 671 (2009) 391–397, arXiv:0810.1502 [hep-ph].
  * [21] B. D. Sáez and K. Ghorbani, “Singlet scalars as dark matter and the muon (g-2) anomaly,” Phys. Lett. B 823 (2021) 136750, arXiv:2107.08945 [hep-ph].
  * [22] A. Djouadi, “The Anatomy of electro-weak symmetry breaking. I: The Higgs boson in the standard model,” Phys. Rept. 457 (2008) 1–216, arXiv:hep-ph/0503172.
  * [23] R. Essig, J. Mardon, and T. Volansky, “Direct Detection of Sub-GeV Dark Matter,” Phys. Rev. D 85 (2012) 076007, arXiv:1108.5383 [hep-ph].
  * [24] J. S. Schwinger, “On Quantum electrodynamics and the magnetic moment of the electron,” Phys. Rev. 73 (1948) 416–417.
  * [25] R. Jackiw and S. Weinberg, “Weak-interaction corrections to the muon magnetic moment and to muonic-atom energy levels,” Phys. Rev. D 5 (May, 1972) 2396–2398. https://link.aps.org/doi/10.1103/PhysRevD.5.2396.
  * [26] Planck Collaboration, N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641 (2020) A6, arXiv:1807.06209 [astro-ph.CO]. [Erratum: Astron.Astrophys. 652, C4 (2021)].
  * [27] Muon g-2 Collaboration, G. W. Bennett et al., “Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL,” Phys. Rev. D 73 (2006) 072003, arXiv:hep-ex/0602035.
  * [28] BaBar Collaboration, J. P. Lees et al., “Search for a muonic dark force at BABAR,” Phys. Rev. D 94 no. 1, (2016) 011102, arXiv:1606.03501 [hep-ex].
  * [29] BaBar Collaboration, J. P. Lees et al., “Search for a Dark Leptophilic Scalar in $e^{+}e^{-}$ Collisions,” Phys. Rev. Lett. 125 no. 18, (2020) 181801, arXiv:2005.01885 [hep-ex].
  * [30] ILC Collaboration, “The International Linear Collider Technical Design Report - Volume 2: Physics,” arXiv:1306.6352 [hep-ph].
  * [31] P. Bambade et al., “The International Linear Collider: A Global Project,” arXiv:1903.01629 [hep-ex].
  * [32] E. J. Chun and T. Mondal, “Leptophilic bosons and muon g-2 at lepton colliders,” JHEP 07 (2021) 044, arXiv:2104.03701 [hep-ph].
  * [33] LDMX Collaboration, T. Åkesson et al., “Light Dark Matter eXperiment (LDMX),” arXiv:1808.05219 [hep-ex].
  * [34] D. Barducci, G. Belanger, J. Bernon, F. Boudjema, J. Da Silva, S. Kraml, U. Laa, and A. Pukhov, “Collider limits on new physics within micrOMEGAs4.3,” Comput. Phys. Commun. 222 (2018) 327–338, arXiv:1606.03834 [hep-ph].