#  GeV Dark Matter in $U(1)_{L_{\mu}-L_{\tau}}$: Role of $(g-2)_{\mu}$ Anomaly

Kuldeep Deka<EMAIL_ADDRESS>Department of Physics and
Astrophysics, University of Delhi, Delhi 110 007, India  Soumya Sadhukhan
<EMAIL_ADDRESS>Ramakrishna Mission Residential College
(Autonomous), Narendrapur, Kolkata 700103, India  Manvinder Pal Singh
<EMAIL_ADDRESS>Department of Physics and Astrophysics, University
of Delhi, Delhi 110 007, India

###### Abstract

In this work, we investigate the connection of the detection possibility of a
light GeV scale fermionic dark matter (DM) with recent anomalous $(g-2)_{\mu}$
measurement with $\sim 4\sigma$ deviation from the SM. The
$U(1)_{L_{\mu}-L_{\tau}}$ scenario readily explains $(g-2)_{\mu}$ anomaly,
allowing only a portion of hitherto allowed parameter space, and restricts the
$Z^{\prime}$ mass in the range of $20-200$ MeV. This constraint impacts the
enhancement of the neutrino floor, the neutrino background in the DM direct
detection experiments. Neutrino floor is enhanced for a lighter $Z^{\prime}$
of the $(g-2)_{\mu}$ allowed spectrum, whereas for a heavier $Z^{\prime}$, the
modification is not significant. For the GeV scale DM, $(g-2)_{\mu}$
restriction results in insignificance of s-channel resonant annihilation in DM
annihilation, therefore resulting in t-channel dependence to produce the
observed DM relic. A t-channel annihilation aided by large couplings can bring
the relic density in the observed range, direct detection cross section of the
GeV DM shoots up. Consequently, super-GeV (with mass $1-10$ GeV) DM gets
almost gets ruled out except for a small parameter region for heavier
$Z^{\prime}$, where as sub-GeV (with mass $0.1-1$ GeV) DM detection
possibility remains bright with more detection possibility for heavier
$Z^{\prime}$. Direct detection constraints when aided by $(g-2)_{\mu}$,
constrains the GeV DM compared to DM indirect detection measurements.

##  I Introduction

One remarkable aspect that motivates us to look for beyond the Standard model
(BSM) signature is the absence of a dark matter candidate inside the SM. The
majority ($\sim 85\%$) of the matter element of the Universe is some inert
non-luminous matter called dark matter (DM), interacting only through
gravitational interaction. Astrophysical observations like galactic rotational
curves and gravitational lensing etc are proof of its presence. Various DM
candidates were incorporated in different beyond the standard model (BSM)
theories like Inert Higgs Doublet, Right handed neutrino and Super-symmetry,
to name a few. But till date, no conclusive observational evidence of of such
a particle is found either in the LHC, specifically designed to probe the TeV
scale physics, or in the DM direct and indirect detection experiments.

Particle candidates for DM are well motivated by the WIMP (weakly interacting
massive particle) miracle, where a DM is expected at the TeV scale with
interaction strength typical to have correct DM relic density. The search for
the DM particles are on through different DM direct detection experiments,
albeit with a renewed vigor directed to find DM particles at a lower mass
scale. The GeV scale DM is particularly interesting from the prospect of their
experimental probe. In this scale DM is too light to be significantly
sensitive to the LHC searches, which in general has played a crucial role in
constraining the WIMPs. Moreover, this GeV mass DM is too heavy to affect any
cosmological considerations. Therefore, in light DM sector at the GeV scale,
constraints were relaxed, compared both to the ultralight DM and the WIMPs. It
is imperative to probe the the GeV regime with innovative probes.

Another measurement that recently gained importance is the anomalous
measurement of the muon magnetic moment. The muon magnetic moment which helps
in its spin-interaction with external magnetic field is predicted in the SM to
have tree level measurement of 2 in the unit of Bohr magnetons. This is called
gyromagnetic ratio ($g=2$) for spin angular momentum. Quantized picture shows
this as a measure of interaction $\mu^{+}\mu^{-}\gamma$, which can take
radiative corrections even inside the SM. All higher order computations are
already performed within the SM. Experimental measurement of $(g-2)$ for
leptons can be a window to probe beyond the SM (BSM) physics. Recently $g-2$
for the muon has been measured very precisely in the Fermilab experiment and
the BSM virtual corrections can well contribute to this observable. The
preferred observable that we work on is defined as $a_{\mu}=\frac{g-2}{2}$.
The Fermilab based experiment measuring muon $(g-2)$ has reported an deviation
from the SM prediction of $\Delta a_{\mu}$ as Abi et al. (2021); Girotti
(2021)

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

, which is equivalent to $4.2\sigma$ deviation away from the SM. This is
already an confirmation of previous similar observation for $(g-2)_{\mu}$ in
Brookhaven experimental observation Bennett et al. (2006). In this work, we
look to probe a GeV scale DM with the help of this anomalous result, that
itself is a motivation to go beyond the SM.

We seek to look into any plausible connection between two approach towards BSM
physics, that on the face of it are not connected to each other. We take
$U(1)_{L_{\mu}-L_{\tau}}$ model, as this model easily accommodates different
type of DM candidates, where the light $Z^{\prime}$ can act as possible
mediator in the DM annihilation. Simultaneously, it can also explain
$(g-2)_{\mu}$ anomalous contribution through its $Z^{\prime}\mu^{+}\mu^{-}$
vertex. We have taken a vectorlike dirac fermion DM candidate ($\chi$) , which
was already explored in the literature. The prospects of this DM candidate is
explored in detail with a specific focus given on the viability of it in the
lighter side of the spectrum, i.e. in the GeV scale. To be precise, the focus
is to explore the DM in the mass range of $0.1-10$ GeV. The reason to choose
the GeV scale is that the most generic GeV-DM is not constrained at all from
different DM search experiments. Another reason GeV scale DM is particularly
interesting, as the DM direct detection in this mass scale is particularly
senisitive to the existing SM background coming from the neutrinos. The
neutrino floor, the neutrino SM background to the DM, gets significantly
modified in the GeV scale DM mass. This interrelation of the neutrino floor
and the GeV scale Dirac DM detection, being modified with the inclusion of the
$(g-2)_{\mu}$ anomalous measurement, ultimately constraining GeV scale DM
phenomenology, are the ensemble of ideas we set to pursue. Even though on the
face of it, the $(g-2)$ and DM direct detection, along with the neutrino floor
in particular, appear as non-correlated aspects of BSM physics, there are some
novel observations that can be made when the DM is a GeV scale Dirac fermion.
These observations in DM phenomenology including the neutrino floor and
intricate connection of DM detection to the anomalous $(g-2)_{\mu}$, within
the $U(1)_{L_{\mu}-L_{\tau}}$ model is the core of our work.

Explanation of anomalous $(g-2)_{\mu}$ in $U(1)_{L_{\mu}-L_{\tau}}$ explored
in detail in the existing literature. Different dark matter phenomenological
implications of the $(g-2)_{\mu}$ observation in this set up were also
discussed already in the literature in Refs. Qi et al. (2021); Hapitas et al.
(2022); Gninenko and Krasnikov (2018); Borah et al. (2021, 2022); Ko et al.
(2021); Holst et al. (2021); Baek (2016); Patra et al. (2017); Foldenauer
(2019); Costa et al. (2022). However, another important DM related phenomenon
of neutrino floor enhancement, explored earlier in another work Sadhukhan and
Singh (2021), was generally overlooked in these papers while correlating the
DM with $(g-2)$. Invocation of neutrino floor in connecting $(g-2)_{\mu}$ with
the DM detection constraints is a novel attempt from our side. One
particularly interesting result shows that the neutrino floor still can get
significant enhancement when $Z^{\prime}$ is in the lower end of the
$(g-2)_{\mu}$ allowed parameter band, while enhancement is reduced due to the
$(g-2)_{\mu}$ constraint when a heavier $Z^{\prime}$ is chosen. This leaves us
with some constrained region to look for the GeV-scale DM candidates through
the direct-detection experiments.

The GeV scale Dirac fermion dark matter phenomenology also gets constrained on
the inclusion of the constraint coming from anomalous $(g-2)_{\mu}$
measurement. The $(g-2)_{\mu}$ constraint restricts the $Z^{\prime}$ mass
below 200 MeV, while putting an upper bound on the BSM $U(1)$ gauge coupling.
These constraints reduces the effectiveness of DM annihilation, putting
stronger constraints to avoid over production of the DM relic density.
Restriction on $Z^{\prime}$ mass bars $Z^{\prime}$ mediated s-channel resonant
DM annihilation to the SM particles. Another annihilation channel $\rm DMDM\to
Z^{\prime}Z^{\prime}$ can have significant annihilation to satisfy DM relic
density constraints which requires larger couplings. Large couplings are
viable when relic density constraints are taken into account, where as it will
invariably lead to large DM-nucleus scattering cross section values. A number
of DM direct detection experiments like Xenon1T, CDMS, DarkSide, CRESST etc
are looking for possible dark matter signals. Any null result in these puts
upper bounds on the DM-nucleus scattering cross section. Here larger direct
detection cross section for the GeV scale DM will either rule them out or will
make them a vaiable candidate to be probed in any future update of the
experiments. Otherwise non-constrained GeV scale Dirac fermion DM case becomes
very predictive in its detection possibility, with the involvement of
deviations of $(g-2)$ of the muons. Within the purview of the
$U(1)_{L_{\mu}-L_{\tau}}$ model, the novelty of this work lies in the study of
the complimentarity between the presence of a GeV scale DM candidate
satisfying muon $g-2$ with its specific dependence on the t channel
annihilation and subsequent modification of the neutrino floor. The hitherto
not so constrained GeV gets more constrained from the uncharacteristic
combination of direct detection with $(g-2)_{\mu}$ constraints compared to
those coming from usual DM indirect detection constraints.

In section II the $U(1)_{L_{\mu}-L_{\tau}}$ model is discussed introducing a
fermionic DM candidate, while already existing constraints are listed. We
explain and exhibit the parameter space where the $(g-2)_{\mu}$ can be
explained in this scenario. The section III is dedicated to the neutrino floor
computation, while pointing out the modification of the neutrino floor while
we take the $(g-2)_{\mu}$ constraint is taken into account. In section IV, we
discuss relic density constraints on the GeV scale Dirac fermion DM, with and
without presence of the constraints coming from $(g-2)_{\mu}$. In the next
section V, we impose further constraints coming from the DM direct detection
experiments along with a lower bound from the neutrino floor to study the
effects of $(g-2)_{\mu}$ constraints on the existing parameter space. Then we
discuss in a very small section VI, the comparison of DM indirect detection
constraints with the constraints discussed before. In final section VII, we
summarize our results and chart possible road ahead.

##  II Muon $g-2$ in $U(1)_{L_{\mu}-L_{\tau}}$

We first discuss the $U(1)_{L_{\mu}-L_{\tau}}$ model briefly outlining the
constraints on this prior to the muon $(g-2)$ anomalous measurement. Then it
is explored how the inclusion of this new constraint can restrict the model
parameter space.

###  $U(1)_{L_{\mu}-L_{\tau}}$ Model: Constraints and DM

In this article we consider the $U(1)_{L_{\mu}-L_{\tau}}$ model to study the
feasibility of simultaneously explaining (g-2)μ, dark matter relic density and
modification in neutrino floor. The $U(1)_{L_{\mu}-L_{\tau}}$ is one of the
three simplest U(1) extension in SM that does not induce any extra gauge
anomalies (Foot, 1991; He et al., 1991a, b; Choudhury et al., 2020). The other
two models being $U(1)_{L_{e}-L_{\mu}}$ and $U(1)_{L_{e}-L_{\tau}}$.

In $U(1)_{L_{\mu}-L_{\tau}}$ model, the new U(1) symmetry can be broken
spontaneously by a new complex scalar S. The symmetry breaking causes the
$U(1)$ associated vector field $Z^{\prime}$ to acquire mass through its
interaction with S. The new Lagrangian terms are given by,

$\displaystyle L_{new}=$
$\displaystyle-\frac{1}{4}Z^{\prime\mu\nu}Z^{\prime}_{\mu\nu}+\sum_{l}\bar{l}\gamma^{\mu}\left(-g_{\mu-\tau}\,Y^{\prime}_{l}\,Z^{\prime}_{\mu}\right)l+\left(D_{\mu}S\right)^{\dagger}\left(D^{\mu}S\right)$
(1)
$\displaystyle+\mu^{2}_{S}S^{\dagger}S+\lambda_{S}\left(S^{\dagger}S\right)^{2}+\lambda_{SH}\left(S^{\dagger}S\right)H^{\dagger}H$

here $\mu^{2}_{S}$ and $\lambda_{S}$ are bilinear and quartic self
interactions for S respectively. S couples with the SM Higgs $H$ via quartic
coupling $\lambda_{SH}$. The $Z^{\prime}$ boson interacts with fermions by
total coupling
$g_{\mu-\tau}Y^{\prime}_{l}=g_{\mu-\tau}\left(L_{\mu}-L_{\tau}\right)$.

In this model $Z^{\prime}$ does not have tree-level interactions with quarks
and electrons. The interactions are mediated by $Z/\gamma-Z^{\prime}$ loop
given by the diagram in Fig. 1.

Figure 1: Lepton loop through which $Z^{\prime}-Z/\gamma$ mixing is induced in
$U(1)_{L_{\mu}-L_{\tau}}$ model.

$\displaystyle\delta^{\mu\nu}_{ij}=\frac{1}{(2\pi^{2})}[-l^{\mu}l^{\nu}+g^{\mu\nu}l^{2}]\int^{1}_{0}dx\,(\log\frac{x(x-1)l^{2}+m^{2}_{l_{i}}}{x(x-1)l^{2}+m^{2}_{l_{j}}})x(1-x)$
(2)

where, $l$ is momentum transfer, $m_{l_{i(j)}}$ is mass of i(j)th flavour
lepton in the loop.

The model can incorporate a vectorlike fermion (VLF), $\chi$, without
introducing an extra gauge anomaly. The DM Lagrangian can take the form:

$\displaystyle L_{DM}=$
$\displaystyle\bar{\chi}\left(i\gamma^{\mu}D_{\mu}-m\right)\chi.$

The $Z^{\prime}$ also acquires interaction with the DM candidate $\chi$
through the covariant derivative
$D_{\mu}\chi=\left(\partial_{\mu}-iq_{\chi}g_{\mu-\tau}\right)\chi$ present in
the DM kinetic term. The interactions between $Z^{\prime}$ and fermions, dark
or otherwise are highlighted by the interaction terms.

$\displaystyle
L_{i}-L_{f}=-g_{\mu-\tau}(q_{\chi}\bar{\chi}\gamma^{\mu}\chi+\bar{l}_{i}\gamma^{\mu}l_{i}-\bar{l}_{j}\gamma^{\mu}l_{j}+\bar{\nu}_{i}\gamma^{\mu}L\nu_{i}-\bar{\nu}_{j}\gamma^{\mu}L\nu_{j})Z^{\prime}_{\mu}\;.$
(3)

Here $q_{\chi}\times g_{\mu-\tau}$ is the total gauge charge of VLF under
U(1)${}_{L_{\mu}-L_{\tau}}$. As $Z^{\prime}$ interacts with both DM candidate
$\chi$ and leptons: $\mu$ and $\tau$, it effectively acts as a portal between
dark matter and SM particles. Therefore DM candidate, $\chi$ contributes to
dark matter relic density through s-channel annihilation process
$\bar{\chi}\,\chi\rightarrow
Z^{\prime}\rightarrow\bar{l}\,\,l\,(\bar{\nu}_{l}\,\,\nu)$ and t-channel
process $\bar{\chi}\,\chi\rightarrow Z^{\prime}\,\,Z^{\prime}$. For
$m_{\chi}\leq m_{Z^{\prime}}$ s-channel is the dominant contribution to DM
relic density whereas for $m_{\chi}\geq m_{Z^{\prime}}$ t-channel remains the
dominant contribution. The annihilation can be detected through final state
radiation (FSR) in form of photon flux in Fermi-LAT experiment et.allá (2017).
Also $\mu$ and $\tau$ leptons along with their anti particles in final states
can further decay to electron and positron before reaching Earth. Electron and
positron excess has been measured at DAMPE Chang et al. (2017) and AMS02
Aguilar (2013) experiments respectively. We shall refrain from discussing
indirect detection of dark matter in detail. We shall discuss the direct
detection of dark matter through a recoil of target nucleus in detail in the
following sections.

We refer to Ref. (Bauer et al., 2018; Chun et al., 2019) for detailed
understanding of $U(1)_{L_{\mu}-L_{\tau}}$ model constraints. In the electron
beam dump experiments (Riordan et al., 1987; Bjorken et al., 1988; Bross et
al., 1991) and the proton beam dump experiments (et al, 1986; Athanassopoulos
et al., 1998; Blumlein and Brunner, 2011), an accelerated beam of electrons
and protons respectively falls on a target material. Secluded photons can be
produced with electron and proton as final state radiations and through the
decay of mesons which are produced when energetic proton hit the target
material. As the $Z^{\prime}$ interactions with electrons and quarks are loop
suppressed in this model, electron and proton beam dump constraints are
weaker. TEXONO (Deniz et al., 2010) experiment measures elastic scattering of
$\bar{\nu}_{e}$ produced at Kuo-sheng Nuclear power reactor with CsI(Tl)
crystal array target. As the incident neutrino is $\bar{\nu}_{e}$, TEXONO
experimental constraints are diminished for the model. Experiments like CCFR
(Mishra et al., 1991), Charm-II (Vilain et al., 1994) provide constraints for
the model derived from the neutrino trident process
$\nu_{\mu}Z\to\nu_{\mu}\mu^{+}\mu^{-}$ . Borexino (Bellini et al., 2011)
measures the scattering of solar neutrinos with a liquid scintillator.
Borexino and Charm-II provide us the most stringent constraints in the region
of interest, i.e. for m${}_{Z^{\prime}}$ range 1-200 MeV. Currently COHERENT
experiment CE$\nu$NS measurements are preliminary and constraints remain
comparatively weaker. In future, with increased exposure COHERENT experiment
will be able to probe currently available parameter space for $(g-2)_{\mu}$.
In addition, constraints derived from astrophysical observations and meson
decays for an ultra light $Z^{\prime}$ (m${}_{Z^{\prime}}\leq 1$ eV), have
been studied in Ref. Dror (2020).

### Muon $g-2$

Figure 2: BSM contribution to the muon (g-2) through a $Z^{\prime}$ mediated
loop diagram.

In $U(1)_{L_{\mu}-L_{\tau}}$ model, we get a contribution to muon-($g-2$) at
one loop level due to the additional $Z^{\prime}$ boson mediated diagram,
which is given by:

$\Delta
a_{\mu}=\frac{g_{\mu-\tau}^{2}}{8\pi^{2}}\int_{0}^{1}dx\frac{2m_{\mu}^{2}x^{2}(1-x)}{x^{2}m_{\mu}^{2}+(1-x)m_{Z^{\prime}}^{2}}.$
(4)

The Brookhaven National laboratory(BNL) E821 experiment had earlier shown a
$\Delta a_{\mu}^{\rm BNL}$ = $279(76)\times 10^{-11}$ Bennett et al. (2006),
thus resulting in a $3.7\sigma$ deviation of from the SM value. Recently
Fermilab National Laboratory(FNAL) has confirmed the BNL results by reporting
a deviation of $\Delta a_{\mu}^{\rm FNAL}$ = $230(69)\times 10^{-11}$. The two
results has thus put a combined discrepancy of 4.2$\sigma$ from the SM with
$\Delta a_{\mu}^{\rm FNAL+BNL}$ = $251(59)\times 10^{-11}$ Abi et al. (2021);
Girotti (2021). This result increases the prospects for a light $Z^{\prime}$
boson in the masses of around 10-100 MeVs. The left plot of Figure 3 shows the
allowed $\Delta a_{\mu}$ band as a function of the $m_{Z^{\prime}}$ mass and
the coupling $g_{\mu-\tau}$ utilizing this combined BNL-FNAL result. In
calculating this $\Delta a_{\mu}$, the Standard Model value of $g-2$ is taken
from the data driven method utilizing perturbative QCD Aoyama et al. (2020).
This particular result, although opens prospects for new physics, however has
to be taken with a grain of salt because of another SM muon g-2 calculation by
Budapest-Marseille-Wuppertal (BMW) collaboration Borsanyi et al. (2021)
utilizing lattice QCD results, providing us with the value, $\Delta
a_{\mu}^{\rm LO-HVP}(\rm lattice)$ = $107(69)\times 10^{-11}$. If this result
is indeed true, the discrepancy of the the combined BNL-FNAL result with the
SM comes within $2\sigma$. This reduces the prospects of looking for new
physics models.

Figure 3: Allowed parameter space of $\Delta a_{\mu}$ in the
$g_{\mu-\tau}$-$M_{Z^{\prime}}$ plane, first including the latest data driven
approach for the SM value of $(g-2)_{\mu}$ (left) and then considering the
latest SM value of $(g-2)_{\mu}$ given by lattice results (right).

For the sake of completeness, we have shown allowed value of parameter space
for such a scenario in the right plot of Figure 3. The $(g-2)_{\mu}$
constraint restricts the $Z^{\prime}$ mass to a small range of $20-200$ MeV
while for higher masses the region explaining the anomaly is already ruled out
from other experiments. When we include the lattice QCD computation of SM
$(g-2)$ values, the deviation from the SM are smaller with larger uncertainty
which prefers smaller $g_{\mu-\tau}$ values along with larger
$m_{Z^{\prime}}$, broadening and lowering in coupling the allowed parameter
range overall.

##  III Neutrino Floor Modification: $(g-2)_{\mu}$ Effects

As we discussed previously, in $U(1)_{L_{\mu}-L_{\tau}}$ model, the additional
$Z^{\prime}$ boson interacts with $\nu_{\mu}$ and $\nu_{\tau}$. Therefore non
standard coherent elastic neutrino nucleus scattering (CE$\nu$NS) can be
induced in the model through $Z^{\prime}-\gamma$ mixing.

### Differential scattering cross-section

First, we look at the case where astrophysical neutrinos are hitting the DM
search detector material causinng a nuclear recoil. Total neutrino-nucleus
differential scattering cross-section in $U(1)_{\mu-\tau}$ is given by,

$\displaystyle\frac{d\sigma_{\mu-\tau}}{dE_{r}}=\frac{d\sigma_{SM}}{dE_{r}}-\frac{{m_{N}}\,G_{f}\,Q_{\nu
N{\mu-\tau}}Q_{\nu
N}\left(1-\frac{{E_{r}}{m_{N}}}{2{E_{\nu}}^{2}}\right)F^{2}(E_{r})}{\sqrt{2}\pi\left(2{E_{r}}{m_{N}}+{m_{Z^{\prime}}}^{2}\right)}+\frac{{m_{N}}\,Q_{\nu
N{\mu-\tau}}^{2}\left(1-\frac{{E_{r}}{m_{N}}}{2{E_{\nu}}^{2}}\right)F^{2}(E_{r})}{2\pi\left(2{E_{r}}{m_{N}}+{m_{Z^{\prime}}}^{2}\right)^{2}},$
(5)

where $\frac{d\sigma_{SM}}{dE_{r}}$ is the contribution from SM, given as,

$\displaystyle\frac{d\sigma_{SM}}{dE_{r}}=G_{f}^{2}\frac{m_{N}}{4\pi}Q^{2}_{\nu
N}\left(1-\frac{E_{r}m_{N}}{2{E_{\nu}}^{2}}\right)F^{2}(E_{r}).$ (6)

Here $G_{f}$ is the Fermi constant, $Q_{\nu N}=N-(1-4\sin^{2}\theta_{w})Z$ is
effective weak hyper-charge in the SM for the target nucleus with $N$ neutrons
and $Z$ protons and $F(E_{r})$ is the Helm form factor (Ref. Lewin and Smith
(1996)). The effective coupling between neutrino and nucleus for
$U(1)_{\mu-\tau}$ model can be written as,

$\displaystyle Q_{\nu
N{\mu-\tau}}=g_{\mu-\tau}^{2}\frac{2\,\alpha_{EM}}{\pi}\delta_{\mu\tau}Z$ (7)

where $g_{\mu-\tau}$ is coupling given in Eq. 3, $\alpha_{EM}$ is the fine
structure constant and $\delta_{\mu\tau}$ is the scalar part of loop factor
given in equation 3.

### Neutrino and Dark Matter Rate equation

Using the differential cross-section of the DM/neutrino scattering and
augmenting that with the source and detector effects, the scattering rates can
be obtained. CE$\nu$NS rate equation is given by,

$\displaystyle\frac{dR_{\nu-N}}{dE_{r}}=\frac{\epsilon}{m_{N}}\int_{E^{min}_{\nu}}\mathcal{A}(E_{r})\left.\frac{d\phi_{\nu}}{dE_{\nu}}\right|_{\nu_{\alpha}}P(\nu_{\alpha}\rightarrow\nu_{\beta},E_{\nu})\frac{d\sigma(E_{\nu},E_{r},\nu_{\beta})}{dE_{r}}dE_{\nu}$
(8)

here $\epsilon$ is the exposure of the experiment measured in units of mass
$\times$ time, $\mathcal{A}(E_{r})$ is the detector efficiency and is set to
one in following calculations. $E^{min}_{\nu}$ is minimum incident neutrino
energy required to produce a detectable recoil for a material nucleus of mass
$m_{N}$ with energy $E_{r}$, which in the limit of $m_{N}>>E_{\nu}$ can be
written as,

$\displaystyle E^{\rm min}_{\nu}=\sqrt{\frac{m_{N}E_{r}}{2}}.$ (9)

Here, $\frac{d\sigma(E_{\nu},E_{r},\nu_{\beta})}{dE_{r}}$ is $\beta$ flavor
dependent neutrino-nucleus differential scattering cross-section and
$\left.\frac{d\phi_{\nu}}{dE_{\nu}}\right|_{\nu_{\alpha}}$ is the incoming
neutrino flux of flavor $\alpha$. The fluxes used in this analysis involve
fluxes from solar, atmospheric, diffuse supernova neutrinos, which can be
found in Refs. Strigari (2009); Billard et al. (2014).

#### Dark matter direct detection rate equation

For the DM, the spin-independent DM-nucleon scattering rate equation is given
by,

$\frac{dR_{DM-N}}{dE_{r}}=\epsilon\,\frac{\rho_{DM}\sigma^{0}_{n}A^{2}}{2m_{DM}\mu^{2}_{n}}F^{2}(E_{r})\int_{v_{min}}\frac{f(v)}{v}d^{3}v$
(10)

Here $\epsilon$ is the exposure of the detector given in units of MT
(mass$\times$time), $m_{DM}$ is the DM mass $\mu_{n}$ is DM-nucleon reduced
mass, $A$ is the mass number of target nuclei, $\sigma^{0}_{n}$ is the DM-
nucleon scattering cross-section at zero momentum transfer. $F(E_{r})$ is the
Helmholtz form factor. $\rho_{DM}$ is the local density of the DM in the
immediate vicinity of the Earth.

$\displaystyle\int_{v_{min}}\frac{f(v)}{v}d^{3}v=\frac{1}{2v_{0}\eta_{E}}\left[erf(\eta_{+})-erf(\eta_{-})\right]-\frac{1}{\pi
v_{0}\eta_{E}}\left(\eta_{+}-\eta_{-}\right)e^{\eta^{2}_{esc}}$ (11)

Here $\eta_{E}=\frac{v_{E}}{v_{0}}$,$\eta_{esc}=\frac{v_{esc}}{v_{0}}$ and
$\eta_{\pm}=min\left(\frac{v_{min}}{v_{0}}\pm\eta_{E},\frac{v_{esc}}{v_{0}}\right)$,
where $v_{0}$ is local galactic rotational velocity, $v_{E}$ velocity of Earth
with respect to galactic center, $v_{esc}$ escape velocity of DM from galaxy.
We have used values $v_{0}=220$km/s, $v_{E}=$232 km/s and $v_{esc}=$ 544 km/s
in above calculations.

### Neutrino Floor

Neutrino floor is defined as the value of DM-nucleon scattering cross-section
at which ratio of no. of scattering events generated by DM-nucleon scattering
to that of events generated by neutrino background is 2.3 is to 1 (at 90% CL).
This establishes a borderline below which contribution of neutrino background
increases and certainty that the observed events if any are from DM-nucleon
scattering decreases. It is calculated by measuring the exposure required to
observe a neutrino scattering event and then for the same exposure calculation
the value of DM-nucleon scattering cross-section for which we get 2.3
scattering events (For more details see Ref. Sadhukhan and Singh (2021) ).

$\displaystyle\int^{E_{DM}^{max}}_{E_{th}}\frac{dR_{DM-N}}{dE_{r}}dE_{r}=\frac{2.3}{1}\int^{E_{r}^{max}}_{E_{th}}\frac{dR}{dE_{r}}dE_{r},$

(a) $m_{Z^{\prime}}=16$ MeV, $g_{\mu-\tau}=6\times 10^{-4}$, with
$(g-2)_{\mu}$

(b) $m_{Z^{\prime}}=25$ MeV, $g_{\mu-\tau}=6.7\times 10^{-4}$. Enhancement is
shown after the inclusion of $(g-2)_{\mu}$ constraint.

(c) $m_{Z^{\prime}}=25$ MeV, $g_{\mu-\tau}=9.3\times 10^{-4}$. This is the
point reporting maximum enhancement of the neutrino floor in the absence of
$(g-2)_{\mu}$ constraint, for a Ge based DM detector.

(d) $m_{Z^{\prime}}=25$ MeV, $g_{\mu-\tau}=9.3\times 10^{-4}$. Maximum
enhancement of the neutrino floor in the absence of $(g-2)_{\mu}$ constraint
is shown, for a Xe based DM detector.

Figure 4: Neutrino floor projected in the $\sigma^{0}_{n}$ vs $m_{DM}$ plane.
Comparison of the neutrino floor for the SM (presented by the blue line) and
that for $U(1)_{\mu-\tau}$ (presented by the red line). For different detector
materials, Ge68 and Xe131 .

This can be reiterated in form of the master equation,

$\displaystyle\sigma^{0}_{n}=$
$\displaystyle\frac{2.3}{1}\left(\int^{E_{max}}_{E_{th}}\frac{1}{m_{N}}\int_{E^{min}_{\nu}}\left.\frac{d\phi_{\nu}}{dE_{\nu}}\right|_{\nu_{\alpha}}P(\nu_{\alpha}\rightarrow\nu_{\beta},E_{\nu})\frac{d\sigma(E_{\nu},E_{r},\nu_{\beta})}{dE_{r}}dE_{\nu}\right)$
(12)
$\displaystyle\times\left(\frac{\rho_{DM}A^{2}}{2m_{DM}\mu^{2}_{n}}\int^{E^{max}_{DM}}_{E_{th}}F^{2}(E_{r})\int_{v_{min}}\frac{f(v)}{v}d^{3}v\right)^{-1}$

Here $E^{max}_{DM}$ is the maximum recoil energy of DM with mass $m_{DM}$ can
produce in a given nucleus. It is written as,

$2\,m_{DM}\left(\frac{m_{N}m_{DM}}{(m_{N}+m_{DM})}\right)v_{esc}^{2}$.

Using the master equation 12, we evaluate the modified neutrino floor for the
allowed region in Fig. 3. It is noticed that enhancement in neutrino floor
increases with increasing value of $m_{Z^{\prime}}$ till 25 MeV for
corresponding maximum value of allowed $g_{\mu-\tau}$ and starts to decrease
with larger masses and becomes negligible around and after $m_{Z^{\prime}}$=
200 MeV. This can be understood from the profile of excluded region seen in
Fig. 3 which shows a sharp kink around $m_{Z^{\prime}}$ = 25 MeV. Also the
beyond standard model contribution to CE$\nu$NS is proportional to
$\frac{g^{2}_{\mu-\tau}}{2m_{N}E_{r}+m^{2}_{Z^{\prime}}}$, therefore neutrino
floor enhancement decreases with larger $m_{Z^{\prime}}$ value and lower
$g_{\mu-\tau}$. We choose the benchmark points
$m_{Z^{\prime}}=\\{16,25,50,200\\}$ MeV to discuss the enhancement of the
neutrino floor at different DM masses in tabulated form in the Appendix .1. As
the chosen region is also important from $(g-2)_{\mu}$ perspective, we also
tabulate neutrino floor enhancement for points with $(g-2)_{\mu}$ allowed
couplings for same benchmarks.

In Fig. 4 we show graphs showing modification of theoretically estimated
neutrino floor on the $\sigma^{0}_{n}-m_{DM}$ plane. Solid lines exhibit the
boundary of DM-nucleon scattering cross-section above which we can be certain
(90% C.L.) that ratio of DM-scattering events to that of neutrino-scattering
events is greater than 2.3:1. Blue contours signify neutrino floor estimated
from Standard Model interactions whereas red contour show neutrino floor for
$U(1)_{\mu-\tau}$ model. It can be noticed that modification in neutrino floor
is most significant for DM mass below 7 Gev and diminishes to a barely
noticeable change for higher masses. This can be understood as below 7 Gev DM
mass $\sigma_{n}^{0}$ is sensitive to threshold recoil energies below 1 kev
and less. Below 1 keV threshold energy CE$\nu$NS rate for U(1)μ-τ show
modification of the same order as seen in neutrino floor below 7 GeV DM mass
and for threshold energy greater than 1 keV CE$\nu$NS rate modification in
U(1)μ-τ rapidly diminishes (For detail see Ref. Sadhukhan and Singh (2021)).
Top left panel 4(a) shows modification in neutrino floor for the benchmark
point $m_{Z^{\prime}}=16$ MeV, $g_{\mu-\tau}=6\times 10^{-4}$ and top right
panel 4(b) shows modification in neutrino floor for the benchmark point
$m_{Z^{\prime}}=25$ MeV, $g_{\mu-\tau}=6.7\times 10^{-4}$. These benchmark
points are also within viable parameter space to explain BNL-FNAL
$(g-2)_{\mu}$ anomaly for U(1)μ-τ model. Modification of around 2 times and
1.6 is seen below DM mass 7 GeV for Germanium and Xenon based detectors
respectively. The bottom panels Fig. 4(c) (left) and and Fig. 4(d) (right)
show neutrino floor contours for the benchmark points $m_{Z^{\prime}}=25$ MeV
and $g_{\mu-\tau}=9.3\times 10^{-4}$ for Germanium and Xenon based experiments
respectively. This point corresponds to the maximum enhancement seen in the
allowed parameter space. Modification by a factor around 2.2 and 1.6 times are
observed respectively in neutrino floor below the DM mass around 7 GeV.

##  IV GeV scale Dark Matter: Role of $(g-2)_{\mu}$ Anomaly

Figure 5: The annihilation diagrams of DM candidates: The $Z^{\prime}$
mediated s-channel annihilation $\chi\chi\to\rm SM\,SM$ (left) and
$Z^{\prime}$ mediated t-channel annihilation $\chi\chi\to
Z^{\prime}Z^{\prime}$ (right)

We have argued before that the 100 MeV to 10 GeV mass scale is a very
favorable parameter space that survives and there we can explore possibility
of a light dark matter (DM) candidate. Here we work with a light Dirac fermion
as a DM candidate. At mass scales around 100 MeV, DM is sufficiently heavy to
contribute anything to cosmological computations. As with $m_{DM}\sim
100~{}$MeV, it cannot remain in thermal equilibrium at the energy scale of Big
bang Nucleosynthesis (BBN), contributing minimally to effective degrees of
freedom as was indicated in Ref. Drees and Zhao (2022). On the other end of
the spectrum, 10 GeV dark matter is light enough to not get significantly
constrained from the ongoing LHC observations, possibly in the jet/lepton plus
missing energy search channels.

In the thermal DM regime, weakly interacting massive particle (WIMP) search
sensitivity rules out a large part of TeV scale DM masses. The constraints on
the GeV scale DM are weaker, compared to those on an ultralight DM and the
WIMP paradigm. Here we try to probe this DM territory with the current
$(g-2)_{\mu}$ measurement in our arsenal, that shows a anomalous deviation
($\sim 4.2\sigma$)from the SM predicted values.

Figure 6: Cross-section ($GeV^{-2}$) for the t-channel and s-channel
annihilation diagrams

### Relic Density Constraints: Role of $(g-2)_{\mu}$ Anomaly

The Dirac fermion DM ($\chi$) introduced here has coupling with
$U(1)_{\mu-\tau}$ gauge fermion through an interaction term in the Lagrangian
like

$g_{\mu-\tau}q_{\chi}\bar{\chi}\gamma^{\mu}Z^{\prime}_{\mu}\chi$

, where we take the effective $Z^{\prime}-\chi$ coupling as
$g_{\chi}=g_{\mu-\tau}q_{\chi}$. The DM annihilation in this scenario happens
through three major channels, namely $\chi\chi\to\ell\ell$ and
$\chi\chi\to\nu\nu$, both through s-channel $Z^{\prime}$ mediated annihilation
and a t-channel $Z^{\prime}$ mediated annihilation process $\chi\chi\to
Z^{\prime}Z^{\prime}$, while $Z^{\prime}$ can decay to the SM muons and tau
leptons. The annihilation processes are depicted in Fig. 5. The s-channel and
t-channel annihilation cross sections are computed following the Re.
Altmannshofer et al. (2016). The s-channel diagram contributions to the DM
annihilation:

$\displaystyle\langle\sigma
v\rangle(\chi\bar{\chi}\to\ell\bar{\ell})\approx\frac{q_{\chi}^{2}Y^{\prime
2}_{\ell}g_{\mu-\tau}^{4}}{2\pi}\sqrt{1-\frac{m_{\ell}^{2}}{m_{\chi}^{2}}}\frac{2m_{\chi}^{2}+m_{\ell}^{2}}{(4m_{\chi}^{2}-m^{2}_{Z^{\prime}})},$
$\displaystyle\langle\sigma
v\rangle(\chi\bar{\chi}\to\nu\bar{\nu})\approx\frac{q_{\chi}^{2}Y^{\prime
2}_{\ell}g_{\mu-\tau}^{4}}{2\pi}\frac{m_{\chi}^{2}}{(4m_{\chi}^{2}-m^{2}_{Z^{\prime}})},$
(13)

where $Y^{\prime}_{\ell}=1\rm-1$ for the muons and tau leptons respectively.
The t-channel contribution is computed as

$\displaystyle\langle\sigma v\rangle(\chi\bar{\chi}\to
Z^{\prime}Z^{\prime})\approx\frac{q_{\chi}^{4}g_{\mu-\tau}^{4}}{16\pi
m_{\chi^{2}}}\left(1-\frac{m_{Z^{\prime}}^{2}}{m_{\chi}^{2}}\right)^{3/2}\left(1-\frac{m_{Z^{\prime}}^{2}}{2m_{\chi}^{2}}\right)^{-2},$
(14)

providing total thermally averaged annihilation cross section as
$\langle\sigma v\rangle=\langle\sigma
v\rangle(\chi\bar{\chi}\to\ell\bar{\ell})+\langle\sigma
v\rangle(\chi\bar{\chi}\to\nu\bar{\nu})+\langle\sigma
v\rangle(\chi\bar{\chi}\to Z^{\prime}Z^{\prime})$.

Initially when the Universe was hot, DM particles were in thermal equilibrium
with other SM particles inside the thermal plasma through $2\to 2$
annihilation processes. As the Universe cools down as it expands, the DM
annihilation does not remain efficient enough to keep them in thermal
equilibrium, where two DM particles cannot come together for an annihilation.
This is called the thermal freeze out. After the freeze out of the dark matter
out of the thermal equilibrium, the remnant density of the dark matter that
remains till today, the DM relic density is given as,

$\displaystyle\Omega h^{2}\approx\frac{1.04\times 10^{9}x_{F}\rm
GeV^{-1}}{\sqrt{g}M_{pl}\langle\sigma v\rangle},$ (15)

where $x_{F}$ dictates the time or temperature of the thermal freeze out given
as $x_{F}=\frac{m_{\chi}}{T_{F}}$, with $T_{F}$ as the freeze out temperature.
For the freeze out point fixation and DM relic density computation we closely
follow the approximate method we had outlined in Ref. Borah et al. (2017). For
GeV scale Dirac fermionic DM, the freeze out temperature is specified by
$x_{F}\sim(17-20)$, and this is obtained solving the Boltzmann equation
governing the thermal evolution of the DM.

The s-channel annihilation processes contribute significantly to the overall
cross section only in the region of resonance i.e. $m_{Z^{\prime}}\approx
2m_{\chi}$. therefore, the MeV scale DM of masses $5-100$ MeV can only
annihilate through an s-channel resonance. The $5-100$ MeV range DM is
plausible through s-channel resonant annihilation even after $(g-2)_{\mu}$,
but still there are not many points for moderate $q_{\chi}$. For the MeV scale
DM, when we increase our $q_{\chi}$ values which will be reflected in
increasing $g_{\chi}$ values, the t-channel contribution to the total
annihilation increases continually reducing dependence on s-channel
annihilation. Without the $(g-2)_{\mu}$ constraint, $Z^{\prime}$ is allowed
for a very light mass range, but $g_{\mu-\tau}$ is strongly constrained from
existing neutrino experiments like Borexino, Charm-II, flavor physics
experiments like BABAR etc. All these renders s-channel not that useful from
the DM perspective.

In this work, we are interested in a light DM at the mass range of $0.1-10$
GeV, which can annihilate resonantly if there is lighter $Z^{\prime}$ to
assist, through the resonant process. However if we impose the muon $(g-2)$
constraint, we are forced to stay in the $Z^{\prime}$ mass range of 10-200
MeV. This implies that we are almost always away from the resonance peak for
s-channel annihilation to be effective. Hence in order to satisfy $(g-2)$ of
the muon, its always the t-channel which plays the major role in DM
annihilation to provide us the right relic density. Comparison of s and t
channel contribution is shown in Fig. 6 for two regions of our interest: I.
sub-GeV dark matter ($m_{\chi}=0.1-1~{}$GeV) II. super-GeV dark matter
($m_{\chi}=1-10~{}$GeV). from the top two plots of Fig. 6, where we plot the t
and s channel contribution to the annihilation cross section as a function of
DM mass by fixing $M_{Z}^{\prime}$ at 25 MeV, $g_{\mu-\tau}$ at 0.00067 with
$q_{\chi}$ of 10 for $m_{\chi}$ in 0.1-1 GeV and $q_{\chi}$ of 20 for
$m_{\chi}$ in 1-10 GeV. With only the t-channel contribution being important
after imposition of $(g-2)_{\mu}$ constraint, then annihilation decreases with
increasing DM mass, pushing the couplings to larger values, which justfies the
higher $q_{\chi}$ being taken for higher DM mass. As shown in the previous
section, this is also incidentally the region where the enhancement of
neutrino floor is significant while simultaneously satisfying the muon $(g-2)$
constraint. The choices of $q_{\chi}$ are dictated by the fact that for a
thermal relic, we require a $\langle\sigma v\rangle$ around the range of
$10^{-10}-10^{-9}~{}GeV^{-2}$ and these values of $q_{\chi}$ helps us achieve
that. As seen from the plot, the s-channel cross section is almost negligible
compared to the t-channel one and as we move towards higher DM mass, the
annihilation cross-section decreases. The annihilation cross sections
increases slighly (both s and t) as we move towards a higher $Z^{\prime}$ mass
because the allowed $g_{\mu-\tau}$ from the muon $(g-2)$ band increases
thereby resulting in higher cross-sections. This can be seen from the bottom
two plots of Fig-6 where the benchmark is chosen to be $M_{Z}^{\prime}$ of 190
MeV and $g_{\mu-\tau}$ of 0.0012 keeping the $q_{\chi}$ values similar to the
top plots. Even though the neutrino-floor is not enhanced in this part of
parameter space, this region has an impact in the direct-detection
experiments, which we will see in the next section. One important point to
note here is that earlier studies have only focussed on the resonant region
where s-channel annihilation is dominant, but here we show that t-channel
plays a very important role in sub-GeV DM annihilation when combined with the
muon $(g-2)$ constraint. The price to pay is the higher value of $q_{\chi}$
compared to the charges for muon or tau, but from a bottom-up phenomenological
point of view, such values are still acceptable.

##  V DM direct detection vs Neutrino Floor: $(g-2)_{\mu}$ Effects

The model in contention being leptophilic, does not have any direct coupling
between the DM particle and the quarks. Hence any scattering off the nuclei of
the form $\chi N\rightarrow\chi N$ can only be rendered through a loop of
charged leptons, where photons emitted by virtual leptons couple to the charge
of the nucleons inside a nucleus as shown in Fig 7.

Figure 7: DM direct detection process through DM-N scattering that can only
happen in this theory through a loop diagram.

The expression for spin-independent cross section for the D-Nucleon scattering
can be written as:

$\sigma_{SI}=\frac{\mu_{N}^{2}}{9\pi
A^{2}}\Big{(}\frac{\alpha_{EM}Zg_{\mu-\tau}^{2}q_{\chi}}{\pi
m_{Z^{\prime}}^{2}}\log\frac{m_{\mu}^{2}}{m_{\tau}^{2}}\Big{)}^{2}$ (16)

where $\mu_{N}$ is the reduced mass of the nucleon-DM system given by
$m_{N}m_{\chi}/(m_{N}+m_{\chi})$, with $m_{N}$ being the nuclear mass, $A$ and
$Z$ being mass number and atomic number respectively. We stick to $Z=54$ and
$m_{N}=129$ GeV consistent with Xenon (Xe) based experiments. As observed
above, the enhancement of the neutrino floor is subdued due to anomalous
$(g-2)_{\mu}$ measurement, even more so when the detector is Xenon material
based. Here, we stick to the SM neutrino floor for the DM direct detection
analysis, as our conclusion is not critically dependent on the neutrino floor
enhancement. The region of primary interest lies in DM with masses ranging
from 0.1-10 GeV, called GeV scale DM. As shown by the purple-shaded region of
the left plot of Fig 10, the primary constraint for sub-GeV DM is provided by
the CRESST Abdelhameed et al. (2019) experiment. For DM in the range 1-10 GeV,
the constraints become severe as we go towards higher mass, with the most
constraining limits coming from Darkside-50 Agnes et al. (2018), XENON1T-2018
Aprile et al. (2018), XENON1T-2019 Aprile et al. (2019) along with SuperCDMS
Agnese et al. (2019), as shown by the purple-shaded region of the right plot
of Fig 10. The neutrino floor is depicted by the grey shaded region in both
the plots, with it becoming less constraining as we move towards high DM
masses. For the points that go below the neutrino floor, the parameter space
associated to them cannot be probed in these set of experiments, where as they
are not necessarily ruled out. Concentrating on the sub-GeV scenario, here we
are more likely to get a parameter space which satisfies the relic density
constraints and the constraints coming from $(g-2)_{\mu}$.

Concentrating on the sub-GeV scenario, we take two benchmark points on the
muon $(g-2)$ band which will sum up the overall behavior of the parameter
space. As already mentioned, we can satisfy the muon $(g-2)$ only in the
$Z^{\prime}$ mass range of $10-200$ MeV. One benchmark point corresponds to
the leftmost part of the $(g-2)_{\mu}$ allowed parameter band corresponding to
low $Z^{\prime}$ mass and the other benchmark point corresponds to the
rightmost part of the band corresponding to a comparatively heavier
$Z^{\prime}$. We vary the value of $q_{\chi}$ between 0.1-50, as different
values satisfy the relic density for different DM masses. We have already seen
in the annihilation cross-section plots that $s$ channel doesn’t have a
significant role to play in our case and it is only the $t$ channel which
dictates the DM behavior. Keeping these in mind, we now proceed to elaborate
on the two benchmark scenarios:

###  V.1 Scenario 1: low Z’ mass

Figure 8: (left) $\sigma_{SI}$ cross-section for the DM of mass 0.1 to 1 GeV,
with parameters as mentioned in the text. (right) Same DM-N cross section for
relatively higher DM masses from 1 to 10 GeV.

This scenario corresponds to a $M_{Z}^{\prime}$ mass of 25 MeV and of gauge
coupling $g_{\mu-\tau}$ of 0.00067. With these values, we have a significant
enhancement of the neutrino floor which has already been discussed in detail
in Section-III. The neutrino floor including its possible enhancement in this
scenario is represented by the cyan colored region in Figure 8. The green
points corresponds to those parameter points which can satisfy at least 10
$\%$ of the relic density whereas the blue points correspond to those
satisfying the entire relic density. As can be seen from the plots, the direct
detection measurements combined with the muon $(g-2)$ constraint rules out the
parameter points which satisfies the entire relic density in the 0.1-1 GeV DM
mass range. However, we do find parameter points which can satisfy at least
10$\%$ of the relic density. This makes an interesting proposition to look for
scenarios where we can also have other DM candidates along with this VLF, or
the DM sector can be more complicated through a multi-component DM scenario.
Such detailed analyis are however beyond the scope of this work and here we
employ a modest approach to study the prospect of this VLF. We propose to
study the more complicated aspects in a future work dedicated only to DM. The
situation however becomes more constraining for a DM mass in the range of 1-10
GeV, where a thermal relic is entirely ruled out by the direct-detection
experiments. This however does not rule out this parameter space in this model
since with these small gauge couplings we still has the option for a freeze-in
DM, which again we postpone for the future dedicated work. For the future
detection prospect, as we develop more and more advanced detectors for these
direct-detection experiments which has very low threshold recoil energies, we
become sensitive to lower direct-detection cross sections and hence can probe
into the white region of the parameter space of the GeV scale. With the
neutrino floor enhanced compared to the SM in this part of parameter space,
such detectors would also eventually start probing the neutrinos coming from
this BSM scenario. However, in order to distinguish them from DM events, we
would also need these detectors to be sensitive to the directionality of the
events. Indirect measurements in future can also provide valuable inputs for
distinguishing the DM from neutrino events in this part of the parameter space
with enhanced neutrino floor.

###  V.2 Scenario 2: high mass

Figure 9: (left) $\sigma_{SI}$ cross-section for the DM of mass 0.1 to 1 GeV,
with parameters as mentioned in the text. (right) Same DM-N cross section for
relatively higher DM masses from 1 to 10 GeV.

As already explained in the Section-III, as we move towards higher
$Z^{\prime}$ mass, the enhancement in neutrino floor decreases. Although the
BSM impact on the neutrino floor decreases, the prospect of a VLF DM detection
brightens as we move towards higher $Z^{\prime}$ mass because the direct-
detection nucleon-DM scattering cross-section decreases and hence becomes less
constrained from the experiments. This can be seen from the two plots of
Figure-9 for a benchmark point towards the rightmost allowed part of the muon
$(g-2)$ band, where the green and blue points span the parameter space in
regions where the direct-detection experiments are not yet sensitive to. As we
will explain in the next section, the indirect detection bounds are much
weaker than that the muon $(g-2)$ in combination with the direct-detection
constraints impose. Hence, this particular region of parameter space is very
important for future DM direct-detection studies. The neutrino floor is
however not very interesting for a heavier $Z^{\prime}$ in the context of the
direct-detection.

Figure 10: (left) $\sigma_{SI}$ cross-section for the DM of mass 0.1 to 1 GeV,
with other parameters as mentioned in the text. (right) Same DM-N cross
section for relatively higher DM masses from 1 to 10 GeV.

As already shown in Fig. 3, a $Z^{\prime}$ mass in the range $10-200$ MeV is
required to satisfy the $(g-2)_{\mu}$ constraint. This ensures that resonant
s-channel gets restricted only to DM mass regime of $m_{\chi}\leq 0.1$ GeV. In
order to decipher the allowed parameter space, we look for direct detection
cross section numbers for the $m_{Z}^{\prime}\sim 0.01-2$ GeV,
$g_{\mu-\tau}\sim 5\times 10^{-3}-5\times 10^{-4}$ and the DM charge
$q_{\chi}$ from $0.1$ to $50$. The green points denote the cross sections
before the imposition of the constraints coming from $(g-2)_{\mu}$ when at
least $10\%$ of the DM relic density is satisfied. The dark-red points on the
top of the green points are the same relic density points after imposing the
$\pm 2\sigma$ $(g-2)_{\mu}$ bounds. The sub-GeV situation is somewhat
different than when the DM masses are higher as depicted in Fig 10.

In $m_{\chi}$ range $0.1-1$ GeV i.e. for a sub-GeV DM, before $(g-2)_{\chi}$
had constrained the parameter space, efficient annihilation puts DM relic
density well within the allowed limit for moderate $g_{\mu-\tau},q_{\chi}$
values. With moderate couplings along with the plausibility of
$m_{Z^{\prime}}$ being as high up to 20 GeV, the direct detection cross
section was mostly in the $10^{-45}-10^{-38}\rm cm^{2}$ range, gradually
increasing with the DM mass. In this range, large portion of parameter space
will have cross section lower than the neutrino floor, therefore evading any
probe.

A large part of the parameter space will present a DM candidate to show up
with a cross section above the neutrino floor and below the most conservative
direct detection bound in this DM mass region that comes from the CRESST
results. Very few points in this region can reach current experimental search
sensitivity and they cannot be ruled out from the experiment. With the
$(g-2)_{\mu}$ bounds being included the $Z^{\prime}$ mass range gets
restricted resulting in absence of s-channel contribution. That requires
larger $q_{\chi}$ values to have a significant contribution from t-channel
process, which as a consequence increases the direct detection cross section.
The $(g-2)_{\mu}$ constrains the overall parameter space only to allow only a
part of it where almost no allowed points have cross section below the
neutrino floor. While this is good from the detection perspective, a
significant number of points now reach experimental search sensitivity and
therefore get ruled out. Any experimental update in the $0.1-0.5$ GeV dark
matter range can probe presence of dark matter with signal cross section
$10^{-40}-10^{-38}\rm cm^{2}$.

For the DM mass range just above a GeV, the $(g-2)_{\mu}$ constraints are even
more tight. Without the $(g-2)_{\mu}$ constraint being imposed there was
certain possibility of DM observations in the cross section range
$10^{-43}-10^{-41}\rm cm^{2}$, while a significant chunk of points were ruled
out by the direct detection experiments like Xenon1T, CDMSLite, DarkSide
experiments. After $(g-2)_{\mu}$ being included the cross section for the
allowed points go further higher due to prevalence of higher $q_{\chi}$
values. This results in all the points allowed by the relic density
constraints, getting ruled out by the direct detection search of various
experiments named above. Only very few points for DM mass below 2 GeV are
allowed. For $m_{\rm DM}\geq 2~{}$GeV, the DM relic density and the constraint
from $g-2$ cannot be satisfied simultaneously. These results in almost ruling
out $1-10$ GeV DM from the $(g-2)_{\mu}$ anomalous results while no tight
bounds were there in this mass range earlier.

##  VI Constraints from Indirect Detection

The indirect detection constraints in general can provide strong bounds for
the dark matter parameter space Leane et al. (2018). Some of the most
stringent bounds come from the Planck measurement of CMB anisotropies Ade et
al. (2016), where injection of energy through ionizing charged particles from
the DM annihilation can perturb the standard thermal history of ionization
involving the hydrogen and helium gas as the constituents of the Universe at
that era. Gamma-ray fluxes from Dwarf Spheroidal Galaxies (dSphs) which is
measured by Fermi-Lat et.allá (2017) can also provide strong bounds to any
photonic final states from DM annihilation. Along with these, the AMS
measurements Aguilar (2013) of the electron and positron flux can also provide
strong constraints for electron and other leptonic final states. Due to the
inefficient annihilation in s-channel to the muons and taus in our case,
constraints coming from bounds on annihilation cross section to a pair of
muons and taus are insignificant for our analysis. The constraints for
annihilation to two electrons or positrons are even weaker due to the absence
of any tree-level coupling of $Z^{\prime}$ to electrons. Due to the presence
of t-channel annihilations to a pair of $Z^{\prime}$s, the strongest limits
emanate from the dSphs measurement of $4\tau$ final states once DM mass goes
above 3 GeV Profumo et al. (2018). Compared to this, the strongest bound from
the $4\mu$ final states coming from the CMB measurements are slightly weaker
and comes into play once we are above 4 GeV DM mass Profumo et al. (2018).
These bounds, although strong, cannot compete with the bounds coming from the
combination of $(g-2)_{\mu}$ and direct detection measurements as shown in
Figure 8 and Figure 9. Hence the constraints coming the the anomalous magnetic
moment of the muon and dark matter direct detection constraints provide a
complementarity to the constraints coming from the indirect detection
experiments.

##  VII Summary and Conclusion

In this article we have reinvestigated viability of $U(1)_{L_{\mu}-L_{\tau}}$
model to simultaneously explain $(g-2)_{\mu}$ and DM relic density, while
adding a crucial part in the DM direct detection through possible modification
in the neutrino floor. We confirm the surviving parameter range of
$m_{Z^{\prime}}$ in $10-200$ MeV with gauge coupling
$g_{\mu-\tau}\sim(0.5-1.0)\times 10^{-3}$, viable to explain anomalous
$(g-2)_{\mu}$ which is measurd by latest FNAL experimental measurement. We
have studied this allowed region to explore possible modification of the
neutrino floor. It is found that with increasing $m_{Z^{\prime}}$ along the
$(g-2)_{\mu}$ viable band, neutrino floor enhancement becomes less
significant. The enhancement peaks by the factors of 1.7-2.7 compared to the
SM background for Germanium and Xenon based experiments, We have presented
neutrino floor modiffication for a benchmark point
$\left(m_{Z^{\prime}}=25\,MeV,g_{\mu-\tau}=9.3\times 10^{-4}\right)$, where
$m_{Z}^{\prime}$ mass is small. Above this mass, floor enhancement gradually
decreases with increasing $m_{Z^{\prime}}$. With the inclusion of
$(g-2)_{\mu}$, the new allowed benchmark point
$\left(m_{Z^{\prime}}=16\,MeV,g_{\mu-\tau}=6.0\times 10^{-4}\right)$ shows the
maximum enhancement of 1.6-2.2 times for different type of DM direct detection
experiments. Next, considering the ultralight DM bound by cosmology, we
venture into a territory exploring the effects of the constraints from
$(g-2)_{\mu}$, investigating the DM mass range from 0.1 to 10 GeV for a VLF,
$\chi$ as dark matter candidate. The DM candidate $\chi$ contributes to dark
matter relic density though the s-channel annihilation process
$\bar{\chi}\,\chi\rightarrow
Z^{\prime}\rightarrow\bar{l}\,\,l\,(\bar{\nu}_{l}\,\,\nu)$ and t-channel
process $\bar{\chi}\,\chi\rightarrow Z^{\prime}\,\,Z^{\prime}$ with t-channel
having the dominant say in almost entire part of the parameter space. For sub-
GeV $m_{\chi}$: $\left(0.1-1\right)$ GeV, without the $(g-2)_{\mu}$ constraint
the parameter space allowed by DM relic density ranges from smaller coupling
values and mass in the range of $m_{Z^{\prime}}=0.01-10$ GeV. With the
inclusion of $(g-2)_{\mu}$ constraint allowed parameter space is more
restricted with $m_{Z^{\prime}}=0.01-0.2$ GeV and $g_{\chi}$ prefering
relatively higher values, when we explore the sub-GeV side of DM mass range.
For super GeV range, we found a similar trend, but with inclusion of
$(g-2)_{\mu}$, parameter space is even more restricted, with $g_{\chi}$ being
confined to a narrow band of values. For sub GeV dark matter CRESST experiment
excludes $\sigma_{SI}$ above $10^{-37}-10^{-38}$ cm2, but still appreciable
parameter space in this range is allowed. In order to elaborate things more,
we include two benchmark scenarios: one towards the left most part of the
allowed $(g-2)_{\mu}$ band and other towards the rightmost part along with the
corresponding behavior of the neutrino floor in those regions. We observe from
the DM parameter space constrained by direct detection experiments, most of
the super-GeV parameter space viable for $(g-2)_{\mu}$ is ruled out by
DarKSide-50, CDMSLite, XENON-1T, with only small parameter space being allowed
for DM mass less than 2 GeV. One important observation is that the constraints
are very different in the different parts of the muon $(g-2)$ allowed region.

In conclusion, the model parameter space viable for $g-2$ experiments can only
account for a tiny part of sub-GeV VLF dark matter, while showing around two
times enhancement of the neutrino floor as well. This region of interest can
be worthwhile to probe in upcoming neutrino experiments like COHERENT. With
increasing exposure, it may become difficult for direct detection experiments
to differentiate between the dark matter signal from the neutrino background.
The conclusion we draw here are valid for the Dirac fermionic DM candidates. A
light Majorana fermion DM candidate can have a new scalar portal for its
annihilation, which can potentially open up most of the parameter space we
restrict due to the muon $(g-2)$ constraint here. Also a scalar dark matter
can be studied in this context. In general GeV scale DM is an interesting
region due to its viability, being not so constrained by other existing
constraints. The recent anomalous $(g-2)$ measurement plays a crucial role
when it is coupled with the DM direct detection constraints, being more
constraining to the GeV scale DM compared to the usual indirect detection
constraints.

## Acknowledgments

We would like to thank Prof. Debajyoti Choudhury for useful discussions. K.D.
acknowledges Council for Scientific and Industrial Research (CSIR), India for
JRF/SRF fellowship with the award letter no. 09/045(1654)/2019-EMR-I. SS
thanks Vivekanada Centre for Research (VCR) for providing me with research
facilities. MPS would like to thank S.P. Singh for partial financial support.

## Appendix

###  .1 Enhancement in CE$\nu$NS event rate and neutrino floor

I. Benchmark chosen: $Z^{\prime}$ mass 16 MeV and coupling
$g_{\mu-\tau}=6.0\times 10^{-4}$

mDM(GeV) | SM Neutrino floor(cm2) | U(1)μ-τ Neutrino floor(cm2) | Enhancement
---|---|---|---
Germanium based experiments
0.5 | 2.36$\times 10^{-43}$ | 5.22$\times 10^{-43}$ | 2.211
1. | 5.23$\times 10^{-44}$ | 1.14$\times 10^{-43}$ | 2.17
5 | 7.05$\times 10^{-45}$ | 1.51$\times 10^{-44}$ | 2.121
10 | 1.42$\times 10^{-47}$ | 1.74$\times 10^{-47}$ | 1.225
Xenon based experiments
0.5 | 3.91$\times 10^{-43}$ | 6.4$\times 10^{-43}$ | 1.63
1. | 8.49$\times 10^{-44}$ | 1.38$\times 10^{-43}$ | 1.62
5 | 1.08$\times 10^{-44}$ | 1.76$\times 10^{-44}$ | 1.62
10 | 9.27$\times 10^{-48}$ | 1.04$\times 10^{-47}$ | 1.12
Table 1:  Neutrino floor versus dark matter mass table highlighting
modification of neutrino floor for Uμ-τ with respect to SM .

II. Benchmark chosen: $Z^{\prime}$ mass 200 MeV and coupling
$g_{\mu-\tau}=1.64\times 10^{-3}$

mDM(GeV) | SM Neutrino floor(cm2) | U(1)μ-τ Neutrino floor(cm2) | Enhancement
---|---|---|---
Germanium based experiments
0.5 | 2.36$\times 10^{-43}$ | 2.46$\times 10^{-43}$ | 1.04
1. | 5.23$\times 10^{-44}$ | 5.48$\times 10^{-43}$ | 1.047
5 | 7.05$\times 10^{-45}$ | 7.44$\times 10^{-44}$ | 1.055
10 | 1.42$\times 10^{-47}$ | 1.49$\times 10^{-47}$ | 1.049
Xenon based experiments
0.5 | 3.91$\times 10^{-43}$ | 4.01$\times 10^{-43}$ | 1.02
1. | 8.49$\times 10^{-44}$ | 8.74$\times 10^{-43}$ | 1.03
5 | 1.08$\times 10^{-44}$ | 1.125$\times 10^{-44}$ | 1.04
10 | 9.27$\times 10^{-48}$ | 9.56$\times 10^{-47}$ | 1.03
Table 2:  Neutrino floor versus dark matter mass table highlighting
modification of neutrino floor for Uμ-τ with respect to SM .

III. Benchmark chosen: $Z^{\prime}$ mass 50 MeV and coupling
$g_{\mu-\tau}=1.1\times 10^{-3}$

mDM(GeV) | SM Neutrino floor(cm2) | U(1)μ-τ Neutrino floor(cm2) | Enhancement
---|---|---|---
Germanium based experiments
0.5 | 2.36$\times 10^{-43}$ | 3.16$\times 10^{-43}$ | 1.33
1. | 5.23$\times 10^{-44}$ | 7.23$\times 10^{-43}$ | 1.38
5 | 7.05$\times 10^{-45}$ | 1.01$\times 10^{-44}$ | 1.43
10 | 1.42$\times 10^{-47}$ | 1.82$\times 10^{-47}$ | 1.28
Xenon based experiments
0.5 | 3.91$\times 10^{-43}$ | 4.67$\times 10^{-43}$ | 1.19
1. | 8.49$\times 10^{-44}$ | 1.03$\times 10^{-43}$ | 1.21
5 | 1.08$\times 10^{-44}$ | 1.35$\times 10^{-44}$ | 1.25
10 | 9.27$\times 10^{-48}$ | 1.08$\times 10^{-47}$ | 1.16
Table 3:  Neutrino floor versus dark matter mass table highlighting
modification of neutrino floor for Uμ-τ with respect to SM .

IV. Benchmark chosen: $Z^{\prime}$ mass 25 MeV and coupling
$g_{\mu-\tau}=0.93\times 10^{-3}$

mDM(GeV) | SM Neutrino floor(cm2) | U(1)μ-τ Neutrino floor(cm2) | Enhancement
---|---|---|---
Germanium based experiments
0.5 | 2.36$\times 10^{-43}$ | 5.06$\times 10^{-43}$ | 2.14
1. | 5.23$\times 10^{-44}$ | 1.16$\times 10^{-43}$ | 2.23
5 | 7.05$\times 10^{-45}$ | 1.61$\times 10^{-44}$ | 2.28
10 | 1.42$\times 10^{-47}$ | 2.05$\times 10^{-47}$ | 1.443
Xenon based experiments
0.5 | 3.91$\times 10^{-43}$ | 6.27$\times 10^{-43}$ | 1.60
1. | 8.49$\times 10^{-44}$ | 1.40 $\times 10^{-43}$ | 1.65
5 | 1.08$\times 10^{-44}$ | 1.84$\times 10^{-44}$ | 1.70
10 | 9.27$\times 10^{-48}$ | 1.16$\times 10^{-47}$ | 1.25
Table 4:  Neutrino floor versus dark matter mass table highlighting
modification of neutrino floor for Uμ-τ with respect to SM .

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