# General Kinetic Mixing in Gauged $U(1)_{L_{\mu}-L_{\tau}}$ Model for
Muon $g-2$ and Dark Matter

Timothy Hapitas<EMAIL_ADDRESS>Douglas Tuckler
<EMAIL_ADDRESS>Yue Zhang<EMAIL_ADDRESS>Department
of Physics, Carleton University, Ottawa, Ontario K1S 5B6, Canada

###### Abstract

The gauged $U(1)_{L_{\mu}-L_{\tau}}$ extension of the Standard Model is a very
simple framework that can alleviate the tension in muon anomalous magnetic
dipole moment, reinforced by the recent Fermilab measurement. We explore
experimental probes of the $(g-2)_{\mu}$ target with a general treatment of
kinetic mixing between the $Z^{\prime}$ gauge boson and the photon. The
physical value of the kinetic mixing depends on a free parameter of the model
and energy scale of a process. We find neutrino constraints on the
$(g-2)_{\mu}$ target including Borexino, CE$\nu$NS, and white dwarfs are
sensitive to this freedom and can be lifted if the kinetic mixing lies in
proximity of zero at low momentum transfer. As a further step, we explore
$L_{\mu}-L_{\tau}$ charged dark matter with a thermal origin and show that the
same scenario of kinetic mixing can relax existing direct detection
constraints and predict novel recoil energy dependence in the upcoming
searches. Future joint effort of neutrino and dark matter experiments and
precision spectral measurement will be the key to test such a theory.

###### Contents

  1. I Introduction
  2. II Gauged $U(1)_{L_{\mu}-L_{\tau}}$ Model
    1. II.1 The $(g-2)_{\mu}$ target
    2. II.2 General kinetic mixing and momentum dependence
  3. III Implication for Neutrino Probes of the $(g-2)_{\mu}$ Target
    1. III.1 Borexino
    2. III.2 CE$\nu$NS
    3. III.3 White dwarf
  4. IV Implication for $L_{\mu}-L_{\tau}$ Charged Dark Matter
    1. IV.1 The thermal dark matter target
    2. IV.2 Upper limit on dark matter mass from direct detection
    3. IV.3 Novel recoil energy spectrum
    4. IV.4 Other Constraints
      1. IV.4.1 Indirect detection using CMB and neutrinos
      2. IV.4.2 Dark matter self interaction
  5. V Conclusion
  6. A Statistical Analysis for CE$\nu$NS

##  I Introduction

The quest for physics beyond the Standard Model (SM) is the central focus of
particle physics nowadays. There exist a number of strong motivations for
pursuing this endeavor, especially the phenomena established experimentally
that the SM cannot adequately explain. It is inspiring and important to
explore potential theories and connections behind some of these puzzles.

A long-standing puzzle at the precision frontier is the discrepancy between
the experimental and theoretical values of the muon anomalous magnetic dipole
moment $a_{\mu}\equiv(g-2)_{\mu}/2$. Earlier this year the Fermilab Muon $g-2$
Collaboration announced the latest measurement of the muon anomalous magnetic
dipole moment Abi  _et al._ (2021) and, when combined with the previous result
from Brookhaven Bennett  _et al._ (2006), exhibits a $4.2\sigma$ deviation
from the SM prediction Aoyama  _et al._ (2020)

$\Delta a_{\mu}=a_{\mu}^{\text{exp}}-a_{\mu}^{\text{SM}}=(251\pm 59)\times
10^{-11}\ .$ (1)

Although this observation has not yet reached a $5\sigma$ significance, it
could happen within the coming years with improved experimental statistics and
refined theoretical calculations. Given that gauge theories have played an
instrumental role in the establishment of the SM, it is particularly
attractive to address the $(g-2)_{\mu}$ tension by postulating new symmetries
of nature. A very simple consideration is to extend the SM with a gauged
$U(1)_{L_{\mu}-L_{\tau}}$ symmetry, where exchange of the gauge boson
$Z^{\prime}$ at one-loop level can shift the theoretical prediction of
$(g-2)_{\mu}$ towards the measured value Baek  _et al._ (2001). A number of
existing and upcoming experiments can be used to constrain the relevant
parameter space of the model Alekhin  _et al._ (2016); Bauer  _et al._ (2018);
Gninenko and Krasnikov (2018); Kahn  _et al._ (2018); Altmannshofer  _et al._
(2019).

At the same time, a burning question at the cosmic frontier is the nature of
dark matter (DM). If DM is a particle and interacts with SM particles through
a new force beyond gravity, a large number of opportunities become available
for probing it in laboratories. With the introduction of the
$U(1)_{L_{\mu}-L_{\tau}}$ gauge interaction, it is natural to further
speculate on its role in the search for DM. If nature does work this way, the
$(g-2)_{\mu}$ results would be considered an indirect glimpse toward the dark
side of the universe. It also provides a strong motivation to explore other
potential signatures of such a DM candidate.

In this work, we explore the phenomenology of the gauged
$U(1)_{L_{\mu}-L_{\tau}}$ model. We pay special attention to neutrino
experiments that could probe the $(g-2)_{\mu}$ target parameter space, as well
as the prospects of $L_{\mu}-L_{\tau}$ charged DM. A common feature shared by
neutrinos and DM in this model is that their low-energy scattering with human-
made detectors must occur through a kinetic mixing between the $Z^{\prime}$
gauge boson and the photon. In general, the total kinetic mixing is given by
the sum of a bare mixing parameter (not calculable) and radiative corrections,
and is a function of the momentum transfer in a particular physical process.
As a result, the interpretation of experimental constraints and prospects
strongly depends on the nature of the kinetic mixing. In the literature, it
was often assumed the total kinetic mixing is zero at energy scales much
higher than the $\tau$ lepton mass Kamada and Yu (2015); Ibe  _et al._ (2017);
Araki  _et al._ (2017); Gninenko and Krasnikov (2018); Bauer  _et al._ (2018);
Asai  _et al._ (2021) and a number of experimental constraints were derived
based on this assumption. A vanishing high-scale kinetic mixing, although may
be realized in specific frameworks Argüelles  _et al._ (2017); Gherghetta  _et
al._ (2019), is by no means general.

We revisit these constraints with a more general treatment of the kinetic
mixing and find the asymptotic values of the kinetic mixing do matter a lot.
First, the constraint on the $(g-2)_{\mu}$ target obtained from solar neutrino
measurements at the Borexino experiment can be lifted if the kinetic mixing
function is suppressed at low momentum transfer (well below the muon mass).
The same applies to the coherent elastic neutrino-nucleus scattering
(CE$\nu$NS) experiments and white dwarf cooling. Second, for
$L_{\mu}-L_{\tau}$ charged DM with a thermal origin, direct detection
constraints on the DM mass scale are also sensitive to the nature of kinetic
mixing. We find that a new viable window of DM mass above GeV scale can be
opened if the kinetic mixing is suppressed at low momentum transfer,
coinciding with the scenario where neutrino constraints are weakened. Notably,
such a mass window has been considered excluded in the literature by assuming
a vanishing kinetic mixing at high scales. Moreover, the DM candidate in such
a mass window offers new predictions at future low-threshold direct detection
experiments. The corresponding DM-nucleus scattering features a novel shape of
the recoil energy spectrum that will serve as a “smoking-gun” signature for
testing the nature of DM in this model.

This article is organized as follows. In Sec. II we present the gauged
$U(1)_{L_{\mu}-L_{\tau}}$ model and derive the momentum dependence of a
general kinetic mixing. It sets the stage for phenomenological discussions of
experimental probes using neutrinos and DM in Sec. III and IV, respectively.
While Sec. III mainly argues that several neutrino constraints on the
$(g-2)_{\mu}$ target can be lifted, in Sec. IV we will show that relaxing the
existing direct search constraints can lead to novel exciting prospects for
future DM experiments. The conclusions are drawn in Sec. V.

##  II Gauged $U(1)_{L_{\mu}-L_{\tau}}$ Model

As the starting point, we present the minimal Lagrangian which extends the SM
with a $U(1)_{L_{\mu}-L_{\tau}}$ gauge symmetry,

$\begin{split}\mathcal{L}_{L_{\mu}-L_{\tau}}=\ &\mathcal{L}_{\rm
SM}-\frac{1}{4}Z^{\prime}_{\alpha\beta}Z^{\prime\alpha\beta}+\frac{1}{2}m^{2}_{Z^{\prime}}Z^{\prime}_{\alpha}Z^{\prime\alpha}+\frac{\varepsilon_{0}}{2}Z^{\prime}_{\alpha\beta}F^{\alpha\beta}\\\
&+g_{\mu\tau}\big{(}\bar{\mu}\gamma^{\alpha}\mu-\bar{\tau}\gamma^{\alpha}\tau+\bar{\nu}_{\mu}\gamma^{\alpha}P_{L}\nu_{\mu}-\bar{\nu}_{\tau}\gamma^{\alpha}P_{L}\nu_{\tau}\big{)}Z^{\prime}_{\alpha}\
,\\\ \end{split}$ (2)

where $Z^{\prime}$ is the gauge boson of the $U(1)_{L_{\mu}-L_{\tau}}$
symmetry under which the lepton doublet $L_{\mu}$ ($L_{\tau}$) carry a
positive (negative) unit charge, and $g_{\mu\tau}$ is the corresponding gauge
coupling. We assume that the $U(1)_{L_{\mu}-L_{\tau}}$ symmetry is already
Higgsed and the $Z^{\prime}$ gauge boson is massive. A renormalizable bare
kinetic mixing parameter $\varepsilon_{0}$ has been introduced between the
$Z^{\prime}$ and photon field strengths. It is a free parameter and not
calculable within this model. We do not address the origin of neutrino masses
and mixing which requires further model building (see Zhou (2021) for recent
studies). The DM sector will be introduced later in Sec. IV.

###  II.1 The $(g-2)_{\mu}$ target

It is well known that the virtual exchange of a $Z^{\prime}$ boson can induce
a one-loop contribution to $a_{\mu}$ Baek  _et al._ (2001); Fayet (2007);
Pospelov (2009); Davoudiasl  _et al._ (2014) given by

$\Delta
a_{\mu}=\frac{(g_{\mu\tau}+e\varepsilon_{0})^{2}}{4\pi^{2}}\int_{0}^{1}dz\frac{m_{\mu}^{2}z^{2}(1-z)}{m^{2}_{Z^{\prime}}(1-z)+m_{\mu}^{2}z^{2}}\
,$ (3)

With a vector current coupling to the muon, this contribution has the correct
sign to shift the theoretical prediction of $a_{\mu}$ towards the
experimentally measured value. Throughout this work, we will restrict to the
hierarchy where $\varepsilon_{0}$ is loop-factor suppressed compared to
$g_{\mu\tau}/e$. In this case, the $Z^{\prime}$ couples just like a
$U(1)_{L_{\mu}-L_{\tau}}$ gauge boson rather than a dark photon.111It is worth
noting that the pure dark photon explanation of the $(g-2)_{\mu}$ has already
been excluded Alexander  _et al._ (2016); Fabbrichesi  _et al._ (2020), which
suggests $\varepsilon_{0}$ cannot be much larger than $g_{\mu\tau}/e$.
Neglecting the $\varepsilon_{0}$ piece of the contribution, Fig. 1 depicts the
parameter space in the $g_{\mu\tau}$ versus $m_{Z^{\prime}}$ plane where the
$(g-2)_{\mu}$ tension is alleviated, along with some existing constraints. For
$m_{Z^{\prime}}\gtrsim 5$ GeV the stongest constraints are from a CMS search
for the $Z^{\prime}$ in the
$pp\to\mu^{+}\mu^{-}Z^{\prime}\to\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ channel
Sirunyan  _et al._ (2019). The region where $Z^{\prime}$ is heavier than twice
the muon mass has been excluded by $e^{+}e^{-}\to 4\mu$ searches at BaBar Lees
_et al._ (2016) and neutrino trident production searches at CCFR and CHARM
Altmannshofer  _et al._ (2014), whereas the region with $Z^{\prime}$ lighter
than $\sim 6$ MeV is excluded by the $\Delta N_{\rm eff}$ constraints from
big-bang nucleosynthesis (BBN) and cosmic microwave background (CMB) Ahlgren
_et al._ (2013); Kelly  _et al._ (2020). The other constraints, such as that
from the Borexino experiment, are not displayed here because of additional
model dependence, as will be clarified below. 222The $\Delta N_{\rm eff}$
constrainted can be modified for a nonzero kinetic mixing but only for values
of $g_{\mu\tau}$ much smaller than shown in Fig. 1 Escudero  _et al._ (2019).
The viable mass window ($5\,{\rm MeV}\lesssim m_{Z^{\prime}}\lesssim
200\,$MeV) for the $(g-2)_{\mu}$ explanation serves as a motivated target for
many future experiments Alekhin  _et al._ (2016); Bauer  _et al._ (2018);
Gninenko and Krasnikov (2018); Kahn  _et al._ (2018); Altmannshofer  _et al._
(2019).

Figure 1: Parameter space for addressing the $(g-2)_{\mu}$ tension in gauged
$U(1)_{L_{\mu}-L_{\tau}}$ model (green band) shown along with experimental
constraints that are independent of $Z^{\prime}$-photon kinetic mixing. The
exclusion regions include BBN (orange shaded region), neutrino trident search
at CCFR (purple shaded region), BaBar search for $e^{+}e^{-}\to 4\mu$ (blue
shaded region), and a CMS search for
$pp\to\mu^{+}\mu^{-}Z^{\prime}\to\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ (red shaded
region). The allowed mass range of the $Z^{\prime}$ boson is $\sim 5-200$ MeV.
The two red five-stars represent the benchmark points to be used for the DM
discussion in Sec. IV.

###  II.2 General kinetic mixing and momentum dependence

The $Z^{\prime}$ gauge boson of $U(1)_{L_{\mu}-L_{\tau}}$ does not directly
couple to electrons or quarks. Therefore, the scattering of muon neutrinos or
$L_{\mu}-L_{\tau}$ charged DM (to be introduced in Sec. IV) in detectors must
occur through the kinetic mixing between the $Z^{\prime}$ and the photon (see
Fig. 2). The total kinetic mixing receives contributions from both the tree-
level mixing term in Eq. (2) and radiative corrections with virtual $\mu$ and
$\tau$ lepton exchange at loop level. It takes the following momentum transfer
dependent form

$\varepsilon_{\rm
tot}(Q)=\varepsilon_{0}-\frac{eg_{\mu\tau}}{2\pi^{2}}\int_{0}^{1}dx\,x(1-x)\log\left[\frac{m_{\tau}^{2}+x(1-x)Q^{2}}{m_{\mu}^{2}+x(1-x)Q^{2}}\right]\
,$ (4)

where $Q=\sqrt{-q^{2}}>0$, with $q^{\mu}$ being the four-momentum transfer in
the scattering process. As mentioned earlier, the bare mixing parameter
$\varepsilon_{0}$ is not calculable.

It is straightforward to verify that at zero or infinitely large momentum
transfer the total kinetic mixing approaches a constant. We denote these
asymptotic values as $\varepsilon_{\rm
IR}=\varepsilon_{0}-\frac{eg_{\mu\tau}}{12\pi^{2}}\log\frac{m_{\tau}^{2}}{m_{\mu}^{2}}$
and $\varepsilon_{\rm UV}=\varepsilon_{0}$, respectively. Within the particle
content of this model (introduced in Eq. (2)), they always satisfy,

$\varepsilon_{\rm UV}-\varepsilon_{\rm
IR}=\frac{eg_{\mu\tau}}{12\pi^{2}}\log\frac{m_{\tau}^{2}}{m_{\mu}^{2}}\ .$ (5)

Expanding to the next order, we find the following momentum transfer
dependence

$\begin{split}\varepsilon_{\rm tot}(Q)\simeq\varepsilon_{\rm
IR}+\frac{eg_{\mu\tau}}{60\pi^{2}}\frac{Q^{2}}{m_{\mu}^{2}}\ ,\quad\quad{\rm
for}\ Q\ll m_{\mu}\ ,\\\ \varepsilon_{\rm tot}(Q)\simeq\varepsilon_{\rm
UV}-\frac{eg_{\mu\tau}}{2\pi^{2}}\frac{m_{\tau}^{2}}{Q^{2}}\ ,\quad\quad{\rm
for}\ Q\gg m_{\tau}\ .\end{split}$ (6)

In contrast, for intermediate momentum transfer with $m_{\mu}<Q<m_{\tau}$, the
$Q$ dependence is approximately logarithmic,

$\frac{d\varepsilon_{\rm tot}}{d\log Q}\simeq\frac{eg_{\mu\tau}}{6\pi^{2}}\ ,$
(7)

which corresponds to a running kinetic mixing.

Figure 2: Feynman diagrams for low energy neutrino or DM detection in gauged
$U(1)_{L_{\mu}-L_{\tau}}$ model. The scattering with proton (or nucleus,
electron) target happens through the kinetic mixing between the gauge boson
$Z^{\prime}$ and the photon. In general, the effective kinetic mixing depends
on the momentum transfer $q^{\mu}$, or $Q=\sqrt{-q^{2}}$.

This exercise shows that the exact form of $\varepsilon_{\rm tot}$ not only
depends on the momentum transfer $Q$, but also its boundary conditions,
$\varepsilon_{\rm IR}$ or $\varepsilon_{\rm UV}$. This can be seen in Fig. 3
which depicts $\varepsilon_{\rm tot}$ as a function of $Q$ for two distinct
choices of boundary values. In the case where $\varepsilon_{\rm IR}=0$ (solid
curve), $\varepsilon_{\rm tot}$ remains close to zero until $Q$ becomes larger
than $m_{\mu}$. For $Q\gg m_{\tau}$, $\varepsilon_{\rm tot}$ approaches
$\frac{eg_{\mu\tau}}{12\pi^{2}}\log({m_{\tau}^{2}}/{m_{\mu}^{2}})\simeq
g_{\mu\tau}/70$. In the second case where $\varepsilon_{\rm UV}=0$ (dashed
curve), $\varepsilon_{\rm tot}$ remains close to zero for $Q\gg m_{\tau}$ and
approaches $\varepsilon_{\rm IR}\simeq-g_{\mu\tau}/70$ for $Q\ll m_{\mu}$. The
latter corresponds to a rather common choice made in the literature Kamada and
Yu (2015); Ibe  _et al._ (2017); Araki  _et al._ (2017); Gninenko and
Krasnikov (2018); Bauer  _et al._ (2018); Asai  _et al._ (2021). More
generally, the boundary conditions can be varied continuously such that they
intersect the above two special scenarios.

Because our knowledge of higher scale physics is limited, in this work we keep
an open mind to all possible boundary choices for the kinetic mixing. Instead
of judging which one is more appealing, we take a phenomenological approach
and quantify their different implications in experiments where the gauged
$U(1)_{L_{\mu}-L_{\tau}}$ model could be tested.

Here is an important observation that sets the stage for the remainder of this
article. In a number of neutrino and DM experiments, where interactions are
mediated through the $Z^{\prime}$-photon kinetic mixing, the typical momentum
transfer $Q$ lies well below the mass of the muon. As a result, the two
scenarios depicted Fig. 3 can predict drastically different scattering rates
and recoil energy spectra. In the first case ($\varepsilon_{\rm IR}=0$) the
detection rates are often highly suppressed and the $Q$ dependence is
important, whereas in the second case ($\varepsilon_{\rm UV}=0$) setting
$\varepsilon_{\rm tot}\simeq-g_{\mu\tau}/70$ is a good approximation at low
$Q<m_{\mu}$. As we will see, these two choices lead to drastically different
interpretations of neutrino and DM results, impacting the prospects of future
experiments.

Figure 3: Momentum dependent kinetic mixing $\varepsilon_{\text{tot}}(Q)$ as a
function of momentum transfer $Q$. The solid (dashed) curve corresponds to the
boundary condition $\varepsilon_{\text{IR}}=0$ ($\varepsilon_{\text{UV}}=0$).
The vertical, dashed purple lines indicate where the momentum transfer is
equal to $m_{\mu}$ and $m_{\tau}$.

##  III Implication for Neutrino Probes of the $(g-2)_{\mu}$ Target

In this section, we revisit low-energy neutrino probes of the $(g-2)_{\mu}$
target in the context of the gauged $U(1)_{L_{\mu}-L_{\tau}}$ model. We derive
constraints for the two kinetic mixing scenarios presented in Fig. 3 of Sec.
II.2. The main point is that these experimental constraints are sensitive to
the free parameter in the kinetic mixing and can be made substantially weaker
than quoted in the literature.

###  III.1 Borexino

The Borexino experiment has obtained a measurement of solar neutrino-electron
scattering. The result was originally shown to set a useful constraint on the
gauged $U(1)_{B-L}$ model where the gauge boson directly couples to both
electrons and neutrinos Harnik  _et al._ (2012). It was later reinterpreted in
the context of the gauged $U(1)_{L_{\mu}-L_{\tau}}$ model Kamada and Yu
(2015); Araki  _et al._ (2017); Gninenko and Gorbunov (2020), where neutrino-
electron scattering can occur through the $Z^{\prime}$-photon kinetic mixing
in addition to the weak interaction. By assuming $\varepsilon_{\rm tot}\simeq-
g_{\mu\tau}/70$, corresponding to the dashed curve in Fig. 3, an upper limit
on the coupling $g_{\mu\tau}$ was derived as a function of the $Z^{\prime}$
mass,

$|g_{\mu\tau}|<6\times 10^{-4}\left({m_{Z^{\prime}}}/{10\,\rm MeV}\right)\ .$
(8)

Contrasting with the $(g-2)_{\mu}$ favored region, this sets the leading lower
bound (around 10 MeV) on the $Z^{\prime}$ mass.

However, this constraint is model dependent due to the ambiguity of the
kinetic mixing. As discussed in Sec. II.2, the general kinetic mixing at
Borexino takes the asymptotic form

$\varepsilon_{\rm tot}\simeq\varepsilon_{\rm
IR}+\frac{eg_{\mu\tau}}{60\pi^{2}}\left(\frac{{\rm MeV}}{m_{\mu}}\right)^{2}\
,$ (9)

where we have used the fact that the typical momentum transfer in solar
neutrino-electron scattering process is around the MeV scale, much lower than
the muon mass. Thus the first line of Eq. (6) applies. Setting the boundary
value $\varepsilon_{\rm IR}=0$, which corresponds to the solid curve in Fig.
3, we obtain

$\varepsilon_{\rm tot}\simeq 5\times 10^{-8}g_{\mu\tau}\ .$ (10)

Compared to the other choice $\varepsilon_{\rm tot}\simeq-g_{\mu\tau}/70$, the
upper bound Eq. (8) can be relaxed by a factor of $\sim 500$, making it
irrelevant for probing the $(g-2)_{\mu}$ target. This can be seen in the top
plot of Fig. 4 where the constraints from Borexino are given by the solid and
dashed red curves for $\varepsilon_{\rm IR}=0$ and
$\varepsilon_{\text{UV}}=0$, respectively. The bounds on $g_{\mu\tau}$ are
placed using measurements of the 7Be neutrino line, which has the most
precisely measured flux Agostini  _et al._ (2019). The bound is significantly
weaker for the boundary condition $\varepsilon_{\rm IR}=0$. This simple
exercise demonstrates that caution should be taken on constraints from low
energy neutrino scattering experiments in the gauged $U(1)_{L_{\mu}-L_{\tau}}$
model. They are not as robust as those that are independent of the kinetic
mixing (as shown in Fig. 1).

Figure 4: Current and future reach (upper bound) on $g_{\mu\tau}$ from
neutrino experiments, including Borexino and white dwarfs (first row) and the
CE$\nu$NS experiments (second row). The dashed curves correspond to the
kinetic mixing scenario with $\varepsilon_{\rm UV}=0$, and the solid curves
correspond to the kinetic mixing scenario with $\varepsilon_{\rm IR}=0$ which
are much weaker.

###  III.2 CE$\nu$NS

Similar to neutrino-electron scattering, the $Z^{\prime}$ of
$U(1)_{L_{\mu}-L_{\tau}}$ can contribute to the CE$\nu$NS processes. A first
measurement of this rare process was made by the COHERENT experiment using a
CsI target Akimov  _et al._ (2017, 2018) and later using a liquid argon (LAr)
target Akimov  _et al._ (2021, 2020a). These results are compatible with SM
predictions and can be used to constrain the parameter space of the
$U(1)_{L_{\mu}-L_{\tau}}$ model.

In CE$\nu$NS experiments, neutrinos are produced from pions that decay at
rest, with energies below the muon mass. The neutrino flux consists of a
prompt component of muon-neutrinos from $\pi^{+}\to\mu^{+}\nu_{\mu}$, and a
delayed component of muon anti-neutrinos and electron neutrinos produced from
the subsequent decay of the muon. The neutrino spectra seen in the detector
are Akimov  _et al._ (2017, 2018)

$\begin{split}\frac{dN_{\bar{\nu}_{\mu}}}{dE_{\nu}}&=\eta\frac{64E^{2}_{\nu}}{m^{3}_{\mu}}\bigg{(}\frac{3}{4}-\frac{E_{\nu}}{m_{\mu}}\bigg{)}\
,\\\
\frac{dN_{\nu_{e}}}{dE_{\nu}}&=\eta\frac{192E^{2}_{\nu}}{m^{3}_{\mu}}\bigg{(}\frac{1}{2}-\frac{E_{\nu}}{m_{\mu}}\bigg{)}\
,\\\
\frac{dN_{\nu_{\mu}}}{dE_{\nu}}&=\eta\delta\bigg{(}E_{\nu}-\frac{m_{\pi}^{2}-m_{\mu}^{2}}{2m_{\pi}}\bigg{)}\
,\end{split}$ (11)

where $\eta=rN_{\text{POT}}/4\pi L^{2}$ is a normalization factor that depends
on the experimental design, with $r$ being the number of neutrinos of a
particular flavor produced per proton-on-target (POT) and $L$ the distance
between the neutrino source and the detector. Here we will consider
constraints from existing COHERENT CsI and LAr experiments, as well as
projections for future LAr experiments. In particular, we will consider a
future 610 kg experiment CENNS-610 Akimov  _et al._ (2020b), a proposed 10-kg
experiment at the European Spallation Source (ESS) Baxter  _et al._ (2020),
and the 7-ton fiducial mass Coherent CAPTAIN-Mills (CCM) experiment
Experiment” ; Aguilar-Arevalo  _et al._ (2021). Details of the experimental
parameters can be found in Table 1 in the appendix.

The number of expected CE$\nu$NS events in a given detector can be calculated
as

$N_{\text{CE$\nu$NS}}=\sum_{\nu_{\alpha}}N_{\text{targ}}\int_{E_{\rm
th}}^{E_{R}^{\rm max}}dE_{R}\int_{E_{\nu}^{\rm min}}^{E_{\nu}^{\rm
max}}dE_{\nu}\frac{dN_{\nu_{\alpha}}}{dE_{\nu}}\epsilon(E_{R})\frac{d\sigma_{\nu_{\alpha}A}}{dE_{R}}\
,$ (12)

where $E_{\nu}^{\rm min}=\sqrt{M_{A}E_{R}/2}$, $E_{\nu}^{\rm max}=m_{\mu}/2$,
$E_{R}^{\rm max}=m_{\mu}^{2}/(2M_{A})$, and $M_{A}$ is the target nucleus
mass. The function $\epsilon(E_{R})$ is the efficiency factor, and the
differential cross section takes the form

$\frac{d\sigma_{\nu_{\mu}N}}{dE_{R}}=\frac{d\sigma_{\bar{\nu}_{\mu}N}}{dE_{R}}=\frac{m_{A}}{2\pi}\left(1-\frac{M_{A}E_{R}}{2E_{\nu}^{2}}\right)\left|\frac{G_{F}Q_{\nu
A}}{2}+\frac{g_{\mu\tau}\varepsilon_{\rm
tot}Ze}{2m_{A}E_{R}+m_{Z^{\prime}}^{2}}\right|^{2}F_{\rm Helm}^{2}(E_{R})\ ,$
(13)

where $Q_{\nu A}=A-(1-4\sin^{2}\theta_{W})Z$. Here, the sign of $g_{\mu\tau}$
matters and for illustration we consider $g_{\mu\tau}>0$. In Eq. (12), for
$\alpha=e$, the leading cross section is SM-like. The efficiency factor
$\epsilon(E_{R})$ for the CsI is taken from Ref. Akimov  _et al._ (2018),
while for all LAr experiments we adopt the efficiency factor from Ref. Akimov
_et al._ (2020a). The Helm form factor takes the form Helm (1956); Duda  _et
al._ (2007); Jungman  _et al._ (1996)

$F(E_{R})=\frac{3j_{1}(Qr_{n})}{Qr_{n}}\exp(-Q^{2}s^{2}/2)\ ,$ (14)

where $Q=\sqrt{2M_{A}E_{R}}$, $r_{n}=1.14A^{1/3}\,{\rm fm}$, and $s\simeq
0.9\,$fm.

To determine the upper bounds on $g_{\mu\tau}$ from current and future
experiments we use a chi-square test, which is detailed in Appendix A. In the
bottom row of Fig. 4, we show the current (left plot) and future (right plot)
constraints on the parameter space of the $U(1)_{L_{\mu}-L_{\tau}}$ gauge
boson. In both plots, the dashed curves correspond to the boundary condition
$\varepsilon_{\text{UV}}=0$, while the solid curves correspond to the boundary
condition $\varepsilon_{\text{IR}}=0$. Clearly, the choice of boundary
condition impacts the constraints strongly.

For $\varepsilon_{\text{UV}}=0$, the current experiments do not yet constrain
the favoured $(g-2)_{\mu}$ parameter space (green shaded band), but the future
experiments will begin probing this target for $m_{Z^{\prime}}\lesssim 100$
MeV. On the other hand, for $\varepsilon_{\text{IR}}=0$, all the constraints
weaken substantially. In the bottom-right plot of Fig. 4, even the sensitivity
of future experiments considered above are orders of magnitude too weak. In
this case, the parameter space that explains the $(g-2)_{\mu}$ anomaly will
remain in tact after those experiments.

###  III.3 White dwarf

Here we briefly comment on the constraint from white dwarf cooling Bauer  _et
al._ (2018); Dreiner  _et al._ (2013). The gauged $U(1)_{L_{\mu}-L_{\tau}}$
model provides a novel cooling process that could cause excessive energy loss
of the star: plasmon decaying into $\nu_{\ell}\bar{\nu}_{\ell}$
($\ell=\mu,\tau$) through the exchange of the $Z^{\prime}$ boson. Because the
temperature of white dwarfs ($\sim 10^{7}\,$K $\sim$ keV) is much lower than
the $Z^{\prime}$ mass scale of interest to $(g-2)_{\mu}$, it is sufficient to
consider the effective Lagrangian,

$\mathcal{L}_{\rm eff}=({g_{\mu\tau}\varepsilon_{\rm
tot}e}/{m_{Z^{\prime}}^{2}})(\bar{e}\gamma^{\alpha}e)(\bar{\nu}_{\ell}\gamma_{\alpha}P_{L}\nu_{\ell})\
.$ (15)

With the rather low momentum transfer, it is clear that choosing a kinetic
mixing with $\varepsilon_{\rm IR}=0$ leads to a much weaker constraint than
the $\varepsilon_{\rm UV}=0$ case, by a factor of $\sim 6\times 10^{5}$. This
is derived using a similar approach as in Sec. III.1. 333For white dwarf
cooling, we have $q^{2}>0$ where the $q$ is the four momentum transfer through
the virtual $Z^{\prime}$ to final state $\nu\bar{\nu}$. However, the momentum
dependence derived in Eq. (6) still holds after an analytic continuation to
the $Q^{2}=-q^{2}<0$ regime. The corresponding constraint curve for the
$\varepsilon_{\rm IR}=0$ case lies outside (above) the range of $g_{\mu\tau}$
plotted in Fig. 4.

We close this section by stressing once again that the momentum dependence in
the $Z^{\prime}$-photon kinetic mixing can strongly impact experimental
constraints in the gauged $U(1)_{L_{\mu}-L_{\tau}}$ model. The model features
an extra parameter — the boundary condition of the kinetic mixing. We have
shown that low energy neutrino processes can be suppressed if the kinetic
mixing is in the proximity of zero at low momentum transfer below the muon
mass. In contrast, constraints that do not involve the kinetic mixing still
apply, including those from CMS, BaBar, CCFR, and BBN shown in Fig. 1. The
freedom of choosing a kinetic mixing value other than $\varepsilon_{\rm
tot}=-g_{\mu\tau}/70$ has been considered in some previous studies Banerjee
and Roy (2019); Escudero  _et al._ (2019); Amaral  _et al._ (2021), but we
think our work is the first to take into account the momentum transfer
dependence when deriving the above experimental constraints.

##  IV Implication for $L_{\mu}-L_{\tau}$ Charged Dark Matter

In this section, we go beyond the minimal gauged $U(1)_{L_{\mu}-L_{\tau}}$
model of Eq. (2) by further considering that the $Z^{\prime}$ gauge boson
serves as a mediator between the visible sector and DM. This is a natural and
well-motivated idea to pursue given the significance of existing $(g-2)_{\mu}$
tension. While there could be various incarnations for $L_{\mu}-L_{\tau}$
charged DM Altmannshofer  _et al._ (2016); Kamada  _et al._ (2018); Foldenauer
(2019); Asai  _et al._ (2021); Borah  _et al._ (2021); Holst  _et al._ (2021);
Drees and Zhao (2021), here we consider a rather minimal setup where the DM
candidate $\chi$ is a vector-like (Dirac) fermion. The Lagrangian is

$\begin{split}\mathcal{L}=\mathcal{L}_{L_{\mu}-L_{\tau}}+i\bar{\chi}\gamma^{\alpha}\partial_{\alpha}\chi+g_{\chi}\bar{\chi}\gamma^{\alpha}\chi\hat{Z}^{\prime}_{\alpha}-m_{\chi}\bar{\chi}\chi\
,\\\ \end{split}$ (16)

where $g_{\chi}$ is the $U(1)_{L_{\mu}-L_{\tau}}$ gauge coupling for DM. In
general, it is different from $g_{\mu\tau}$ introduced earlier, which
acknowledges that the $L_{\mu}-L_{\tau}$ charge of $\chi$ can be different
from those of $\mu$ and $\tau$ leptons.

We will explore the DM phenomenology in this model by always making the
following assumptions:

  * •

Throughout the analysis, we always assume that the $(g-2)_{\mu}$ anomaly is
addressed by the $Z^{\prime}$ boson, which fixes the coupling $g_{\mu\tau}$ as
a function of $m_{Z^{\prime}}$.

  * •

We will assume that DM is a thermal relic of the early universe, which fixes
the DM-$Z^{\prime}$ coupling $g_{\chi}$ as a function of DM mass $m_{\chi}$.

###  IV.1 The thermal dark matter target

We consider the DM candidate $\chi$ with a thermal origin, obtaining the
observed relic abundance via the freeze out mechanism. There are two classes
of channels that can fix the relic abundance of DM today: annihilation into
$Z^{\prime}$ bosons, or annihilation into SM leptons via off-shell
$Z^{\prime}$ exchange. We continue to assume the hierarchy, $g_{\mu\tau}\gg
e\varepsilon_{0}$, so that the gauge coupling $g_{\mu\tau}$ plays the dominant
role over the kinetic mixing. The corresponding annihilation cross sections
are

$\begin{split}(\sigma v)_{\chi\bar{\chi}\to
Z^{\prime}Z^{\prime}}&=\frac{g_{\chi}^{4}}{16\pi
m_{\chi}^{2}}\left(1-\frac{m_{Z^{\prime}}^{2}}{m_{\chi}^{2}}\right)^{3/2}\left(1-\frac{m_{Z^{\prime}}^{2}}{2m_{\chi}^{2}}\right)^{-2}\
,\\\ (\sigma
v)_{\chi\bar{\chi}\to\nu_{\ell}\bar{\nu}_{\ell}}&=\frac{g_{\chi}^{2}g_{\mu\tau}^{2}m_{\chi}^{2}}{2\pi[(4m_{\chi}^{2}-m_{Z^{\prime}}^{2})^{2}+m_{Z^{\prime}}^{2}\Gamma_{Z^{\prime}}^{2}]}\
,\\\ (\sigma
v)_{\chi\bar{\chi}\to\ell^{+}\ell^{-}}&=\frac{g_{\chi}^{2}g_{\mu\tau}^{2}m_{\chi}^{2}}{\pi[(4m_{\chi}^{2}-m_{Z^{\prime}}^{2})^{2}+m_{Z^{\prime}}^{2}\Gamma_{Z^{\prime}}^{2}]}\left(1+\frac{m_{\ell_{a}}^{2}}{2m_{\chi}^{2}}\right)\sqrt{1-\frac{m_{\ell_{a}}^{2}}{m_{\chi}^{2}}}\
,\end{split}$ (17)

where the flavor index $\ell=\mu$ or $\tau$. The $Z^{\prime}$ decay width is
Kelly  _et al._ (2020)

$\Gamma_{Z^{\prime}}=\frac{g_{\mu\tau}^{2}m_{Z^{\prime}}}{12\pi}\left[1+\sum_{\alpha=\mu}^{\tau}(1+2r_{\alpha})(1-4r_{\alpha})^{1/2}\Theta(1-4r_{\alpha})\right]\
,$ (18)

where $r_{\alpha}=m_{\alpha}^{2}/m_{Z^{\prime}}^{2}$ and $\Theta$ is the
Heaviside theta function.

In Fig. 5, the solid red and blue curves depict the parameter space where the
DM relic density is successfully produced. In this work, we are primarily
interested in the direct detection prospects of DM in traditional (large)
detectors, thus the focus is on DM heavier than several hundred MeV.
Numerically, we find that the correct relic density favors $g_{\chi}\gg
g_{\mu\tau}$ where the latter is used to fix the $(g-2)_{\mu}$ anomaly.
Therefore, the first annihilation channel in Eq. (17) dominates the thermal
freeze out. Roughly, for $m_{\chi}\gg m_{Z^{\prime}}$:

$g_{\chi}\simeq 0.025\sqrt{\frac{m_{\chi}}{\rm GeV}}\ .$ (19)

Figure 5: Parameter space of DM in the $g_{\chi}$ versus $m_{\chi}$ plane that
depicts where the observed relic density is successfully explained, for
$m_{Z^{\prime}}=$ 10 MeV (red curve) and $m_{Z^{\prime}}=$ 100 MeV (blue
curve). For $m_{\chi}\gg 100\,$MeV, the annihilation channel
$\chi\bar{\chi}\to Z^{\prime}Z^{\prime}$ plays the dominant role. The kink on
the curves around $m_{\chi}\simeq 4$ GeV is mainly due to the large number of
degree of freedom change when the universe was at the QCD phase transition
temperature.

###  IV.2 Upper limit on dark matter mass from direct detection

There is a strong interplay between thermal relic density and DM direct
detection in this model. Since the DM does not couple directly to quarks, its
scattering with nuclei in a detector must occur through the
$Z^{\prime}$-photon kinetic mixing and is thus subject to similar model
dependence as the neutrino scattering experiments discussed in Sec. II and
III. With $g_{\mu\tau}$ fixed such that the $(g-2)_{\mu}$ anomaly is
explained, the DM scattering rate is controlled by the coupling $g_{\chi}$ and
the choice of kinetic mixing.

At the nucleon level, the low-energy DM scattering cross section takes the
form

$\sigma_{\chi p}\approx\frac{g_{\chi}^{2}\varepsilon_{\rm
tot}^{2}e^{2}\mu_{\chi p}^{2}}{\pi m_{Z^{\prime}}^{4}}\ ,$ (20)

where $\mu_{\chi p}=m_{\chi}m_{p}/(m_{\chi}+m_{p})$ is the reduced mass of DM-
proton system. An approximation has been made by neglecting the momentum
transfer in the $Z^{\prime}$ propagator, which is potentially important for
light $Z^{\prime}$.

To account for the momentum transfer dependence, we consider the nucleus level
DM scattering rate derived using the standard halo model

$R=\frac{N_{A}M_{A}\rho_{\odot}g_{\chi}^{2}Z^{2}e^{2}}{2\pi
m_{\chi}}\int_{E_{\rm th}}^{E_{R}^{\rm max}}dE_{R}\frac{\varepsilon^{2}_{\rm
tot}(Q)F_{\rm Helm}^{2}(Q)}{(Q^{2}+m_{Z^{\prime}}^{2})^{2}}\int_{v_{\rm
min}(E_{R})}^{v_{\rm esc}}d^{3}v\frac{f(\vec{v})}{v}\ ,$ (21)

where $N_{A}$ is the number of target nucleus, $M_{A}$ and $Ze$ are the mass
and electric charge of the nucleus, $F_{\rm Helm}$ is the Helm form factor,
and $Q\simeq\sqrt{2M_{A}E_{R}}$ for non-relativistic scattering. The local DM
mass density is $\rho_{\odot}=0.3\,{\rm GeV/cm^{3}}$ and the velocity
distribution is Maxwellian
$f(\vec{v})=C\exp(-|\vec{v}+\vec{v}_{\odot}|^{2}/v_{0}^{2})\,\Theta(v_{\rm
esc}-|\vec{v}+\vec{v}_{\odot}|)$, where $\vec{v}$ is the DM velocity in the
rest frame of the solar system, $v=|\vec{v}|$, $v_{\odot}=220\,{\rm km/s}$,
$v_{0}=235\,{\rm km/s}$, $v_{\rm esc}=550\,{\rm km/s}$ Lin (2019), and $C$ is
a normalization factor such that $\int d^{3}vf(\vec{v})=1$. The Helm form
factor is given by Eq. (14). The limits of integration over velocity and
recoil energy in Eq. (21) are

$v_{\rm min}(E_{R})=\sqrt{\frac{M_{A}E_{R}}{2\mu_{\chi A}^{2}}}\ ,\quad\quad
E_{R}^{\rm max}=\frac{2\mu_{\chi A}^{2}(v_{\rm esc}+v_{\odot})^{2}}{M_{A}}\ ,$
(22)

where $\mu_{\chi A}=m_{\chi}M_{A}/(m_{\chi}+M_{A})$ is the reduced mass of DM-
nucleus system, and $E_{\rm th}$ is the energy threshold of the detector under
consideration.

Figure 6:  Thermal relic targets (black solid curves) and existing direct
detection constraints (solid red curves) in the $L_{\mu}-L_{\tau}$ charged DM
parameter space (coupling versus mass). The direct detection limits depend on
the choice of kinetic mixing and plots in the left and right columns
correspond to $\varepsilon_{\rm UV}=0$ and $\varepsilon_{\rm IR}=0$,
respectively. We consider two benchmark $Z^{\prime}$ masses, 20 MeV (plots in
the first row) and 100 MeV (second row), with the corresponding gauge coupling
$g_{\mu\tau}$ fixed to address the $(g-2)_{\mu}$ tension. In addition, the
blue shaded region is an indirect constraint on DM annihilation during the
recombination epoch.

To determine the upper bound on $g_{\chi}$ from direct detection experiments,
we begin with the $\varepsilon_{\rm UV}=0$ case. We first consider the heavy
$Z^{\prime}$ limit where Eq. (20) applies. This allows us to directly
translate the experimental upper limits on the DM-nucleon scattering cross
section into limits on $g_{\chi}$ as a function of $m_{\chi}$ for given value
of $m_{Z^{\prime}}$. We take into account existing results from CRESST-III
Abdelhameed  _et al._ (2019), Darkside-50 Agnes  _et al._ (2018), CDSMLite
Agnese  _et al._ (2019), XENON1T Aprile  _et al._ (2018, 2019, 2021), and the
very recent PandaX-4T result Meng  _et al._ (2021). Next, to restore the
momentum transfer dependence in the $Z^{\prime}$ propagator, we rescale this
bound by a factor

$\sqrt{\frac{R^{Q=0}_{\varepsilon_{\text{\sc
uv=0}}}}{R_{\varepsilon_{\text{\sc uv=0}}}}}$ (23)

where $R_{\varepsilon_{\text{\sc uv=0}}}$ is the direct detection event rate
given in Eq. (21), evaluated with a kinetic mixing that satisfies a boundary
condition $\varepsilon_{\rm UV}=\varepsilon_{\rm tot}(Q\to\infty)\to 0$.
$R^{Q=0}_{\varepsilon_{\text{\sc uv=0}}}$ corresponds to further replacing the
factor $(Q^{2}+m_{Z^{\prime}}^{2})^{2}$ by $m_{Z^{\prime}}^{4}$ in Eq. (21).
The quantity in Eq. (23) is always larger than 1.

The resulting upper bounds on $g_{\chi}$ as a function of the DM mass
$m_{\chi}$ are depicted in the top left and bottom left plots of Fig. 6 for
$m_{Z^{\prime}}=10$ MeV and $m_{Z^{\prime}}=100$ MeV, respectively. The
corresponding coupling values $g_{\mu\tau}=5.04\times 10^{-4}$ and $9.5\times
10^{-4}$ are used to fix the $(g-2)_{\mu}$ tension. For the thermal DM target,
we find that the DM mass is constrained by existing experiments to be less
than $\sim 0.4$ (1) GeV for $m_{Z^{\prime}}=10$ (100) MeV. These results are
roughly in agreement with the findings of Ref. Asai  _et al._ (2021). We also
checked that the direct DM search limit using electron recoil Aprile  _et al._
(2016) is too weak and does not place additional constraints. However, future
experiments using electron recoils could potentially constrain the thermal
relic target of the model for the $\varepsilon_{\text{UV}}=0$ case Battaglieri
_et al._ (2017).

However, it cannot be overemphasized that such a conclusion is derived by
assuming a boundary condition $\varepsilon_{\text{UV}}=0$ for the kinetic
mixing. As already stated earlier, the above upper bounds for $g_{\chi}$ and
DM mass are not general. They could be modified significantly with other
choices of the kinetic mixing. As a comparison, we consider the other boundary
condition in Fig. 3, with $\varepsilon_{\rm IR}=0$ and explore the
corresponding DM phenomenology. We will comment on the more general choices
near the end of this section.

The momentum transfer dependence in the kinetic mixing affects the DM
scattering rate through the $E_{R}$ integral of Eq. (21). For direct
detection, the typical momentum transfer $Q$ lies below the muon mass, thus we
have approximately,

$\varepsilon_{\rm tot}(Q)\simeq 5\times
10^{-4}g_{\mu\tau}\left(\frac{Q}{m_{\mu}}\right)^{2}\ ,$ (24)

in the $\varepsilon_{\rm IR}=0$ case, which is much smaller than
$\varepsilon_{\rm tot}(Q)\simeq-g_{\mu\tau}/70$ for
$\varepsilon_{\text{UV}}=0$. Fig. 7 (left) depicts the differential scattering
rate $dR/dE_{R}$ in these two cases, for a set of model parameters,
$m_{Z^{\prime}}=10\,$MeV and $m_{\chi}=3\,$GeV. The solid curves correspond to
the case with $\varepsilon_{\rm IR}=0$ whereas the dashed curves correspond to
$\varepsilon_{\rm UV}=0$. Clearly, such a sizable suppression in the
scattering rate implies much weaker direct detection limits for
$\varepsilon_{\rm IR}=0$.

Figure 7: Left: Differential scattering rates of DM with common detector
materials under both considerations of the boundary conditions on
$\varepsilon_{\rm tot}$, for $m_{\chi}=$ 3 GeV and $m_{Z^{\prime}}=$ 10 MeV.
It can be seen that for $\varepsilon_{\rm IR}=0$ there is strong suppression
in the associated rate $dR/dE_{R}$ relative to the $\varepsilon_{\rm UV}=0$
case. Right: Relaxation factor for the dark coupling $g_{\chi}$ (defined in
Eq. (25)) as a function of $m_{\chi}$ for the experiments XENON1T (red
curves), DarkSide-50 (blue curves), and CDMSLite (green curves). We use
$m_{Z^{\prime}}=10$ (solid curves) and $m_{Z^{\prime}}$ = 100 MeV (dashed
curves) as benchmark values. The recoil energy thresholds of detectors we used
are 2 keV for XENON1T, 0.6 keV for DarkSide, and 0.07 keV for CDMSLite.

For more general choices of DM and $Z^{\prime}$ masses, Fig. 7 (right) shows
the relaxation factor for the upper bound on $g_{\chi}$ in the
$\varepsilon_{\rm IR}=0$ case compared to the $\varepsilon_{\rm UV}=0$ case,
as a function of $m_{\chi}$. The relaxation factor is defined as

$\sqrt{\frac{R_{\varepsilon_{\text{\sc uv=0}}}}{R_{\varepsilon_{\text{\sc
ir=0}}}}}$ (25)

where $R$ is the direct detection event rate given in Eq. (21) evaluated with
a boundary condition for $\varepsilon_{\rm tot}(Q)$. In this comparison, the
upper bound on $g_{\chi}$ can be relaxed by as large as 4 orders of magnitude
in the case of a GeV scale DM scattering off of an argon target.

We are now able to derive the upper bound on $g_{\chi}$ in the
$\varepsilon_{\rm IR}=0$ scenario by rescaling the limits in the left panels
of Fig. 6 with the relaxation factor, Eq. (25). The results are shown in the
top right and bottom right plots of Fig. 6. We find the direct detection
constraints indeed weaken substantially in this case. At the same time, the
neutrino floors also rescale along with the direct detection limits. The
neutrino floor for different target materials (Ar, Xe, Ge) rescale by
different amount because the rescaling factor Eq. (25) is detector dependent.
Interestingly, this results in a new interplay with the thermal DM target. For
the $m_{Z^{\prime}}=100\,$MeV case, the portion of the thermal relic curve not
excluded by existing direct detection constraints is fully buried under the
neutrino floor. This is because existing limits from XENON1T and PandaX-4T
already are strong enough to touch the neutrino floor around 5–6 GeV DM mass,
whereas the thermal relic curve happens to pass across this region. In
contrast, for $m_{Z^{\prime}}=10$ MeV case, we find there is still a window of
viable thermal DM mass (around $\sim 1-4$ GeV) that survives above the
neutrino floors. Such a new window (associated with a light $Z^{\prime}$)
serves as a nice target of upcoming direct detection experiments, such as
DarkSide-LowMass Collaboration ; Ajaj  and SuperCDMS-Ge Agnese  _et al._
(2017) that will be running at SNOLAB (see Fig. 9 below).

Figure 8: Upper bounds on the DM mass in the gauged $U(1)_{L_{\mu}-L_{\tau}}$
model for two scenarios of the kinetic mixing, $\varepsilon_{\rm UV}=0$
(dashed curve) and $\varepsilon_{\rm IR}=0$ (solid curve).

The upper bound on the DM mass $m_{\chi}$ can be determined as the
intersection point of the relic density curve with the edges of direct
detection excluded region. Fig. 8 quantifies this upper bound by scanning the
${Z^{\prime}}$ mass for both the $\varepsilon_{\rm UV}=0$ and
$\varepsilon_{\rm IR}=0$ scenarios. We find that the upper bound on $m_{\chi}$
can be relaxed up to $\sim 6$ GeV with $\varepsilon_{\rm IR}=0$, in contrast
to $m_{\chi}\lesssim 1$ GeV in the $\varepsilon_{\rm UV}=0$ case. The mass of
$Z^{\prime}$ has an effect on these results, which could be inferred from Fig.
6. In the $\varepsilon_{\rm UV}=0$ case, direct detection limit on the thermal
target curve is mainly set by CRESST-III, which gets stronger for lighter
$Z^{\prime}$ (the propagator effect). As a result, the upper bound on
$m_{\chi}$ reduces for lighter $Z^{\prime}$, which agrees with the behavior of
the dashed curve in Fig. 8. In the $\varepsilon_{\rm IR}=0$ case, we observe
similar behavior but it is PandaX-4T and XENON1T that dictate the upper bound
on $m_{\chi}$.

###  IV.3 Novel recoil energy spectrum

As discussed above, for a kinetic mixing satisfying $\varepsilon_{\rm IR}=0$
together with a light $Z^{\prime}$ boson, we open up a new window of thermal
DM mass above GeV scale that is consistent with existing direct detection
limits and can be probed by the upcoming experiments. To highlight the
interplay, we show Fig. 9, as a zoomed-in version of Fig. 6 (right) for the
case of $m_{Z^{\prime}}=10\,$MeV. On top of it, we include the future
sensitivities of DarkSide-LowMass and SuperCDMS-Ge experiments as purple and
green curves, respectively. Clearly, they stand in very good position to cover
the remaining DM target above the neutrino floor. Note, that we show
additional existing constraints from one tonne-year exposer of XENON1T Aprile
_et al._ (2018) (dashed red curve labelled “XENON1T”) and the XENON1T results
from secondary scintillation (S2) only analysis Aprile  _et al._ (2019)
(dashed red curve labelled “XENON1T-S2”).

Figure 9:  A zoomed-in version of Fig. 6 (right) for the case of
$m_{Z^{\prime}}=10\,$MeV. Again, the black curve is the the thermal DM target
in the light of $(g-2)_{\mu}$. The red shaded regions are existing
constraints. Future reaches by the DarkSide-LowMass and SuperCDMS-Ge
experiments are shown as the purple and green curves, respectively.

In this subsection, we derive another important consequence of the momentum
dependence in the kinetic mixing, by exploring the recoil energy spectra of DM
scattering in the DarkSide-LowMass and SuperCDMS experiments. To proceed, we
choose a set of benchmark parameters with

$m_{\chi}=3\,{\rm GeV},\quad m_{Z^{\prime}}=10\,{\rm MeV},\quad
g_{\chi}=0.04,\quad g_{\mu\tau}=5.04\times 10^{-4}\ ,$ (26)

which corresponds to the blue five-star in Fig. 9.

Fig. 7 already shows that the differential DM scattering rate $dR/dE_{R}$ is
strongly suppressed by the small total kinetic mixing evaluated at low $Q$, in
the case of $\varepsilon_{\rm IR}=0$. Eq. (24) further tells us that such a
suppression effect is the strongest at $Q=0$ where $\varepsilon_{\rm tot}(Q)$
vanishes completely. For higher recoil energies, the differential rate first
increases following the behavior of $\varepsilon_{\rm tot}$ and eventually
shuts off when the scattering runs out of phase space. This interplay predicts
a peak in the $dR/dE_{R}$ curve at intermediate recoil energies.

All the above features are captured by the solid curves in the first row of
Fig. 10. The shape of recoil spectrum differs from normal elastic scattering
with constant couplings (i.e. the $\varepsilon_{\rm UV}=0$ case) whose
differential rate $dR/dE_{R}$ peaks at zero recoil energy, as shown by the
dashed curves in Fig. 10. Instead, for $\varepsilon_{\rm IR}=0$ the recoil
spectrum peaks at a nonzero recoil energy. Such a novel shape of the recoil
energy spectrum can serve as an important “smoking-gun” signal for upcoming
direct detection experimental searches, especially those featuring low
detector thresholds, DarkSide-LowMass ($E_{\rm th}=0.6$ keV) Collaboration
and SuperCDMS-Ge ($E_{\rm th}=40$ eV) Agnese  _et al._ (2017).

It is worth emphasizing again that for the benchmark parameters considered in
Eq. (26), only the $\varepsilon_{\rm IR}=0$ case allows for a simultaneous
explanation of $(g-2)_{\mu}$ and thermal DM relic abundance without being
excluded by existing direct detection limits. The $\varepsilon_{\rm UV}=0$
case fails to do so. The dashed curves in Fig. 10 are depicted with arbitrary
normalization for comparison and could only work if one of the assumptions is
relaxed.

We also derive the annual modulated recoil spectrum in the two types of
detectors, which is given by the difference when the parameter $v_{\odot}$ in
DM velocity distribution (see Eq. (21)) is replaced by $v_{\odot}\pm
v_{\oplus}$, where $v_{\oplus}\simeq 30\,{\rm km/s}$ is the velocity of the
earth around the sun. The annual modulation spectra are displayed in the
bottom row of Fig. 10. For $\varepsilon_{\rm IR}=0$, they inherit a similar
peaked shape like the averaged spectrum, driven by the kinetic mixing effect.
The recoil energies associated with the peaks of the spectra are much higher
for $\varepsilon_{\rm IR}=0$ than the case with $\varepsilon_{\rm UV}=0$ where
DM scatters via nearly constant couplings. This feature adds another handle to
the smoking-gun signature.

Figure 10: Averaged and annual modulated DM recoil spectra at the future
DarkSide-LM and SuperCDMS experiments for two scenarios of the kinetic mixing,
$\varepsilon_{\rm UV}=0$ (dashed curves) and $\varepsilon_{\rm IR}=0$ (solid
curves). The normalization of the $\varepsilon_{\rm UV}=0$ curves is chosen so
that it gives the same total scattering rate as the $\varepsilon_{\rm IR}=0$
case.

To summarize, the DM recoil energy spectrum encodes important information of
$L_{\mu}-L_{\tau}$ charged DM and the nature of kinetic mixing. It serves as a
well motivated target for the upcoming search at DarkSide-LM and SuperCDMS
experiments. For general kinetic mixing scenarios beyond the $\varepsilon_{\rm
IR}=0$, $\varepsilon_{\rm UV}=0$ cases, the novel $E_{R}$ (or $Q$) dependence
in the kinetic mixing could still be tested with a precision measurement of
the DM recoil energy spectrum at low threshold detectors. Having multiple
detectors and target materials will be helpful for testing the $E_{R}$
dependence.

###  IV.4 Other Constraints

####  IV.4.1 Indirect detection using CMB and neutrinos

Going beyond direct detection, we comment on other approaches to probe the
$L_{\mu}-L_{\tau}$ charged DM. For $m_{\chi}>m_{\mu,\tau}$ DM can annihilate
into these visible particles which leads to indirect signals. The annihilation
cross section is given by the third line of Eq. (17). Energy injection due to
DM annihilation during the recombination era is constrained by the observed
cosmic microwave background (CMB) spectrum Slatyer (2016); Kawasaki  _et al._
(2021). Assuming $\chi$, $\bar{\chi}$ make up 100% of DM in the universe, the
current CMB exclusion of the model parameter space is shown by the blue shaded
regions in Fig. 6. We find it is not strong enough to touch the thermal relic
curve, primarily because $\chi\bar{\chi}\to\mu^{+}\mu^{-}$ and
$\tau^{+}\tau^{-}$ are subdominant annihilation channels.

There are also indirect searches for galactic DM annihilating into neutrinos,
which occurs through $\chi\bar{\chi}\to Z^{\prime}Z^{\prime}$ followed by
$Z^{\prime}\to\nu\bar{\nu}$ in the parameter space we consider. The
corresponding constraints are yet to reach the thermal target Argüelles  _et
al._ (2019). These indirect constraints are robust and do not depend on
assumptions of the kinetic mixing.

####  IV.4.2 Dark matter self interaction

The DM model setup considered here features a light mediator $Z^{\prime}$,
which allows DM to self interact. For GeV scale thermal DM and the mass range
of $Z^{\prime}$ of interest to $(g-2)_{\mu}$, it is sufficient to estimate the
self interaction using the Born approximation Feng  _et al._ (2010). The
momentum transfer cross section satisfies

$\sigma_{T}/m_{\chi}\simeq\frac{g_{\chi}^{4}m_{\chi}}{4\pi
m_{Z^{\prime}}^{4}}\simeq 0.02\,{\rm cm^{2}/g}\left(\frac{m_{\chi}}{\rm
3\,GeV}\right)\left(\frac{10\,\rm MeV}{m_{Z^{\prime}}}\right)^{4}\ .$ (27)

This implies that the model parameter space explored in this work is
compatible with DM self-interaction constraints such as the bullet cluster
Randall  _et al._ (2008).

##  V Conclusion

In this work, we explore neutrino and DM phenomenology in the gauged
$U(1)_{L_{\mu}-L_{\tau}}$ model that addresses the $(g-2)_{\mu}$ tension. A
number of experimental probes are controlled by the kinetic mixing between the
new gauge boson $Z^{\prime}$ and the photon, which depends on the bare kinetic
mixing $\varepsilon_{0}$ as a free parameter. In general, the effective
kinetic mixing in a process depends on the typical momentum transfer. By
comparing different scenarios for the kinetic mixing, we find that they
predict drastically different signal rates at low-energy neutrino experiments.
More concretely we find that, the constraint on neutrino-electron scattering
set by the Borexino experiment, constraints on neutrino-nucleus scattering set
by the COHERENT experiments, as well as the white dwarf cooling constraint,
all can be substantially weakened if the asymptotic value of the kinetic
mixing vanishes at zero momentum transfer. In this case, they do not impose
any constraint on the $(g-2)_{\mu}$ favored target parameter space of the
model.

As a further step, we explore the impact of the kinetic mixing effect on
$L_{\mu}-L_{\tau}$ charged DM. We make the minimal assumption where the DM is
a Dirac fermion and obtain the correct relic abundance by annihilating into
$Z^{\prime}$ bosons. We find direct detection constraints can also be relaxed
with the same choice of kinetic mixing that allows the above neutrino
constraints to be lifted. This helps to open up a mass window for the DM above
the GeV scale which can be further explored by future low-threshold DM direct
searches including DarkSide and SuperCDMS. The interplay with momentum
dependence in the kinetic mixing leads to novel recoil energy spectrum in DM
scattering.

Our work demonstrates the phenomenological importance of the kinetic mixing in
gauged $U(1)_{L_{\mu}-L_{\tau}}$ extension of the Standard Model motivated by
the muon $g-2$ anomaly. A joint effort of future neutrino and DM experiments
and precision spectral measurement using low threshold detectors will be the
key to test such a theory.

## Acknowledgment

We thank Ning Zhou for useful discussions. This work is supported by the
Arthur B. McDonald Canadian Astroparticle Physics Research Institute.

##  Appendix A Statistical Analysis for CE$\nu$NS

In this appendix, we describe the statistical analysis used to derive the
constraints from CE$\nu$NS shown in Fig. 4. We follow closely the discussion
in Banerjee  _et al._ (2021). To determine upper limits on the parameter space
of the $U(1)_{L_{\mu}-L_{\tau}}$ model we use a chi-squared test given by

$\chi^{2}=\underset{\alpha}{\text{Min}}\Bigg{[}\frac{\big{(}N_{\text{exp}}-(1+\alpha)N_{\text{th}}\big{)}^{2}}{\sigma^{2}_{\text{stat}}}+\bigg{(}\frac{\alpha}{\sigma_{\alpha}}\bigg{)}^{2}\Bigg{]}$
(28)

where $N_{\text{exp}}$ is the number of CE$\nu$NS events observed by a given
experiment, $N_{\text{th}}$ is the predicted number of CE$\nu$NS events in the
model given by Eq. (12), $\sigma_{\text{stat}}$ is the statistical uncertainty
in the $N_{\text{exp}}$, and $\alpha$ is an overall normalization factor that
accounts for the systematic uncertainty $\sigma_{\alpha}$.

For the COHERENT CsI and CENNS-10 experiments, the statistical uncertainty in
the number of observed CE$\nu$NS events is given by
$\sigma_{\text{stat}}=\sqrt{N_{\text{exp}}+N_{\text{SS}}+N_{\text{BRN}}+N_{\text{NIN}}}$,
where $N_{\text{SS}}$ is the observed number of steady-state background
events, $N_{\text{BRN}}$ is the number of beam-related neutron events, and
$N_{\text{NIN}}$ is the number of neutrino-induced neutrons. The systematic
uncertainties for CsI and argon experiments are $\sigma_{\alpha}=0.13$ and
$\sigma_{\alpha}=0.085$, respectively. Note, that in our chi-square test we
use the fit results of Analysis A of the CENNS-10 data Akimov  _et al._
(2021).

For future liquid argon experiments, we assume that the observed number of
events is consistent with SM predictions, and use a similar chi-square test as
in Eq. (28) with $N_{\text{exp}}$ replaced by the number of CE$\nu$NS events
predicted in the SM $N_{\text{SM}}$. The statistical uncertainty is given by
$\sigma_{\text{stat}}=\sqrt{N_{\text{SM}}+N_{\text{bkg}}}$ where we assume
that $N_{\text{bkg}}=N_{\text{SM}}/10$. In addition, we assume a systematic
uncertainty $\sigma_{\alpha}=0.05$.

After minimizing Eq. (28) with respect to $\alpha$, the 95% exclusion limits
from CE$\nu$NS is found by requiring that
$\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\text{SM}}=5.99$ ($\chi^{2}_{\text{SM}}$ is
found by making the replacement $N_{\text{th}}\to N_{\text{SM}}$ in Eq. (28)).
This requirement will give us a value of $N_{\text{th}}$ that can then be used
to find the upper limit on the coupling $g_{\mu\tau}$, shown in the bottom row
Fig. 4.

Note that this limit depends on the sign of $g_{\mu\tau}$ for the scenario
with the boundary condition $\varepsilon_{\text{UV}}=0$. For $g_{\mu\tau}>0$
there is destructive interference between the SM and $Z^{\prime}$
contributions to CE$\nu$NS since from Eq. (12) we have

$\frac{d\sigma_{\nu_{\mu}N}}{dE_{R}}=\frac{d\sigma_{\bar{\nu}_{\mu}N}}{dE_{R}}\propto\left|\frac{G_{F}Q_{\nu
A}}{2}+\frac{g_{\mu\tau}\varepsilon_{\rm
tot}Ze}{2m_{A}E_{R}+m_{Z^{\prime}}^{2}}\right|^{2}.$ (29)

In some regions of parameter space this destructive interference predicts a
value of $N_{\text{CE}\nu\text{NS}}$ that is much lower than the experimental
value. There will be a lower and upper limit on the value of $N_{\text{th}}$
that satisfies $\Delta\chi^{2}=5.99$, as shown in the left plot of Fig 11.

In practice, we find this only happens for the $\varepsilon_{\text{UV}}=0$
case. In the right plot of Fig. 11 we show $\Delta\chi^{2}$ as a function of
$g_{\mu\tau}$ for $m_{Z^{\prime}}=100$ MeV. For the boundary condition
$\varepsilon_{\text{IR}}=0$ (solid curve) we find that there is a single value
of $g_{\mu\tau}$ that satisfies $\Delta\chi^{2}=5.99$, corresponding to the
maximum value of $N_{\text{th}}$ that is compatible with the expected number
of observed events i.e this scenario only adds to the SM prediction. On the
other hand, for $\varepsilon_{\text{UV}}=0$ (dashed curve) there are multiple
values of $g_{\mu\tau}$ that satisfy $\Delta\chi^{2}=5.99$. The strongest
constraint on $g_{\mu\tau}$ in this case is due to the destructive
interference between the SM and $Z^{\prime}$, corresponding to the minimum
value of $N_{\text{th}}$ that is compatible with the expected number of
observed events.

Figure 11: $\Delta\chi^{2}$ as a function of theory prediction (left) and gauge coupling $g_{\mu\tau}$ (right) for a CE$\nu$NS experiment in the gauged $U(1)_{L_{\mu}-L_{\tau}}$ model. We use the proposed CENNS-610 experiment for illustration and $m_{Z^{\prime}}=100$ MeV. Experiment | Mass [kg] |  $E_{\text{th}}$ [keVnr] | $N_{\text{POT}}$ | $r$ | $L~{}[m]$
---|---|---|---|---|---
COHERENT CsI Akimov  _et al._ (2017, 2018) | 14.6 | 4.25 | $1.76\times 10^{23}$ | 0.08 | 19.3
COHERENT LAr Akimov  _et al._ (2021, 2020a) | 24.4 | 20 | $1.38\times 10^{23}$ | 0.09 | 27.5
CENNS-610 Akimov  _et al._ (2020b) | 610 | 20 | $1.5\times 10^{23}$ | 0.08 | 28.4
ESS-10 Baxter  _et al._ (2020) | 10 | 0.1 | $2.8\times 10^{23}$ | 0.3 | 20
CCM Experiment” ; Aguilar-Arevalo  _et al._ (2021) | 7000 | 10 | $0.177\times 10^{23}$ | 0.0425 | 20
Table 1: Experimental parameters for current and future CE$\nu$NS experiments.
For current CsI and LAr COHERENT experiments (first two rows) we quote the
total number of protons-on-target during their full run time.; for future
experiments we quote the number of protons-on-target $N_{\text{POT}}$ assuming
one year of running.

## References

  * Abi  _et al._ (2021) B. Abi _et al._ (Muon g-2), “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm,” Phys. Rev. Lett. 126, 141801 (2021), arXiv:2104.03281 [hep-ex] .
  * Bennett  _et al._ (2006) G. W. Bennett _et al._ (Muon g-2), “Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL,” Phys. Rev. D 73, 072003 (2006), arXiv:hep-ex/0602035 .
  * Aoyama  _et al._ (2020) T. Aoyama _et al._ , “The anomalous magnetic moment of the muon in the Standard Model,” Phys. Rept. 887, 1–166 (2020), arXiv:2006.04822 [hep-ph] .
  * Baek  _et al._ (2001) Seungwon Baek, N. G. Deshpande, X. G. He, and P. Ko, “Muon anomalous g-2 and gauged L(muon) - L(tau) models,” Phys. Rev. D 64, 055006 (2001), arXiv:hep-ph/0104141 .
  * Alekhin  _et al._ (2016) Sergey Alekhin _et al._ , “A facility to Search for Hidden Particles at the CERN SPS: the SHiP physics case,” Rept. Prog. Phys. 79, 124201 (2016), arXiv:1504.04855 [hep-ph] .
  * Bauer  _et al._ (2018) Martin Bauer, Patrick Foldenauer, and Joerg Jaeckel, “Hunting All the Hidden Photons,” JHEP 07, 094 (2018), arXiv:1803.05466 [hep-ph] .
  * Gninenko and Krasnikov (2018) S. N. Gninenko and N. V. Krasnikov, “Probing the muon $g_{\mu}$ \- 2 anomaly, $L_{\mu}-L_{\tau}$ gauge boson and Dark Matter in dark photon experiments,” Phys. Lett. B 783, 24–28 (2018), arXiv:1801.10448 [hep-ph] .
  * Kahn  _et al._ (2018) Yonatan Kahn, Gordan Krnjaic, Nhan Tran, and Andrew Whitbeck, “M3: a new muon missing momentum experiment to probe $(g-2)_{\mu}$ and dark matter at Fermilab,” JHEP 09, 153 (2018), arXiv:1804.03144 [hep-ph] .
  * Altmannshofer  _et al._ (2019) Wolfgang Altmannshofer, Stefania Gori, Justo Martín-Albo, Alexandre Sousa, and Michael Wallbank, “Neutrino Tridents at DUNE,” Phys. Rev. D 100, 115029 (2019), arXiv:1902.06765 [hep-ph] .
  * Kamada and Yu (2015) Ayuki Kamada and Hai-Bo Yu, “Coherent Propagation of PeV Neutrinos and the Dip in the Neutrino Spectrum at IceCube,” Phys. Rev. D 92, 113004 (2015), arXiv:1504.00711 [hep-ph] .
  * Ibe  _et al._ (2017) Masahiro Ibe, Wakutaka Nakano, and Motoo Suzuki, “Constraints on $L_{\mu}-L_{\tau}$ gauge interactions from rare kaon decay,” Phys. Rev. D 95, 055022 (2017), arXiv:1611.08460 [hep-ph] .
  * Araki  _et al._ (2017) Takeshi Araki, Shihori Hoshino, Toshihiko Ota, Joe Sato, and Takashi Shimomura, “Detecting the $L_{\mu}-L_{\tau}$ gauge boson at Belle II,” Phys. Rev. D 95, 055006 (2017), arXiv:1702.01497 [hep-ph] .
  * Asai  _et al._ (2021) Kento Asai, Shohei Okawa, and Koji Tsumura, “Search for U(1)${}_{L_{\mu}-L_{\tau}}$ charged Dark Matter with neutrino telescope,” JHEP 03, 047 (2021), arXiv:2011.03165 [hep-ph] .
  * Argüelles  _et al._ (2017) Carlos A. Argüelles, Xiao-Gang He, Grigory Ovanesyan, Tao Peng, and Michael J. Ramsey-Musolf, “Dark Gauge Bosons: LHC Signatures of Non-Abelian Kinetic Mixing,” Phys. Lett. B 770, 101–107 (2017), arXiv:1604.00044 [hep-ph] .
  * Gherghetta  _et al._ (2019) Tony Gherghetta, Jörn Kersten, Keith Olive, and Maxim Pospelov, “Evaluating the price of tiny kinetic mixing,” Phys. Rev. D 100, 095001 (2019), arXiv:1909.00696 [hep-ph] .
  * Zhou (2021) Shun Zhou, “Neutrino Masses, Leptonic Flavor Mixing and Muon $(g-2)$ in the Seesaw Model with the $U(1)_{L_{\mu}-L_{\tau}}$ Gauge Symmetry,” (2021), arXiv:2104.06858 [hep-ph] .
  * Fayet (2007) Pierre Fayet, “U-boson production in e+ e- annihilations, psi and Upsilon decays, and Light Dark Matter,” Phys. Rev. D 75, 115017 (2007), arXiv:hep-ph/0702176 .
  * Pospelov (2009) Maxim Pospelov, “Secluded U(1) below the weak scale,” Phys. Rev. D 80, 095002 (2009), arXiv:0811.1030 [hep-ph] .
  * Davoudiasl  _et al._ (2014) Hooman Davoudiasl, Hye-Sung Lee, and William J. Marciano, “Muon $g-2$, rare kaon decays, and parity violation from dark bosons,” Phys. Rev. D 89, 095006 (2014), arXiv:1402.3620 [hep-ph] .
  * Alexander  _et al._ (2016) Jim Alexander _et al._ , “Dark Sectors 2016 Workshop: Community Report,” (2016) arXiv:1608.08632 [hep-ph] .
  * Fabbrichesi  _et al._ (2020) Marco Fabbrichesi, Emidio Gabrielli, and Gaia Lanfranchi, “The Dark Photon,”  (2020), 10.1007/978-3-030-62519-1, arXiv:2005.01515 [hep-ph] .
  * Sirunyan  _et al._ (2019) Albert M Sirunyan _et al._ (CMS), “Search for an $L_{\mu}-L_{\tau}$ gauge boson using Z$\to 4\mu$ events in proton-proton collisions at $\sqrt{s}=$ 13 TeV,” Phys. Lett. B 792, 345–368 (2019), arXiv:1808.03684 [hep-ex] .
  * Lees  _et al._ (2016) J. P. Lees _et al._ (BaBar), “Search for a muonic dark force at BABAR,” Phys. Rev. D 94, 011102 (2016), arXiv:1606.03501 [hep-ex] .
  * Altmannshofer  _et al._ (2014) Wolfgang Altmannshofer, Stefania Gori, Maxim Pospelov, and Itay Yavin, “Neutrino Trident Production: A Powerful Probe of New Physics with Neutrino Beams,” Phys. Rev. Lett. 113, 091801 (2014), arXiv:1406.2332 [hep-ph] .
  * Ahlgren  _et al._ (2013) Björn Ahlgren, Tommy Ohlsson, and Shun Zhou, “Comment on “Is Dark Matter with Long-Range Interactions a Solution to All Small-Scale Problems of $\Lambda$ Cold Dark Matter Cosmology?”,” Phys. Rev. Lett. 111, 199001 (2013), arXiv:1309.0991 [hep-ph] .
  * Kelly  _et al._ (2020) Kevin J. Kelly, Manibrata Sen, Walter Tangarife, and Yue Zhang, “Origin of sterile neutrino dark matter via secret neutrino interactions with vector bosons,” Phys. Rev. D 101, 115031 (2020), arXiv:2005.03681 [hep-ph] .
  * Escudero  _et al._ (2019) Miguel Escudero, Dan Hooper, Gordan Krnjaic, and Mathias Pierre, “Cosmology with A Very Light Lμ $-$ Lτ Gauge Boson,” JHEP 03, 071 (2019), arXiv:1901.02010 [hep-ph] .
  * Harnik  _et al._ (2012) Roni Harnik, Joachim Kopp, and Pedro A. N. Machado, “Exploring nu Signals in Dark Matter Detectors,” JCAP 07, 026 (2012), arXiv:1202.6073 [hep-ph] .
  * Gninenko and Gorbunov (2020) Sergei Gninenko and Dmitry Gorbunov, “Refining constraints from Borexino measurements on a light $Z^{\prime}$-boson coupled to $L_{\mu}$-$L_{\tau}$ current,” (2020), arXiv:2007.16098 [hep-ph] .
  * Agostini  _et al._ (2019) M. Agostini _et al._ (Borexino), “First Simultaneous Precision Spectroscopy of $pp$, 7Be, and $pep$ Solar Neutrinos with Borexino Phase-II,” Phys. Rev. D 100, 082004 (2019), arXiv:1707.09279 [hep-ex] .
  * Akimov  _et al._ (2017) D. Akimov _et al._ (COHERENT), “Observation of Coherent Elastic Neutrino-Nucleus Scattering,” Science 357, 1123–1126 (2017), arXiv:1708.01294 [nucl-ex] .
  * Akimov  _et al._ (2018) D. Akimov _et al._ (COHERENT), “COHERENT Collaboration data release from the first observation of coherent elastic neutrino-nucleus scattering,”  (2018), 10.5281/zenodo.1228631, arXiv:1804.09459 [nucl-ex] .
  * Akimov  _et al._ (2021) D. Akimov _et al._ (COHERENT), “First Measurement of Coherent Elastic Neutrino-Nucleus Scattering on Argon,” Phys. Rev. Lett. 126, 012002 (2021), arXiv:2003.10630 [nucl-ex] .
  * Akimov  _et al._ (2020a) D. Akimov _et al._ (COHERENT), “COHERENT Collaboration data release from the first detection of coherent elastic neutrino-nucleus scattering on argon,”  (2020a), 10.5281/zenodo.3903810, arXiv:2006.12659 [nucl-ex] .
  * Akimov  _et al._ (2020b) D. Akimov _et al._ (COHERENT), “Sensitivity of the COHERENT Experiment to Accelerator-Produced Dark Matter,” Phys. Rev. D 102, 052007 (2020b), arXiv:1911.06422 [hep-ex] .
  * Baxter  _et al._ (2020) D. Baxter _et al._ , “Coherent Elastic Neutrino-Nucleus Scattering at the European Spallation Source,” JHEP 02, 123 (2020), arXiv:1911.00762 [physics.ins-det] .
  * (37) ”Coherent Captain-Mills (CCM) Experiment”, https://p25ext.lanl.gov/7̃Elee/CaptainMills/.
  * Aguilar-Arevalo  _et al._ (2021) A. A. Aguilar-Arevalo _et al._ (CCM), “First Dark Matter Search Results From Coherent CAPTAIN-Mills,” (2021), arXiv:2105.14020 [hep-ex] .
  * Helm (1956) Richard H. Helm, “Inelastic and Elastic Scattering of 187-Mev Electrons from Selected Even-Even Nuclei,” Phys. Rev. 104, 1466–1475 (1956).
  * Duda  _et al._ (2007) Gintaras Duda, Ann Kemper, and Paolo Gondolo, “Model Independent Form Factors for Spin Independent Neutralino-Nucleon Scattering from Elastic Electron Scattering Data,” JCAP 04, 012 (2007), arXiv:hep-ph/0608035 .
  * Jungman  _et al._ (1996) Gerard Jungman, Marc Kamionkowski, and Kim Griest, “Supersymmetric dark matter,” Phys. Rept. 267, 195–373 (1996), arXiv:hep-ph/9506380 .
  * Dreiner  _et al._ (2013) Herbert K. Dreiner, Jean-François Fortin, Jordi Isern, and Lorenzo Ubaldi, “White Dwarfs constrain Dark Forces,” Phys. Rev. D 88, 043517 (2013), arXiv:1303.7232 [hep-ph] .
  * Banerjee and Roy (2019) Heerak Banerjee and Sourov Roy, “Signatures of supersymmetry and $L_{\mu}-L_{\tau}$ gauge bosons at Belle-II,” Phys. Rev. D 99, 035035 (2019), arXiv:1811.00407 [hep-ph] .
  * Amaral  _et al._ (2021) D. W. P. Amaral, D. G. Cerdeño, A. Cheek, and P. Foldenauer, “Distinguishing $U(1)_{L_{\mu}-L_{\tau}}$ from $U(1)_{L_{\mu}}$ as a solution for $(g-2)_{\mu}$ with neutrinos,” (2021), arXiv:2104.03297 [hep-ph] .
  * Altmannshofer  _et al._ (2016) Wolfgang Altmannshofer, Stefania Gori, Stefano Profumo, and Farinaldo S. Queiroz, “Explaining dark matter and B decay anomalies with an $L_{\mu}-L_{\tau}$ model,” JHEP 12, 106 (2016), arXiv:1609.04026 [hep-ph] .
  * Kamada  _et al._ (2018) Ayuki Kamada, Kunio Kaneta, Keisuke Yanagi, and Hai-Bo Yu, “Self-interacting dark matter and muon $g-2$ in a gauged U$(1)_{L_{\mu}-L_{\tau}}$ model,” JHEP 06, 117 (2018), arXiv:1805.00651 [hep-ph] .
  * Foldenauer (2019) Patrick Foldenauer, “Light dark matter in a gauged $U(1)_{L_{\mu}-L_{\tau}}$ model,” Phys. Rev. D 99, 035007 (2019), arXiv:1808.03647 [hep-ph] .
  * Borah  _et al._ (2021) Debasish Borah, Manoranjan Dutta, Satyabrata Mahapatra, and Narendra Sahu, “Muon $(g-2)$ and XENON1T Excess with Boosted Dark Matter in $L_{\mu}-L_{\tau}$ Model,” (2021), arXiv:2104.05656 [hep-ph] .
  * Holst  _et al._ (2021) Ian Holst, Dan Hooper, and Gordan Krnjaic, “The Simplest and Most Predictive Model of Muon $g-2$ and Thermal Dark Matter,” (2021), arXiv:2107.09067 [hep-ph] .
  * Drees and Zhao (2021) Manuel Drees and Wenbin Zhao, “$U(1)_{L_{\mu}-L_{\tau}}$ for Light Dark Matter, $g_{\mu}-2$, the $511$ keV excess and the Hubble Tension,” (2021), arXiv:2107.14528 [hep-ph] .
  * Lin (2019) Tongyan Lin, “Dark matter models and direct detection,” PoS 333, 009 (2019), arXiv:1904.07915 [hep-ph] .
  * Abdelhameed  _et al._ (2019) A. H. Abdelhameed _et al._ (CRESST), “First results from the CRESST-III low-mass dark matter program,” Phys. Rev. D 100, 102002 (2019), arXiv:1904.00498 [astro-ph.CO] .
  * Agnes  _et al._ (2018) P. Agnes _et al._ (DarkSide), “Low-Mass Dark Matter Search with the DarkSide-50 Experiment,” Phys. Rev. Lett. 121, 081307 (2018), arXiv:1802.06994 [astro-ph.HE] .
  * Agnese  _et al._ (2019) R. Agnese _et al._ (SuperCDMS), “Search for Low-Mass Dark Matter with CDMSlite Using a Profile Likelihood Fit,” Phys. Rev. D 99, 062001 (2019), arXiv:1808.09098 [astro-ph.CO] .
  * Aprile  _et al._ (2018) E. Aprile _et al._ (XENON), “Dark Matter Search Results from a One Ton-Year Exposure of XENON1T,” Phys. Rev. Lett. 121, 111302 (2018), arXiv:1805.12562 [astro-ph.CO] .
  * Aprile  _et al._ (2019) E. Aprile _et al._ (XENON), “Light Dark Matter Search with Ionization Signals in XENON1T,” Phys. Rev. Lett. 123, 251801 (2019), arXiv:1907.11485 [hep-ex] .
  * Aprile  _et al._ (2021) E. Aprile _et al._ (XENON), “Search for Coherent Elastic Scattering of Solar 8B Neutrinos in the XENON1T Dark Matter Experiment,” Phys. Rev. Lett. 126, 091301 (2021), arXiv:2012.02846 [hep-ex] .
  * Meng  _et al._ (2021) Yue Meng _et al._ , “Dark Matter Search Results from the PandaX-4T Commissioning Run,” (2021), arXiv:2107.13438 [hep-ex] .
  * Aprile  _et al._ (2016) E. Aprile _et al._ (XENON), “Low-mass dark matter search using ionization signals in XENON100,” Phys. Rev. D 94, 092001 (2016), [Erratum: Phys.Rev.D 95, 059901 (2017)], arXiv:1605.06262 [astro-ph.CO] .
  * Battaglieri  _et al._ (2017) Marco Battaglieri _et al._ , “US Cosmic Visions: New Ideas in Dark Matter 2017: Community Report,” in  _U.S. Cosmic Visions: New Ideas in Dark Matter_ (2017) arXiv:1707.04591 [hep-ph] .
  * (61) The Global Argon Dark Matter Collaboration, “Future Dark Matter Searches with Low-Radioactivity Argon,” .
  * (62) Rahaf Ajaj, “Low Radioactivity Argon for Dark Matter and Rare Event Searches,” .
  * Agnese  _et al._ (2017) R. Agnese _et al._ (SuperCDMS), “Projected Sensitivity of the SuperCDMS SNOLAB experiment,” Phys. Rev. D 95, 082002 (2017), arXiv:1610.00006 [physics.ins-det] .
  * Slatyer (2016) Tracy R. Slatyer, “Indirect dark matter signatures in the cosmic dark ages. I. Generalizing the bound on s-wave dark matter annihilation from Planck results,” Phys. Rev. D 93, 023527 (2016), arXiv:1506.03811 [hep-ph] .
  * Kawasaki  _et al._ (2021) Masahiro Kawasaki, Hiromasa Nakatsuka, Kazunori Nakayama, and Toyokazu Sekiguchi, “Revisiting CMB constraints on dark matter annihilation,” (2021), arXiv:2105.08334 [astro-ph.CO] .
  * Argüelles  _et al._ (2019) Carlos A. Argüelles, Alejandro Diaz, Ali Kheirandish, Andrés Olivares-Del-Campo, Ibrahim Safa, and Aaron C. Vincent, “Dark Matter Annihilation to Neutrinos,” (2019), arXiv:1912.09486 [hep-ph] .
  * Feng  _et al._ (2010) Jonathan L. Feng, Manoj Kaplinghat, and Hai-Bo Yu, “Halo Shape and Relic Density Exclusions of Sommerfeld-Enhanced Dark Matter Explanations of Cosmic Ray Excesses,” Phys. Rev. Lett. 104, 151301 (2010), arXiv:0911.0422 [hep-ph] .
  * Randall  _et al._ (2008) Scott W. Randall, Maxim Markevitch, Douglas Clowe, Anthony H. Gonzalez, and Marusa Bradac, “Constraints on the Self-Interaction Cross-Section of Dark Matter from Numerical Simulations of the Merging Galaxy Cluster 1E 0657-56,” Astrophys. J. 679, 1173–1180 (2008), arXiv:0704.0261 [astro-ph] .
  * Banerjee  _et al._ (2021) Heerak Banerjee, Bhaskar Dutta, and Sourov Roy, “Probing $L_{\mu}-L_{\tau}$ models with CE$\nu$NS: A new look at the combined COHERENT CsI and Ar data,” (2021), arXiv:2103.10196 [hep-ph] .