#  Opportunities for Probing $U(1)_{T3R}$ with Light Mediators

Bhaskar<EMAIL_ADDRESS>Mitchell Institute for Fundamental
Physics and Astronomy, Department of Physics and Astronomy,
Texas A$\&$M University, College Station, Texas 77843-4242, USA  Sumit
<EMAIL_ADDRESS>Mitchell Institute for Fundamental Physics and
Astronomy, Department of Physics and Astronomy,
Texas A$\&$M University, College Station, Texas 77843-4242, USA  Jason
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Hawaii, Honolulu, Hawaii 96822, USA

We consider strategies for using new datasets to probe scenarios in which
light right-handed SM fermions couple to a new gauge group, $U(1)_{T3R}$. This
scenario provides a natural explanation for the light flavor sector scale, and
a motivation for sub-GeV dark matter. There is parameter space which is
currently allowed, but we find that much of it can be probed with future
experiments. In particular, cosmological and astrophysical observations,
neutrino experiments and experiments which search for displaced visible decay
or invisible decay can all play a role. Still, there is a small region of
parameter space which even these upcoming experiments will not be able to
probe. This model can explain the observed 2.4-3$\sigma$ excess of events at
the COHERENT experiment in the parameter space allowed by current laboratory
experiments, but the ongoing/upcoming laboratory experiments will decisively
probe this possibility.

3cm(13cm,-6.5cm)  MI-TH-2021, UH511-1315-2020

##  I Introduction

A well-motivated scenario for new physics beyond the Standard Model (BSM) is
the existence of new gauge groups. One well-studied gauge group, first
considered in the context of left-right-models [1, 2, 3], is $U(1)_{T3R}$. In
this scenario, a set of right-handed Standard Model (SM) fermions are charged
under the new gauge group, while left-handed SM fermions remain uncharged. It
was recently pointed out [4] that, if there is a low symmetry-breaking scale,
then this scenario naturally lead to sub-GeV dark matter because the dark
sector mass scale is tied to that of the light SM fermions.

There are a variety of experimental and observational bounds on this scenario,
but there is still open parameter space which evades all current constraints,
and in which the dark matter candidate can achieve the correct relic density.
In this work, we will consider prospects for future datasets to definitively
probe the entirety of the open parameter space.

In this scenario, some right-handed first- and/or second-generation fermions
are charged under $U(1)_{T3R}$, and this symmetry protects their masses. If
the symmetry-breaking scale is $\lesssim{\cal O}(10)$ GeV, then these SM
fermions naturally obtain sub-GeV masses. Moreover, this symmetry-breaking
scale can naturally feed into the dark sector, yielding sub-GeV dark matter
which interacts with the Standard Model through the processes mediated by a
low-mass dark photon ($A^{\prime}$) or dark Higgs ($\phi^{\prime}$).

We will find that, under certain assumptions, this scenario can be tightly
constrained by cosmological and astrophysical observations. Collider, fixed-
target, beam-dump, and coherent elastic neutrino-nucleus scattering
(CE$\nu$NS) experiments can produce large fluxes of the dark photon and dark
Higgs, which can be effectively probed by searching either for visible decays
at distant detectors, or evidence for invisible decays. Invisible decays of
the dark photon and dark Higgs can also produce a flux of either sterile
neutrinos or of dark matter, which can be searched for at distant detectors
which look for scattering. We will see that models in the allowed parameter
space can explain the COHERENT excess, which is associated with dark
matter/sterile neutrinos emerging from the decays of the dark photon [5].
Finally, this scenario can also produce non-standard interactions (NSI) of
active neutrinos, mediated by the $A^{\prime}$ or $\phi^{\prime}$, which are
being probed in various types of neutrino experiments.

These new searches can potentially rule out or find evidence for models in
much of the allowed parameter space. But there are still regions of the
parameter space which will evade bounds from upcoming experiments; for these
models, the mediators decay rapidly into SM particles, leaving no signals at
displaced detectors and signals at nearby detectors which are difficult to
distinguish from background.

The outline of this paper is as follows. In Sec. II, we describe the model,
and the interactions of the dark photon and dark Higgs. In Sec. III, we
describe constraints arising from cosmological and astrophysical observables.
In Sec. IV, we describe constraints arising from visible decays at displaced
detectors. We describe constraints arising from decays at nearby detectors in
Sec. V. In Sec. VI, we describe constraints on dark matter or sterile neutrino
scattering at detectors displaced from a beam source. In Sec. VII, we describe
constraints on non-standard active neutrino interactions. We conclude with a
discussion of our results including a summary table in Sec. VIII.

##  II Model

The details of this scenario are described in Ref. [4], but we briefly review
the relevant properties here. A set of right-handed Standard Model fermions
are charged under $U(1)_{T3R}$, with up-type fermions having charge $+2$ and
down-type fermions having charge $-2$. If a full generation of right-handed SM
fermions (including a right-handed neutrino) are charged under $U(1)_{T3R}$,
then all gauge and gravitational anomalies cancel. The gauge boson of
$U(1)_{T3R}$ is the dark photon, $A^{\prime}$, with gauge coupling $g_{T3R}$.

We assume that the fermions charged under $U(1)_{T3R}$ are first- or second-
generation, with sub-GeV masses. Note that, to cancel anomalies, it is
necessary for a right-handed up-type quark, down-type quark, charged lepton
and neutrino to be charged under $U(1)_{T3R}$, but they need not all be in the
same generation. As shown in [6], it is technically natural for one quark (the
up-type quark, for example) and the charged lepton to be mass eigenstates. But
if second-generation quarks are charged under $U(1)_{T3R}$, then there are
tight constraints arising from measurements of anomalous Kaon decay. We will
therefore assume that the right-handed $u$\- and $d$-quarks are charged under
$U(1)_{T3R}$. We will also see that there are tight bounds on the scenario
where electrons are charged under $U(1)_{T3R}$, arising from cosmological
observables [7] and atomic parity violation experiments [8]. We will therefore
assume that the right-handed muon is charged under $U(1)_{T3R}$. We also add a
left-handed and right-handed fermion pair $\eta_{L}$ and $\eta_{R}$, which are
SM singlets, but are charged under $U(1)_{T3R}$ with charges $\pm 1$. The
$U(1)_{T3R}$ charge assignments are summarized in Table. 1

Table 1:  The charges of the fields which transform under $U(1)_{T3R}$. For the fermionic fields, the shown charges are for the left-handed component of each Weyl spinor. field | $u_{R}$ | $d_{R}$ | $\mu_{R}$ | $\nu_{R}$ | $\eta_{L}$ | $\eta_{R}$ | $\phi$
---|---|---|---|---|---|---|---
$q_{T3R}$ | -2 | 2 | 2 | -2 | 1 | -1 | 2

With the given field content, the interaction Lagrangian can be written as,

$\displaystyle\mathcal{L}=$ $\displaystyle-$
$\displaystyle\frac{\lambda_{u}}{\Lambda}\tilde{H}\phi^{*}\bar{Q}_{L}u_{R}-\frac{\lambda_{d}}{\Lambda}H\phi\bar{Q}_{L}d_{R}-\frac{\lambda_{\nu}}{\Lambda}\tilde{H}\phi^{*}\bar{L}_{L}\nu_{R}-\frac{\lambda_{\mu}}{\Lambda}H\phi\bar{L}_{L}\mu_{R}-m_{D}\bar{\eta}_{R}\eta_{L}-\frac{1}{2}\lambda_{L}\phi\bar{\eta}^{c}_{L}\eta_{L}$
(1) $\displaystyle-$
$\displaystyle\frac{1}{2}\lambda_{R}\phi^{*}\bar{\eta}^{c}_{R}\eta_{R}-\mu_{\phi}^{2}\phi^{*}\phi-\lambda_{\phi}(\phi^{*}\phi)^{2}+H.c.,$

where $Q_{L}$ and $L_{L}$ denote the left-handed SM quark and lepton doublet,
respectively; $H$ is the SM Higgs doublet; and $\tilde{H}$ is defined as
$\tilde{H}$$=$$i\tau_{2}H^{*}$.

$U(1)_{T3R}$ is broken by the condensation of a field $\phi$, which has charge
$+2$. This breaks $U(1)_{T3R}$ down to a parity, under which SM fermions are
even. We can then express this field as $\phi=V+\phi^{\prime}/\sqrt{2}$, where
$V$ is the vacuum expectation value of $\phi$, and $\phi^{\prime}$ is the dark
Higgs with mass $m_{\phi^{\prime}}$$=$$2\lambda_{\phi}^{1/2}V$. The mass
matrix for $\eta$ contains both Dirac terms, $m_{D}$, and Majorana terms,
$m_{M}$, the latter of which are necessarily proportional to $V$ as
$m_{M}=\lambda_{M}V$, where we assume that $\lambda_{L}=\lambda_{R}$. We
assume that the Dirac terms are small compared to the Majorana terms, leaving
us with two Majorana fermions, $\eta_{1}$ and $\eta_{2}$, with masses
$m_{1}=m_{M}-m_{D}$ and $m_{2}=m_{M}+m_{D}$ respectively. The mass splitting
is very small, $\Delta m=2m_{D}$. The $\eta_{1,2}$ are odd under the surviving
parity, and the lighter one, $\eta_{1}$ is a dark matter candidate. In the
low-energy effective field theory below the electroweak symmetry-breaking
scale, SM fermions have a Yukawa coupling term and the mass term of the form

$\displaystyle{\cal L}=$ $\displaystyle-$ $\displaystyle
m_{u}\bar{u}_{L}u_{R}-m_{d}\bar{d}_{L}d_{R}-m_{\nu
D}\bar{\nu}_{L}\nu_{R}-m_{\mu}\bar{\mu}_{L}\mu_{R}-\frac{1}{2}m_{1}\bar{\eta}_{1}{\eta}_{1}-\frac{1}{2}m_{2}\bar{\eta}_{2}{\eta}_{2}-\frac{m_{u}}{V\sqrt{2}}\bar{u}_{L}u_{R}\phi^{\prime}$
(2) $\displaystyle-$
$\displaystyle\frac{m_{d}}{V\sqrt{2}}\bar{d}_{L}d_{R}\phi^{\prime}-\frac{m_{\nu
D}}{V\sqrt{2}}\bar{\nu}_{L}\nu_{R}\phi^{\prime}-\frac{m_{\mu}}{V\sqrt{2}}\bar{\mu}_{L}\mu_{R}\phi^{\prime}-\frac{1}{2\sqrt{2}}\frac{m_{1}}{V}\bar{\eta}_{1}{\eta}_{1}\phi^{\prime}-\frac{1}{2\sqrt{2}}\frac{m_{2}}{V}\bar{\eta}_{2}{\eta}_{2}\phi^{\prime}+H.c.,$

The mass matrix for $\nu_{L,R}$ contains a Dirac mass term, $m_{\nu_{D}}$,
which is proportional to $V$, and can contain a Majorana mass for $\nu_{R}$
which scales as $\propto V^{2}/\Lambda$, where $\Lambda$ is some high-energy
scale. As such, we expect the Majorana mass to be smaller than $V$. The
diagonalization of the squared mass matrix will yield two mass eigenstates,
$\nu_{A}$ and $\nu_{S}$. We will assume that the active neutrino $\nu_{A}$ is
mostly $\nu_{L}$, with only a small mixing of $\nu_{R}$.

The interactions in the gauge sector can be explored by defining the covariant
derivative as,

$\displaystyle
D_{\mu}\mathbb{I}={\partial}_{\mu}\mathbb{I}+i\frac{g}{2}{\tau}_{a}W_{\mu
a}+ig^{\prime}YB_{\mu}+i\frac{g_{T_{3R}}}{2}Q_{T_{3R}}A^{\prime}_{\mu}.$ (3)

where $g$, $g^{\prime}$ and $g_{T_{3R}}$ represent the coupling constants of
the $SU(2)_{L}$, $U(1)_{Y}$ and $U(1)_{T_{3R}}$ groups, respectively. And the
respective gauge bosons are denoted by $W_{\mu}$, $B_{\mu}$ and
$A^{\prime}_{\mu}$. The mass of the dark photon, $A^{\prime}$, can be obtained
from $|D_{\mu}\phi|^{2}$ as $m_{A^{\prime}}^{2}=2g_{T_{3R}}^{2}V^{2}$. The
interactions involving the gauge boson $A^{\prime}$ are then given by,

$\displaystyle\mathcal{L}_{\text{gauge}}$ $\displaystyle=$
$\displaystyle\frac{m_{A^{\prime}}}{4\sqrt{2}V}A^{\prime}_{\mu}(\bar{\eta}_{1}\gamma^{\mu}\eta_{2}-\bar{\eta}_{2}\gamma^{\mu}\eta_{1})+\frac{m_{A^{\prime}}^{2}}{V\sqrt{2}}\phi^{\prime}A^{\prime}_{\mu}A^{\prime}{}^{\mu}+\frac{m_{A^{\prime}}^{2}}{4V^{2}}\phi^{\prime}\phi^{\prime}A^{\prime}_{\mu}{A^{\prime}}^{\mu}-\frac{m_{A^{\prime}}}{2\sqrt{2}V}j^{\mu}_{A^{\prime}}{A^{\prime}}_{\mu}.$
(4)

where the interaction current for the SM fermions is defined as,
$j^{\mu}_{A^{\prime}}=\sum\limits_{f}Q_{T_{3R}}^{f}\bar{f}\gamma^{\mu}\left(\frac{1+\gamma_{5}}{2}\right)f$.
Note that the $\eta$ fields have only off-diagonal vector interaction with
$A^{\prime}$.

All the SM fermion masses and the DM masses are proportional to the symmetry
breaking scale $V$, and their masses are $\lesssim V$. If we assume that there
is no other suppression due to any other flavor physics, $V=\mathcal{O}(1)$
GeV would naturally give rise to sub-GeV masses for the fermions with
$\mathcal{O}(1)$ Yukawa couplings. But this would be ruled out by the current
constraints. Therefore we choose the symmetry breaking scale to be $V=10$ GeV,
leading to couplings which are moderately smaller than $\mathcal{O}(1)$. The
dark photon and the dark Higss masses will also be $\leq\mathcal{O}(1)$ GeV.

If $m_{\nu_{S}}>2m_{\mu}$, then the tree-level decay process
$\nu_{S}\rightarrow\mu^{+}\mu^{-}\nu_{A}$ will occur rapidly. For
$m_{\nu_{S}}<2m_{\mu}$, the sterile neutrino $\nu_{S}$ can decay via the
process $\nu_{S}\rightarrow\nu_{A}\gamma\gamma$, with a rate

$\displaystyle\Gamma_{\nu_{S}}$ $\displaystyle\propto$
$\displaystyle\alpha_{em}^{2}\frac{m_{\nu_{S}}^{7}m_{\nu_{D}}^{2}}{m_{\phi^{\prime}}^{4}V^{4}}.$
(5)

For $V=10\,{\rm GeV}$, $m_{\phi^{\prime}}\sim 100\,{\rm MeV}$,
$m_{\nu_{S}}=10~{}\,{\rm MeV}$, $m_{\nu_{D}}=10^{-3}~{}\,{\rm MeV}$ we find
$\tau_{\nu_{S}}\sim{\cal O}(10^{13})\,{\rm s}$. So we may essentially assume
that the light sterile neutrino is stable for the purpose of laboratory
experiments, though it need not be cosmologically stable. For points in
parameter space at which the sterile neutrino is very light, it will also be a
dark matter component. Note, the $\nu_{S}\rightarrow\nu_{A}\gamma$ decay
process is also possible through a transition dipole interaction, but this
process arises at two loop level and is therefore highly suppressed. This
decay cannot proceed through a vector interaction, as a result of gauge
invariance.

###  II.I Corrections to $g-2$

Note that the muon anomalous magnetic moment will receive corrections arising
from diagrams in which either $\phi^{\prime}$ or $A^{\prime}$ run in the loop.
The correction to $a_{\mu}=(g_{\mu}-2)/2$ due to one-loop diagrams involving
$A^{\prime}$ and $\phi^{\prime}$ is given by [9]

$\displaystyle\Delta a_{\mu}$ $\displaystyle=$
$\displaystyle\frac{m_{\mu}^{4}}{16\pi^{2}V^{2}}\int_{0}^{1}dx\frac{(1-x)^{2}(1+x)}{(1-x)^{2}m_{\mu}^{2}+xm_{\phi^{\prime}}^{2}}+\frac{m_{\mu}^{2}}{32\pi^{2}V^{2}}\int_{0}^{1}dx\frac{2x(1-x)(x-2)m_{A^{\prime}}^{2}-2x^{3}m_{\mu}^{2}}{x^{2}m_{\mu}^{2}+(1-x)m_{A^{\prime}}^{2}}.$

But it is important to note that $g_{\mu}-2$ can also receive corrections for
high-scale physics which is disconnected from the dark sector. As such,
$g_{\mu}-2$ really constrains the amount of fine-tuning which is needed in
order to match the data. As shown in [4], the allowed parameter space for this
model would require a fine-tuning against high-scale physics at the 1% level
if $V=10~{}\,{\rm GeV}$. This would be reduced to 10% if we instead adopted
$V=30~{}\,{\rm GeV}$.

###  II.II $A^{\prime}$ Interactions and Decays

The dark photon has a tree-level coupling to some right-handed SM fermions
($u_{R}$, $d_{R}$, $\mu_{R}$ and $\nu_{R}$), with coupling strength given by,

$\displaystyle g_{T3R}=\frac{m_{A^{\prime}}}{\sqrt{2}V}.$ (7)

The relation between $g_{T3R}$ and $m_{A^{\prime}}$ is shown in Fig. 1 for
various choices of $V$.

Figure 1:  The relation between the coupling constant $g_{T3R}$ and the gauge
boson mass $m_{A^{\prime}}$ for three different values of $V=1,10,30$ GeV. For
phenomenological study we set the value $V=10$ GeV in rest of the paper.

The $A^{\prime}$ has a vector coupling to all other charged SM fermions, with
coupling given by $\epsilon e$, where $\epsilon$ is a kinetic mixing
parameter. The kinetic mixing parameter receives a one-loop contribution from
the right-handed fermions charged under $U(1)_{T3R}$ ($\sim
g_{T3R}\sqrt{\alpha_{em}/4\pi^{3}}$), as shown in Fig. 2.

Figure 2:  The one loop diagrams which give the mixing induced coupling
between the SM fields to $A^{\prime}$. Here, $f_{R}=\mu_{R},u_{R},d_{R}$.

But $\epsilon$ can also receive a tree-level contribution in the low-energy
effective field theory. One source of these contributions could be the
integrating out of heavy degrees of freedom which are also charged under
$U(1)_{T3R}$. We will thus consider the kinetic mixing parameter $\epsilon$ to
be a free parameter.

$A^{\prime}$ also has an off-diagonal coupling to the vector current

$\displaystyle j_{\eta}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\bar{\eta}_{1}\gamma^{\mu}\eta_{2}-\bar{\eta}_{2}\gamma^{\mu}\eta_{1}).$
(8)

Note, this coupling can only be off-diagonal, because it descends from a
vector coupling to $\eta_{L,R}$. This Dirac fermion splits into two Majorana
fermions as a result of symmetry-breaking, but the diagonal vector current for
a Majorana fermion vanishes identically.

Since we take the $A^{\prime}$ to be lighter than $2m_{\mu}$ in order to avoid
bounds from BaBar [10, 11], the only potentially allowed two-body final states
are $\eta_{1,2}\eta_{2,1}$, $\nu\nu$, and $e^{+}e^{-}$, of which only the last
one is visible. Note, the decay $A^{\prime}\rightarrow\gamma\gamma$ is
forbidden by the Landau-Yang theorem [12, 13]. The relevant $A^{\prime}$ decay
rates are

$\displaystyle\Gamma_{\eta_{1}\eta_{2}}^{A^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{m_{A^{\prime}}^{3}}{96\pi
V^{2}}\left(1-\frac{4m_{\eta}^{2}}{m_{A^{\prime}}^{2}}\right)^{1/2}\left(1+\frac{2m_{\eta}^{2}}{m_{A^{\prime}}^{2}}\right),$
$\displaystyle\Gamma_{\nu_{S}\nu_{S}}^{A^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{m_{A^{\prime}}^{3}}{12\pi
V^{2}}\left(1-\frac{4m_{\nu_{S}}^{2}}{m_{A^{\prime}}^{2}}\right)^{3/2},$
$\displaystyle\Gamma_{e^{+}e^{-}}^{A^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\epsilon^{2}\alpha_{em}m_{A^{\prime}}}{3}\left(1-\frac{4m_{e}^{2}}{m_{A^{\prime}}^{2}}\right)^{1/2}\left(1+\frac{2m_{e}^{2}}{m_{A^{\prime}}^{2}}\right).$
(9)

where we have assumed that $m_{\eta_{2}}-m_{\eta_{1}}$ is negligible. Note,
$A^{\prime}$ has no visible two-body decays if $m_{A^{\prime}}<2m_{e}$. If
either the $\eta_{1}\eta_{2}$ or $\nu_{S}\nu_{S}$ final states are
kinematically allowed, then those tree-level decays will dominate the
branching fraction. Moreover, they will be prompt unless $m_{A^{\prime}}$ is
very small.

If neither of those states are kinematically allowed, then the dominant decays
will be to either $\nu_{S}\nu_{A}$, $\nu_{A}\nu_{A}$, or $e^{+}e^{-}$. The
first two of these are suppressed by powers of the mixing angle, while the
last is suppressed by the kinetic mixing parameter.

###  II.III $\phi^{\prime}$ Interactions and Decays

$\phi^{\prime}$ couples to the SM fermions charged under $U(1)_{T3R}$, as well
as to $\eta_{1}$ and $\eta_{2}$, with a coupling given by $m_{f}/\sqrt{2}V$.
$\phi^{\prime}$ couples to $\nu_{L}\nu_{R}$ with a coupling given by
$m_{\nu_{D}}/\sqrt{2}V$, where $m_{\nu_{D}}$ is the neutrino Dirac mass.

$\phi^{\prime}$ can decay to $\mu^{+}\mu^{-}$, $\eta\eta$, $\nu\nu$,
$A^{\prime}A^{\prime}$ and $\gamma\gamma$. The first and the last of these are
visible, and the last one occurs only at one-loop. Decays to $\nu_{S}$ or
$A^{\prime}$ can also produce visible energy, if those states in turn decay to
SM particles. Tree-level decays to hadronic states are also possible if
$m_{\phi^{\prime}}>2m_{\pi}$, but the branching fraction to these states is
negligible compared to $\mu^{+}\mu^{-}$, because the coupling to first-
generation quarks is so small.

The decay rates are

$\displaystyle\Gamma^{\phi^{\prime}}_{A^{\prime}A^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{m_{\phi^{\prime}}^{3}}{128\pi
V^{2}}\left(1-\frac{4m_{A^{\prime}}^{2}}{m_{\phi^{\prime}}^{2}}\right)^{1/2}\left(1+12\frac{m_{A^{\prime}}^{4}}{m_{\phi^{\prime}}^{4}}-4\frac{m_{A^{\prime}}^{2}}{m_{\phi^{\prime}}^{2}}\right),$
$\displaystyle\Gamma_{\mu^{+}\mu^{-}}^{\phi^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{m_{\mu}^{2}m_{\phi^{\prime}}}{16\pi
V^{2}}\left(1-\frac{4m_{\mu}^{2}}{m_{\phi^{\prime}}^{2}}\right)^{3/2},$
$\displaystyle\Gamma_{\eta_{i}\eta_{i}}^{\phi^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{m_{\eta_{i}}^{2}m_{\phi^{\prime}}}{32\pi
V^{2}}\left(1-\frac{4m_{\eta_{i}}^{2}}{m_{\phi^{\prime}}^{2}}\right)^{3/2},$
$\displaystyle\Gamma_{\nu_{S}\nu_{A}}^{\phi^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{m_{\nu_{D}}^{2}m_{\phi^{\prime}}}{16\pi
V^{2}}\left(1-\frac{m_{\nu_{S}}^{2}}{m_{\phi^{\prime}}^{2}}\right)^{2},$
$\displaystyle\Gamma_{\gamma\gamma}^{\phi^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{em}^{2}m_{\mu}^{4}}{8\pi^{3}m_{\phi^{\prime}}V^{2}}\left[1+\left(1+\frac{4m_{\mu}^{2}}{m_{\phi^{\prime}}^{2}}\right)\left(\sin^{-1}\frac{m_{\phi^{\prime}}}{2m_{\mu}}\right)^{2}\right]^{2},$

where we have computed $\Gamma^{\phi^{\prime}}_{\gamma\gamma}$ only under the
assumption $m_{\phi^{\prime}}<2m_{\mu}$ (otherwise, this decay is negligible
compared to the $\mu^{+}\mu^{-}$ channel). Note that $\phi^{\prime}$ will
always decay very promptly. In particular, the decay to $\gamma\gamma$ is
always kinematically allowed, and (for $m_{\phi^{\prime}}=100~{}\,{\rm MeV}$)
will occur at a rate $\sim{\cal O}(10^{12})~{}\,{\rm s}^{-1}$.

Note also that, if $m_{\phi^{\prime}}>2m_{A^{\prime}}$, then $\phi^{\prime}$
can decay promptly to $A^{\prime}$. But if this decay channel is dominant,
then $\phi^{\prime}$ production in a beam experiment is essentially no
different from $A^{\prime}$ production, and can be searched for using
strategies for detecting $A^{\prime}$ production.

Figure 3:  Constraints arising from cosmological and astrophysical
observables, assuming that the $A^{\prime}$ and $\phi^{\prime}$ decay to
invisible states. These include constraints on $\Delta N_{eff}$ (green region)
[7], on excess cooling of stars [14, 15] and globular clusters (gray region)
[14], and excess cooling of supernovae (light green region) [16, 17]. The
astrophysical bounds, however, are model dependent as mentioned in the text.

###  II.IV Longitudinal Polarization of $A^{\prime}$

When the gauge group is $U(1)_{T3R}$, which couples to chiral fermions, there
is one aspect of the $A^{\prime}$ coupling to matter which is qualitatively
different from other cases (such as $U(1)_{B-L}$, $U(1)_{L_{i}-L_{j}}$,
$U(1)_{X}$ [18, 19, 20, 21, 22]) and which can have a major impact on
experimental sensitivity. In particular, there can be an enhancement in the
production of the $A^{\prime}$ longitudinal polarization.

The longitudinal polarization vector is $\propto
E_{A^{\prime}}/m_{A^{\prime}}\propto E_{A^{\prime}}/g_{T3R}V$. This yields an
enhancement to the matrix element for processes wherein the longitudinal mode
is produced at high-boost. For such processes, by the Goldstone Equivalence
theorem, the matrix element is similar to that for production of the Goldstone
boson of $U(1)_{T3R}$ symmetry-breaking, with a coupling to fermions which
goes as $m_{f}/\sqrt{2}V$. For small $m_{A^{\prime}}$, the coupling of a SM
fermion to the longitudinal polarization is enhanced with respect to the
transverse polarizations by a factor $m_{f}/m_{A^{\prime}}$. But this
enhancement cannot be arbitrarily large, as it is limited by perturbative
unitarity. Since $m_{f}<V=10\,{\rm GeV}$, our scenario is perturbative.

Note that this enhancement only comes into play because the $A^{\prime}$
couples to chiral SM fermions. The vector part of the interaction vanishes
identically for a longitudinally-polarized $A^{\prime}$, due to the Ward
identity, and the enhanced matrix element arises entirely from the axial part.
As a result, the enhancement in $A^{\prime}$ production occurs only for chiral
models such as $U(1)_{T3R}$, not vector-like models such as $B-L$,
$L_{i}-L_{j}$, etc.

As a result, the $A^{\prime}$ production cross section is only enhanced if the
$A^{\prime}$ is produced at tree-level. If $A^{\prime}$ is produced through
kinetic mixing, then the contribution from longitudinal polarization will
again vanish identically due to the Ward Identity. Although there is an
enhancement of processes where $A^{\prime}$ is produced through a coupling to
$u$-/$d$-quarks, one can see from the Goldstone Equivalence theorem that, even
for small $m_{A^{\prime}}$, this process can be approximated by the production
of a massless pseudoscalar with coupling $m_{q}/\sqrt{2}V\lesssim
10^{-3}-10^{-4}$. The most dramatic effect will be on production of the
$A^{\prime}$ through a coupling to muons as the coupling to $u-/d-$quarks is
suppressed by close to two orders of magnitude, compared to the coupling to
muons. This will be relevant for cosmological production (via
$\mu^{+}\mu^{-}\rightarrow\gamma A^{\prime}$), production in supernovae (which
have non-negligible muon content), and from future experiments involving the
invisible decays of light $A^{\prime}$ coupling directly to muons, such as
NA64$\mu$ and LDMX-M3.

##  III Cosmological and Astrophysical Constraints

There are a variety of constraints on new physics models which arise from
cosmological and astrophysical observables.

For example, if the Universe reheats to a temperature $\gtrsim 100~{}\,{\rm
MeV}$, then models with $m_{A^{\prime}}<1~{}\,{\rm MeV}$ and $V\sim{\cal
O}(10)~{}\,{\rm GeV}$ are ruled out, because they would lead to a number of
effective neutrinos ($N_{eff}$) [7] which is inconsistent with CMB
measurements [23]. But this bound is circumvented if the Universe reheats to a
lower temperature.

Note that if the right-handed electron were charged under $U(1)_{T3R}$, then
models with $m_{A^{\prime}}<1~{}\,{\rm MeV}$ would similarly be ruled out by
constraints on $N_{eff}$. But this constraint cannot be evaded by reheating to
a lower temperature; to avoid this constraint, one would have to reheat to a
temperature below $1\,{\rm MeV}$, but this is ruled out by BBN.

Similarly, a variety of new constraints have recently been shown to arise from
bounds on supernovae cooling [16, 17]. Essentially, the temperature of
supernovae is large enough that a non-negligible population of muons is
produced, and if they couple to new scalars or gauge bosons which decay
invisibly, then there will be an anomalous rate of supernova cooling which is
ruled out by observations of SN1987A.

An axion-like particle (ALP) with a mass as large as $200~{}\,{\rm MeV}$ and a
coupling to muons $\sim{\cal O}(10^{-9})$ would be ruled out if it decayed
entirely to a new, invisible particle. This would effectively rule out
scenarios in which the dominant decay of $A^{\prime}$ was to dark matter or
sterile neutrinos. But if the dominant $A^{\prime}$ decays are to
$e^{+}e^{-}$, then the $A^{\prime}$ will decay back to SM particles long
before escaping.

Similarly, these supernova cooling bounds would also rule out models in which
the decays $\phi^{\prime}\rightarrow\eta\eta,\nu_{S}\nu_{S}$ are kinematically
allowed, provided $m_{\phi^{\prime}}\lesssim 200\,{\rm MeV}$.

White dwarf (WD) cooling constraints are negligible if
$m_{\eta},m_{\nu_{s}}\geq 0.1$ MeV, in which case they will be in equilibrium
with the plasmons inside the WD and can not escape. The other possible final
states are $e^{+}e^{-}$ which is not allowed kinematically and
$\nu_{A}\nu_{A}$, which is mixing angle suppressed [24]. We show constraint
from solar cooling [14, 15] and cooling of stars in Globular clusters [14]

If Fig. 3, we plot these cosmological and astrophysical bounds on the
$(m_{A^{\prime}},m_{\phi^{\prime}})$ parameter space, in the case where
$A^{\prime}$ and $\phi^{\prime}$ decay invisibly. Note that there are no
constraints plotted in the case where
$m_{A^{\prime}},m_{\phi^{\prime}}>200\,{\rm MeV}$, because in this case,
decays to $\mu^{+}\mu^{-}$ necessarily occur at tree-level. However, in this
case, there can still be constraints arising from excess supernova cooling,
since there can still be an ${\cal O}(1)$ branching fraction to invisible
states. To develop these constraints is a detailed question, but it is not
necessary for our purposes, since this region of parameter space is already
ruled out by BaBar, as we will see later.

All the astrophysical constraints, however, can be evaded by assuming dark
photon to be chameleon-type field with its mass depending on the environmental
matter density [25, 26, 27, 14].

Figure 4:  Regions of parameter space excluded by current laboratory
experiments, assuming that $A^{\prime}$ and $\phi^{\prime}$ decay dominantly
to SM particles. Included are bounds from BaBar [10, 11, 28], E137 [29, 30,
31], Orsay [32, 28], U70/NuCal [33, 32, 28] and from fifth force experiments
[34, 14].

##  IV Visible Decays at Displaced Detectors

One experimental strategy consists of producing light mediators at a proton
collider, fixed-target, or beam dump experiment and searching for visible
decays of this mediator at a distant detector. Upcoming experiments of this
type include FASER [35, 36, 37, 38, 39], SHiP [40, 41], LDMX [42, 43, 44, 45,
46], and proposed modifications of SeaQuest [47, 48].

These experiments can only probe a model if the mediator is long-lived, and if
it decays visibly. As such, these detectors cannot be used to search for the
$\phi^{\prime}$ of our scenario, which decays rapidly; for all of the
experiments listed above, its decay length will be too short for an
appreciable number of particles to reach the detector. However, if the
dominant decay of the $A^{\prime}$ is to $e^{+}e^{-}$ through kinetic mixing,
then the decay length may be long enough for the decay to occur within the
detector.

For most of these experiments, the production of $A^{\prime}$ will occur
through $p$-bremsstrahlung and meson decay, where the $A^{\prime}$ couples at
tree-level to $u$\- and $d$-quarks. Since the $A^{\prime}$ can be produced
with a significant boost, one might wonder if the enhancement to the
production of the longitudinal polarization will be relevant. But the
enhancement in the coupling to the Goldstone mode, relative to the transverse
polarizations, scales as $m_{f}/m_{A^{\prime}}$, and is only large when
$m_{A^{\prime}}$ is small. But visible decays are only possible for
$m_{A^{\prime}}>1\,{\rm MeV}$. In any case, we will see that the limit of the
sensitivity range for upcoming experiments will be $m_{A^{\prime}}={\cal
O}(100)\,{\rm MeV}$, and for such high masses, the enhancement to the
production rate of the longitudinal polarization is minimal. Indeed we will
eventually see that the production enhancement for the longitudinal mode is
most dramatic for invisible decays of light $A^{\prime}$.

But although $A^{\prime}$ production is a tree-level process, $A^{\prime}$
decay occurs at one-loop through kinetic mixing. As a result, the sensitivity
to our model can be estimated by considering the estimated sensitivity of
these experiments to models where $A^{\prime}$ only couples to the SM via
kinetic mixing, but with the number of events enhanced by the factor
$(\pi/\alpha_{em}f)^{2}$, to account for the fact that $A^{\prime}$ production
is a tree-level process. Here, $f$ is the factor by which the kinetic mixing
parameter exceeds that obtained only from one-loop diagrams with SM fermions
in the loop.

The sensitivity of experiments of this type have a ceiling and floor; below
the floor, the coupling is too weak for enough $A^{\prime}$ to be produced,
while above the ceiling, the $A^{\prime}$ decays too rapidly to reach the
detector. If $\epsilon_{+}^{kin}(m_{A^{\prime}})$ is the ceiling of the
sensitivity region of an experiment for the secluded model (where all dark
photon interactions proceed through kinetic mixing), then the ceiling of the
$U(1)_{T3R}$ model is given by the solution to the equation

$\displaystyle\left(\frac{\pi}{\alpha_{em}f}\right)^{2}\left(\frac{\epsilon_{+}^{T3R}}{\epsilon_{+}^{kin}}\right)^{2}\exp\left[-\left(\frac{\epsilon_{+}^{T3R}}{\epsilon_{+}^{kin}}\right)^{2}\right]$
$\displaystyle=$ $\displaystyle 1,$ (11)

where the quadratic factor accounts for the change in $A^{\prime}$ production
and exponential factor accounts for the change in the decay length, in the
limit where the decay length is longer than the distance to the detector.

Similarly, if $\epsilon_{-}^{kin}(m_{A^{\prime}})$ is the floor of the
sensitivity region of an experiment for the secluded model, then the floor of
the $U(1)_{T3R}$ model is given by the solution to the equation

$\displaystyle\left(\frac{\pi}{\alpha_{em}f}\right)^{2}\left(\frac{\epsilon_{-}^{T3R}}{\epsilon_{-}^{kin}}\right)^{4}$
$\displaystyle=$ $\displaystyle 1,$ (12)

where one quadratic factor arises from the rescaling of the $A^{\prime}$
production cross section, and the second factor arises from the rescaling of
the rate of decay within the detector, in the limit where the decay length is
much less than the distance to the detector.

But if visible decays are kinematically allowed at all, then we must have
$m_{A^{\prime}}>1~{}\,{\rm MeV}$; taking $V=10\,{\rm GeV}$, we find that we
must have $g_{T3R}\gtrsim 10^{-4}$.

Figure 5:  The sensitivity of upcoming laboratory experiments to $A^{\prime}$,
$\phi^{\prime}$ decay at displaced detectors. Shown are the sensitivities of
FASER [39] (purple region), FASER 2/SHiP [39] (dark green region), and
SeaQuest [47] (light green region). Also shown are constraints from current
laboratory experiments (light gray region), reproduced from Fig. 4.

If we assume that kinetic mixing is generated at one-loop only by SM fermions
(that is, $f=1$) then we find that the following bounds (from U70/NuCal) and
future sensitivities :

  * •

U70/NuCal: Ruled out if $1~{}\,{\rm MeV}\lesssim m_{A^{\prime}}\lesssim
93~{}\,{\rm MeV}$

  * •

FASER: Probed if $1~{}\,{\rm MeV}\lesssim m_{A^{\prime}}\lesssim 140~{}\,{\rm
MeV}$

  * •

FASER-2 and SHiP (their sensitivities are similar): Probed if $1~{}\,{\rm
MeV}\lesssim m_{A^{\prime}}\lesssim 161~{}\,{\rm MeV}$

  * •

SeaQuest: Probed if $1~{}\,{\rm MeV}\lesssim m_{A^{\prime}}\lesssim
180~{}\,{\rm MeV}$

Note that, since the gauge coupling scales as $\propto m_{A^{\prime}}/V$,
increasing $V$ actually increases the mass reach, by increasing the decay
length. In fact, if $V$ is increased to 30 GeV, then U70/NuCal would already
rule out the entire currently available parameter space for the scenario in
which the $A^{\prime}$ decays to $e^{+}e^{-}$.

There are also a variety of current electron beam dump experiments which can
constrain this scenario [4]. For these experiments, one-loop processes can
result in the production of either $A^{\prime}$ (kinetic-mixing) or
$\phi^{\prime}$ (Primakoff production), with subsequent one-loop decays to SM
particles at a displaced detector.

There are some other constraints on this scenario from current laboratory
experiments, which were discussed in [4]. In particular, BaBar [10, 11]
provides tight constraints on the regions of parameter space where
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}(A^{\prime},\phi^{\prime}\rightarrow\mu^{+}\mu^{-})$
is kinematically accessible. Regions of parameter space in which either
$\phi^{\prime}$ or $A^{\prime}$ are extremely light are also tightly
constrained by fifth force experiments [34]. Note that, although
$g_{T3R}\propto m_{A^{\prime}}$, regions of parameter space with very small
$m_{A^{\prime}}$ are still tightly constrained by fifth force experiments;
although the transverse modes of the $A^{\prime}$ decouple as
$m_{A^{\prime}}\rightarrow 0$, the Goldstone mode still contributes to the
fifth force.

In Fig. 4, we plot constraints on this scenario from current laboratory
experiments in the $(m_{A^{\prime}},m_{\phi^{\prime}})$-plane, where we assume
that $A^{\prime}$, $\phi^{\prime}$ predominantly decay to SM particles. In
particular, we plot constraints from BaBar [10, 11, 28], E137 [29, 30, 31],
Orsay [32, 28], U70/NuCal [33, 32, 28] and from fifth force experiments [34,
14]. If they decay instead dominantly to invisible states, then these bounds
are weakened considerably. In Fig. 5, we plot projected bounds arising from
visible decay at displaced detectors such as FASER, FASER-2 [35, 36, 37, 38,
39], SHiP [40, 41] and SeaQuest [47, 48] in the
$(m_{A^{\prime}},m_{\phi^{\prime}})$-plane.

##  V Visible and Invisible Decays at Nearby detectors

Figure 6:  Regions of parameter space excluded by current laboratory
experiments, assuming that $A^{\prime}$ and $\phi^{\prime}$ decay dominantly
to invisible final states. Included are bounds from COHERENT [49, 50, 51, 52,
53](light blue) and Crystal Barrels [54, 55] (light purple) experiments.

The Crystal Barrel (CB) [54, 55] detector can give constraints on
$m_{A^{\prime}}$ when $A^{\prime}$ predominantly decays to invisible final
states. CB set an upper limit of the branching ratios (Br) for the process
$P\rightarrow\gamma X$, where $P=\pi^{0},\eta,\eta^{\prime}$ and $X$ is a
boson which is either long-lived, or decays invisibly. We consider
$X=A^{\prime}$, the dark photon. The parameter space probed for the $\eta$ and
$\eta^{\prime}$ decay will be ruled out by other experiments. We mainly look
at $\pi^{0}$ decay. The bound from CB is [54, 55],

$\text{Br}(\pi^{0}\rightarrow\gamma A^{\prime})\leq 2.8\times
10^{-4},~{}~{}~{}~{}~{}~{}m_{A^{\prime}}\leq 65~{}\text{MeV},$ (13)

and,

$\text{Br}(\pi^{0}\rightarrow\gamma A^{\prime})\leq 6.0\times
10^{-5},~{}~{}~{}~{}~{}~{}65~{}\text{MeV}\leq m_{A^{\prime}}\leq
125~{}\text{MeV},$ (14)

The branching fraction for our model is given by

$\text{Br}(\pi^{0}\rightarrow\gamma
A^{\prime})=\frac{m_{A^{\prime}}^{2}}{4\pi\alpha
V^{2}}\left(1-\frac{m_{A^{\prime}}^{2}}{m_{\pi^{0}}^{2}}\right)^{3}$ (15)

Therefore the region of paramter space,
$55~{}\text{MeV}<m_{A^{\prime}}<120~{}\text{MeV}$ will be ruled out by CB.

Proposed detectors such as NA64$\mu$ [56, 57] and LDMX-M3 [58, 46] (a proposed
muon beam version of LDMX [43]) can probe this scenario in the case where
either $A^{\prime}$ or $\phi^{\prime}$ has a significant decay rate to
invisible states. NA64$\mu$ proposes to collide a muon beam with a target, and
search for interactions with missing energy. In the case of a scalar mediator
which does not decay to visible energy within the detector, it is estimated
that NA64$\mu$ could probe muon-scalar couplings $\sim{\cal O}(10^{-5})$,
largely independent of the scalar mass [56]. The coupling of $\phi^{\prime}$
to muons is ${\cal O}(10^{-2})$, implying that this scenario can be probed by
NA64$\mu$ for any $m_{\phi^{\prime}}$, provided the branching fraction to
invisible states satisfies $Br(invisible)>10^{-6}$. Note that, in the scenario
in which $\phi^{\prime}$ couples to muons at tree-level, LDMX-M3 Phase 1 will
probe any $m_{\phi^{\prime}}$, provided $Br(invisible)>10^{-4}$, while Phase 2
will have a greater sensitivity than NA64$\mu$ [46].

Since the decay $\phi^{\prime}\rightarrow\gamma\gamma$ is one-loop suppressed,
whereas the decays $\phi^{\prime}\rightarrow\nu_{S}\nu_{A},\eta\eta$ occur at
tree-level, these invisible decays will dominate if kinematically allowed. In
fact, even if the decay $\phi^{\prime}\rightarrow\mu^{+}\mu^{-}$ is
kinematically allowed, the invisible decays will still have a branching
fraction of at least ${\cal O}(10^{-4})$, provided
$m_{\eta,\nu_{S}}>1~{}\,{\rm MeV}$. All these scenarios can thus be probed by
NA64$\mu$ and LDMX-M3.

Note that the sensitivity of NA64$\mu$ to $A^{\prime}$ is roughly similar.
Even for arbitrarily light $A^{\prime}$ (with arbitrarily weak coupling), the
longitudinal mode couples to muons approximately the same as a pseudoscalar
with coupling $\sim{\cal O}(10^{-2})$. We thus find that NA64$\mu$ and LDMX-M3
will be able to probe the entire parameter space, provided
$m_{A^{\prime},\phi^{\prime}}>2m_{\eta}$ or $2\nu_{S}$.

Even if the $\eta\eta$ and $\nu_{S}\nu_{S}$ final states are not kinematically
allowed, and the dominant decay of $A^{\prime}$ is to $e^{+}e^{-}$, NA64$\mu$
and LDMX-M3 will still be sensitive if the decay length of the $A^{\prime}$ is
long enough that a significant number of $A^{\prime}$ leave the detector
without decaying.

The sensitivity of NA64$\mu$ to a minimally flavor-violating (MFV) scalar
which couples to both muons and electrons was considered in [56], and this
case is essentially the same as for the $A^{\prime}$. But in that study it was
assumed that the coupling of the mediator to electrons was suppressed relative
to the coupling to muons by the factor $(m_{e}/m_{\mu})$, whereas we instead
assume that is is suppressed by the kinetic mixing factor
$(\alpha_{em}/\pi)f$.

As with displaced detectors, the NA64$\mu$ sensitivity region has a floor
(below which not enough $A^{\prime}$ are produced) and a ceiling (above which
the $A^{\prime}$ decays to visible states within the detector). But because
the coupling of the longitudinal polarization to muons is never smaller than
${\cal O}(10^{-2})$, our model is never below the floor for any choice of
$m_{A^{\prime}}.$

The ceiling of the NA64$\mu$ sensitivity to the MFV scalar model can be
translated into a sensitivity to the $U(1)_{T3R}$ model by rescaling the
coupling by the factor $(m_{e}/m_{\mu})(\pi/\alpha_{em}f)\sim 2.08f^{-1}$;
this rescaling keeps the coupling to electrons (and thus the decay length)
fixed, while increasing the $A^{\prime}$ production rate by an ${\cal O}(1)$
factor. Since decreasing the decay length causes an exponential suppression to
the number of events, this simple rescaling is a good approximation. Applying
this rescaling to the limits found in [56], we find that NA64$\mu$ is
sensitive to $U(1)_{T3R}$ models for which $m_{A^{\prime}}<77~{}\,{\rm MeV}$.
But if $e^{+}e^{-}$ is the dominant final state, then the range $1\,{\rm
MeV}<m_{A^{\prime}}<77~{}\,{\rm MeV}$ is already ruled out by U70/NuCal. But
if $m_{A^{\prime}}<1~{}\,{\rm MeV}$, then no two-body visible decays are
allowed, and both NA64$\mu$ and LDMX-M3 will probe this scenario.

Finally, we note that LDMX-M3 may have sensitivity even if dark matter decays
to visible states within the target. Whereas NA64$\mu$ relies entirely on
calorimetry, LDMX relies on tracking, and the location of energy deposition
within the calorimeter. Even if prompt decays, such as
$\phi^{\prime}\rightarrow\gamma\gamma$ occur within the target, LDMX may be
able to use information from the tracker to distinguish this event from
Standard Model background.

Figure 7:  The sensitivity of upcoming laboratory experiments to the parameter
space assuming that $A^{\prime}$, $\phi^{\prime}$ decay dominantly to
invisible final states. Shown are the sensitivities of NA64$\mu$ [56, 57], and
LDMX-$M^{3}$ [58, 46] (light blue region). Also shown are constraints from
current laboratory experiments (light gray region), reproduced from Fig. 6.

###  V.I Belle-II: $e^{+}e^{-}\rightarrow\gamma+$ invisible

Since the dark photon can kinetically mix with the photon, Belle-II [59, 60,
28] can study the process $e^{+}e^{-}\rightarrow\gamma A^{\prime}$. Its
sensitivity is best if the $A^{\prime}$ decays invisibly [60, 28], since in
that case the Standard Model rate is relatively small. For
$m_{A^{\prime}}\lesssim 200\,{\rm MeV}$, Belle-II will be able to probe models
with $g_{T3R}\gtrsim 10^{-3}$, corresponding to $m_{A^{\prime}}\gtrsim
30\,{\rm MeV}$.

##  VI Dark Matter and Sterile Neutrino Scattering at Displaced Detectors

There are a variety of stopped pion based experiments e.g., COHERENT [49, 50,
51, 52, 53], CCM [61, 62], JSNS2 [63, 64, 65, 66, 67], etc. which are designed
to produce neutrinos from a proton beam hitting a target, and search for the
neutral current scattering of these neutrinos at a distant detector. Among
these experiments, the ongoing COHERENT and CCM experiments are CE$\nu$NS [68,
69] experiments. The COHERENT experiment has observed 6.7$\sigma$ (at CsI
detector [49]) and 3.8$\sigma$(at LAr [53]) evidences of CE$\nu$NS type
events. However these experiments also produce tremendous amount of photons
from proton, electron bremsstrahlung and meson decays [70]. These photons can
then produce $A^{\prime}$. If the decays
$A^{\prime}\rightarrow\eta\eta,~{}\nu_{S}\nu_{S}$ are kinematically allowed,
then they will occur promptly and dominate the $A^{\prime}$ branching
fraction. The scattering of the dark matter or sterile neutrinos against
nuclei can then be probed. Unlike NA64$\mu$ and LDMX-M3, these neutrino
experiments probe the appearance of dark matter/sterile neutrinos at the
detector, which makes these neutrino experiments complimentary to the beam
dump searches described before. The neutrinos from pion and muon decays
produce backgrounds for such searches. However, utilizing the pulsed nature of
the beam and the timing and the energy spectra of the recoiling nucleus, it
has been shown recently that one can extract the dark matter signal from the
neutrino background [5]. The current result from the COHERENT experiment
actually yields a tighter constraint on the dark matter parameter space
compared to MiniBooNE [71, 72, 73, 74], LSND [71, 75, 76], NA64 [56, 57] etc.
In fact COHERENT CsI-data shows some excess ($\sim 2.4-3\sigma$) for
$A^{\prime}$ mass $\lesssim 100$ MeV where the $A^{\prime}$ decays promptly to
dark matter or sterile neutrino (in this model) [5]. In Ref. [5], photons from
$\pi^{0}$ decays were studied, while in Ref. [70] bremsstrahlung photons were
also included to investigate dark matter emerging from dark photon decay, with
similar results.

The search for light dark matter/sterile neutrinos relies on kinematic
features to distinguish the scattering of $\nu_{S}$ or $\eta$ from that of
active neutrinos produced from the beam via SM processes. SM processes will
dominantly produce active neutrinos via stopped pion decay, yielding neutrinos
with an energy of $30~{}\,{\rm MeV}$. But $A^{\prime}$ can decay in flight to
$\nu_{S}\nu_{S}$ or $\eta\eta$, yielding much higher energy particles. This
type of search was considered in detail in [70], and we can apply their
general results to our scenario.

Note that if sterile neutrino scattering is mediated by the $\phi^{\prime}$,
then it is exothermic, as the outgoing neutrino is active. But as the sterile
neutrino will already be boosted, the change in the event rate is an ${\cal
O}(1)$ factor.

To rescale the limits found in [70] to our scenario, we need only note that
the dominant method for $A^{\prime}$ production in our scenario will be from
direct coupling to $u$\- or $d$-quarks. As such, the event rate is
proportional to two powers of the coupling of $A^{\prime}$ to dark matter
(from the squared scattering matrix element) and four powers of the coupling
of the $A^{\prime}$ to first generation quarks (two from the squared
scattering matrix element, and two from the squared $A^{\prime}$ production
matrix element). The exception is JSNS2, which looks for scattering against
electrons; for this experiment, two powers of the coupling to first-generation
quarks are replaced with the coupling to electrons.

With these rescalings, the event rate for the models considered in [70] can be
directly related to the event rate in our scenario. To facilitate comparison
with [70], we take $m_{\eta}/m_{A^{\prime}},m_{\nu_{S}}/m_{A^{\prime}}=1/3$,
though deviations from this assumption will not affect sensitivity
significantly.

If dark matter or sterile neutrino scattering is dominantly mediated by the
$A^{\prime}$, then from the analysis of [5], we find that the COHERENT excess
can be reproduced for $g_{T3R}\sim 0.002$. For $V=10\,{\rm GeV}$, this
corresponds to $m_{A^{\prime}}\sim 30\,{\rm MeV}$. This parameter space is not
ruled out by any other available laboratory based experimental result. Since
the event rate scales as $g_{T3R}^{6}$, the dark photon mass range
significantly above this benchmark mass is excluded by COHERENT data.

Note that the enhancement in coupling to the longitudinal mode is not relevant
for spin-independent scattering, which is mediated by a purely vector
interaction.

If dark matter scattering is instead mediated by the $\phi^{\prime}$, then the
event rate is suppressed by the factor
$g_{T3R}^{-4}m_{\eta}^{2}m_{u,d,e}^{2}/2V^{4}$; as long as
$m_{\eta},m_{A^{\prime}}\gtrsim 30~{}\,{\rm MeV}$, these models can also ruled
out by COHERENT. Finally, if the $A^{\prime}$ predominantly decays to sterile
neutrinos, which scatter off nuclei dominantly through $\phi^{\prime}$
exchange, then COHERENT rules out models for which
$m_{\nu_{D}}m_{A^{\prime}}\gtrsim(30\,{\rm MeV})^{2}$.

Note that there are regions of parameter space for which these bounds can be
weakened. For example, changing $m_{\eta,\nu_{S}}/m_{A^{\prime}}$ can effect
the sensitivity by an ${\cal O}(1)$ factor, which could open up regions of
parameter space at the edge of COHERENT’s sensitivity. Similarly, if
$m_{\eta,\nu_{S}}/m_{A^{\prime}}\sim 1/2$, then for relatively heavy
$m_{A^{\prime}}$, scattering at the detector may be non-relativistic, leading
to a suppression of the event rate. But note that CCM and JSNS2 will have
sensitivity which should improve on COHERENT. This dark photon invisible decay
parameter space also can be probed at DUNE [77, 78, 79, 80, 81, 82].

Bounds from Crystal Barrel and COHERENT are shown in the
$(m_{A^{\prime}},m_{\phi^{\prime}})$ parameter space in Fig. 6, assuming that
the $A^{\prime}/\phi^{\prime}$ dominantly decay to invisible states. We also
show the future sensitivities of NA64$\mu$ and LDMX-$M^{3}$ on the
$(m_{A^{\prime}},m_{\phi^{\prime}})$ parameter space in Fig. 7.

##  VII Non-Standard Interactions for Active Neutrinos

The $A^{\prime}$ and $\phi^{\prime}$ can mediate non-standard interactions of
active neutrinos with nuclei. A variety of constraints on such interactions
have been found from oscillation effects, short and long baseline experiments,
CE$\nu$NS experiments etc. Constraints can be found by using a large family of
NSI, but considering one or two of them at a time [83], by reparameterizing
the NSI down to a more phenomenological and pragmatically manageable subset
based on model assumptions (for example, in Refs. [84, 85, 86]), or by
considering all the NSIs from a large set at the same time [87].

If the momentum transfer in the scattering process is much smaller than the
mediator mass, then the effect of the $A^{\prime}$ and $\phi^{\prime}$
couplings can be approximated by dimension-6 effective operators:

$\displaystyle{\cal O}_{A^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\sin^{2}\theta}{2V^{2}}(\bar{\nu}_{A}\gamma^{\mu}P_{L}\nu_{A})(\bar{q}\gamma_{\mu}P_{R}q),$
$\displaystyle{\cal O}_{\phi^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{m_{q}^{2}\sin\theta}{2V^{2}m_{\phi^{\prime}}^{2}}(\bar{\nu}_{A}P_{L}\nu_{A})(\bar{q}P_{R}q),$
(16)

where $\theta$ is the neutrino mixing angle. Note that the coefficient of
${\cal O}_{A^{\prime}}$ has two powers of the neutrino mixing angle, since
$A^{\prime}$ couples to two right-handed neutrinos, whereas the coefficient of
${\cal O}_{\phi^{\prime}}$ has only one power, as $\phi^{\prime}$ couples to a
right-handed and a left-handed neutrino.

The coefficients of these operators are bounded by current experiments to be
of $\lesssim{\cal O}(10^{-5})~{}\,{\rm GeV}^{-2}$ [87], and future experiments
could improve this bound by an order of magnitude. For ${\cal
O}_{A^{\prime}}$, current experiments require $\sin^{2}\theta\lesssim{\cal
O}(10^{-3})$ [87], while future experiments could probe values of
$\sin^{2}\theta$ which are an order of magnitude smaller. For ${\cal
O}_{\phi^{\prime}}$, current experiments require [87]

$\displaystyle\sin\theta\lesssim\left[{\cal
O}(10^{-3})\right]\left(\frac{m_{\phi^{\prime}}}{5\,{\rm MeV}}\right)^{2},$
(17)

with the sensitivity of upcoming experiments, e.g., DUNE [77, 78, 79, 80, 81,
82], Hyper-K [88, 89, 90, 91, 92, 93] etc. to $\sin\theta$ being up to an
order of magnitude higher.

The sterile neutrino, $\nu_{s}$, is mostly composed of the right-handed
neutrino ($\nu_{R}$), but with a small mixing with the active neutrino.
Because of this mixing the sterile neutrino can be produced in laboratory
experiments. $\nu_{s}$ can be searched for at the HUNTER [94] experiment by
kinematic reconstruction of the electron capture decay of the radioactive atom
131Cs. Similarly, the $\nu_{s}$ can potentially be searched for with the
TRISTAN project of the KATRIN [95] experiment, where it can yield a kink-like
distortion in the tritium beta decay spectrum if
$m_{\nu_{s}}\sim\mathcal{O}(\,{\rm keV})$.

##  VIII Conclusion

We have considered the scenario in which right-handed light SM fermions are
charged under a new gauge group, $U(1)_{T3R}$. This scenario is of particular
interest because it can tie the symmetry-breaking scale of $U(1)_{T3R}$ to
that of the light SM fermions and of the dark sector. This scenario thus
naturally leads to a new set of sub-GeV particles, including the dark matter,
a sterile neutrino, a dark photon, and a dark Higgs. In this paper, we have
focused on ways of probing this scenario with new data sets. We have focused
on the case where the symmetry-breaking scale is taken to be $10~{}\,{\rm
GeV}$, but the results do not change qualitatively if that scale is increased
to $\sim 30~{}\,{\rm GeV}$, unless the $A^{\prime}$ decays primarily to
visible states.

We have found that the optimal probe of this scenario depends on the details
of the model. Our main results are shown in Figures 3-7, and a summary of
these sensitivities is presented in Table 2 . One distinct feature of this
scenario which has an impact on search strategies is that the dark photon has
a chiral coupling to some SM fermions, yielding an enhancement to tree-level
production of the longitudinal polarization.

If the dark photon or dark Higgs have kinematically-allowed decays to dark
matter or to sterile neutrinos, then those tree-level decay processes will be
prompt. In that case, excellent sensitivity arises for the ongoing experiments
such as COHERENT, CCM and JSNS2, in which the $A^{\prime}$ is produced at a
target hit by a proton beam, and the invisible decay products scatter off
nuclei in a distant detector. Indeed, the current $2.4-3\sigma$ excess in the
event rate at COHERENT could be explained by a 30 MeV dark photon which is
produced from photons at the target and decays to either dark matter or
sterile neutrinos, and which also mediates the scattering of these invisible
particles with nuclei in the target. Constraints from COHERENT rule out larger
masses ($m_{A^{\prime}}\gtrsim 30\,{\rm MeV}$). One can also find very fine-
tuned regions of parameter space where COHERENT’s sensitivity is weakened
because the dark matter is slow-moving when it reaches the detector. But CCM
and JSNS2 will improve on the current COHERENT sensitivity. This dark photon
parameter space also can be searched at DUNE.

Moreover, excellent detection prospects also lie with experiments such as
NA64$\mu$ and LDMX-M3, in which a muon beam is collided with a target, and one
searches for invisible decays. These experiments can probe the entirety of
currently available parameter space in which the $\phi^{\prime}$ or
$A^{\prime}$ decay invisibly, including the parameter space region
$m_{A^{\prime}}\lesssim 30\,{\rm MeV}$. The searches for dark photon at the
neutrino experiments are however complimentary to the searches at NA64$\mu$
and LDMX-M3 since the former investigates the appearance of dark
matter/sterile neutrinos at the detector compared to the disappearance
searches at the latter facilities. Belle-II can also probe models with
$m_{A^{\prime}}\gtrsim 30\,{\rm MeV}$.

If the dark photon or dark Higgs decay largely to visible states, then the
best prospects lie with beam experiments which search for visible decays in a
distant detector, such as FASER, SeaQuest, and SHiP. These models provide
excellent detection prospects, provided the mediating particle has a decay
length long enough to reach the detector. As such, these experiments will have
difficulty in seeing the dark Higgs, but can probe dark photons in the ${\cal
O}(1-100~{}\,{\rm MeV})$ range, though there is still open parameter space at
relatively large dark photon mass which these upcoming experiments cannot
probe. However, if $A^{\prime}$ decays primarily to visible states, then even
modestly increasing $V$ to 30 GeV would result in the entire allowed parameter
space being ruled out by these experiments.

Cosmological and astrophysical observables can also play an important role.
These constraints are espeically interesting because, even for arbitrarily
small gauge coupling, the longitudinal polarization of the dark photon has a
large coupling to muons. In particular, if the Universe reheats to a
temperature greater than ${\cal O}(100\,{\rm MeV})$, then the entire parameter
space with $m_{\phi^{\prime},A^{\prime}}\lesssim 1\,{\rm MeV}$ can be ruled
out. Similarly, observations of SN1987A can rule all currently allowed
scenarios for which the dominant decay of either $\phi^{\prime}$ or
$A^{\prime}$ is dark matter or sterile neutrinos, though these supernovae
constraints are subject to large systematic uncertainties, and can be weakened
by chameleon effects, or other features of a more complicated dark sector.

We see that there is a rich and interesting phenomenology associated with
scenarios in which light right-handed SM fermions are charged under a new
gauge group, $U(1)_{T3R}$, with low-mass mediators. This scenario is tightly
constrained, yet there are still unexplored regions of parameter space.

The region of parameter space which will not be tested with upcoming
experiments is where the dark photon and dark Higgs have dominantly visible
decays, but with a decay length which is too short to reach upcoming displaced
detectors. It would be interesting to consider new strategies for closing this
remaining window, including shorter decay regions.

Table 2:  A summary of the various experiments/probes considered here, their
methods for producing and detecting the mediating particles, and the resulting
sensitivities.

Type of experiments Name of the experiment Production of
$A^{\prime}/\phi^{\prime}$ Final states Results Electron beam dump experiments
E137, Orsay $A^{\prime}$ : electron bremsstrahlung through kinetic mixing at
one-loop, $\phi^{\prime}$ : Primakoff production at one-loop. Both
$A^{\prime},\phi^{\prime}$ decay predominantly to visible SM states
$e^{+}e^{-}$. $\phi^{\prime}$ decay is rapid. E137 rules out : 1 MeV $\leq
m_{A^{\prime}}\leq$ 20 MeV, 1 MeV$\leq m_{\phi^{\prime}}\leq$ 65 MeV. Orsay
rules out : 1 MeV $\leq m_{A^{\prime}}\leq$ 40 MeV. Proton beam dump
experiments U70/NuCal, FASER SHiP, SeaQuest (displaced detector)
$p$-bremsstrahlung or meson decay at tree level $A^{\prime}\rightarrow
e^{+}e^{-}$ through kinetic mixing. $\phi^{\prime}\rightarrow\gamma\gamma$
$\phi^{\prime}$ decays rapidly hence cannot be probed. U70/NuCal rules out : 1
MeV $\leq m_{A^{\prime}}\leq$ 93 MeV. FASER can probe : 1 MeV $\leq
m_{A^{\prime}}\leq$ 140 MeV. FASER 2/SHiP can probe : 1 MeV $\leq
m_{A^{\prime}}\leq$ 161 MeV. SeaQuest can probe : 1 MeV $\leq
m_{A^{\prime}}\leq$ 180 MeV. $e^{+}e^{-}$ collider experiments BaBar, Belle-II
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}+A^{\prime}/\phi^{\prime}$,
$e^{+}e^{-}\rightarrow\gamma A^{\prime}$ 4$\mu$ final states, $\gamma$ \+
invisible BaBar rules out for ($4\mu$ final states) : 200 MeV $\leq
m_{A^{\prime}}\leq$ 1.3 GeV, 290 MeV $\leq m_{\phi^{\prime}}\leq$ 3 GeV.
Belle-II can probe ($\gamma$ \+ invisible): $m_{A^{\prime}}\geq 30$ MeV.
$\bar{p}p$ collider experiments Crystal Barrel
$\bar{p}p\rightarrow\pi^{0}\pi^{0}\pi^{0}$, $\pi^{0}\rightarrow\gamma
A^{\prime}$ invisible states The parameter space is ruled out for:
$55~{}\text{MeV}<m_{A^{\prime}}<120~{}\text{MeV}$ Fifth force searches
experiments Precision tests of gravitational Casimir, and van der Waals forces
Relevant for extremely light $A^{\prime}/\phi^{\prime}$. For
$m_{A^{\prime}}\rightarrow 0$ limit, the Longitudinal mode will contribute.
n/a The parameter space is ruled out for:
$m_{A^{\prime}}/m_{\phi^{\prime}}\leq 1$ eV. Astrophysical probes SN1987A,
Cooling of Sun and globular clusters, White dwarfs $\gamma+\mu\rightarrow
A^{\prime}+\mu$, $\mu+p\rightarrow\mu+p+A^{\prime}$,
$\mu^{+}\mu^{-}\rightarrow A^{\prime}$ at tree level, $e^{+}e^{-}\rightarrow
A^{\prime}$ through kinetic mixing.
$A^{\prime}\rightarrow\eta\eta,\nu_{s}\nu_{s}$ (if decays to
$\nu\nu,e^{+}e^{-}$ then can not escape),
$\phi^{\prime}\rightarrow\eta\eta,\nu\nu$ SN1987A rules out :
$m_{A^{\prime}},m_{\phi^{\prime}}\leq 200$ MeV. Stellar cooling rules out:
$m_{A^{\prime}},m_{\phi^{\prime}}\leq 1$ MeV. WD constraint are negligible if
$m_{\eta},m_{\nu_{s}}\geq 0.1$ MeV. (All these astrophysical bounds can be
evaded using chameleon effect.) Cosmological probes $\Delta N_{eff}$ value
$\mu^{+}\mu^{-}\rightarrow\gamma A^{\prime}$, production of longitudinal mode
get enhanced due to axial vector coupling. invisible states If the Universe
reheat at a temperature $\geq 100$ MeV, $m_{A^{\prime}},m_{\phi^{\prime}}\leq
1$ MeV is ruled out. (Can be evaded if reheat occurs at a lower temperature.)

Type of experiments Name of the experiment Production of
$A^{\prime}/\phi^{\prime}$ Final states Results Muon beam experiments
NA64$\mu$, LDMX-M3 (nearby detectors) $\mu-$bremsstrahlung Can probe when
$A^{\prime}/\phi^{\prime}$ has a significant decay rate to invisible states
such as $\nu\nu,\eta\eta$ NA64$\mu$, LDMX-M3 can probe the entire parameter
space if $m_{A^{\prime},\phi^{\prime}}>2m_{\eta,\nu_{s}}$ with
Br(invisible)$>10^{-4}$, even if
$A^{\prime}/\phi^{\prime}\rightarrow\mu^{+}\mu^{-}$ is allowed still
Br(invisible)$>10^{-4}$ provided $m_{\eta,\nu_{s}}>1$ MeV. Neutrino
experiments COHERENT, CCM JSNS2 $p/e$\- bremsstrahlung, meson decay
$A^{\prime}\rightarrow\nu_{s}\nu_{s}/\eta\eta$,
$\nu_{s}/\eta_{i}+N\rightarrow\nu_{s}/\eta_{j}+N$ generate nuclear recoil,
$\nu_{s}/\eta_{i}+e\rightarrow\nu_{s}/\eta_{j}+e$ generate electron recoil Can
be probed by looking at nuclear/electron recoil. $m_{A^{\prime}}\sim 30$ MeV
can explain the 2.4-3$\sigma$ excess found by COHERENT, $m_{A^{\prime}}\gtrsim
30$ MeV is ruled out. CCM and JSNS2 will improve the sensitivity.

Acknowledgments We are grateful to Asher Berlin and Jeremy Sakstein for useful
discussions. We acknowledge Shannon Kumar for her hospitality. The work of BD
and SG are supported in part by the DOE Grant No. DE-SC0010813. The work of JK
is supported in part by DOE Grant No. DE-SC0010504.

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