# Probing light mediators and $(g-2)_{\mu}$ through detection of coherent
elastic neutrino nucleus scattering at COHERENT

M. Atzori Corona  Dipartimento di Fisica, Università degli Studi di Cagliari,
Complesso Universitario di Monserrato - S.P. per Sestu Km 0.700, 09042
Monserrato (Cagliari), Italy  INFN, Sezione di Cagliari, Complesso
Universitario di Monserrato - S.P. per Sestu Km 0.700, 09042 Monserrato
(Cagliari), Italy  M. Cadeddu  Istituto Nazionale di Fisica Nucleare (INFN),
Sezione di Cagliari, Complesso Universitario di Monserrato - S.P. per Sestu Km
0.700, 09042 Monserrato (Cagliari), Italy  N. Cargioli  Dipartimento di
Fisica, Università degli Studi di Cagliari, Complesso Universitario di
Monserrato - S.P. per Sestu Km 0.700, 09042 Monserrato (Cagliari), Italy
INFN, Sezione di Cagliari, Complesso Universitario di Monserrato - S.P. per
Sestu Km 0.700, 09042 Monserrato (Cagliari), Italy  F. Dordei  Istituto
Nazionale di Fisica Nucleare (INFN), Sezione di Cagliari, Complesso
Universitario di Monserrato - S.P. per Sestu Km 0.700, 09042 Monserrato
(Cagliari), Italy  C. Giunti  Istituto Nazionale di Fisica Nucleare (INFN),
Sezione di Torino, Via P. Giuria 1, I–10125 Torino, Italy  Y.F. Li  Institute
of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China  E. Picciau  Dipartimento di Fisica, Università degli
Studi di Cagliari, Complesso Universitario di Monserrato - S.P. per Sestu Km
0.700, 09042 Monserrato (Cagliari), Italy  INFN, Sezione di Cagliari,
Complesso Universitario di Monserrato - S.P. per Sestu Km 0.700, 09042
Monserrato (Cagliari), Italy  C. A. Ternes  Istituto Nazionale di Fisica
Nucleare (INFN), Sezione di Torino, Via P. Giuria 1, I–10125 Torino, Italy
Y.Y. Zhang<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China  School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China

( , )

###### Abstract

We present the constraints on the parameters of several light boson mediator
models obtained from the analysis of the current data of the COHERENT
CE$\nu$NS experiment. We consider a variety of vector boson mediator models:
the so-called universal, the $B-L$ and other anomaly-free $U(1)^{\prime}$
gauge models with direct couplings of the new vector boson with neutrinos and
quarks, and the anomaly-free $L_{e}-L_{\mu}$, $L_{e}-L_{\tau}$, and
$L_{\mu}-L_{\tau}$ gauge models where the coupling of the new vector boson
with the quarks is generated by kinetic mixing with the photon at the one-loop
level. We consider also a model with a new light scalar boson mediator that is
assumed, for simplicity, to have universal coupling with quarks and leptons.
Since the COHERENT CE$\nu$NS data are well-fitted with the cross section
predicted by the Standard Model, the analysis of the data yields constraints
for the mass and coupling of the new boson mediator that depend on the charges
of quarks and neutrinos in each model under consideration. We compare these
constraints with the limits obtained in other experiments and with the values
that can explain the muon $g-2$ anomaly in the models where the muon couples
to the new boson mediator.

##  I Introduction

The discovery of coherent elastic neutrino-nucleus scattering (CE$\nu$NS) in
cesium-iodide (CsI) by the COHERENT Collaboration Akimov  _et al._ (2017,
2018) sparked a flood of research into a variety of physical processes, with
substantial implications for particle physics, astrophysics, nuclear physics,
and beyond Cadeddu  _et al._ (2018a); Papoulias  _et al._ (2020); Coloma  _et
al._ (2017); Liao and Marfatia (2017); Lindner  _et al._ (2017); Giunti
(2020); Denton  _et al._ (2018); Aristizabal Sierra  _et al._ (2018); Cadeddu
_et al._ (2018b); Papoulias and Kosmas (2018); Cadeddu  _et al._ (2020a);
Papoulias (2020); Khan and Rodejohann (2019); Dutta  _et al._ (2019);
Aristizabal Sierra  _et al._ (2018); Cadeddu and Dordei (2019); Dutta  _et
al._ (2020); Abdullah  _et al._ (2018); Ge and Shoemaker (2018). After the
discovery in 2017 of CE$\nu$NS with the CsI detector, the COHERENT
Collaboration accomplished in 2020 the first observation of CE$\nu$NS in argon
(Ar) Akimov  _et al._ (2021a, 2020) and updated in 2021 the results obtained
with the CsI detector Akimov  _et al._ (2021b). By combining the greater
CE$\nu$NS statistics with a refined quenching factor estimation for the CsI
measurement and by virtue of the complementary role of two different target
nuclei, more stringent tests of nuclear physics, neutrino properties,
electroweak interactions, and new physics beyond the Standard Model (SM) have
been performed Cadeddu  _et al._ (2020b); Miranda  _et al._ (2020); Cadeddu
_et al._ (2021a, b); Banerjee  _et al._ (2021).

The CE$\nu$NS process happens when the momentum transfer between the incoming
neutrino and the target nucleus is so small that the wavelength of the boson
which mediates the interaction is larger than the nuclear radius, so that the
neutrino interacts with the nucleus as a whole and the cross section is
proportional to the square of the number of nucleons participating to the
process. The CE$\nu$NS process is a pure neutral current interaction which is
mediated by the exchange of the $Z$ vector boson in the SM, but it can also
receive contributions from other hypothetical neutral bosons in theories
beyond the SM. Therefore, it turns out to be a powerful tool to probe new
physics interactions beyond the SM Coloma  _et al._ (2017); Liao and Marfatia
(2017); Lindner  _et al._ (2017); Giunti (2020); Denton  _et al._ (2018);
Aristizabal Sierra  _et al._ (2018).

In this paper we test new physics models with interactions mediated by a light
vector or scalar boson that contribute to the CE$\nu$NS process by analyzing
the recently released 2021 CsI data Akimov  _et al._ (2021b) and the 2020 Ar
data Akimov  _et al._ (2021a, 2020) of the COHERENT experiment. For each
model, we present the constraints on the mass and coupling of the light vector
or scalar boson mediator that we obtained from the separate and combined fits
of the CsI and Ar COHERENT CE$\nu$NS data. Comparing with the previous
publication in Ref. Cadeddu  _et al._ (2021a), we have considered a larger
variety of vector mediator models, we have included the scalar mediator model,
and we have updated the analysis using the recently released 2021 CsI data
Akimov  _et al._ (2021b).

We also consider the possible explanation of the $4.2\sigma$ difference
between the SM prediction Aoyama  _et al._ (2020) of the value of the muon
anomalous magnetic moment $(g-2)_{\mu}$ and the combination of the values
measured at the Brookhaven National Laboratory Bennett  _et al._ (2006) and
recently at the Fermi National Laboratory Abi  _et al._ (2021). This so-called
$(g-2)_{\mu}$ anomaly is a putative signal of physics beyond the SM, which has
been studied in many papers (see, e.g., Refs. Athron  _et al._ (2021); Lindner
_et al._ (2018)). Interestingly, several light vector mediator models are
regarded as candidate solutions (see, e.g., Refs. Baek  _et al._ (2001); Ma
_et al._ (2002); Altmannshofer  _et al._ (2016); Amaral  _et al._ (2021);
Cadeddu  _et al._ (2021c); Zhou (2021); Ko  _et al._ (2021); Hapitas  _et al._
(2021); Cheng  _et al._ (2021)), but also a light scalar mediator have the
potential to solve the anomaly Jegerlehner and Nyffeler (2009). Among the
models that we consider in this paper, those in which the muon interacts with
the new light boson mediator can explain the $(g-2)_{\mu}$ anomaly. For these
models, we compare the constraints on the mass and coupling of the new light
boson mediator obtained from the analysis of the COHERENT CE$\nu$NS data and
those obtained by other experiments focusing on the parameter region that can
solve the $(g-2)_{\mu}$ anomaly. There are many studies of extensions of the
SM with the addition of a $U(1)^{\prime}$ gauge group with an associated
neutral vector gauge boson $Z^{\prime}$ (see, e.g., the review in Ref.
Langacker (2009)). The models differ in the charges of the fermions, which
determine the contributions to CE$\nu$NS of the interactions mediated by the
$Z^{\prime}$ vector boson. These contributions add coherently to the SM weak
neutral current interactions which are mediated by the $Z$ vector boson. The
effects are quantified by additional terms in the weak charge of the nucleus.
Note that the effects on the CE$\nu$NS process of the interactions mediated by
a light boson are different from those induced by the so-called non-standard
interactions (NSI), that arise in an effective four-fermion theory in which
the heavy mediator has been integrated out. In the case of NSI there is a
global rescaling of the CE$\nu$NS cross section that depends on the
interaction parameters of the NSI, whereas a light boson mediator can alter
the nuclear recoil energy spectrum through the boson propagator that depends
on the momentum transfer. This effect generates distinct spectral features
that can be probed with the experimental observations.

In this paper we first consider the so-called universal $Z^{\prime}$ model in
which all the standard fermions have the same charge Liao and Marfatia (2017);
Papoulias and Kosmas (2018); Billard  _et al._ (2018); Papoulias (2020); Khan
and Rodejohann (2019); Cadeddu  _et al._ (2021a); Bertuzzo  _et al._ (2021).
This model is not anomaly-free per se, but it can be extended with new non-
standard particles to make it anomaly-free. Then, we consider several
$U(1)^{\prime}$ models in which quarks and leptons have appropriate non-zero
charges that cancel the quantum anomalies (e.g., the popular $B-L$ model
Langacker (2009); Mohapatra (2014); Okada (2018), where $B$ is the baryon
number and $L$ is the total lepton number). Since in these models the
$Z^{\prime}$ vector boson interacts directly with neutrinos and nucleons, the
CE$\nu$NS process occurs at tree level and it is possible to obtain stringent
constraints on the mass and coupling of the new vector boson from the COHERENT
CE$\nu$NS data.

We also consider the anomaly-free $L_{e}-L_{\mu}$, $L_{e}-L_{\tau}$, and
$L_{\mu}-L_{\tau}$ $U(1)^{\prime}$ models Foot (1991); Foot  _et al._ (1990);
He  _et al._ (1991a); Foot  _et al._ (1993) (where $L_{\alpha}$ are the lepton
generation numbers, for $\alpha=e,\mu,\tau$) in which the charges are
exclusively leptonic. However, in these models there are contributions to the
CE$\nu$NS process, which occur through the kinetic mixing of the $Z^{\prime}$
boson with the photon, that is generated at one-loop level Altmannshofer  _et
al._ (2019); Banerjee and Roy (2019); Banerjee  _et al._ (2021), and the
interaction of the photon with the protons in the target nuclei. Therefore, we
can constrain the mass and coupling of the vector boson in these models using
the COHERENT CE$\nu$NS data, albeit less tightly than in the models with
direct quark-$Z^{\prime}$ interactions, because of the weaker one-loop
interaction.

We consider also contributions to the CE$\nu$NS process of interactions
mediated by a light scalar boson Lindner  _et al._ (2017); Cerdeño  _et al._
(2016); Farzan  _et al._ (2018); Aristizabal Sierra  _et al._ (2018, 2019),
which differ from those mediated by a light vector boson for the following two
fundamental reasons. First, the helicity-flipping interactions mediated by a
scalar boson contribute incoherently to the CE$\nu$NS process with respect to
the helicity-conserving SM contribution, contrary to the helicity-conserving
interactions mediated by a new vector boson that contribute coherently.
Therefore, in the scalar case, the new contribution consists in an addition to
the cross section, not to the amplitude of the process as in the vector case.
Second, the scalar charges of the nucleons are not simply given by the sum of
the charges of the valence quarks as in the vector case, because the scalar
currents are not conserved as the vector currents. Therefore, the scalar
charges of the nucleons must be calculated and the results have large
theoretical uncertainties.

The paper is organized as follows. In Section II the method of the COHERENT
data analysis is described. In Section III, we present the cross section of
the CE$\nu$NS process and summarize the models of light mediators and the
corresponding effects on the CE$\nu$NS cross section. In Section IV, the
COHERENT CE$\nu$NS constraints on the allowed parameter space of the light
mediator models are presented and compared with the $(g-2)_{\mu}$ allowed
regions and other current limits. Finally, we conclude and summarize our
results in Section V.

##  II COHERENT data analysis

The CE$\nu$NS event energy spectra in the COHERENT experiment depend on the
neutrino flux produced by the pion decay. The total differential neutrino flux
is given by the sum of the three neutrino components, where the first prompt
component is coming from the pion decay
($\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$), and the second two delayed components
are coming from the subsequent muon decay ($\mu^{+}\rightarrow
e^{+}+\nu_{e}+\bar{\nu}_{\mu}$)

$\displaystyle\frac{dN_{\nu_{\mu}}}{dE}$
$\displaystyle=\eta\,\delta\\!\left(E-\frac{m_{\pi}^{2}-m_{\mu}^{2}}{2m_{\pi}}\right),$
(1) $\displaystyle\frac{dN_{\nu_{\tilde{\mu}}}}{dE}$
$\displaystyle=\eta\,\frac{64E^{2}}{m_{\mu}^{3}}\left(\frac{3}{4}-\frac{E}{m_{\mu}}\right),$
(2) $\displaystyle\frac{dN_{\nu_{e}}}{dE}$
$\displaystyle=\eta\,\frac{192E^{2}}{m_{\mu}^{3}}\left(\frac{1}{2}-\frac{E}{m_{\mu}}\right).$
(3)

Here, $E$ is the neutrino energy, $m_{\pi}$ and $m_{\mu}$ are the pion and
muon masses, and $\eta=rN_{\mathrm{POT}}/4\pi L^{2}$ is the normalization
factor, where $r$ is the number of neutrinos per flavor produced for each
proton-on-target (POT), $N_{\mathrm{POT}}$ is the number of POT, and $L$ is
the baseline between the source and the detector. For the COHERENT Ar
detector, called CENNS-10, we use $r=0.09$, $N_{\mathrm{POT}}=13.7\cdot
10^{22}$ and $L=27.5~{}\mathrm{m}$ Akimov  _et al._ (2020). For the COHERENT
CsI detector, we use $r=0.0848$, $N_{\mathrm{POT}}=3.198\cdot 10^{23}$ and
$L=19.3~{}\mathrm{m}$ Akimov  _et al._ (2021b).

The theoretical CE$\nu$NS event number $N^{\mathrm{CE\nu NS}}_{i}$ in each
nuclear-recoil energy-bin $i$ is given by

$N_{i}^{\mathrm{CE}\nu\mathrm{NS}}=N(\mathcal{N})\int_{T_{\mathrm{nr}}^{i}}^{T_{\mathrm{nr}}^{i+1}}dT_{\mathrm{nr}}A(T_{\mathrm{nr}})\int_{0}^{T^{\prime\text{max}}_{\text{nr}}}dT^{\prime}_{\text{nr}}\,R(T_{\text{nr}},T^{\prime}_{\text{nr}})\int_{E_{\text{min}}(T^{\prime}_{\text{nr}})}^{E_{\text{max}}}dE\sum_{\nu=\nu_{e},\nu_{\mu},\bar{\nu}_{\mu}}\frac{dN_{\nu}}{dE}(E)\frac{d\sigma_{\nu-\mathcal{N}}}{dT_{\mathrm{nr}}}(E,T^{\prime}_{\mathrm{nr}}),$
(4)

where $T_{\text{nr}}$ is the reconstructed nuclear recoil kinetic energy,
$T^{\prime}_{\text{nr}}$ is the true nuclear recoil kinetic energy,
$A(T_{\text{nr}})$ is the energy-dependent detector efficiency,
$R(T_{\text{nr}},T^{\prime}_{\text{nr}})$ is the energy resolution function,
$T^{\prime\text{max}}_{\text{nr}}=2E_{\text{max}}^{2}/M$,
$E_{\text{max}}=m_{\mu}/2\sim 52.8$ MeV,
$E_{\text{min}}(T^{\prime}_{\text{nr}})=\sqrt{MT^{\prime}_{\text{nr}}/2}$,
$m_{\mu}$ being the muon mass, $M$ the nuclear mass, and $N(\mathcal{N})$ the
number of $\mathcal{N}$ atoms in the detector. We obtained information on
these quantities from Refs. Akimov  _et al._ (2021a, 2020) for the Ar data and
from Ref. Akimov  _et al._ (2021b) for the CsI data. The number of
$\mathcal{N}$ atoms in each detector is given by
$N(\mathcal{N})=N_{\mathrm{A}}M_{\mathrm{det}}/M_{\mathrm{\mathcal{N}}}$,
where $N_{\mathrm{A}}$ is the Avogadro number, $M_{\mathrm{det}}$ is the
detector active mass ($M_{\mathrm{det}}=24~{}\mathrm{kg}$ for Ar and
$M_{\mathrm{det}}=14.6~{}\mathrm{kg}$ for CsI), and $M_{\mathrm{\mathcal{N}}}$
is the molar mass ($M_{\mathrm{Ar}}=39.96~{}\mathrm{g/mol}$ and
$M_{\mathrm{CsI}}=259.8~{}\mathrm{g/mol}$). The differential CE$\nu$NS cross
section $d\sigma_{\nu-\mathcal{N}}/dT_{\mathrm{nr}}$ is discussed in Section
III.

Due to the quenching effect, the energy actually observed is the electron-
equivalent recoil energy $T_{ee}$, which is transformed into the nuclear
recoil energy $T_{\mathrm{nr}}$ by inverting the relation

$T_{ee}=f_{Q}\left(T_{\mathrm{nr}}\right)T_{\mathrm{nr}},$ (5)

where $f_{Q}$ is the quenching factor, which is given in Refs. Akimov  _et
al._ (2021b, c) for the CsI detector and in Ref. Akimov  _et al._ (2020) for
the Ar detector.

An important characteristic of the neutrino beam in the COHERENT experiment is
the time dependence of the neutrino flavor components: the prompt
$\nu_{\mu}$’s produced in fast pion decay ($\tau_{\pi^{\pm}}\simeq
26~{}\text{ns}$) arrive within about $1~{}\mu\text{s}$ from the on-beam
trigger, whereas the delayed $\nu_{e}$’s and $\bar{\nu}_{\mu}$’s produced in
the slower muon decay ($\tau_{\mu^{\pm}}\simeq 2.2~{}\mu\text{s}$) arrive in a
time interval which tails out at about $10~{}\mu\text{s}$. Therefore, taking
into account the time evolution of the data is useful for distinguishing the
interactions of the two neutrino flavors. We implemented the analyses of the
COHERENT CsI and Ar data using the timing information provided by the COHERENT
Collaboration Akimov  _et al._ (2020, 2021b, 2021c) and distributing the
theoretical CE$\nu$NS event numbers $N^{\text{CE$\nu$NS}}_{i}$ in Eq. (4) in
time bins that are calculated from the exponential decay laws of the
generating pions and muons. With this procedure we obtained the theoretical
CE$\nu$NS event numbers $N^{\text{CE$\nu$NS}}_{ij}$, where $i$ is the index of
the energy bins and $j$ is the index of the time bins.

We performed the analysis of the COHERENT CsI data in the energy and time bins
considered in Ref. Akimov  _et al._ (2021b). Since in some energy-time bins
the number of events is zero, we used the Poissonian least-squares function
Baker and Cousins (1984); Zyla  _et al._ (2020)

$\displaystyle\chi^{2}_{\mathrm{CsI}}=\hbox{}$
$\displaystyle\hbox{}2\sum_{i=1}^{9}\sum_{j=1}^{11}\left[\sum_{z=1}^{4}(1+\eta_{z})N_{ij}^{z}-N_{ij}^{\text{exp}}+N_{ij}^{\text{exp}}\ln\left(\frac{N_{ij}^{\text{exp}}}{\sum_{z=1}^{4}(1+\eta_{z})N_{ij}^{z}}\right)\right]+\sum_{z=1}^{4}\left(\dfrac{\eta_{z}}{\sigma_{z}}\right)^{2},$
(6)

where the indices $i$ and $j$ denote, respectively, the energy and time bins,
and the indices $z=1,2,3,4$ stand for CE$\nu$NS, beam-related neutron (BRN),
neutrino-induced neutron (NIN), and steady-state (SS) backgrounds,
respectively. In our notation, $N_{ij}^{\text{exp}}$ is the experimental event
number obtained from coincidence (C) data, $N_{ij}^{\text{CE$\nu$NS}}$ is the
predicted number of CE$\nu$NS events that depends on the physics model under
consideration, $N_{ij}^{\text{BRN}}$ is the estimated BRN background,
$N_{ij}^{\text{NIN}}$ is the estimated NIN background, and
$N_{ij}^{\text{SS}}$ is the SS background obtained from the anti-coincidence
(AC) data. We took into account the systematic uncertainties described in Ref.
Akimov  _et al._ (2021b) with the nuisance parameters $\eta_{z}$ and the
corresponding uncertainties $\sigma_{\text{CE$\nu$NS}}=0.12$ (which is the
systematic uncertainty of the signal rate considering the effects of the 10%,
3.8%, 4.1%, and 3.4% uncertainties of the neutrino flux, quenching factor,
CE$\nu$NS efficiency, and neutron form factors, respectively),
$\sigma_{\text{BRN}}=0.25$, $\sigma_{\text{NIN}}=0.35$, and
$\sigma_{\text{SS}}=0.021$.

We performed the analysis of the COHERENT Ar data in the energy and time bins
given in the data release Akimov  _et al._ (2020) with the least-squares
function

$\displaystyle\chi^{2}_{\mathrm{Ar}}=\hbox{}$
$\displaystyle\hbox{}\sum_{i=1}^{12}\sum_{j=1}^{10}\left(\dfrac{N_{ij}^{\text{exp}}-\sum_{z=1}^{4}(1+\eta_{z}+\sum_{l}\eta^{\mathrm{sys}}_{zl,ij})N_{ij}^{z}}{\sigma_{ij}}\right)^{2}+\sum_{z=1}^{4}\left(\dfrac{\eta_{z}}{\sigma_{z}}\right)^{2}+\sum_{z,l}\left(\epsilon_{zl}\right)^{2},$
(7)

where $i$ is the index of the energy bins and $j$ is the index of the time
bins. Here $z=1,2,3,4$ stands for the theoretical prediction of CE$\nu$NS,
Steady-State (SS), Prompt Beam-Related Neutron (PBRN) and Delayed Beam-Related
Neutron (DBRN) backgrounds, and $N_{ij}^{z}$ is the number of observed events
in each energy and time bin. The statistical uncertainty ${\sigma_{ij}}$ is
given by

$(\sigma_{ij}^{\mathrm{}})^{2}=(\sigma_{ij}^{\mathrm{exp}})^{2}+(\sigma_{ij}^{\text{SS}})^{2},$
(8)

where $\sigma_{ij}^{\mathrm{exp}}=\sqrt{N_{ij}^{\text{exp}}}$ and
$\sigma_{ij}^{\mathrm{SS}}=\sqrt{{N_{ij}^{\text{SS}}}/{5}}$. The factor 1/5 is
due to the 5 times longer sampling time of the SS background with respect to
the signal time window. The nuisance parameters $\eta_{z}$ quantify the
systematic uncertainties of the event rate for the theoretical prediction of
CE$\nu$NS, SS, PBRN, and DBRN backgrounds, with the corresponding
uncertainties $\sigma_{\mathrm{CE\nu NS}}=0.13$,
$\sigma_{\mathrm{PBRN}}=0.32$, $\sigma_{\mathrm{DBRN}}=1$, and
$\sigma_{\mathrm{SS}}=0.0079$. We considered also the systematic uncertainties
of the shapes of CE$\nu$NS and PBRN spectra using the information in the
COHERENT data release Akimov  _et al._ (2020). This is done in Eq. (7) through
the nuisance parameters $\epsilon_{zl}$ and the terms
$\eta^{\mathrm{sys}}_{zl,ij}$ given by

$\eta^{\mathrm{sys}}_{zl,ij}=\epsilon_{zl}\,\frac{N_{zl,ij}^{\mathrm{sys}}-N_{zl,ij}^{\mathrm{CV}}}{N_{zl,ij}^{\mathrm{CV}}},$
(9)

where $l$ is the index of the source of the systematic uncertainty. Here
$N_{zl,ij}^{\mathrm{sys}}$ and $N_{zl,ij}^{\mathrm{CV}}$ are, respectively,
$1\sigma$ probability distribution functions (PDFs) described in Table 3 of
Ref. Akimov  _et al._ (2020) and the central-value (CV) SM predictions
described in Table 2 of Ref. Akimov  _et al._ (2020). For the theoretical
prediction of CE$\nu$NS ($z=1$), the sources of systematic shape uncertainties
are the $F_{90}$ energy dependence and the mean time to trigger
($t_{\mathrm{trig}}$) distribution. For the PBRN background ($z=2$), the
sources of systematic shape uncertainties are the energy, $t_{\mathrm{trig}}$
mean, and $t_{\mathrm{trig}}$ width distributions.

##  III CE$\nu$NS process and light mediators

In the SM, the differential cross section as a function of the nuclear kinetic
recoil energy $T_{\text{nr}}$ of the CE$\nu$NS process with a neutrino
$\nu_{\ell}$ ($\ell=e,\mu,\tau$) and a nucleus $\mathcal{N}$ is given by
Drukier and Stodolsky (1984); Barranco  _et al._ (2005); Patton  _et al._
(2012)

$\dfrac{d\sigma_{\nu_{\ell}\text{-}\mathcal{N}}}{dT_{\text{nr}}}(E,T_{\text{nr}})=\dfrac{G_{\text{F}}^{2}M}{\pi}\left(1-\dfrac{MT_{\text{nr}}}{2E^{2}}\right)(Q^{V}_{\ell,\mathrm{SM}})^{2},$
(10)

where $G_{\text{F}}$ is the Fermi constant and

$Q^{V}_{\ell,\mathrm{SM}}=\left[g_{V}^{p}\left(\nu_{\ell}\right)ZF_{Z}(|\vec{q}|^{2})+g_{V}^{n}NF_{N}(|\vec{q}|^{2})\right],$
(11)

is the weak charge of the nucleus. Here, $Z$ and $N$ are the numbers of
protons and neutrons in the nucleus, respectively, and $g_{V}^{p}$ and
$g_{V}^{n}$ are the neutrino-proton and neutrino-neutron couplings,
respectively. Taking into account radiative corrections in the
$\overline{\mathrm{MS}}$ scheme Erler and Su (2013), accurate values of the
vector couplings can be derived as Cadeddu  _et al._ (2020b)

$\displaystyle g_{V}^{p}(\nu_{e})=0.0401,\quad\quad
g_{V}^{p}(\nu_{\mu})=0.0318,\quad\quad g_{V}^{n}=-0.5094\,.$ (12)

In Eq. (11), $F_{Z}(|\vec{q}|^{2})$ and $F_{N}(|\vec{q}|^{2})$ are,
respectively, the form factors of the proton and neutron distributions in the
nucleus, which are the Fourier transforms of the corresponding nucleon
distribution in the nucleus and describe the loss of coherence for large
values of the momentum transfer $|\vec{q}|$. We use an analytic expression,
namely the Helm parameterization Helm (1956), for the form factors, that gives
practically equivalent results to the other two well known parameterizations,
i.e., the symmetrized Fermi Piekarewicz  _et al._ (2016) and Klein-Nystrand
Klein and Nystrand (1999) ones. The proton rms radii can be obtained from the
muonic atom spectroscopy experiments Fricke  _et al._ (1995); Angeli and
Marinova (2013) as explained in Ref. Cadeddu  _et al._ (2020b)

$\displaystyle R_{p}(\mathrm{Cs})=4.821~{}\mathrm{fm},\quad\quad
R_{p}(\mathrm{I})=4.766~{}\mathrm{fm},\quad\quad
R_{p}(\mathrm{Ar})=3.448~{}\mathrm{fm}.$ (13)

On the other hand, there is a poor knowledge of the values of the
${}^{133}\text{Cs}$, ${}^{127}\text{I}$ and ${}^{40}\text{Ar}$ neutron rms
radii obtained from the analyses of the COHERENT data Cadeddu  _et al._
(2018a); Papoulias  _et al._ (2020); Cadeddu  _et al._ (2018b); Huang and Chen
(2019); Papoulias (2020); Khan and Rodejohann (2019); Cadeddu  _et al._
(2020b, a). The values of these neutron rms radii can, however, be estimated
with theoretical calculations based on different nuclear models Hoferichter
_et al._ (2020); Cadeddu  _et al._ (2020b, 2021b). Here, we consider the
following values obtained from the recent nuclear shell model estimate of the
corresponding neutron skins (i.e. the differences between the neutron and the
proton rms radii) in Ref. Hoferichter  _et al._ (2020)

$\displaystyle R_{n}(\mathrm{Cs})\simeq 5.09~{}\mathrm{fm},\quad\quad
R_{n}(\mathrm{I})\simeq 5.03~{}\mathrm{fm},\quad\quad R_{n}(\mathrm{Ar})\simeq
3.55~{}\mathrm{fm}.$ (14)

Following the COHERENT Collaboration Akimov  _et al._ (2021b, a, 2020), we
take into account the effect of the uncertainty of the values of the neutron
rms radii by considering 3.4% and 2% uncertainties for the CsI and Ar
CE$\nu$NS rates, respectively.

The SM CE$\nu$NS differential event rates that are predicted for the COHERENT
Ar and CsI detectors are shown in Fig. 1 as functions of $T_{\text{nr}}$. One
can see that there are kinks at $T_{\text{nr}}\approx 50~{}\text{keV}$ for Ar
and $T_{\text{nr}}\approx 15~{}\text{keV}$ for CsI. The steeper slope of the
SM differential event rates below these values of $T_{\text{nr}}$ is due to
the coherency condition $T_{\text{nr}}\lesssim 1/2MR^{2}$.

The CE$\nu$NS cross section is modified if there is a new massive mediator
which couples to the SM leptons and quarks. In this work, we focus on two
mediator types that have been considered in several previous works Liao and
Marfatia (2017); Papoulias and Kosmas (2018); Papoulias  _et al._ (2018,
2019); Khan and Rodejohann (2019); Dutta  _et al._ (2019); Bertuzzo  _et al._
(2017); Abdullah  _et al._ (2018); Billard  _et al._ (2018); Han  _et al._
(2019); Flores  _et al._ (2020); Papoulias (2020); Miranda  _et al._ (2020);
Cadeddu  _et al._ (2021a); de la Vega  _et al._ (2021); Bertuzzo  _et al._
(2021); Bonet  _et al._ (2021): an additional vector mediator $Z^{\prime}$
with mass $M_{Z^{\prime}}$ associated to a new $U(1)^{\prime}$ gauge group and
an additional scalar mediator $\phi$ with mass $M_{\phi}$. The phenomenology
of CE$\nu$NS in the specific models that we consider is briefly described in
the following two Subsections.

###  III.1 Light vector mediator

The interaction of a $Z^{\prime}$ vector boson with neutrinos and quarks is
described by the generic Lagrangian

$\mathcal{L}_{Z^{\prime}}^{V}=-Z^{\prime}_{\mu}\left[\sum_{\ell=e,\mu,\tau}g_{Z^{\prime}}^{\nu_{\ell}V}\,\overline{\nu_{\ell
L}}\gamma^{\mu}\nu_{\ell
L}+\sum_{q=u,d}g_{Z^{\prime}}^{qV}\,\overline{q}\gamma^{\mu}q\right],$ (15)

where $g_{Z^{\prime}}^{qV}$ and $g_{Z^{\prime}}^{\nu_{\ell}V}$ are the
couplings constants.

In the case of a vector mediator associated with a new $U(1)^{\prime}$ gauge
group, the coupling constants are proportional to the charges $Q^{\prime}_{q}$
and $Q^{\prime}_{\ell}$ of quarks and neutrinos under the new gauge symmetry:
$g_{Z^{\prime}}^{qV}=g_{Z^{\prime}}Q^{\prime}_{q}$ and
$g_{Z^{\prime}}^{\nu_{\ell}V}=g_{Z^{\prime}}Q^{\prime}_{\ell}$, where
$g_{Z^{\prime}}$ is the coupling constant of the symmetry group. Since both
the SM and the $Z^{\prime}$ interactions are of vector type, they contribute
coherently to the CE$\nu$NS cross section. Moreover, since the vector current
is conserved, the proton and neutron coupling are given by the sums of the
couplings of their valence quarks. Therefore, the total cross section is
obtained by replacing the SM weak charge $Q^{V}_{\ell,\mathrm{SM}}$ with the
new total weak charge (see Appendix A)

$Q^{V}_{\ell,\mathrm{SM+V}}=Q^{V}_{\ell,\mathrm{SM}}+\frac{g^{2}_{Z^{\prime}}Q_{\ell}^{\prime}}{\sqrt{2}G_{F}\left(|\vec{q}|^{2}+M_{Z^{\prime}}^{2}\right)}\left[\left(2Q_{u}^{\prime}+Q_{d}^{\prime}\right)ZF_{Z}(|\vec{q}|^{2})+\left(Q_{u}^{\prime}+2Q_{d}^{\prime}\right)NF_{N}(|\vec{q}|^{2})\right],$
(16)

with $|\vec{q}|^{2}\simeq 2MT_{\text{nr}}$.

In this work we consider the models listed in Table 1. There are many models
beyond the SM with an additional massive $Z^{\prime}$ vector boson associated
with a new $U(1)^{\prime}$ gauge symmetry (see, e.g., the review in Ref.
Langacker (2009)). A necessary requirement is that the theory is anomaly-free.
However, it is possible to consider effective anomalous models that describe
the interactions of SM fermions with the implicit requirement that the
anomalies are canceled by the contributions of the non-standard fermions of
the full theory. This is the case of the first model that we consider: a
$Z^{\prime}$ boson which couples universally to all SM fermions Liao and
Marfatia (2017); Papoulias and Kosmas (2018); Billard  _et al._ (2018);
Papoulias (2020); Khan and Rodejohann (2019); Cadeddu  _et al._ (2021a);
Bertuzzo  _et al._ (2021). In this case
$Q^{\prime}_{\ell}=Q^{\prime}_{u}=Q^{\prime}_{d}=1$, and the coupling is same
for all the fermions.

Model | $Q^{\prime}_{u}$ | $Q^{\prime}_{d}$ | $Q^{\prime}_{e}$ | $Q^{\prime}_{\mu}$ | $Q^{\prime}_{\tau}$
---|---|---|---|---|---
universal | 1 | 1 | 1 | 1 | 1
$B-L$ | $1/3$ | $1/3$ | $-1$ | $-1$ | $-1$
$B-3L_{e}$ | $1/3$ | $1/3$ | $-3$ | 0 | 0
$B-3L_{\mu}$ | $1/3$ | $1/3$ | 0 | $-3$ | 0
$B-2L_{e}-L_{\mu}$ | $1/3$ | $1/3$ | $-2$ | $-1$ | 0
$B-L_{e}-2L_{\mu}$ | $1/3$ | $1/3$ | $-1$ | $-2$ | 0
$B_{y}+L_{\mu}+L_{\tau}$ | $1/3$ | $1/3$ | 0 | 1 | 1
$L_{e}-L_{\mu}$ | 0 | 0 | 1 | $-1$ | 0
$L_{e}-L_{\tau}$ | 0 | 0 | 1 | 0 | $-1$
$L_{\mu}-L_{\tau}$ | 0 | 0 | 0 | 1 | $-1$
Table 1: The $U(1)^{\prime}$ charges of quarks and leptons in the vector
mediator models considered in this work.

Other models that we consider are anomaly-free if the SM is extended with the
introduction of three right-handed neutrinos (see, e.g., Ref. Allanach  _et
al._ (2019)), which are also beneficial for the generation of the neutrino
masses that are necessary for the explanation of the oscillations of neutrinos
observed in many experiments (see, e.g., Refs. Giunti and Kim (2007); Zyla
_et al._ (2020)). In this case, there is an infinite set of anomaly-free
$U(1)^{\prime}$ gauge groups generated by

$G(c_{1},c_{2},c_{3},c_{e},c_{\mu},c_{\tau})=c_{1}B_{1}+c_{2}B_{2}+c_{3}B_{3}-c_{e}L_{e}-c_{\mu}L_{\mu}-c_{\tau}L_{\tau},$
(17)

where $B_{1}$, $B_{2}$, and $B_{3}$ are the baryon numbers of the three
generations and $L_{\alpha}$ are the lepton numbers for $\alpha=e,\mu,\tau$.
We assume that for each generation the $U(1)^{\prime}$ couplings of the right-
handed neutrino is the same as that of the left-handed neutrino in order to
have vectorial $U(1)^{\prime}$ interactions. Therefore, when we extend the SM
gauge group $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ to $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}\times U(1)^{\prime}$, there are no
$[SU(3)_{C}]^{2}U(1)_{Y}$, $[U(1)^{\prime}]^{3}$ and
$[\text{gravity}]^{2}U(1)^{\prime}$ anomalies, because of the vectorial
character of the involved interactions. The $[U(1)^{\prime}]^{2}U(1)_{Y}$
anomaly cancels because for each generation the difference of the $Y$ charges
of left-handed and right-handed quarks (leptons) is zero. The remaining
$[SU(2)_{L}]^{2}U(1)^{\prime}$ and $[U(1)_{Y}]^{2}U(1)^{\prime}$ anomalies are
canceled with the constraint

$c_{1}+c_{2}+c_{3}-c_{e}-c_{\mu}-c_{\tau}=0.$ (18)

It is often assumed that the quark charges are universal, in order to avoid
unobserved flavor-changing neutral currents in the quark sector. In this case,
we have

$G_{B}(c_{B},c_{e},c_{\mu},c_{\tau})=c_{B}B-c_{e}L_{e}-c_{\mu}L_{\mu}-c_{\tau}L_{\tau},$
(19)

with the constraint (see, e.g., Refs. Lee and Ma (2010); Araki  _et al._
(2012))

$3c_{B}-c_{e}-c_{\mu}-c_{\tau}=0.$ (20)

Here $B=B_{1}+B_{2}+B_{3}$ is the usual baryon number.

We consider the following anomaly-free models that correspond to different
choices of the coefficients in Eq. (17) or (19) and contribute to CE$\nu$NS
interactions of $\nu_{e}$ and $\nu_{\mu}$:

$\bm{B-L=G_{B}(1,1,1,1)}$

Here $L=L_{e}+L_{\mu}+L_{\tau}$ is the total lepton number. This is the most
popular $Z^{\prime}$ model, with a huge literature (see, e.g., the reviews in
Refs. Langacker (2009); Mohapatra (2014); Okada (2018)). It was considered
recently in several CE$\nu$NS phenomenological analyses, e.g. those in Refs.
Miranda  _et al._ (2020); Cadeddu  _et al._ (2021a); Coloma  _et al._ (2021);
de la Vega  _et al._ (2021); Bertuzzo  _et al._ (2021). Note that, since there
are no $\nu_{\tau}$’s in the COHERENT neutrino beam, bounds on the coupling
constant in the anomaly-free model generated by

$G_{B}(1,3/2,3/2,0)=B-\frac{3}{2}\left(L_{e}+L_{\mu}\right),$ (21)

considered, e.g., in Ref. Han  _et al._ (2019), can be obtained from the
bounds on the coupling constant $g_{Z^{\prime}}$ in the $B-L$ model by
rescaling it by the factor $\sqrt{2/3}$, because the $\nu_{e}$ and $\nu_{\mu}$
couplings are changed by the same factor $3/2$.

$\bm{B_{y}+L_{\mu}+L_{\tau}=G(1,-y,y-3,0,-1,-1)}$

In this model, proposed in Ref. Farzan (2015) and considered, e.g., in Ref.
Coloma  _et al._ (2021), $B_{y}=B_{1}-yB_{2}+(y-3)B_{3}$.

$\bm{B-3L_{e}=G_{B}(1,3,0,0)}$

This model was considered, e.g., in Refs. Han  _et al._ (2019); Heeck  _et
al._ (2019); Coloma  _et al._ (2021); de la Vega  _et al._ (2021). In this
case, only the $\nu_{e}$ CE$\nu$NS cross section is affected by the new
$Z^{\prime}$-mediated interaction. Moreover, since there are no $\nu_{\tau}$’s
in the COHERENT neutrino beam, the bounds on the coupling constant
$g_{Z^{\prime}}$ obtained in this model can be extended to all the anomaly-
free models generated by

$G_{B}(1,3w_{e},0,3(1-w_{e}))=B-3w_{e}L_{e}-3(1-w_{e})L_{\tau}$ (22)

through a rescaling of the coupling constant by a factor $1/\sqrt{w_{e}}$.

$\bm{B-3L_{\mu}=G_{B}(1,0,3,0)}$

This model was considered, e.g., in Refs. Heeck  _et al._ (2019); Coloma  _et
al._ (2021); de la Vega  _et al._ (2021). In this case, only the $\nu_{\mu}$
CE$\nu$NS cross section is affected by the new $Z^{\prime}$-mediated
interaction and, in analogy with the argument in the previous item, the bounds
on the coupling constant $g_{Z^{\prime}}$ obtained in this model can be
extended to all the anomaly-free models generated by

$G_{B}(1,0,3w_{\mu},3(1-w_{\mu}))=B-3w_{\mu}L_{\mu}-3(1-w_{\mu})L_{\tau}$ (23)

through a rescaling of the coupling constant by a factor $1/\sqrt{w_{\mu}}$.
For example, the $B-(3/2)(L_{\mu}+L_{\tau})$ considered in Refs. Heeck  _et
al._ (2019); Coloma  _et al._ (2021) is obtained with $w_{\mu}=1/2$.

$\bm{B-2L_{e}-L_{\mu}=G_{B}(1,2,1,0)}$

This model was considered, e.g., in Ref. de la Vega  _et al._ (2021). In
analogy with the discussion in the previous items, the bounds on the coupling
constant $g_{Z^{\prime}}$ obtained in this model can be extended to all the
anomaly-free models generated by

$G_{B}(1,2w_{1},w_{1},3(1-w_{1}))=B-2w_{1}L_{e}-w_{1}L_{\mu}-3(1-w_{1})L_{\tau}$
(24)

through a rescaling of the coupling constant by a factor $1/\sqrt{w_{1}}$.

$\bm{B-L_{e}-2L_{\mu}=G_{B}(1,1,2,0)}$

This model, was considered, e.g., in Ref. de la Vega  _et al._ (2021). Again,
in analogy with the discussion in the previous items, the bounds on the
coupling constant $g_{Z^{\prime}}$ obtained in this model can be extended to
all the anomaly-free models generated by

$G_{B}(1,w_{2},2w_{2},3(1-w_{2}))=B-w_{2}L_{e}-2w_{2}L_{\mu}-3(1-w_{2})L_{\tau}$
(25)

through a rescaling of the coupling constant by a factor $1/\sqrt{w_{2}}$.

Figure 1: Predicted CE$\nu$NS differential event rates corresponding to the
experimental configuration and data taking time of the COHERENT Ar (a, c) and
CsI (b, d) detectors in the vector mediator models considered in this work.

The effects of the models above on the CE$\nu$NS differential event rates that
are predicted for the COHERENT Ar and CsI detectors are illustrated,
respectively, in Figs. 1 and 1. In these figures we choose
$g_{Z^{\prime}}=10^{-4}$ and $M_{Z^{\prime}}=10~{}\text{MeV}$ and we compared
the model predictions with the SM one. One can see that the effects of the
light mediator are similar for the Ar and CsI detectors and the vector boson
mediator contribution increases for small values of
$T_{\text{nr}}\simeq|\vec{q}|^{2}/2M$ because of the propagator in Eq. (16).
The different scales of $T_{\text{nr}}$ in Figs. 1 and 1 are obviously due to
the different masses of the nuclei.

In the case of the universal $Z^{\prime}$ model there is a deep dip due to a
cancellation between the negative SM and the positive $Z^{\prime}$
contributions to the weak charge in Eq. (16). This occurs only in the
universal model because only in this case all the quark and lepton charges are
positive and both $\nu_{e}$ and $\nu_{\mu}$ interact with the $Z^{\prime}$.
Indeed, there is a cancellation for

$T_{\text{nr}}=-\frac{1}{2M}\left(\frac{3g_{Z^{\prime}}^{2}}{\sqrt{2}G_{F}}\,\frac{ZF_{Z}(|\vec{q}|^{2})+NF_{N}(|\vec{q}|^{2})}{g_{V}^{p}ZF_{Z}(|\vec{q}|^{2})+g_{V}^{n}NF_{N}(|\vec{q}|^{2})}+M_{Z^{\prime}}^{2}\right),$
(26)

which occurs at $T_{\text{nr}}\simeq 92~{}\text{keV}$ for Ar in Fig. 1 and
$T_{\text{nr}}\simeq 27~{}\text{keV}$ for CsI in Fig 1.

There is a cancellation for $\nu_{\mu}$ also in the $B_{y}+L_{\mu}+L_{\tau}$
model, since the quarks and $\nu_{\mu}$ have positive charges (see Table 1).
The cancellation occurs at

$T_{\text{nr}}=-\frac{1}{2M}\left(\frac{g_{Z^{\prime}}^{2}}{\sqrt{2}G_{F}}\,\frac{ZF_{Z}(|\vec{q}|^{2})+NF_{N}(|\vec{q}|^{2})}{g_{V}^{p}ZF_{Z}(|\vec{q}|^{2})+g_{V}^{n}NF_{N}(|\vec{q}|^{2})}+M_{Z^{\prime}}^{2}\right),$
(27)

which corresponds to $T_{\text{nr}}\simeq 29~{}\text{keV}$ for Ar in Fig. 1
and $T_{\text{nr}}\simeq 8~{}\text{keV}$ for CsI in Fig. 1. Since in this case
there is no cancellation of the SM contribution of $\nu_{e}$, which does not
interact with the $Z^{\prime}$, there are only shallow dips at these energies
in Figs. 1 and 1 for this model. Note that the total differential rate is
smaller than the SM differential rate for energies above the dip, because the
positive and smaller $Z^{\prime}$ contribution to $Q^{V}_{\mu,\mathrm{SM+V}}$
is added to the dominant negative SM contribution, decreasing the absolute
value of $Q^{V}_{\mu,\mathrm{SM+V}}$.

In all the other models above the quarks and leptons have opposite charges
(see Table 1) and the $Z^{\prime}$ contribution to the weak charge in Eq. (16)
is negative as the SM contribution. Therefore, the total differential rate is
larger than the SM rate for all values of $T_{\text{nr}}$, as shown in Figs. 1
and 1.

We also consider the following three possible $L_{\alpha}-L_{\beta}$ models
that are anomaly-free and can be gauged without extending the SM content with
right-handed neutrinos Foot (1991); Foot  _et al._ (1990); He  _et al._
(1991a); Foot  _et al._ (1993):

$\bm{L_{e}-L_{\mu}=G_{B}(0,-1,1,0)}$

This model, obtained from Eq. (19) with $c_{B}=0$, $c_{e}=-1$ $c_{\mu}=1$, and
$c_{\tau}=0$, was considered, e.g., in Refs. He  _et al._ (1991a, b); Coloma
_et al._ (2021).

$\bm{L_{e}-L_{\tau}=G_{B}(0,-1,0,1)}$

This model, obtained from Eq. (19) with $c_{B}=0$, $c_{e}=-1$ $c_{\mu}=0$, and
$c_{\tau}=1$, was considered, e.g., in Refs. He  _et al._ (1991a, b); Coloma
_et al._ (2021).

$\bm{L_{\mu}-L_{\tau}=G_{B}(0,0,-1,1)}$

This model, obtained from Eq. (19) with $c_{B}=0$, $c_{e}=0$ $c_{\mu}=-1$, and
$c_{\tau}=1$, was considered in many papers, e.g., in Refs. He  _et al._
(1991a); Baek  _et al._ (2001); Altmannshofer  _et al._ (2019); Banerjee and
Roy (2019); Gninenko and Gorbunov (2021); Cadeddu  _et al._ (2021a); Banerjee
_et al._ (2021); Bertuzzo  _et al._ (2021).

Since in these models the $Z^{\prime}$ vector boson does not couple to quarks,
there are no tree-level interactions that contribute to CE$\nu$NS (assuming
the absence of tree-level kinetic mixing). However, there is kinetic mixing of
the $Z^{\prime}$ and the photon at the one-loop level that induces a
contribution to CE$\nu$NS through the photon interaction with quarks
Altmannshofer  _et al._ (2019); Banerjee and Roy (2019); Banerjee  _et al._
(2021). The CE$\nu$NS cross section in these three models is Altmannshofer
_et al._ (2019); Cadeddu  _et al._ (2021a)111 We correct here the sign of the
$Z^{\prime}$ contribution with respect to that used in Ref. Cadeddu  _et al._
(2021a). Let us also note that in the analysis in Ref. Banerjee  _et al._
(2021) the $Z^{\prime}$ contribution has the correct sign, but there is an
additional factor $1/2$ that is incorrect, as shown in Appendix A.

$\displaystyle\left(\frac{d\sigma}{dT_{\mathrm{nr}}}\right)_{L_{\alpha}-L_{\beta}}^{\nu_{\ell}-\mathcal{N}}$
$\displaystyle\left(E,T_{\mathrm{nr}}\right)=\frac{G_{F}^{2}M}{\pi}\left(1-\frac{MT_{\mathrm{nr}}}{2E^{2}}\right)$
$\displaystyle\times$
$\displaystyle\left\\{\left[g_{V}^{p}\left(\nu_{\ell}\right)+\frac{\sqrt{2}\alpha_{\mathrm{EM}}g_{Z^{\prime}}^{2}\left(\delta_{\ell\alpha}\varepsilon_{\beta\alpha}(|\vec{q}|)+\delta_{\ell\beta}\varepsilon_{\alpha\beta}(|\vec{q}|)\right)}{\pi
G_{F}\left(|\vec{q}|^{2}+M_{Z^{\prime}}^{2}\right)}\right]ZF_{Z}(|\vec{q}|^{2})+g_{V}^{n}NF_{N}(|\vec{q}|^{2})\right\\}^{2},$
(28)

where $\alpha_{\mathrm{EM}}$ is the electromagnetic fine-structure constant
and $\varepsilon_{\beta\alpha}(|\vec{q}|)$ is the one-loop kinetic mixing
coupling, that is given by Banerjee and Roy (2019); Banerjee  _et al._ (2021)

$\varepsilon_{\beta\alpha}(|\vec{q}|)=\int_{0}^{1}x(1-x)\ln\left(\frac{m_{\beta}^{2}+x(1-x)|\vec{q}|^{2}}{m_{\alpha}^{2}+x(1-x)|\vec{q}|^{2}}\right)dx\,,$
(29)

where $m_{\beta}$ and $m_{\alpha}$ are the charged lepton masses and we took
into account that for CE$\nu$NS
$q^{2}\simeq-|\vec{q}|^{2}\simeq-2MT_{\mathrm{nr}}$. Note that the
$Z^{\prime}$ contribution is invariant for $\alpha\leftrightarrows\beta$, as
it should be, since $L_{\alpha}-L_{\beta}$ and $L_{\beta}-L_{\alpha}$ are
physically equivalent. Note also that the sign of the loop contribution of the
$i$ charged lepton to $\nu_{\ell}$ scattering is given by
$-Q^{\prime}_{i}Q^{\prime}_{\ell}$, where the minus comes from the negative
electric charge of the charged lepton propagating in the loop. Therefore, the
mass of the charged lepton with the same flavor $\ell$ of the scattering
neutrino is always at the denominator of the logarithm in Eq. (29) and the
mass of the other charged lepton taking part to the new symmetry is always at
the numerator. Figure 2 shows the value of
$\varepsilon_{\beta\alpha}(|\vec{q}|)$ for each of the three
$L_{\alpha}-L_{\beta}$ symmetries as a function of $|\vec{q}|$ in the range of
the COHERENT CE$\nu$NS. One can see that only $\varepsilon_{\tau\mu}$ is
almost constant, because $|\vec{q}|\ll m_{\tau}$ and $|\vec{q}|<m_{\mu}$. In
this case it is possible to approximate
$\varepsilon_{\tau\mu}\simeq\ln(m_{\tau}^{2}/m_{\mu}^{2})/6$, as done in Refs.
Altmannshofer  _et al._ (2019); Cadeddu  _et al._ (2021a); Bertuzzo  _et al._
(2021). On the other hand, for the symmetries $L_{e}-L_{\mu}$ and
$L_{e}-L_{\tau}$ the $|\vec{q}|$ dependence of $\varepsilon_{\beta\alpha}$ on
$|\vec{q}|$ must be taken into account, because $|\vec{q}|\gg m_{e}$.

Figure 2:  Values of $\varepsilon_{\beta\alpha}$ in Eq. (29) for each of the
three $L_{\alpha}-L_{\beta}$ symmetries as a function of
$q=|\vec{q}|\simeq\sqrt{2MT_{\mathrm{nr}}}$ in the range of the COHERENT
CE$\nu$NS data.

Figures 1 and 1 illustrate the effects of the $Z^{\prime}$ contribution to the
CE$\nu$NS differential event rates that are predicted for the COHERENT Ar and
CsI detectors in the $L_{\alpha}-L_{\beta}$ models. In these figures we choose
$g_{Z^{\prime}}=2\times 10^{-3}$ and $M_{Z^{\prime}}=10~{}\text{MeV}$ and we
compared the model predictions with that of the SM. One can see that, as for
the models in Figs. 1 and 1 discussed above, the effects of the light mediator
are similar for the Ar and CsI detectors and the vector boson mediator
contribution increases for small values of
$T_{\text{nr}}\simeq|\vec{q}|^{2}/2M$ because of the propagator in Eq. (16).

In the case of the $L_{\mu}-L_{\tau}$ model the $Z^{\prime}$ contribution to
$Q^{V}_{\mu,\mathrm{SM+V}}$ is positive and there can be a cancellation with
the negative SM contribution. The cancellation occurs at

$T_{\text{nr}}=-\frac{1}{2M}\left(\frac{\alpha_{\mathrm{EM}}g_{Z^{\prime}}^{2}}{3\pi\sqrt{2}G_{F}}\,\ln\\!\left(\frac{m_{\tau}^{2}}{m_{\mu}^{2}}\right)\,\frac{ZF_{Z}(|\vec{q}|^{2})}{g_{V}^{p}ZF_{Z}(|\vec{q}|^{2})+g_{V}^{n}NF_{N}(|\vec{q}|^{2})}+M_{Z^{\prime}}^{2}\right),$
(30)

which corresponds to $T_{\text{nr}}\simeq 23~{}\text{keV}$ for Ar in Fig. 1
and $T_{\text{nr}}\simeq 6~{}\text{keV}$ for CsI in Fig. 1. Since there is no
cancellation of the SM contribution of $\nu_{e}$, which does not interact with
the $Z^{\prime}$, there are only shallow dips at these energies in Figs. 1 and
1 for this model. The total differential rate is smaller than the SM
differential rate for energies above the dip for the same reason that has been
discussed above for the $B_{y}+L_{\mu}+L_{\tau}$ model.

In the case of the $L_{e}-L_{\tau}$ model, there can be a cancellation of the
positive $Z^{\prime}$ contribution to $Q^{V}_{e,\mathrm{SM+V}}$ with the
negative SM contribution, but it is difficult to estimate for which value of
$T_{\text{nr}}$ because of the strong dependence of $\varepsilon_{\tau e}$ on
$T_{\mathrm{nr}}\simeq|\vec{q}|^{2}/2M$ shown in Fig. 2. However, one can see
from Figs. 1 and 1 that there are shallow dips of the differential rates at
values of $T_{\mathrm{nr}}$ that are larger than in the $L_{\mu}-L_{\tau}$
model, because $\varepsilon_{\tau e}>\varepsilon_{\tau\mu}$, as shown in Fig.
2. The dip is more shallow than in the $L_{\mu}-L_{\tau}$ model because the
$\nu_{e}$ contribution to the CE$\nu$NS event rate is smaller than the sum of
the $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ contributions.

In the case of the $L_{e}-L_{\mu}$ model, the situation is more complicated,
because the $Z^{\prime}$ contribution to $Q^{V}_{e,\mathrm{SM+V}}$ is
positive, since $\varepsilon_{\mu e}>0$, but the $Z^{\prime}$ contribution to
$Q^{V}_{\mu,\mathrm{SM+V}}$ is negative, since $\varepsilon_{e\mu}<0$.
Therefore, the $Z^{\prime}$ contributions of the dominant $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ fluxes enhance the CE$\nu$NS differential event rate with
respect to the SM prediction, whereas the subdominant $\nu_{e}$ flux generate
a decrease for sufficiently large values of $T_{\mathrm{nr}}$ (about 40 keV
for Ar in Fig. 1 and 15 keV for CsI in Fig. 1) As a result of these opposite
contributions, the total CE$\nu$NS differential rates of the $L_{e}-L_{\mu}$
model shown in Figs. 1 and 1 are only slightly larger than the SM rates in the
large-$T_{\mathrm{nr}}$ parts of the figures.

Figure 3: Predicted CE$\nu$NS differential event rates corresponding to the
experimental configuration and data taking time of the COHERENT Ar (a) and CsI
(b) detectors in the universal scalar mediator model.

###  III.2 Light scalar mediator

Non-standard neutrino interactions mediated by a scalar boson $\phi$ are
possible if the SM fermion content is extended with the addition of right-
handed neutrinos. The generic Lagrangian that describes the interaction of
$\phi$ with neutrinos and quarks is

$\mathcal{L}_{\phi}^{S}=-\phi\left[\sum_{\ell=e,\mu,\tau}g_{\phi}^{\nu_{\ell}}\,\overline{\nu_{\ell}}\,\nu_{\ell}+\sum_{q=u,d}g_{\phi}^{q}\,\overline{q}\,q\right],$
(31)

where $\nu_{\ell}=\nu_{\ell L}+\nu_{\ell R}$ and $g_{\phi}^{\nu_{\ell}}$ and
$g_{\phi}^{q}$ are the coupling constants. The contribution of the scalar
boson interaction to the CE$\nu$NS cross section adds incoherently to the SM
cross section Lindner  _et al._ (2017); Cerdeño  _et al._ (2016); Farzan  _et
al._ (2018); Aristizabal Sierra  _et al._ (2018, 2019)

$\dfrac{d\sigma_{\nu_{\ell}\text{-}\mathcal{N}}}{dT_{\text{nr}}}=\left(\dfrac{d\sigma_{\nu_{\ell}\text{-}\mathcal{N}}}{dT_{\text{nr}}}\right)_{\mathrm{SM}}+\left(\dfrac{d\sigma_{\nu_{\ell}\text{-}\mathcal{N}}}{dT_{\text{nr}}}\right)_{\mathrm{scalar}},$
(32)

with

$\left(\frac{d\sigma_{\nu_{\ell}\text{-}\mathcal{N}}}{dT_{\text{nr}}}\right)_{\text{scalar}}=\frac{M^{2}T_{\text{nr}}}{4\pi
E^{2}}\,\frac{(g_{\phi}^{\nu_{\ell}})^{2}\mathcal{Q}_{\phi}^{2}}{(|\vec{q}|^{2}+M_{\phi}^{2})^{2}},$
(33)

where $\mathcal{Q}_{\phi}$ is the scalar charge of the nucleus, given by

$\mathcal{Q}_{\phi}=ZF_{Z}(|\vec{q}|^{2})\sum_{q=u,d}g_{\phi}^{q}\langle
p|\bar{q}q|p\rangle+NF_{N}(|\vec{q}|^{2})\sum_{q=u,d}g_{\phi}^{q}\langle
n|\bar{q}q|n\rangle.$ (34)

It is sometimes written as Cerdeño  _et al._ (2016); Farzan  _et al._ (2018);
Aristizabal Sierra  _et al._ (2018, 2019)

$\mathcal{Q}_{\phi}=ZF_{Z}(|\vec{q}|^{2})\sum_{q=u,d}g_{\phi}^{q}\frac{m_{p}}{m_{q}}f_{q}^{p}+NF_{N}(|\vec{q}|^{2})\sum_{q=u,d}g_{\phi}^{q}\frac{m_{n}}{m_{q}}f_{q}^{n},$
(35)

with the quark contributions to the nucleon masses

$f_{q}^{\mathbb{N}}=\frac{m_{q}}{m_{n}}\,\langle\mathbb{N}|\bar{q}q|\mathbb{N}\rangle,$
(36)

for $\mathbb{N}=p,n$. Since the scalar currents are not conserved, the scalar
charges of the nucleons are not simply given by the sums of the charges of
their valence quarks, as in the case of a vector boson mediator (see Eq.
(16)). The proton and neutron matrix elements of the scalar quark current must
be calculated (see, e.g., the recent Refs. Hoferichter  _et al._ (2015); Durr
_et al._ (2016); Ellis  _et al._ (2018); Alexandrou  _et al._ (2020)). For
simplicity, we consider equal couplings for the $u$ and $d$ quarks and equal
couplings for $\nu_{e}$ and $\nu_{\mu}$

$g_{\phi}^{u}=g_{\phi}^{d}=g_{\phi}^{q}\quad\text{and}\quad
g_{\phi}^{\nu_{e}}=g_{\phi}^{\nu_{\mu}}=g_{\phi}^{\nu}.$ (37)

Then, we have

$\mathcal{Q}_{\phi}=g_{\phi}^{q}\left[ZF_{Z}(|\vec{q}|^{2})\langle
p|\bar{u}u+\bar{d}d|p\rangle+NF_{N}(|\vec{q}|^{2})\langle
n|\bar{u}u+\bar{d}d|n\rangle\right].$ (38)

Considering the isospin approximation, we obtain

$\langle p|\bar{u}u+\bar{d}d|p\rangle=\langle
n|\bar{u}u+\bar{d}d|n\rangle=\langle
N|\bar{u}u+\bar{d}d|N\rangle=\frac{\sigma_{\pi N}}{\overline{m}_{ud}},$ (39)

where $\overline{m}_{ud}=(m_{u}+m_{d})/2$ and $\sigma_{\pi N}$ is the pion-
nucleon $\sigma$-term that has been determined in different ways in the
literature (see the recent review in Ref. Alarcón (2021)). Recent values have
been obtained from pionic atoms and pion-nucleon scattering Hoferichter  _et
al._ (2015); Ruiz de Elvira  _et al._ (2018); Friedman and Gal (2019) and from
lattice calculations Durr  _et al._ (2016); Alexandrou  _et al._ (2020). Since
there are large uncertainties on the values of $\sigma_{\pi N}$ and
$\overline{m}_{ud}$, we choose a reference value for $\sigma_{\pi
N}/\overline{m}_{ud}$ given by the ratio of the central value of $\sigma_{\pi
N}$ determined in Ref. Hoferichter  _et al._ (2015) ($\sigma_{\pi
N}=59.1\,\text{MeV}$) and the central PDG values Zyla  _et al._ (2020)
$m_{u}=2.16\,\text{MeV}$ $m_{d}=4.67\,\text{MeV}$, that gives

$\left(\frac{\sigma_{\pi N}}{\overline{m}_{ud}}\right)_{\text{ref}}=17.3,$
(40)

that allows us to write the scalar cross section (33) as

$\left(\frac{d\sigma_{\nu_{\ell}\text{-}\mathcal{N}}}{dT_{\text{nr}}}\right)_{\text{scalar}}=\frac{M^{2}T_{\text{nr}}}{4\pi
E^{2}}\,\frac{\tilde{g}_{\phi}^{4}}{(|\vec{q}|^{2}+M_{\phi}^{2})^{2}}\left(\frac{\sigma_{\pi
N}}{\overline{m}_{ud}}\right)_{\text{ref}}^{2}\left[ZF_{Z}(|\vec{q}|^{2})+NF_{N}(|\vec{q}|^{2})\right]^{2},$
(41)

with

$\tilde{g}_{\phi}^{2}=g_{\phi}^{\nu_{\ell}}g_{\phi}^{q}\,\frac{\sigma_{\pi
N}/\overline{m}_{ud}}{\left(\sigma_{\pi
N}/\overline{m}_{ud}\right)_{\text{ref}}}.$ (42)

In this way the results of other calculations can be compared with our results
by appropriate rescaling of $\tilde{g}_{\phi}$ according with the assumptions.
We guess that $\tilde{g}_{\phi}$ is practically equal to $g_{\phi}$ in Ref.
Miranda  _et al._ (2020), where the expression (35) was used for the scalar
charge of the nucleus, with the values of the $f_{q}^{\mathbb{N}}$’s given in
Ref. Hoferichter  _et al._ (2015), although the assumed values of the quark
masses are not specified. Indeed, the values of the $f_{q}^{\mathbb{N}}$’s in
Ref. Hoferichter  _et al._ (2015) have been obtained from the value of
$\sigma_{\pi N}$ using Eq. (13) of Ref. Crivellin  _et al._ (2014), which
implies

$\sum_{q=u,d}\frac{m_{p}}{m_{q}}f_{q}^{p}=\sum_{q=u,d}\frac{m_{n}}{m_{q}}f_{q}^{n}=\frac{\sigma_{\pi
N}}{\overline{m}_{ud}}.$ (43)

On the other hand, our approach is different from that in Refs. Cerdeño  _et
al._ (2016); Khan and Rodejohann (2019); Suliga and Tamborra (2021), which
considered different values for the proton and neutron matrix elements in Eq.
(38): $\langle p|\bar{u}u+\bar{d}d|p\rangle=15.1$ and $\langle
n|\bar{u}u+\bar{d}d|n\rangle=14$. These values correspond to a rather large
$8\%$ violation of the isospin symmetry.

Let us also note that our treatment neglected the contribution of the strange
and heavier quarks, whose contributions to the nucleon mass have very large
uncertainties (see, e.g., Table 4 of Ref. Cirelli  _et al._ (2013)). If one
wants to consider them, their contributions can be taken into account by
rescaling appropriately $\tilde{g}_{\phi}$, assuming that the coupling of
$\phi$ with all quarks is the same.

Figure 3 illustrates the effect of the scalar boson mediator on the CE$\nu$NS
differential event rates that are predicted for the COHERENT Ar and CsI
detectors for $\tilde{g}_{\phi}=10^{-4}$ and $M_{\phi}=50~{}\text{MeV}$. One
can see that the total CE$\nu$NS rates are larger than the SM rates for all
values of $T_{\text{nr}}$, because the scalar boson cross section adds
incoherently to the SM cross section, according to Eq. (32). In the two panels
of Fig.3 one can also notice that the total CE$\nu$NS rates represented by the
red-dashed lines have small discontinuities at
$T_{\text{nr}}=47.7~{}\text{keV}$ for Ar and $T_{\text{nr}}\simeq
15~{}\text{keV}$ for CsI. These values correspond to the maximum nuclear
kinetic energy $T_{\text{nr}}^{\text{max}}=2E^{2}/M$ for the monoenergetic
$\nu_{\mu}$ from pion decay ($E=29.8~{}\text{MeV}$), as shown by the green-
dashed lines that represent the $\nu_{\mu}$ contributions. One can see that
there is an effect also for the SM differential event rates, which change
slope at the same values of $T_{\text{nr}}$. The effect for the scalar boson
contribution is larger because it is enhanced by the $T_{\text{nr}}$ in the
numerator of the scalar cross section, see Eq. (33). Such a dependence causes
also the decrease of the scalar contribution for very low values of
$T_{\text{nr}}$ that is visible in Fig. 3.

##  IV Constraints on light mediator models

In this Section we present the results of the analyses of the COHERENT CsI and
Ar data with the light-mediator models described in Section III. Since the
data are fitted well by the SM CE$\nu$NS prediction, we obtain constraints on
the mass and coupling of the light mediator in each model. Let us note that
the constraints that can be obtained with previous COHERENT CsI and Ar data
have been presented in Refs. Liao and Marfatia (2017); Denton  _et al._
(2018); Dutta  _et al._ (2019); Abdullah  _et al._ (2018); Cadeddu  _et al._
(2021a); Banerjee  _et al._ (2021) for the more popular universal, $B-L$, and
$L_{\mu}-L_{\tau}$ models and in Ref. Coloma  _et al._ (2021) for the
$B-3L_{e}$, $B-3L_{\mu}$, and $B_{y}+L_{\mu}+L_{\tau}$ models.

In the following subsections, we present the $2\sigma$ (95.45% C.L.) limits
obtained from the COHERENT Ar and CsI data for the models discussed in Section
III and we compare them with the constraints of other experiments by using the
darkcast Ilten  _et al._ (2018) code for recasting the limits in the different
models under consideration. In particular, we compare the constraints on the
light vector boson mediator obtained from the COHERENT data with the excluded
regions obtained from searches of visible dark photon decays in beam dump
(E141 Riordan  _et al._ (1987), E137 Bjorken  _et al._ (1988), E774 Bross  _et
al._ (1991), KEK Konaka  _et al._ (1986), Orsay Davier and Nguyen Ngoc (1989);
Bjorken  _et al._ (2009); Andreas  _et al._ (2012), $\nu$-CAL I Blumlein  _et
al._ (1991, 1992); Blumlein and Brunner (2011); Blümlein and Brunner (2014),
CHARM Bergsma  _et al._ (1985); Gninenko (2012a), NOMAD Astier  _et al._
(2001), and PS191 Bernardi  _et al._ (1986); Gninenko (2012b)), fixed target
(A1 Merkel  _et al._ (2014) and APEX Abrahamyan  _et al._ (2011)), collider
(BaBar Lees  _et al._ (2014), KLOE Archilli  _et al._ (2012); Anastasi  _et
al._ (2016), LHCb Aaij  _et al._ (2018)), and rare-meson-decay (NA48/2 Batley
_et al._ (2015)) experiments, and searches of invisible dark photons decays in
the NA64 Banerjee  _et al._ (2018) and BaBar Lees  _et al._ (2017)
experiments.

Figure 4: Excluded regions (2$\sigma$) in the
$M_{Z^{\prime}}$-$g_{Z^{\prime}}$ plane for the universal vector mediator
model.

###  IV.1 Universal $Z^{\prime}$ model

Figure 4 shows the $2\sigma$ limits that we obtained from the COHERENT Ar and
CsI data for the universal $Z^{\prime}$ model Liao and Marfatia (2017);
Papoulias and Kosmas (2018); Billard  _et al._ (2018); Papoulias (2020); Khan
and Rodejohann (2019); Cadeddu  _et al._ (2021a); Bertuzzo  _et al._ (2021).
The black line delimits the $2\sigma$ allowed regions obtained from the
combined analysis of the CsI and Ar data, while the blue and red lines delimit
the excluded regions obtained from the CsI and Ar data, respectively.

Considering the combined analysis of the CsI and Ar data, one can see that in
the low-mass region the black line, which represents the upper boundary of the
$2\sigma$ allowed region, flattens due to the fact that the contribution of
the $Z^{\prime}$ boson to $Q^{V}_{\ell,\mathrm{SM+V}}$ is small. This happens
for $M_{Z^{\prime}}\ll 100~{}\text{MeV}$, because $g_{Z^{\prime}}$ is small
and the boundary does not depend on $M_{Z^{\prime}}$ since $|\vec{q}|\gg
M_{Z^{\prime}}$ in the $Z^{\prime}$ boson propagator. On the other hand, for
higher masses the contribution of the $Z^{\prime}$ boson is suppressed by a
large $M_{Z^{\prime}}$, which is dominant in the propagator, and the boundary
is given by a diagonal line proportional to $M_{Z^{\prime}}$. The numerical
values of the $2\sigma$ limits in these two simple cases are given in Table 2.

In the upper-middle part of Fig. 4, one can see that another black line
delimits a thin diagonal strip, where
$Q^{V}_{\ell,\mathrm{SM+V}}\simeq-Q^{V}_{\mathrm{SM}}$, corresponding to a
degeneracy with the SM cross section, as explained in Ref. Cadeddu  _et al._
(2021a). Neglecting the form factors and the small proton SM contribution, one
can find that the thin allowed strip corresponds to

$(g_{Z^{\prime}}^{\text{univ}})_{\text{strip}}\simeq\sqrt{\frac{N}{A}\,\frac{\sqrt{2}G_{F}M_{Z^{\prime}}^{2}}{3}}\simeq
1.8\times 10^{-3}\,\frac{M_{Z^{\prime}}}{\text{GeV}},$ (44)

taking into account that $(N/A)_{\text{Ar}}\simeq(N/A)_{\text{CsI}}\simeq
0.58$. Note that the existence of the allowed strip in the universal model is
related to the possibility to have a cancellation of the CE$\nu$NS
differential event rate discussed in Section III (see Eq. (26)) because it is
a consequence of the different signs of the SM and $Z^{\prime}$ contributions
to $Q^{V}_{\ell,\mathrm{SM+V}}$. Indeed, all the models that can have a
cancellation of the CE$\nu$NS differential as discussed in Section III (i.e.
the universal, $B_{y}+L_{\mu}+L_{\tau}$, $L_{e}-L_{\tau}$, and
$L_{\mu}-L_{\tau}$ models) have an allowed strip, as discussed in the
following. The cancellation occurs in the excluded parameter space between the
lower allowed region and the thin allowed strip for

$(g_{Z^{\prime}}^{\text{univ}})_{\text{canc}}\simeq\sqrt{\frac{N}{A}\,\frac{\sqrt{2}G_{F}M_{Z^{\prime}}^{2}}{6}}\simeq
1.3\times 10^{-3}\,\frac{M_{Z^{\prime}}}{\text{GeV}},$ (45)

where we neglected the form factors and the small proton SM contribution.

| Ar | CsI | CsI+Ar
---|---|---|---
model |  $g_{Z^{\prime}}$(low $M_{Z^{\prime}}$) |  $\dfrac{g_{Z^{\prime}}}{M_{Z^{\prime}}}$(high $M_{Z^{\prime}}$) |  $g_{Z^{\prime}}$(low $M_{Z^{\prime}}$) |  $\dfrac{g_{Z^{\prime}}}{M_{Z^{\prime}}}$(high $M_{Z^{\prime}}$) |  $g_{Z^{\prime}}$(low $M_{Z^{\prime}}$) |  $\dfrac{g_{Z^{\prime}}}{M_{Z^{\prime}}}$(high $M_{Z^{\prime}}$)
universal | $3.91\times 10^{-5}$ | $0.82\times 10^{-3}$ | $2.36\times 10^{-5}$ | $0.53\times 10^{-3}$ | $2.07\times 10^{-5}$ | $0.48\times 10^{-3}$
$B-L$ | $5.35\times 10^{-5}$ | $1.67\times 10^{-3}$ | $5.27\times 10^{-5}$ | $1.00\times 10^{-3}$ | $4.42\times 10^{-5}$ | $0.99\times 10^{-3}$
$B_{y}+L_{\mu}+L_{\tau}$ | $10.4\times 10^{-5}$ | $3.58\times 10^{-3}$ | $4.97\times 10^{-5}$ | $1.14\times 10^{-3}$ | $4.47\times 10^{-5}$ | $1.04\times 10^{-3}$
$B-3L_{e}$ | $4.91\times 10^{-5}$ | $1.55\times 10^{-3}$ | $5.16\times 10^{-5}$ | $0.96\times 10^{-3}$ | $4.34\times 10^{-5}$ | $0.95\times 10^{-3}$
$B-3L_{\mu}$ | $3.45\times 10^{-5}$ | $1.09\times 10^{-3}$ | $3.21\times 10^{-5}$ | $0.64\times 10^{-3}$ | $2.76\times 10^{-5}$ | $0.63\times 10^{-3}$
$B-2L_{e}-L_{\mu}$ | $4.62\times 10^{-5}$ | $1.48\times 10^{-3}$ | $4.79\times 10^{-5}$ | $0.89\times 10^{-3}$ | $3.95\times 10^{-5}$ | $0.88\times 10^{-3}$
$B-L_{e}-2L_{\mu}$ | $3.97\times 10^{-5}$ | $1.28\times 10^{-3}$ | $3.86\times 10^{-5}$ | $0.75\times 10^{-3}$ | $3.26\times 10^{-5}$ | $0.74\times 10^{-3}$
$L_{e}-L_{\mu}$ | $161\times 10^{-5}$ | $54.2\times 10^{-3}$ | $166\times 10^{-5}$ | $36.1\times 10^{-3}$ | $137\times 10^{-5}$ | $34.9\times 10^{-3}$
$L_{e}-L_{\tau}$ | $204\times 10^{-5}$ | $71.1\times 10^{-3}$ | $140\times 10^{-5}$ | $29.9\times 10^{-3}$ | $125\times 10^{-5}$ | $26.6\times 10^{-3}$
$L_{\mu}-L_{\tau}$ | $234\times 10^{-5}$ | $80.9\times 10^{-3}$ | $116\times 10^{-5}$ | $26.6\times 10^{-3}$ | $103\times 10^{-5}$ | $24.2\times 10^{-3}$
|  $\tilde{g}_{\phi}$(low $M_{\phi}$) |  $\dfrac{\tilde{g}_{\phi}}{M_{\phi}}$(high $M_{\phi}$) |  $\tilde{g}_{\phi}$(low $M_{\phi}$) |  $\dfrac{\tilde{g}_{\phi}}{M_{\phi}}$(high $M_{\phi}$) |  $\tilde{g}_{\phi}$(low $M_{\phi}$) |  $\dfrac{\tilde{g}_{\phi}}{M_{\phi}}$(high $M_{\phi}$)
scalar | $2.30\times 10^{-5}$ | $0.58\times 10^{-3}$ | $1.80\times 10^{-5}$ | $0.31\times 10^{-3}$ | $1.68\times 10^{-5}$ | $0.30\times 10^{-3}$
Table 2: The $2\sigma$ (95.45% C.L.) upper bounds on the coupling of the new
boson mediator obtained from the separate and combined analyses of the Ar and
CsI COHERENT CE$\nu$NS data for low and high values of the boson mass in the
models considered in this paper. $g_{Z^{\prime}}/M_{Z^{\prime}}$ and
$\tilde{g}_{\phi}/M_{\phi}$ are in units of $\mathrm{GeV}^{-1}$.

One can see from Fig. 4 that the limits obtained from the CsI data are
stricter than those obtained from the Ar data and are close to those of the
combined fit. The limits obtained from the analysis of the Ar data are more
complicated and one can see that there are three corresponding red dashed
lines in Fig. 4. The lowest one represents the upper boundary of the $2\sigma$
allowed region where the contribution of the $Z^{\prime}$ boson to
$Q^{V}_{\ell,\mathrm{SM+V}}$ is small, similarly to the blue-dashed line below
and the black line further below that correspond to the CsI fit and the
combined fit, respectively. The two red-dashed lines above delimit the strip
in which the Ar data are well-fitted
$Q^{V}_{\ell,\mathrm{SM+V}}\simeq-Q^{V}_{\mathrm{SM}}$, as discussed above for
the combined fit. However, since the Ar data are less constraining, the strip
is wider than those obtained from the CsI and combined analyses and it extends
to small values of $M_{Z^{\prime}}$.

In Fig. 4 we compared the limits obtained from the COHERENT CE$\nu$NS data
with those of non-CE$\nu$NS experiments and those of the CONNIE reactor
CE$\nu$NS experiment Aguilar-Arevalo  _et al._ (2020). Figure 4 shows also the
$(g-2)_{\mu}$ $2\sigma$ allowed band which can explain the anomalous magnetic
moment of the muon in this model Abi  _et al._ (2021); Jegerlehner and
Nyffeler (2009) (see Appendix B). One can see that the explanation of the
$(g-2)_{\mu}$ anomaly with the universal model is excluded by the combination
of the non-CE$\nu$NS exclusion limits in Fig. 4, by the CONNIE CE$\nu$NS
bounds alone, and by the COHERENT CE$\nu$NS limits alone, which confirm and
extend the CONNIE CE$\nu$NS bounds. Moreover, the COHERENT CE$\nu$NS limits
extend the total exclusion region by covering a previously not-excluded area
for $20~{}\text{MeV}\lesssim M_{Z^{\prime}}\lesssim 200~{}\text{MeV}$ and
$2\times 10^{-5}\lesssim g_{Z^{\prime}}\lesssim 10^{-4}$. The new COHERENT
CE$\nu$NS limits are consistent with those obtained in Ref. Cadeddu  _et al._
(2021a) using the first COHERENT CsI data and slightly extend the COHERENT
CE$\nu$NS exclusion region.

Figure 5: Excluded regions (2$\sigma$) in the
$M_{Z^{\prime}}$-$g_{Z^{\prime}}$ plane for the $B-L$ 5 and
$B_{y}+L_{\mu}+L_{\tau}$ 5 models.

###  IV.2 $B-L$ model

The gauged $B-L$ model is the most popular $Z^{\prime}$ model (see, e.g., the
reviews in Refs. Langacker (2009); Mohapatra (2014); Okada (2018)) and its
effects in CE$\nu$NS have been studied in Refs. Miranda  _et al._ (2020);
Cadeddu  _et al._ (2021a); Coloma  _et al._ (2021); de la Vega  _et al._
(2021); Bertuzzo  _et al._ (2021) using previous COHERENT data. Figure 5 shows
the $2\sigma$ limits that we obtained from the COHERENT Ar and CsI data,
compared with the limits obtained from other experiments and the $(g-2)_{\mu}$
$2\sigma$ allowed band in this model. One can see that the bounds obtained by
experiments using only leptonic probes are the same as those for the universal
model in Fig. 4, because of the same magnitudes of the lepton charges (see
Tab. 1). The coupling $g_{Z^{\prime}}$ is well constrained by the accelerator
experiments for large values of $M_{Z^{\prime}}$ and fixed target experiments
for small values of $M_{Z^{\prime}}$. Note also that the allowed region for
$(g-2)_{\mu}$ is the same as that in the universal model, because the magnetic
moment of the muon is not dependent on the couplings of quarks.

On the other hand, the CE$\nu$NS bounds are different from the universal
model, because the $Z^{\prime}$ contribution to $Q^{V}_{\mu,\mathrm{SM+V}}$ is
negative and adds to the negative SM contribution. Therefore, in Fig. 5 there
are only the upper bounds shown by the blue-dashed, red-dashed, and black-
solid lines that we obtained from the CsI, Ar, and combined analyses,
respectively. These limits have the same behaviour as the corresponding ones
discussed in Subsection IV.1 for the universal model, but are weaker because
the quark charges are smaller by a factor of 3, as shown in Tab. 1. The
numerical values of the limits for small and large values of $M_{Z^{\prime}}$
are given in Table 2.

Figure 5 shows that, as in the universal model, the COHERENT CE$\nu$NS limit
confirms the exclusion of the explanation of the $(g-2)_{\mu}$ anomaly with
the $B-L$ model and extends the total exclusion region of non-CE$\nu$NS
experiments by covering a previously not-excluded area for
$10~{}\text{MeV}\lesssim M_{Z^{\prime}}\lesssim 200~{}\text{MeV}$ and $5\times
10^{-5}\lesssim g_{Z^{\prime}}\lesssim 3\times 10^{-4}$. Also in this case,
the new COHERENT CE$\nu$NS limits are consistent with those obtained in Ref.
Cadeddu  _et al._ (2021a) using the first COHERENT CsI data and slightly
extend the COHERENT CE$\nu$NS exclusion region.

###  IV.3 $B_{y}+L_{\mu}+L_{\tau}$ model

The $2\sigma$ limits that we obtained for $g_{Z^{\prime}}$ and
$M_{Z^{\prime}}$ in the $B_{y}+L_{\mu}+L_{\tau}$ model Farzan (2015); Coloma
_et al._ (2021) from the COHERENT Ar and CsI data are shown in Fig. 5. One can
see that the result of the analyses of the CsI and combined Ar and CsI data
are qualitatively similar to those discusses in Subsection IV.1 for the
universal model: there is a lower curve that represents the upper boundary of
the $2\sigma$ allowed region where the contribution of the $Z^{\prime}$ boson
to $Q^{V}_{\ell,\mathrm{SM+V}}$ is small and a thin allowed strip where
$Q^{V}_{\ell,\mathrm{SM+V}}\simeq-Q^{V}_{\mathrm{SM}}$, leading to a
degeneracy with the SM cross section that can fit well the data. Neglecting
the form factors and the small proton SM contribution, one can find that in
the case of the $B_{y}+L_{\mu}+L_{\tau}$ model the thin allowed strip lies at

$(g_{Z^{\prime}}^{B_{y}+L_{\mu}+L_{\tau}})_{\text{strip}}\simeq\sqrt{\frac{N}{A}\,\sqrt{2}G_{F}M_{Z^{\prime}}^{2}}\simeq
3.1\times 10^{-3}\,\frac{M_{Z^{\prime}}}{\text{GeV}}.$ (46)

Under the same approximations, one can find that the cancellation between the
SM and $Z^{\prime}$ contributions to $Q^{V}_{\ell,\mathrm{SM+V}}$ occurs in
the parameter space between the lower upper bound curve and the thin allowed
strip for

$(g_{Z^{\prime}}^{B_{y}+L_{\mu}+L_{\tau}})_{\text{canc}}\simeq\sqrt{\frac{N}{A}\,\frac{\sqrt{2}G_{F}M_{Z^{\prime}}^{2}}{2}}\simeq
2.2\times 10^{-3}\,\frac{M_{Z^{\prime}}}{\text{GeV}}.$ (47)

Since the Ar data are less constraining than the CsI data, the $2\sigma$
allowed region in Fig. 5 is that below the upper red-dashed line, with the
exception of the excluded thin strip that corresponds to the cancellation
condition, see Eq. (47).

Figure 5 shows also the LHCb Aaij  _et al._ (2018) limits on $g_{Z^{\prime}}$
in the $B_{y}+L_{\mu}+L_{\tau}$ model and the $(g-2)_{\mu}$ $2\sigma$ allowed
band. One can see that the LHCb bounds exclude the $(g-2)_{\mu}$ allowed band
only for some ranges of values of $M_{Z^{\prime}}$ above about 200 MeV. On the
other hand, the bounds that we obtained from the analysis of the COHERENT
CE$\nu$NS data exclude all the $(g-2)_{\mu}$ allowed band, leading to the
rejection of the explanation of the $(g-2)_{\mu}$ anomaly with the
$B_{y}+L_{\mu}+L_{\tau}$ model.

###  IV.4 $B-3L_{e}$ model

Figure 6 shows the $2\sigma$ limits that we obtained from the COHERENT Ar and
CsI data in the $B-3L_{e}$ model Han  _et al._ (2019); Heeck  _et al._ (2019);
Coloma  _et al._ (2021); de la Vega  _et al._ (2021), compared with the limits
obtained from non-CE$\nu$NS experiments, which are quite strong, because there
are many experiments that probe the interactions of electrons and their
coupling with the $Z^{\prime}$ boson in this model is three times stronger
than that in the $B-L$ model. Strict limits are especially derived from
$e^{+}e^{-}$ collider data.

Note that in Fig. 6 obviously there is no $(g-2)_{\mu}$ allowed region,
because in this model the $Z^{\prime}$ boson does not interact with muonic
flavor. On the other hand, there is the $(g-2)_{e}$ obtained from the
measurement of the magnetic moment of the electron Hanneke  _et al._ (2008,
2011) which is compatible with the prediction at $1.6\sigma$ level taking into
account the recent determination of the fine structure constant Morel  _et
al._ (2020).

These limits that we obtained from the combined analysis of the COHERENT CsI
and Ar CE$\nu$NS data have the same behaviour as the corresponding ones for
the $B-L$ model. They have also similar magnitudes, because the lack of
interaction with $Z^{\prime}$ of the dominant $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ fluxes is compensated by the threefold increase of the
$\nu_{e}$ coupling. The numerical values of the limits for small and large
values of $M_{Z^{\prime}}$ are given in Table 2.

Figure 6 shows that the COHERENT CsI and Ar CE$\nu$NS data allow us to extend
the total exclusion region of non-CE$\nu$NS by covering a previously not-
excluded area for $10~{}\text{MeV}\lesssim M_{Z^{\prime}}\lesssim
100~{}\text{MeV}$ and $5\times 10^{-5}\lesssim g_{Z^{\prime}}\lesssim 2\times
10^{-4}$.

Figure 6: Excluded regions (2$\sigma$) in the
$M_{Z^{\prime}}$-$g_{Z^{\prime}}$ plane for the $B-3L_{e}$ 6 and $B-3L_{\mu}$
6 models.

###  IV.5 $B-3L_{\mu}$ model

Figure 6 shows the $2\sigma$ limits that we obtained from the COHERENT Ar and
CsI data in the $B-3L_{\mu}$ model Heeck  _et al._ (2019); Coloma  _et al._
(2021); de la Vega  _et al._ (2021), compared with the limits obtained from
the LHCb Aaij  _et al._ (2018) experiment ($Z^{\prime}\to\mu^{+}\mu^{-}$),
which exist and are relatively strong only for $M_{Z^{\prime}}\gtrsim
200~{}\text{MeV}$. The figure shows also the $(g-2)_{\mu}$ $2\sigma$ allowed
band in this model, which is not excluded by the LHCb bounds for
$M_{Z^{\prime}}\lesssim 200~{}\text{MeV}$, but it is completely excluded by
the bounds that we obtained from the analysis of the COHERENT CE$\nu$NS data.

Figure 7: Excluded regions (2$\sigma$) in the
$M_{Z^{\prime}}$-$g_{Z^{\prime}}$ plane for the $B-2L_{e}-L_{\mu}$ 7 and
$B-L_{e}-2L_{\mu}$ 7 models.

###  IV.6 $B-2L_{e}-L_{\mu}$ model

In this model de la Vega  _et al._ (2021) both $\nu_{e}$ and $\nu_{\mu}$
interact with the $Z^{\prime}$ boson as in the $B-L$ model, but the
interaction of the subdominant $\nu_{e}$ flux is twice stronger. Therefore the
bounds that we obtained from the analyses of the COHERENT CE$\nu$NS data,
shown in Fig. 7 are similar and slightly stronger than those in the $B-L$
model (see also Tab. 2). From Fig. 7 one can also see that the $(g-2)_{\mu}$
$2\sigma$ allowed band in this model is excluded by the total exclusion limits
of non-CE$\nu$NS experiments. The analysis of the COHERENT CsI and Ar
CE$\nu$NS data allows us to extend the total exclusion region of non-CE$\nu$NS
experiments by covering a previously not-excluded area for
$10~{}\text{MeV}\lesssim M_{Z^{\prime}}\lesssim 100~{}\text{MeV}$ and $5\times
10^{-5}\lesssim g_{Z^{\prime}}\lesssim 2\times 10^{-4}$.

###  IV.7 $B-L_{e}-2L_{\mu}$ model

The phenomenology of this model de la Vega  _et al._ (2021) is similar to that
of the $B-2L_{e}-L_{\mu}$ model, with the difference that the bounds obtained
from the COHERENT CE$\nu$NS data are stronger, because the interactions with
the $Z^{\prime}$ boson of the dominant $\nu_{\mu}$ and $\bar{\nu}_{\mu}$
fluxes are twice stronger than those of the subdominant $\nu_{e}$ flux, as one
can see from Fig. 7 and Tab. 2. One can also see from Fig. 7 that the limits
from non-CE$\nu$NS are weaker than those in Fig. 7 for the $B-2L_{e}-L_{\mu}$
model, whereas those obtained in $\nu_{\mu}$ experiments are stronger. As a
result, the $(g-2)_{\mu}$ $2\sigma$ allowed band in this model is not
completely excluded by the results of non-CE$\nu$NS experiments, but it is
completely excluded by the bounds that we obtained from the analysis of the
COHERENT CsI and Ar CE$\nu$NS data. Moreover, we extend the total exclusion
region of non-CE$\nu$NS experiments by covering a previously not-excluded area
for $10~{}\text{MeV}\lesssim M_{Z^{\prime}}\lesssim 200~{}\text{MeV}$ and
$3\times 10^{-5}\lesssim g_{Z^{\prime}}\lesssim 3\times 10^{-4}$.

Figure 8: Excluded regions (2$\sigma$) in the mass-coupling plane for the
$L_{e}-L_{\tau}$ 8, $L_{e}-L_{\mu}$ 8, $L_{\mu}-L_{\tau}$ 8, and scalar 8
models.

###  IV.8 $L_{e}-L_{\mu}$ model

Figure 8 shows the $2\sigma$ limits that we obtained from the COHERENT Ar and
CsI data in the $L_{e}-L_{\mu}$ model He  _et al._ (1991a, b); Coloma  _et
al._ (2021). As for all the $L_{\alpha}-L_{\beta}$ models the constraints that
we can obtain from CE$\nu$NS data are weaker than those in the previous
models, because the interaction with quarks occurs only at loop level, and
hence it is weaker. This is also shown by the values in Table 2 where one can
see that the bounds in the $L_{\alpha}-L_{\beta}$ are more than one order of
magnitude weaker than those corresponding to the models that we considered in
the previous Subsections. Moreover, in spite of the fact that all the neutrino
fluxes ($\nu_{e}$, $\nu_{\mu}$, and $\bar{\nu}_{\mu}$) interact with the
$Z^{\prime}$ boson in this model, the $Z^{\prime}$ contribution to the
CE$\nu$NS event rate is suppressed by the opposite signs of the $\nu_{e}$ and
$\nu_{\mu}$ contributions to $Q^{V}_{\mu,\mathrm{SM+V}}$ explained at the end
of Subsection III.1 and illustrated by the red-dashed curves in Figs. 1 and 1.

One can see from Fig. 8 that the bounds obtained from the current COHERENT
CE$\nu$NS data are not competitive with those obtained from non-CE$\nu$NS
experiments and do not contribute to the exclusion of the $(g-2)_{\mu}$
$2\sigma$ allowed band in this model. Let us note that most of this band is
excluded by non-CE$\nu$NS experiments, but there is a small non-excluded part
at $M_{Z^{\prime}}\approx 20-30~{}\text{MeV}$ and
$g_{Z^{\prime}}\approx(4-7)\times 10^{-4}$.

###  IV.9 $L_{e}-L_{\tau}$ model

Since in the $L_{e}-L_{\tau}$ model He  _et al._ (1991a, b); Coloma  _et al._
(2021) the dominant $\nu_{\mu}$ in the COHERENT experiment is not interacting
with the $Z^{\prime}$ boson, the bounds on the parameters of the model are
rather weak. From Fig. 8 and Table 2, one can see that they are comparable
with the bounds in the $L_{e}-L_{\mu}$ model, with the difference that there
is an allowed diagonal strip for $M_{Z^{\prime}}\gtrsim 50~{}\text{MeV}$. This
occurs in the $L_{e}-L_{\tau}$ model because of the different signs of the SM
and $Z^{\prime}$ contributions to $Q^{V}_{\ell,\mathrm{SM+V}}$ discussed in
Subsection III.1. The allowed strip is the region of the parameters where
$Q^{V}_{\ell,\mathrm{SM+V}}\simeq-Q^{V}_{\mathrm{SM}}$, leading to a
degeneracy with the SM cross section, as for the similar strips in Fig. 4 for
the universal model and Fig. 5 for the $B_{y}+L_{\mu}+L_{\tau}$ model.
Neglecting the form factors and the small proton SM contribution, this
degeneracy occurs for

$(g_{Z^{\prime}}^{L_{e}-L_{\tau}})_{\text{strip}}\approx\sqrt{\frac{N}{Z}\,\frac{\pi
G_{F}M_{Z^{\prime}}^{2}}{\sqrt{2}\alpha_{\mathrm{EM}}\varepsilon_{\tau
e}}}\approx 6\times 10^{-2}\,\frac{M_{Z^{\prime}}}{\text{GeV}},$ (48)

where we considered $N/Z\approx 1.3$ and $\varepsilon_{\tau e}\approx 1.5$.
One can see from Fig. 8 that the allowed diagonal strip lies along the line
given by Eq. (48).

The non-CE$\nu$NS bounds in Fig. 8 are the same as the bounds in Fig. 8 that
have been obtained with electron-interaction experiments, including that from
$(g-2)_{e}$ value Hanneke  _et al._ (2008, 2011); Morel  _et al._ (2020) that
we already mentioned above in Subsection IV.4 for the $B-3L_{e}$ model. From
Fig. 8 one can see that in the $L_{e}-L_{\tau}$ model the bounds obtained from
the current COHERENT CE$\nu$NS data are not competitive with those obtained
from non-CE$\nu$NS experiments and the non-CE$\nu$NS experiments exclude the
CE$\nu$NS allowed diagonal strip discussed above.

###  IV.10 $L_{\mu}-L_{\tau}$ model

Figure 8 shows the $2\sigma$ limits that we obtained from the COHERENT Ar and
CsI data in the popular $L_{\mu}-L_{\tau}$ model He  _et al._ (1991a); Baek
_et al._ (2001); Altmannshofer  _et al._ (2019); Banerjee and Roy (2019);
Gninenko and Gorbunov (2021); Cadeddu  _et al._ (2021a); Banerjee  _et al._
(2021); Bertuzzo  _et al._ (2021). From Fig. 8 and the values in Table 2, one
can see that the bounds obtained in this model from the COHERENT CE$\nu$NS
data are the strongest among the $L_{\alpha}-L_{\beta}$ models. This is due to
the interaction with the $Z^{\prime}$ boson of the dominant $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ fluxes that is not suppressed by the opposite contribution
of the $\nu_{e}$ flux as in the $L_{e}-L_{\mu}$ model.

From Fig. 8, one can also see that there is an allowed diagonal strip that is
the region of the parameters where
$Q^{V}_{\ell,\mathrm{SM+V}}\simeq-Q^{V}_{\mathrm{SM}}$, which is due to the
different signs of the SM and $Z^{\prime}$ contributions to
$Q^{V}_{\ell,\mathrm{SM+V}}$, as discussed above for other models. Since
$\varepsilon_{\tau\mu}\simeq\ln(m_{\tau}^{2}/m_{\mu}^{2})/6$, as discussed in
Subsection III.1, the allowed diagonal strip corresponds to

$(g_{Z^{\prime}}^{L_{\mu}-L_{\tau}})_{\text{strip}}\approx\sqrt{\frac{N}{Z}\,\frac{6\pi
G_{F}M_{Z^{\prime}}^{2}}{\sqrt{2}\alpha_{\mathrm{EM}}\ln(m_{\tau}^{2}/m_{\mu}^{2})}}\approx
7\times 10^{-2}\,\frac{M_{Z^{\prime}}}{\text{GeV}},$ (49)

where we considered $N/Z\approx 1.3$

One can see form Fig. 8 that in the $L_{\mu}-L_{\tau}$ model there are several
non-CE$\nu$NS constraints whose combination is more stringent than those given
by the current COHERENT CE$\nu$NS data: CMS Sirunyan  _et al._ (2019) ($Z\to
Z^{\prime}\mu\mu\to 4\mu$), BaBar Lees  _et al._ (2016) ($e^{+}e^{-}\to
Z^{\prime}\mu\mu\to 4\mu$), CCFR Mishra  _et al._ (1991); Altmannshofer  _et
al._ (2014) (neutrino trident production), and Borexino Bellini  _et al._
(2011); Kamada and Yu (2015); Gninenko and Gorbunov (2021)
($Z^{\prime}$-mediated solar neutrino interactions). These non-CE$\nu$NS
constraints exclude the allowed diagonal strip corresponding to Eq. (49). On
the other hand, they do not completely exclude the $(g-2)_{\mu}$ $2\sigma$
allowed band in this model, that is shown in Fig. 8. One can see that the part
of this band for $10~{}\text{MeV}\lesssim M_{Z^{\prime}}\lesssim
200~{}\text{MeV}$ and $3\times 10^{-4}\lesssim g_{Z^{\prime}}\lesssim 10^{-3}$
eludes the exclusions.

###  IV.11 Scalar model

Figure 8 shows the $2\sigma$ limits that we obtained from the COHERENT Ar and
CsI data in the scalar boson mediator model described in Subsection III.2. The
figure shows also the $(g-2)_{\mu}$ $2\sigma$ allowed band in this model and
the constraints obtained from the measurement of neutrons scattering on a
208Pb target Barbieri and Ericson (1975); Schmiedmayer  _et al._ (1988); Leeb
and Schmiedmayer (1992), the measurement of $\tau$, mesons, and $Z$ decays
Berryman  _et al._ (2018); Bilenky and Santamaria (1999); Lessa and Peres
(2007); Pasquini and Peres (2016); Krnjaic  _et al._ (2020); Brdar  _et al._
(2020), and double-beta decay experiments Agostini  _et al._ (2015); Gando
_et al._ (2012); Berryman  _et al._ (2018); Blum  _et al._ (2018) (see also
the summary in Ref.Suliga and Tamborra (2021)).

One can see from Fig. 8 that the COHERENT CE$\nu$NS constraints are much more
stringent than the non-CE$\nu$NS bounds for $M_{\phi}\gtrsim 2~{}\text{MeV}$
and they exclude the explanation of the $(g-2)_{\mu}$ anomaly in the scalar
boson mediator model.

##  V Conclusions

In this paper we analyzed the recent CE$\nu$NS data obtained by the COHERENT
Collaboration with the CsI and Ar detectors and we derived constraints on the
coupling and mass of a non-standard light vector or scalar boson mediator
considering several models that have been studied in the literature. We
presented the results obtained from the separate analyses of the CsI and Ar
data and those obtained from the combined analysis of the two datasets.

We considered several models with a light vector boson $Z^{\prime}$: the
anomalous model with universal coupling of the $Z^{\prime}$ vector boson with
all SM fermions (assuming that the quantum anomalies are canceled by the
contributions of the non-standard fermions of an extended full theory),
several anomaly-free models with gauged $U(1)^{\prime}$ symmetries, as the
popular $B-L$ symmetry, in which the $Z^{\prime}$ vector boson couples
directly to quarks and leptons, and the anomaly-free models with gauged
$L_{e}-L_{\mu}$, $L_{e}-L_{\tau}$, and $L_{\mu}-L_{\tau}$ symmetries, in which
the $Z^{\prime}$ vector boson couples directly to the involved lepton flavors
and indirectly to nucleons at the one-loop level.

We compared the constraints obtained from the COHERENT CsI and Ar CE$\nu$NS
data with those obtained from several non-CE$\nu$NS experiments. We showed
that the COHERENT CE$\nu$NS data allow us to extend the excluded regions of
the parameters in the models in which the $Z^{\prime}$ vector boson couples
directly to quarks and in the universal scalar mediator model. In particular,
the total excluded region is extended to smaller values of the coupling
constant $g_{Z^{\prime}}$ for $10~{}\text{MeV}\lesssim M_{Z^{\prime}}\lesssim
100~{}\text{MeV}$ in the universal, $B-L$, $B-3L_{e}$, $B-2L_{e}-L_{\mu}$, and
$B-L_{e}-2L_{\mu}$ models. The regions in the
$M_{Z^{\prime}}$-$g_{Z^{\prime}}$ plane that are excluded by non-CE$\nu$NS
experiments for the $B_{y}+L_{\mu}+L_{\tau}$ and $B-3L_{\mu}$ models are
limited to $M_{Z^{\prime}}\gtrsim 200~{}\text{MeV}$. Therefore, for these
models the COHERENT CE$\nu$NS data allow us to obtain a large extension of the
total excluded region for $M_{Z^{\prime}}\lesssim 200~{}\text{MeV}$.

The models in which the $Z^{\prime}$ couples to muons can explain the
$(g-2)_{\mu}$ anomaly Bennett  _et al._ (2006); Aoyama  _et al._ (2020); Abi
_et al._ (2021), and the allowed band in the $M_{Z^{\prime}}$-$g_{Z^{\prime}}$
plane is tested by non-CE$\nu$NS experiments, as shown in Figs. 4, 5, 6, and
7. The results of our analysis of the COHERENT CE$\nu$NS data exclude the
explanation of the $(g-2)_{\mu}$ anomaly in the models in which the
$Z^{\prime}$ vector boson couples directly to quarks by confirming the
excluded regions of non-CE$\nu$NS experiments and extending the coverage of
the $(g-2)_{\mu}$ allowed band for the $B_{y}+L_{\mu}+L_{\tau}$, $B-3L_{\mu}$,
and $B-L_{e}-2L_{\mu}$ models.

The constraints that we obtained for the $L_{e}-L_{\mu}$, $L_{e}-L_{\tau}$,
and $L_{\mu}-L_{\tau}$ are less stringent because the one-loop interactions of
the $Z^{\prime}$ vector boson with the nucleons is weaker than the direct
interaction. For these models the current COHERENT CE$\nu$NS data allow us to
confirm the exclusion of part of the parameter space that is already covered
by non-CE$\nu$NS experiments, but cannot probe the $(g-2)_{\mu}$ allowed band
in the $L_{e}-L_{\mu}$ and $L_{\mu}-L_{\tau}$ models.

We finally considered CE$\nu$NS interactions mediated by a light scalar boson
$\phi$ assuming for simplicity a universal coupling with the quarks and
neutrinos involved in the CE$\nu$NS processes measured in the COHERENT
experiment. We obtained the strong constraints on the mass $M_{\phi}$ and
coupling of the scalar boson shown in Fig. 8 that greatly extend the region
excluded by non-CE$\nu$NS experiments and rejects the explanation of the
$(g-2)_{\mu}$ anomaly in this model.

###### Acknowledgements.

We would like to thank A. Konovalov and D. Pershey for the useful information
provided for the analysis of the COHERENT data. The work of C. Giunti and C.A.
Ternes is supported by the research grant ”The Dark Universe: A Synergic
Multimessenger Approach” number 2017X7X85K under the program PRIN 2017 funded
by the Ministero dell’Istruzione, Università e della Ricerca (MIUR). The work
of Y.F. Li and Y.Y. Zhang is supported by the National Natural Science
Foundation of China under Grant No. 12075255, 12075254 and No. 11835013. Y.F.
Li is also grateful for the support by the CAS Center for Excellence in
Particle Physics (CCEPP). The work of Y.Y. Zhang is also supported by China
Postdoctoral Science Foundation under Grant No. 2021T140669.

Figure 9:  Feynman diagrams describing 9 the vector part of the Standard Model
neutral-current interaction and 9 the $Z^{\prime}$ vector interaction of left-
handed neutrinos with quarks.

##  Appendix A $Z^{\prime}$ coupling

There is some confusion on the value of the coefficient of the contribution of
a new $Z^{\prime}$ vector boson mediator in Eq. (16) that is obtained assuming
the interaction Lagrangian in Eq. (15). For example, in Refs. Liao and
Marfatia (2017); Papoulias and Kosmas (2018); Papoulias  _et al._ (2018,
2019); Khan and Rodejohann (2019); Dutta  _et al._ (2019) the coefficient is
half of that in Eq. (16). On the other hand, the coefficient in Refs. Bertuzzo
_et al._ (2017); Abdullah  _et al._ (2018); Billard  _et al._ (2018); Han  _et
al._ (2019); Flores  _et al._ (2020); Papoulias (2020); Miranda  _et al._
(2020); Cadeddu  _et al._ (2021a); de la Vega  _et al._ (2021); Bertuzzo  _et
al._ (2021); Bonet  _et al._ (2021) agrees with that in Eq. (16). In this
Appendix we prove that the coefficient in Eq. (16) is the right one.

Let us start by considering the relevant vector part of the Standard Model
neutral-current weak interaction Lagrangian (see, e.g., Refs. Giunti and Kim
(2007); Zyla  _et al._ (2020))

$\mathcal{L}_{Z}^{V}=-\frac{g}{2\cos\vartheta_{\text{W}}}\,Z_{\mu}\left[2g_{V}^{\nu}\sum_{\ell=e,\mu,\tau}\overline{\nu_{\ell
L}}\gamma^{\mu}\nu_{\ell
L}+\sum_{q=u,d}g_{V}^{q}\,\overline{q}\gamma^{\mu}q\right],$ (50)

with the tree-level couplings

$g_{V}^{\nu}=\frac{1}{2},\quad
g_{V}^{u}=\frac{1}{2}-\frac{4}{3}\,\sin^{2}\vartheta_{\text{W}},\quad\text{and}\quad
g_{V}^{d}=-\frac{1}{2}+\frac{2}{3}\,\sin^{2}\vartheta_{\text{W}}.$ (51)

Confronting Eq. (50) with the Lagrangian (15), one can see that the
$Z^{\prime}$ vector interaction of left-handed neutrinos with quarks is
obtained from the vector part of the Standard Model neutral-current
interaction with the substitutions

$\frac{g}{2\cos\vartheta_{\text{W}}}\,2g_{V}^{\nu}\to
g_{Z^{\prime}}^{\nu_{\ell}V},\quad\frac{g}{2\cos\vartheta_{\text{W}}}\,g_{V}^{q}\to
g_{Z^{\prime}}^{qV},\quad\text{and}\quad m_{Z}\to m_{Z^{\prime}}.$ (52)

This correspondence is shown in Fig. 9, where we depicted the two Feynman
diagrams that describe the neutrino-quarks interactions that contribute to
CE$\nu$NS at tree level. The total amplitude is given by the sum of the two
diagrams

$A\propto\frac{g^{2}}{4\cos^{2}\vartheta_{\text{W}}}\,\frac{2g_{V}^{\nu}g_{V}^{q}}{q^{2}-m_{Z}^{2}}+\frac{g_{Z^{\prime}}^{\nu_{\ell}V}g_{Z^{\prime}}^{qV}}{q^{2}-m_{Z^{\prime}}^{2}}.$
(53)

Taking into account that $g_{V}^{\nu}=1/2$ and

$\frac{g^{2}}{4\cos^{2}\vartheta_{\text{W}}m_{Z}^{2}}=\sqrt{2}G_{\text{F}},$
(54)

for $q^{2}\ll m_{Z}^{2}$ we obtain

$A\propto
g_{V}^{q}+\frac{g_{Z^{\prime}}^{\nu_{\ell}V}g_{Z^{\prime}}^{qV}}{\sqrt{2}G_{\text{F}}\left(q^{2}-m_{Z^{\prime}}^{2}\right)}.$
(55)

This relation leads to Eq. (16), taking into account that the conservation of
the vector current implies that

$g_{Z^{\prime}}^{p}=2g_{Z^{\prime}}^{uV}+g_{Z^{\prime}}^{dV}\quad\text{and}\quad
g_{Z^{\prime}}^{n}=g_{Z^{\prime}}^{uV}+2g_{Z^{\prime}}^{dV}.$ (56)

In conclusion of this Appendix, let us note that the results of the analyses
in Refs. Liao and Marfatia (2017); Papoulias and Kosmas (2018); Papoulias  _et
al._ (2018, 2019); Khan and Rodejohann (2019); Dutta  _et al._ (2019), where
the $Z^{\prime}$ contribution to the weak charge in CE$\nu$NS is half of that
in Eq. (16), must be reinterpreted by rescaling their $Z^{\prime}$ coupling
$g_{Z^{\prime}}$ by a factor $\sqrt{2}$.

##  Appendix B Muon $g-2$

Recently, the Fermilab Muon $g-2$ experiment Abi  _et al._ (2021) confirmed
the value of the muon anomalous magnetic moment $(g-2)_{\mu}$ that was
measured in 2006 in the Muon E821 experiment at Brookhaven National Laboratory
Bennett  _et al._ (2006), leading to the combined $4.2\sigma$ deviation from
the Standard Model prediction

$\Delta a_{\mu}=(25.1\pm 5.9)\times 10^{-10},$ (57)

where $a_{\mu}=(g-2)_{\mu}/2$. This $(g-2)_{\mu}$ anomaly may be due to new
physics beyond the SM (see the reviews in Refs. Jegerlehner and Nyffeler
(2009); Keshavarzi  _et al._ (2022); Li  _et al._ (2021)).

In theories beyond the SM, an additional neutral boson $B$ with mass $M_{B}$,
which interacts with muons with coupling $g_{B}$, contributes to the muon
anomalous magnetic moment with Brodsky and De Rafael (1968)

$\delta
a_{\mu}^{B}=\frac{g_{B}^{2}}{8\pi^{2}}\int_{0}^{1}dx\,\frac{Q(x)}{x^{2}+\left(1-x\right)M_{B}^{2}/m_{\mu}^{2}}$
(58)

where $Q(x)$ depends on the scalar or vector nature of the neutral boson $B$:

$Q(x)=\left\\{\begin{array}[]{rl}\displaystyle
x^{2}\left(2-x\right)&\quad\text{(scalar)},\\\ \displaystyle
2x^{2}\left(1-x\right)&\quad\text{(vector)}.\end{array}\right.$ (59)

## References

  * 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), arXiv:1804.09459 [nucl-ex] .
  * Cadeddu  _et al._ (2018a) M. Cadeddu, C. Giunti, Y. F. Li, and Y. Y. Zhang, “Average CsI neutron density distribution from COHERENT data,” Phys. Rev. Lett. 120, 072501 (2018a), arXiv:1710.02730 [hep-ph] .
  * Papoulias  _et al._ (2020) D. K. Papoulias, T. S. Kosmas, R. Sahu, V. K. B. Kota, and M. Hota, “Constraining nuclear physics parameters with current and future COHERENT data,” Phys. Lett. B 800, 135133 (2020), arXiv:1903.03722 [hep-ph] .
  * Coloma  _et al._ (2017) Pilar Coloma, M. C. Gonzalez-Garcia, Michele Maltoni, and Thomas Schwetz, “COHERENT Enlightenment of the Neutrino Dark Side,” Phys. Rev. D 96, 115007 (2017), arXiv:1708.02899 [hep-ph] .
  * Liao and Marfatia (2017) Jiajun Liao and Danny Marfatia, “COHERENT constraints on nonstandard neutrino interactions,” Phys. Lett. B 775, 54–57 (2017), arXiv:1708.04255 [hep-ph] .
  * Lindner  _et al._ (2017) Manfred Lindner, Werner Rodejohann, and Xun-Jie Xu, “Coherent Neutrino-Nucleus Scattering and new Neutrino Interactions,” JHEP 03, 097 (2017), arXiv:1612.04150 [hep-ph] .
  * Giunti (2020) C. Giunti, “General COHERENT constraints on neutrino nonstandard interactions,” Phys. Rev. D 101, 035039 (2020), arXiv:1909.00466 [hep-ph] .
  * Denton  _et al._ (2018) Peter B. Denton, Yasaman Farzan, and Ian M. Shoemaker, “Testing large non-standard neutrino interactions with arbitrary mediator mass after COHERENT data,” JHEP 07, 037 (2018), arXiv:1804.03660 [hep-ph] .
  * Aristizabal Sierra  _et al._ (2018) D. Aristizabal Sierra, Valentina De Romeri, and N. Rojas, “COHERENT analysis of neutrino generalized interactions,” Phys. Rev. D 98, 075018 (2018), arXiv:1806.07424 [hep-ph] .
  * Cadeddu  _et al._ (2018b) M. Cadeddu, C. Giunti, K.A. Kouzakov, Y. F. Li, A.I. Studenikin, and Y. Y. Zhang, “Neutrino Charge Radii from COHERENT Elastic Neutrino-Nucleus Scattering,” Phys.Rev. D98, 113010 (2018b), arXiv:1810.05606 [hep-ph] .
  * Papoulias and Kosmas (2018) D. K. Papoulias and T. S. Kosmas, “COHERENT constraints to conventional and exotic neutrino physics,” Phys. Rev. D 97, 033003 (2018), arXiv:1711.09773 [hep-ph] .
  * Cadeddu  _et al._ (2020a) M. Cadeddu, F. Dordei, C. Giunti, Y. F. Li, and Y. Y. Zhang, “Neutrino, electroweak, and nuclear physics from COHERENT elastic neutrino-nucleus scattering with refined quenching factor,” Phys. Rev. D 101, 033004 (2020a), arXiv:1908.06045 [hep-ph] .
  * Papoulias (2020) Dimitrios K. Papoulias, “COHERENT constraints after the COHERENT-2020 quenching factor measurement,” Phys. Rev. D 102, 113004 (2020), arXiv:1907.11644 [hep-ph] .
  * Khan and Rodejohann (2019) Amir N. Khan and Werner Rodejohann, “New physics from COHERENT data with an improved quenching factor,” Phys. Rev. D 100, 113003 (2019), arXiv:1907.12444 [hep-ph] .
  * Dutta  _et al._ (2019) Bhaskar Dutta, Shu Liao, Samiran Sinha, and Louis E. Strigari, “Searching for Beyond the Standard Model Physics with COHERENT Energy and Timing Data,” Phys. Rev. Lett. 123, 061801 (2019), arXiv:1903.10666 [hep-ph] .
  * Cadeddu and Dordei (2019) Matteo Cadeddu and Francesca Dordei, “Reinterpreting the weak mixing angle from atomic parity violation in view of the Cs neutron rms radius measurement from COHERENT,” Phys. Rev. D 99, 033010 (2019), arXiv:1808.10202 [hep-ph] .
  * Dutta  _et al._ (2020) Bhaskar Dutta, Doojin Kim, Shu Liao, Jong-Chul Park, Seodong Shin, and Louis E. Strigari, “Dark matter signals from timing spectra at neutrino experiments,” Phys. Rev. Lett. 124, 121802 (2020), arXiv:1906.10745 [hep-ph] .
  * Abdullah  _et al._ (2018) Mohammad Abdullah, James B. Dent, Bhaskar Dutta, Gordon L. Kane, Shu Liao, and Louis E. Strigari, “Coherent elastic neutrino nucleus scattering as a probe of a Z´ through kinetic and mass mixing effects,” Phys. Rev. D 98, 015005 (2018), arXiv:1803.01224 [hep-ph] .
  * Ge and Shoemaker (2018) Shao-Feng Ge and Ian M. Shoemaker, “Constraining Photon Portal Dark Matter with Texono and Coherent Data,” JHEP 11, 066 (2018), arXiv:1710.10889 [hep-ph] .
  * Akimov  _et al._ (2021a) D. Akimov _et al._ (COHERENT), “First Measurement of Coherent Elastic Neutrino-Nucleus Scattering on Argon,” Phys. Rev. Lett. 126, 012002 (2021a), arXiv:2003.10630 [nucl-ex] .
  * Akimov  _et al._ (2020) D. Akimov _et al._ (COHERENT), “COHERENT Collaboration data release from the first detection of coherent elastic neutrino-nucleus scattering on argon,”  (2020), 10.5281/zenodo.3903810, arXiv:2006.12659 [nucl-ex] .
  * Akimov  _et al._ (2021b) D. Akimov _et al._ , “Measurement of the Coherent Elastic Neutrino-Nucleus Scattering Cross Section on CsI by COHERENT,” (2021b), arXiv:2110.07730 [hep-ex] .
  * Cadeddu  _et al._ (2020b) M. Cadeddu, F. Dordei, C. Giunti, Y. F. Li, E. Picciau, and Y. Y. Zhang, “Physics results from the first COHERENT observation of coherent elastic neutrino-nucleus scattering in argon and their combination with cesium-iodide data,” Phys. Rev. D 102, 015030 (2020b), arXiv:2005.01645 [hep-ph] .
  * Miranda  _et al._ (2020) O. G. Miranda, D. K. Papoulias, G. Sanchez Garcia, O. Sanders, M. Tórtola, and J. W. F. Valle, “Implications of the first detection of coherent elastic neutrino-nucleus scattering (CEvNS) with Liquid Argon,” JHEP 05, 130 (2020), [Erratum: JHEP 01, 067 (2021)], arXiv:2003.12050 [hep-ph] .
  * Cadeddu  _et al._ (2021a) M. Cadeddu, N. Cargioli, F. Dordei, C. Giunti, Y. F. Li, E. Picciau, and Y. Y. Zhang, “Constraints on light vector mediators through coherent elastic neutrino nucleus scattering data from COHERENT,” JHEP 01, 116 (2021a), arXiv:2008.05022 [hep-ph] .
  * Cadeddu  _et al._ (2021b) M. Cadeddu, N. Cargioli, F. Dordei, C. Giunti, Y. F. Li, E. Picciau, C. A. Ternes, and Y. Y. Zhang, “New insights into nuclear physics and weak mixing angle using electroweak probes,” (2021b), arXiv:2102.06153 [hep-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,” Phys. Rev. D 104, 015015 (2021), arXiv:2103.10196 [hep-ph] .
  * 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] .
  * 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 .
  * 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] .
  * Athron  _et al._ (2021) Peter Athron, Csaba Balázs, Douglas HJ Jacob, Wojciech Kotlarski, Dominik Stöckinger, and Hyejung Stöckinger-Kim, “New physics explanations of $a_{\mu}$ in light of the FNAL muon $g-2$ measurement,” JHEP 09, 080 (2021), arXiv:2104.03691 [hep-ph] .
  * Lindner  _et al._ (2018) Manfred Lindner, Moritz Platscher, and Farinaldo S. Queiroz, “A Call for New Physics : The Muon Anomalous Magnetic Moment and Lepton Flavor Violation,” Phys. Rept. 731, 1–82 (2018), arXiv:1610.06587 [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 .
  * Ma  _et al._ (2002) Ernest Ma, D. P. Roy, and Sourov Roy, “Gauged L(mu) - L(tau) with large muon anomalous magnetic moment and the bimaximal mixing of neutrinos,” Phys. Lett. B 525, 101–106 (2002), arXiv:hep-ph/0110146 .
  * Altmannshofer  _et al._ (2016) Wolfgang Altmannshofer, Marcela Carena, and Andreas Crivellin, “$L_{\mu}-L_{\tau}$ theory of Higgs flavor violation and $(g-2)_{\mu}$,” Phys. Rev. D 94, 095026 (2016), arXiv:1604.08221 [hep-ph] .
  * Amaral  _et al._ (2021) D. W. P. Amaral, D. G. Cerdeno, A. Cheek, and P. Foldenauer, “Confirming $U(1)_{L_{\mu}-L_{\tau}}$ as a solution for $(g-2)_{\mu}$ with neutrinos,” Eur. Phys. J. C 81, 861 (2021), arXiv:2104.03297 [hep-ph] .
  * Cadeddu  _et al._ (2021c) M. Cadeddu, N. Cargioli, F. Dordei, C. Giunti, and E. Picciau, “Muon and electron g-2 and proton and cesium weak charges implications on dark Zd models,” Phys. Rev. D 104, 011701 (2021c), arXiv:2104.03280 [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), 10.1088/1674-1137/ac2a25, arXiv:2104.06858 [hep-ph] .
  * Ko  _et al._ (2021) P. Ko, Takaaki Nomura, and Hiroshi Okada, “Muon $g-2$, $B\to K^{(*)}\mu^{+}\mu^{-}$ anomalies, and leptophilic dark matter in $U(1)_{\mu-\tau}$ gauge symmetry,” (2021), arXiv:2110.10513 [hep-ph] .
  * Hapitas  _et al._ (2021) Timothy Hapitas, Douglas Tuckler, and Yue Zhang, “General Kinetic Mixing in Gauged $U(1)_{L_{\mu}-L_{\tau}}$ Model for Muon $g-2$ and Dark Matter,” (2021), arXiv:2108.12440 [hep-ph] .
  * Cheng  _et al._ (2021) Yu Cheng, Xiao-Gang He, and Jin Sun, “Widening the $U(1)_{L_{\mu}-L_{\tau}}$ $Z^{\prime}$ mass range for resolving the muon $g-2$ anomaly,” (2021), arXiv:2112.09920 [hep-ph] .
  * Jegerlehner and Nyffeler (2009) Fred Jegerlehner and Andreas Nyffeler, “The Muon g-2,” Phys. Rept. 477, 1–110 (2009), arXiv:0902.3360 [hep-ph] .
  * Langacker (2009) Paul Langacker, “The Physics of Heavy $Z^{\prime}$ Gauge Bosons,” Rev. Mod. Phys. 81, 1199–1228 (2009), arXiv:0801.1345 [hep-ph] .
  * Billard  _et al._ (2018) Julien Billard, Joseph Johnston, and Bradley J. Kavanagh, “Prospects for exploring New Physics in Coherent Elastic Neutrino-Nucleus Scattering,” JCAP 11, 016 (2018), arXiv:1805.01798 [hep-ph] .
  * Bertuzzo  _et al._ (2021) Enrico Bertuzzo, Giovanni Grilli di Cortona, and Lucas Magno D. Ramos, “Probing light vector mediators with coherent scattering at future facilities,” (2021), arXiv:2112.04020 [hep-ph] .
  * Mohapatra (2014) Rabindra N. Mohapatra, “From Old Symmetries to New Symmetries: Quarks, Leptons and B-L,” Int. J. Mod. Phys. A 29, 1430066 (2014), arXiv:1409.7557 [hep-ph] .
  * Okada (2018) Satomi Okada, “$Z^{\prime}$ Portal Dark Matter in the Minimal $B-L$ Model,” Adv. High Energy Phys. 2018, 5340935 (2018), arXiv:1803.06793 [hep-ph] .
  * Foot (1991) Robert Foot, “New Physics From Electric Charge Quantization?” Mod. Phys. Lett. A 6, 527–530 (1991).
  * Foot  _et al._ (1990) Robert Foot, Girish C. Joshi, H. Lew, and R. R. Volkas, “Charge quantization in the standard model and some of its extensions,” Mod. Phys. Lett. A 5, 2721–2732 (1990).
  * He  _et al._ (1991a) X. G. He, Girish C. Joshi, H. Lew, and R. R. Volkas, “New-$Z^{\prime}$ phenomenology,” Phys. Rev. D 43, R22–R24 (1991a).
  * Foot  _et al._ (1993) Robert Foot, H. Lew, and R. R. Volkas, “Electric charge quantization,” J. Phys. G 19, 361–372 (1993), [Erratum: J.Phys.G 19, 1067 (1993)], arXiv:hep-ph/9209259 .
  * 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] .
  * 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] .
  * Cerdeño  _et al._ (2016) David G. Cerdeño, Malcolm Fairbairn, Thomas Jubb, Pedro A. N. Machado, Aaron C. Vincent, and Céline Bœhm, “Physics from solar neutrinos in dark matter direct detection experiments,” JHEP 05, 118 (2016), [Erratum: JHEP 09, 048 (2016)], arXiv:1604.01025 [hep-ph] .
  * Farzan  _et al._ (2018) Yasaman Farzan, Manfred Lindner, Werner Rodejohann, and Xun-Jie Xu, “Probing neutrino coupling to a light scalar with coherent neutrino scattering,” JHEP 05, 066 (2018), arXiv:1802.05171 [hep-ph] .
  * Aristizabal Sierra  _et al._ (2019) D. Aristizabal Sierra, Bhaskar Dutta, Shu Liao, and Louis E. Strigari, “Coherent elastic neutrino-nucleus scattering in multi-ton scale dark matter experiments: Classification of vector and scalar interactions new physics signals,” JHEP 12, 124 (2019), arXiv:1910.12437 [hep-ph] .
  * Akimov  _et al._ (2021c) D. Akimov _et al._ (COHERENT), “Measurement of scintillation response of CsI[Na] to low-energy nuclear recoils by COHERENT,” (2021c), arXiv:2111.02477 [physics.ins-det] .
  * Baker and Cousins (1984) Steve Baker and Robert D. Cousins, “Clarification of the Use of Chi Square and Likelihood Functions in Fits to Histograms,” Nucl. Instrum. Meth. 221, 437–442 (1984).
  * Zyla  _et al._ (2020) P. A. Zyla _et al._ (Particle Data Group), “Review of Particle Physics,” PTEP 2020, 083C01 (2020).
  * Drukier and Stodolsky (1984) A. Drukier and Leo Stodolsky, “Principles and Applications of a Neutral Current Detector for Neutrino Physics and Astronomy,” Phys. Rev. D 30, 2295 (1984).
  * Barranco  _et al._ (2005) J. Barranco, O. G. Miranda, and T. I. Rashba, “Probing new physics with coherent neutrino scattering off nuclei,” JHEP 12, 021 (2005), arXiv:hep-ph/0508299 .
  * Patton  _et al._ (2012) Kelly Patton, Jonathan Engel, Gail C. McLaughlin, and Nicolas Schunck, “Neutrino-nucleus coherent scattering as a probe of neutron density distributions,” Phys. Rev. C 86, 024612 (2012), arXiv:1207.0693 [nucl-th] .
  * Erler and Su (2013) Jens Erler and Shufang Su, “The Weak Neutral Current,” Prog. Part. Nucl. Phys. 71, 119–149 (2013), arXiv:1303.5522 [hep-ph] .
  * Helm (1956) Richard H. Helm, “Inelastic and Elastic Scattering of 187-Mev Electrons from Selected Even-Even Nuclei,” Phys. Rev. 104, 1466–1475 (1956).
  * Piekarewicz  _et al._ (2016) J. Piekarewicz, A. R. Linero, P. Giuliani, and E. Chicken, “Power of two: Assessing the impact of a second measurement of the weak-charge form factor of 208Pb,” Phys. Rev. C 94, 034316 (2016), arXiv:1604.07799 [nucl-th] .
  * Klein and Nystrand (1999) Spencer Klein and Joakim Nystrand, “Exclusive vector meson production in relativistic heavy ion collisions,” Phys. Rev. C 60, 014903 (1999), arXiv:hep-ph/9902259 .
  * Fricke  _et al._ (1995) G. Fricke, C. Bernhardt, K. Heilig, L. A. Schaller, L. Schellenberg, E. B. Shera, and C. W. de Jager, “Nuclear Ground State Charge Radii from Electromagnetic Interactions,” Atom. Data Nucl. Data Tabl. 60, 177–285 (1995).
  * Angeli and Marinova (2013) I. Angeli and K. P. Marinova, “Table of experimental nuclear ground state charge radii: An update,” Atom. Data Nucl. Data Tabl. 99, 69–95 (2013).
  * Huang and Chen (2019) Xu-Run Huang and Lie-Wen Chen, “Neutron Skin in CsI and Low-Energy Effective Weak Mixing Angle from COHERENT Data,” Phys. Rev. D 100, 071301 (2019), arXiv:1902.07625 [hep-ph] .
  * Hoferichter  _et al._ (2020) Martin Hoferichter, Javier Menéndez, and Achim Schwenk, “Coherent elastic neutrino-nucleus scattering: EFT analysis and nuclear responses,” Phys. Rev. D 102, 074018 (2020), arXiv:2007.08529 [hep-ph] .
  * Papoulias  _et al._ (2018) D. K. Papoulias, R. Sahu, T. S. Kosmas, V. K. B. Kota, and B. Nayak, “Novel neutrino-floor and dark matter searches with deformed shell model calculations,” Adv. High Energy Phys. 2018, 6031362 (2018), arXiv:1804.11319 [hep-ph] .
  * Papoulias  _et al._ (2019) D. K. Papoulias, T. S. Kosmas, and Y. Kuno, “Recent probes of standard and non-standard neutrino physics with nuclei,” Front. in Phys. 7, 191 (2019), arXiv:1911.00916 [hep-ph] .
  * Bertuzzo  _et al._ (2017) Enrico Bertuzzo, Frank F. Deppisch, Suchita Kulkarni, Yuber F. Perez Gonzalez, and Renata Zukanovich Funchal, “Dark Matter and Exotic Neutrino Interactions in Direct Detection Searches,” JHEP 04, 073 (2017), arXiv:1701.07443 [hep-ph] .
  * Han  _et al._ (2019) Tao Han, Jiajun Liao, Hongkai Liu, and Danny Marfatia, “Nonstandard neutrino interactions at COHERENT, DUNE, T2HK and LHC,” JHEP 11, 028 (2019), arXiv:1910.03272 [hep-ph] .
  * Flores  _et al._ (2020) L. J. Flores, Newton Nath, and Eduardo Peinado, “Non-standard neutrino interactions in U(1)’ model after COHERENT data,” JHEP 06, 045 (2020), arXiv:2002.12342 [hep-ph] .
  * de la Vega  _et al._ (2021) Leon M. G. de la Vega, L. J. Flores, Newton Nath, and Eduardo Peinado, “Complementarity between dark matter direct searches and CE$\nu$NS experiments in U(1)’ models,” JHEP 09, 146 (2021), arXiv:2107.04037 [hep-ph] .
  * Bonet  _et al._ (2021) H. Bonet _et al._ (CONUS), “Novel constraints on neutrino physics beyond the standard model from the CONUS experiment,” (2021), arXiv:2110.02174 [hep-ph] .
  * Allanach  _et al._ (2019) B. C. Allanach, Joe Davighi, and Scott Melville, “An Anomaly-free Atlas: charting the space of flavour-dependent gauged $U(1)$ extensions of the Standard Model,” JHEP 02, 082 (2019), [Erratum: JHEP 08, 064 (2019)], arXiv:1812.04602 [hep-ph] .
  * Giunti and Kim (2007) Carlo Giunti and Chung W. Kim,  _Fundamentals of Neutrino Physics and Astrophysics_ (2007).
  * Lee and Ma (2010) Hye-Sung Lee and Ernest Ma, “Gauged $B-x_{i}L$ origin of $R$ Parity and its implications,” Phys. Lett. B 688, 319–322 (2010), arXiv:1001.0768 [hep-ph] .
  * Araki  _et al._ (2012) Takeshi Araki, Julian Heeck, and Jisuke Kubo, “Vanishing Minors in the Neutrino Mass Matrix from Abelian Gauge Symmetries,” JHEP 07, 083 (2012), arXiv:1203.4951 [hep-ph] .
  * Coloma  _et al._ (2021) Pilar Coloma, M. C. Gonzalez-Garcia, and Michele Maltoni, “Neutrino oscillation constraints on U(1)’ models: from non-standard interactions to long-range forces,” JHEP 01, 114 (2021), arXiv:2009.14220 [hep-ph] .
  * Farzan (2015) Yasaman Farzan, “A model for large non-standard interactions of neutrinos leading to the LMA-Dark solution,” Phys. Lett. B 748, 311–315 (2015), arXiv:1505.06906 [hep-ph] .
  * Heeck  _et al._ (2019) Julian Heeck, Manfred Lindner, Werner Rodejohann, and Stefan Vogl, “Non-Standard Neutrino Interactions and Neutral Gauge Bosons,” SciPost Phys. 6, 038 (2019), arXiv:1812.04067 [hep-ph] .
  * He  _et al._ (1991b) Xiao-Gang He, Girish C. Joshi, H. Lew, and R. R. Volkas, “Simplest $Z^{\prime}$ model,” Phys. Rev. D 44, 2118–2132 (1991b).
  * Gninenko and Gorbunov (2021) Sergei Gninenko and Dmitry Gorbunov, “Refining constraints from Borexino measurements on a light Z’-boson coupled to L$\mu$-L$\tau$ current,” Phys. Lett. B 823, 136739 (2021), arXiv:2007.16098 [hep-ph] .
  * Hoferichter  _et al._ (2015) Martin Hoferichter, J. Ruiz de Elvira, Bastian Kubis, and Ulf-G. Meißner, “High-Precision Determination of the Pion-Nucleon $\sigma$ Term from Roy-Steiner Equations,” Phys. Rev. Lett. 115, 092301 (2015), arXiv:1506.04142 [hep-ph] .
  * Durr  _et al._ (2016) S. Durr _et al._ , “Lattice computation of the nucleon scalar quark contents at the physical point,” Phys. Rev. Lett. 116, 172001 (2016), arXiv:1510.08013 [hep-lat] .
  * Ellis  _et al._ (2018) John Ellis, Natsumi Nagata, and Keith A. Olive, “Uncertainties in WIMP Dark Matter Scattering Revisited,” Eur. Phys. J. C 78, 569 (2018), arXiv:1805.09795 [hep-ph] .
  * Alexandrou  _et al._ (2020) C. Alexandrou, S. Bacchio, M. Constantinou, J. Finkenrath, K. Hadjiyiannakou, K. Jansen, G. Koutsou, and A. Vaquero Aviles-Casco, “Nucleon axial, tensor, and scalar charges and $\sigma$-terms in lattice QCD,” Phys. Rev. D 102, 054517 (2020), arXiv:1909.00485 [hep-lat] .
  * Alarcón (2021) J. M. Alarcón, “Brief history of the pion–nucleon sigma term,” Eur. Phys. J. ST 230, 1609–1622 (2021).
  * Ruiz de Elvira  _et al._ (2018) Jacobo Ruiz de Elvira, Martin Hoferichter, Bastian Kubis, and Ulf-G. Meißner, “Extracting the $\sigma$-term from low-energy pion-nucleon scattering,” J. Phys. G 45, 024001 (2018), arXiv:1706.01465 [hep-ph] .
  * Friedman and Gal (2019) E. Friedman and A. Gal, “The pion-nucleon $\sigma$ term from pionic atoms,” Phys. Lett. B 792, 340–344 (2019), arXiv:1901.03130 [nucl-th] .
  * Crivellin  _et al._ (2014) Andreas Crivellin, Martin Hoferichter, and Massimiliano Procura, “Accurate evaluation of hadronic uncertainties in spin-independent WIMP-nucleon scattering: Disentangling two- and three-flavor effects,” Phys. Rev. D 89, 054021 (2014), arXiv:1312.4951 [hep-ph] .
  * Suliga and Tamborra (2021) Anna M. Suliga and Irene Tamborra, “Astrophysical constraints on nonstandard coherent neutrino-nucleus scattering,” Phys. Rev. D 103, 083002 (2021), arXiv:2010.14545 [hep-ph] .
  * Cirelli  _et al._ (2013) Marco Cirelli, Eugenio Del Nobile, and Paolo Panci, “Tools for model-independent bounds in direct dark matter searches,” JCAP 10, 019 (2013), arXiv:1307.5955 [hep-ph] .
  * Ilten  _et al._ (2018) Philip Ilten, Yotam Soreq, Mike Williams, and Wei Xue, “Serendipity in dark photon searches,” JHEP 06, 004 (2018), arXiv:1801.04847 [hep-ph] .
  * Riordan  _et al._ (1987) E. M. Riordan _et al._ , “A Search for Short Lived Axions in an Electron Beam Dump Experiment,” Phys. Rev. Lett. 59, 755 (1987).
  * Bjorken  _et al._ (1988) J. D. Bjorken, S. Ecklund, W. R. Nelson, A. Abashian, C. Church, B. Lu, L. W. Mo, T. A. Nunamaker, and P. Rassmann, “Search for Neutral Metastable Penetrating Particles Produced in the SLAC Beam Dump,” Phys. Rev. D 38, 3375 (1988).
  * Bross  _et al._ (1991) A. Bross, M. Crisler, Stephen H. Pordes, J. Volk, S. Errede, and J. Wrbanek, “A Search for Shortlived Particles Produced in an Electron Beam Dump,” Phys. Rev. Lett. 67, 2942–2945 (1991).
  * Konaka  _et al._ (1986) A. Konaka _et al._ , “Search for Neutral Particles in Electron Beam Dump Experiment,” Phys. Rev. Lett. 57, 659 (1986).
  * Davier and Nguyen Ngoc (1989) M. Davier and H. Nguyen Ngoc, “An Unambiguous Search for a Light Higgs Boson,” Phys. Lett. B 229, 150–155 (1989).
  * Bjorken  _et al._ (2009) James D. Bjorken, Rouven Essig, Philip Schuster, and Natalia Toro, “New Fixed-Target Experiments to Search for Dark Gauge Forces,” Phys. Rev. D 80, 075018 (2009), arXiv:0906.0580 [hep-ph] .
  * Andreas  _et al._ (2012) Sarah Andreas, Carsten Niebuhr, and Andreas Ringwald, “New Limits on Hidden Photons from Past Electron Beam Dumps,” Phys. Rev. D 86, 095019 (2012), arXiv:1209.6083 [hep-ph] .
  * Blumlein  _et al._ (1991) J. Blumlein _et al._ , “Limits on neutral light scalar and pseudoscalar particles in a proton beam dump experiment,” Z. Phys. C 51, 341–350 (1991).
  * Blumlein  _et al._ (1992) J. Blumlein _et al._ , “Limits on the mass of light (pseudo)scalar particles from Bethe-Heitler e+ e- and mu+ mu- pair production in a proton - iron beam dump experiment,” Int. J. Mod. Phys. A 7, 3835–3850 (1992).
  * Blumlein and Brunner (2011) Johannes Blumlein and Jurgen Brunner, “New Exclusion Limits for Dark Gauge Forces from Beam-Dump Data,” Phys. Lett. B 701, 155–159 (2011), arXiv:1104.2747 [hep-ex] .
  * Blümlein and Brunner (2014) Johannes Blümlein and Jürgen Brunner, “New Exclusion Limits on Dark Gauge Forces from Proton Bremsstrahlung in Beam-Dump Data,” Phys. Lett. B 731, 320–326 (2014), arXiv:1311.3870 [hep-ph] .
  * Bergsma  _et al._ (1985) F. Bergsma _et al._ (CHARM), “Search for Axion Like Particle Production in 400-GeV Proton - Copper Interactions,” Phys. Lett. B 157, 458–462 (1985).
  * Gninenko (2012a) S. N. Gninenko, “Constraints on sub-GeV hidden sector gauge bosons from a search for heavy neutrino decays,” Phys. Lett. B 713, 244–248 (2012a), arXiv:1204.3583 [hep-ph] .
  * Astier  _et al._ (2001) P. Astier _et al._ (NOMAD), “Search for heavy neutrinos mixing with tau neutrinos,” Phys. Lett. B 506, 27–38 (2001), arXiv:hep-ex/0101041 .
  * Bernardi  _et al._ (1986) G. Bernardi _et al._ , “Search for Neutrino Decay,” Phys. Lett. B 166, 479–483 (1986).
  * Gninenko (2012b) S. N. Gninenko, “Stringent limits on the $\pi^{0}\to\gamma X,X\to e+e-$ decay from neutrino experiments and constraints on new light gauge bosons,” Phys. Rev. D 85, 055027 (2012b), arXiv:1112.5438 [hep-ph] .
  * Merkel  _et al._ (2014) H. Merkel _et al._ , “Search at the Mainz Microtron for Light Massive Gauge Bosons Relevant for the Muon g-2 Anomaly,” Phys. Rev. Lett. 112, 221802 (2014), arXiv:1404.5502 [hep-ex] .
  * Abrahamyan  _et al._ (2011) S. Abrahamyan _et al._ (APEX), “Search for a New Gauge Boson in Electron-Nucleus Fixed-Target Scattering by the APEX Experiment,” Phys. Rev. Lett. 107, 191804 (2011), arXiv:1108.2750 [hep-ex] .
  * Lees  _et al._ (2014) J. P. Lees _et al._ (BaBar), “Search for a Dark Photon in $e^{+}e^{-}$ Collisions at BaBar,” Phys. Rev. Lett. 113, 201801 (2014), arXiv:1406.2980 [hep-ex] .
  * Archilli  _et al._ (2012) F. Archilli _et al._ (KLOE-2), “Search for a vector gauge boson in $\phi$ meson decays with the KLOE detector,” Phys. Lett. B 706, 251–255 (2012), arXiv:1110.0411 [hep-ex] .
  * Anastasi  _et al._ (2016) A. Anastasi _et al._ (KLOE-2), “Limit on the production of a new vector boson in $\mathrm{e^{+}e^{-}}\rightarrow{\rm U}\gamma$, U$\rightarrow\pi^{+}\pi^{-}$ with the KLOE experiment,” Phys. Lett. B 757, 356–361 (2016), arXiv:1603.06086 [hep-ex] .
  * Aaij  _et al._ (2018) Roel Aaij _et al._ (LHCb), “Search for Dark Photons Produced in 13 TeV $pp$ Collisions,” Phys. Rev. Lett. 120, 061801 (2018), arXiv:1710.02867 [hep-ex] .
  * Batley  _et al._ (2015) J. R. Batley _et al._ (NA48/2), “Search for the dark photon in $\pi^{0}$ decays,” Phys. Lett. B 746, 178–185 (2015), arXiv:1504.00607 [hep-ex] .
  * Banerjee  _et al._ (2018) D. Banerjee _et al._ (NA64), “Search for vector mediator of Dark Matter production in invisible decay mode,” Phys. Rev. D 97, 072002 (2018), arXiv:1710.00971 [hep-ex] .
  * Lees  _et al._ (2017) J. P. Lees _et al._ (BaBar), “Search for Invisible Decays of a Dark Photon Produced in ${e}^{+}{e}^{-}$ Collisions at BaBar,” Phys. Rev. Lett. 119, 131804 (2017), arXiv:1702.03327 [hep-ex] .
  * Aguilar-Arevalo  _et al._ (2020) Alexis Aguilar-Arevalo _et al._ (CONNIE), “Search for light mediators in the low-energy data of the CONNIE reactor neutrino experiment,” JHEP 04, 054 (2020), arXiv:1910.04951 [hep-ex] .
  * Hanneke  _et al._ (2008) D. Hanneke, S. Fogwell, and G. Gabrielse, “New Measurement of the Electron Magnetic Moment and the Fine Structure Constant,” Phys. Rev. Lett. 100, 120801 (2008), arXiv:0801.1134 [physics.atom-ph] .
  * Hanneke  _et al._ (2011) D. Hanneke, S. Fogwell Hoogerheide, and G. Gabrielse, “Cavity Control of a Single-Electron Quantum Cyclotron: Measuring the Electron Magnetic Moment,” Phys. Rev. A 83, 052122 (2011), arXiv:1009.4831 [physics.atom-ph] .
  * Morel  _et al._ (2020) Léo Morel, Zhibin Yao, Pierre Cladé, and Saïda Guellati-Khélifa, “Determination of the fine-structure constant with an accuracy of 81 parts per trillion,” Nature 588, 61–65 (2020).
  * 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] .
  * Mishra  _et al._ (1991) S. R. Mishra _et al._ (CCFR), “Neutrino tridents and W Z interference,” Phys. Rev. Lett. 66, 3117–3120 (1991).
  * 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] .
  * Bellini  _et al._ (2011) G. Bellini _et al._ , “Precision measurement of the 7Be solar neutrino interaction rate in Borexino,” Phys. Rev. Lett. 107, 141302 (2011), arXiv:1104.1816 [hep-ex] .
  * 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] .
  * Barbieri and Ericson (1975) Riccardo Barbieri and Torleif Erik Oskar Ericson, “Evidence Against the Existence of a Low Mass Scalar Boson from Neutron-Nucleus Scattering,” Phys. Lett. B 57, 270–272 (1975).
  * Schmiedmayer  _et al._ (1988) J. Schmiedmayer, H. Rauch, and P. Riehs, “Measurement of the Electric Polarizability of the Neutron,” Phys. Rev. Lett. 61, 1065–1068 (1988).
  * Leeb and Schmiedmayer (1992) H. Leeb and J. Schmiedmayer, “Constraint on hypothetical light interacting bosons from low-energy neutron experiments,” Phys. Rev. Lett. 68, 1472–1475 (1992).
  * Berryman  _et al._ (2018) Jeffrey M. Berryman, André De Gouvêa, Kevin J. Kelly, and Yue Zhang, “Lepton-Number-Charged Scalars and Neutrino Beamstrahlung,” Phys. Rev. D 97, 075030 (2018), arXiv:1802.00009 [hep-ph] .
  * Bilenky and Santamaria (1999) Mikhail S. Bilenky and Arcadi Santamaria, “’Secret’ neutrino interactions,” in  _Neutrino Mixing: Meeting in Honor of Samoil Bilenky’s 70th Birthday_ (1999) pp. 50–61, arXiv:hep-ph/9908272 .
  * Lessa and Peres (2007) A. P. Lessa and O. L. G. Peres, “Revising limits on neutrino-Majoron couplings,” Phys. Rev. D 75, 094001 (2007), arXiv:hep-ph/0701068 .
  * Pasquini and Peres (2016) P. S. Pasquini and O. L. G. Peres, “Bounds on Neutrino-Scalar Yukawa Coupling,” Phys. Rev. D 93, 053007 (2016), [Erratum: Phys.Rev.D 93, 079902 (2016)], arXiv:1511.01811 [hep-ph] .
  * Krnjaic  _et al._ (2020) Gordan Krnjaic, Gustavo Marques-Tavares, Diego Redigolo, and Kohsaku Tobioka, “Probing Muonphilic Force Carriers and Dark Matter at Kaon Factories,” Phys. Rev. Lett. 124, 041802 (2020), arXiv:1902.07715 [hep-ph] .
  * Brdar  _et al._ (2020) Vedran Brdar, Manfred Lindner, Stefan Vogl, and Xun-Jie Xu, “Revisiting neutrino self-interaction constraints from $Z$ and $\tau$ decays,” Phys. Rev. D 101, 115001 (2020), arXiv:2003.05339 [hep-ph] .
  * Agostini  _et al._ (2015) M. Agostini _et al._ , “Results on $\beta\beta$ decay with emission of two neutrinos or Majorons in76 Ge from GERDA Phase I,” Eur. Phys. J. C 75, 416 (2015), arXiv:1501.02345 [nucl-ex] .
  * Gando  _et al._ (2012) A. Gando _et al._ (KamLAND-Zen), “Limits on Majoron-emitting double-beta decays of Xe-136 in the KamLAND-Zen experiment,” Phys. Rev. C 86, 021601 (2012), arXiv:1205.6372 [hep-ex] .
  * Blum  _et al._ (2018) Kfir Blum, Yosef Nir, and Michal Shavit, “Neutrinoless double-beta decay with massive scalar emission,” Phys. Lett. B 785, 354–361 (2018), arXiv:1802.08019 [hep-ph] .
  * Keshavarzi  _et al._ (2022) Alex Keshavarzi, Kim Siang Khaw, and Tamaki Yoshioka, “Muon g-2: A review,” Nucl. Phys. B 975, 115675 (2022), arXiv:2106.06723 [hep-ex] .
  * Li  _et al._ (2021) Song Li, Yang Xiao, and Jin Min Yang, “A pedagogical review on muon $g-2$,” Physics 4, 40–47 (2021), arXiv:2110.04673 [hep-ph] .
  * Brodsky and De Rafael (1968) Stanley J. Brodsky and Eduardo De Rafael, “SUGGESTED BOSON - LEPTON PAIR COUPLINGS AND THE ANOMALOUS MAGNETIC MOMENT OF THE MUON,” Phys. Rev. 168, 1620–1622 (1968).