# Sensitivities on dark photon from the Forward Physics Experiments

Kingman Cheunga,b,c, C.J. Ousepha,b a Department of Physics, National Tsing
Hua University, Hsinchu 30013, Taiwan
b Center for Theory and Computation, National Tsing Hua University, Hsinchu
30013, Taiwan
c Division of Quantum Phases and Devices, School of Physics, Konkuk
University, Seoul 143-701, Republic of Korea

(August 28, 2024)

###### Abstract

Neutrino-electron scattering experiments can explore the potential presence of
a light gauge boson $A^{\prime}$ which arises from an additional $U(1)_{B-L}$
group, or a dark photon $A^{\prime}$ which arises from a dark sector and has
kinetic mixing with the SM hypercharge gauge field. We generically call it a
dark photon. In this study, we investigate the effect of the dark photon on
neutrino-electron scattering $\nu e^{-}\rightarrow\nu e^{-}$ at the newly
proposed forward physics experiments such as FASER$\nu$, FASER$\nu 2$, SND@LHC
and FLArE(10 tons). We estimate the anticipated sensitivities to the
$U(1)_{B-L}$ gauge coupling in a wide range of the dark photon mass
$M_{A^{\prime}}$. We compare the sensitivities of the proposed forward physics
experiments with the current limits from TEXONO, GEMMA, BOREXINO, LSND, and
CHARM II as well as NA64e experiments. We also extend the calculation to
obtain the sensitivities on the kinetic mixing parameter $\epsilon$ in a wide
range of dark photon mass $M_{A^{\prime}}$. We demonstrate that the
sensitivities do not improve for $M_{A^{\prime}}<1$ GeV at the Forward Physics
Facilities.

##  I Introduction

Neutrino-charged-lepton scattering and neutrino-nucleon scattering have a long
history of establishing the standard model (SM) Workman:2022ynf , e.g., the
discovery of neutral current in $\nu N$ scattering in 1973 Freedman:1973yd ,
the measurement of the weak-mixing angle, as well as constraining new physics
that violates parity Workman:2022ynf . In the last two decades, neutrino
physics mainly concerned with the oscillations of flavors of neutrinos and now
the oscillation parameters are reasonably well determined, except for the CP-
violation parameter, the mass hierarchy of neutrino masses and
$\theta_{23}$-octant degeneracy. A number of new experiments are proposed to
solve these subtleties, such as DUNE, JUNO, Hyper-K, etc.

On the other hand, neutrino scattering received renew interests in a number of
proposals, including the near detector of DUNE making use of an off-axis
neutrino beam, and FASER$\nu$ making use of the neutrinos coming from the
interaction point (IP) of the ATLAS experiment. FASER$\nu$ is indeed running
and taking data talk-2022 ; talk-2022(2)  The most distinct feature of the
neutrinos coming off the IP of ATLAS is that the energy range can be as high
as TeV, which provides an unprecedented energy scale of studying neutrino-
nucleon and neutrino-electron scattering. The FASER$\nu$ experiment can cover
a unique energy range that other experiments cannot cover. The ICECUBE focuses
on very high-energy neutrinos with energy 10 TeV to 1 PeV. On the other hand,
the short- and long-baseline experiments cover mostly around MeV up to a few
GeV. There are no precise measurements of neutrino scattering in a few tens of
GeV to a few hundreds of GeV region. FASER$\nu$ based on the neutrino flux
coming off the LHC opens such a unique window in this energy range. A further
proposal is to build the Forward Physics Facilities (FPF) Feng:2022inv  that
can house a number of proposed experiments within the same location, including
FASER2, Advanced SND, and FORMOSA. Thus, such experiments open an avenue to
search for new physics associated with the neutrinos, which are not easily
tested at the $pp$ collisions of the LHC experiments.

In this work, we investigate the potential of detecting a light $U(1)_{B-L}$
gauge boson or a hidden $U(1)$ gauge boson (aka. dark photon) at such proposed
neutrino-scattering experiments. Generically, a hidden $U(1)$ does not couple
to the SM particles, but it can mix with the SM $U(1)_{Y}$ gauge field through
the $B^{\mu\nu}F^{\prime}_{\mu\nu}$ term. Throughout the work we use the
$U(1)_{B-L}$ gauge boson as illustration and at the end we also extend the
calculation to obtain the sensitivities on the kinetic mixing $\epsilon$.

The organization is as follows. In the next section, we describe the
$U(1)_{B-L}$ model and the relation to the hidden $U(1)$ model (dark photon).
In Sec. III, we calculate the neutrino-electron scattering in the SM and the
$U(1)_{B-L}$ model. In Sec. IV, we estimate the sensitivity reach at
FASER$\nu$, FASER$\nu 2$, SND@LHC, and FLArE. We compare the results of Sec.
IV to other experiments in Sec. V. We conclude in Sec. VI.

##  II The $U(1)_{B-L}$ Model

We first introduce the $U(1)_{B-L}$ gauge interactions of a new gauge boson,
denoted by $A^{\prime}$, to the SM fermions with the charge proportional to
quantum number $B-L$ of the fermions, where $B$ and $L$ are the baryon and
lepton number, respectively. The interaction Lagrangian is given by

$\mathcal{L}_{B-L}\supset
g_{B-L}\biggr{[}-\bar{l}\gamma^{\mu}A^{\prime}_{\mu}l-\bar{\nu}_{\alpha}\gamma^{\mu}A^{\prime}_{\mu}\nu_{\alpha}+\frac{1}{3}\bar{q}\gamma^{\mu}A^{\prime}_{\mu}q\biggr{]}\;,$
(1)

where $g_{B-L}$ is the $U(1)_{B-L}$ coupling strength and $q,l$ and $\nu$ are
quark, charged lepton and neutrino fields, respectively. Here the gauge boson
$A^{\prime}$ couples vectorially to the SM fermions and the gauge boson
$A^{\prime}$ receives the mass by either a spontaneous breaking of the
$U(1)_{B-L}$ or Stueckelberg mechanism, which is not our concern here. Thus,
the model is characterized by the mass $M_{A^{\prime}}$ and the gauge coupling
strength $g_{B-L}$.

Hidden-sector $U(1)^{\prime}$ models belong to a class of hidden-valley models
Strassler:2006im . It is often used to build models of dark matter. Such
$U(1)^{\prime}$ models can be connected to the SM via a small kinetic mixing
though it is hidden from our world. The mass of $A^{\prime}$ and mixing
parameter $\epsilon$ can be tuned to satisfy the relic density of the
Universe. The Lagrangian describing the kinetic mixing with the SM hypercharge
field is given by

$\mathcal{L}=-\frac{1}{4}B^{\prime 2}_{\mu\nu}-\frac{1}{4}F^{\prime\prime
2}_{\mu\nu}+\frac{1}{2}\epsilon
B^{\prime}_{\mu\nu}F^{\prime\prime\mu\nu}+\frac{1}{2}M^{2}_{A^{\prime}}A^{\prime\prime
2}_{\mu}-g_{Y}\frac{Y}{2}B^{\prime}_{\mu}\bar{f}\gamma^{\mu}f\,,$ (2)

where
$B^{\prime}_{\mu\nu}=\partial_{\mu}B^{\prime}_{\nu}-\partial_{\nu}B^{\prime}_{\mu}$
and
$F^{\prime\prime}_{\mu\nu}=\partial_{\mu}A^{\prime\prime}_{\nu}-\partial_{\nu}A^{\prime\prime}_{\mu}$,
and here the primed fields $B^{\prime}$ and $A^{\prime\prime}$ denote the
fields before rotating into physical fields. The last term denotes the
interaction of $B^{\prime}_{\mu}$ with SM fermions. The kinetic mixing term
can be eliminated by rotating $(B^{\prime}_{\mu},A^{\prime\prime}_{\mu})$ into
physical fields $(B_{\mu},A^{\prime}_{\mu})$ by a field redefinition:

$\displaystyle B^{\prime}_{\mu}$ $\displaystyle\simeq$ $\displaystyle
B_{\mu}+\epsilon A^{\prime}_{\mu}$ $\displaystyle A^{\prime\prime}_{\mu}$
$\displaystyle\simeq$ $\displaystyle A^{\prime}_{\mu}\,.$

Then the Lagrangian becomes

$\mathcal{L}=-\frac{1}{4}B^{2}_{\mu\nu}-\frac{1}{4}F^{\prime
2}_{\mu\nu}+\frac{1}{2}M^{2}_{A^{\prime}}A^{\prime
2}_{\mu}-g_{Y}\frac{Y}{2}(B_{\mu}+\epsilon
A^{\prime}_{\mu})\,\bar{f}\gamma^{\mu}f\,.$ (3)

Thus, the hidden $U(1)^{\prime}$ gauge boson $A^{\prime}$ can now couple to
the SM fermion with an interaction strength $\epsilon g_{Y}(Y/2)$.

Conventionally, the term ”dark photon” is used to refer to a $U(1)^{\prime}$
gauge boson that is coupled to the SM only through kinetic mixing with the
photon. Strictly speaking such a dark photon does not couple to neutrinos
because of zero electric charge of the neutrino. However, as shown above the
kinetic mixing with the $B_{\mu}$ does create an interaction of $A^{\prime}$
with neutrinos proportional to the hypercharge of the neutrinos.
Simultaneously, it also mixes with the SM $Z$ boson and thus the mixing
parameter $\epsilon$ is constrained to be small by the LEP data
Workman:2022ynf . Nevertheless, we will continue to use the term dark photon
in a more general sense for the gauge bosons coupled to SM particles only
through the kinetic mixing.

The $U(1)_{B-L}$ gauge boson couples to the left- and right-handed fermions
with the same quantum number, and thus the interaction is vector-like.

On the other hand, the hidden $U(1)^{\prime}$ gauge boson discussed above
couples to SM fermions with quantum number equal to the hypercharge of the
fermion: $-\epsilon g_{Y}(Y/2)\bar{f}\gamma^{\mu}f\,A^{\prime}_{\mu}$. Thus,
for the left-handed electron and neutrino the couplings are $-\epsilon
g_{Y}(-1/2)\bar{\nu}\gamma\nu A^{\prime}_{\mu}$ and $-\epsilon
g_{Y}(-1/2)\overline{e_{L}}\gamma e_{L}A^{\prime}_{\mu}$, respectively, while
the coupling of the right-handed electron is $-\epsilon
g_{Y}(-1)\overline{e_{R}}\gamma e_{R}A^{\prime}_{\mu}$.

##  III Neutrino-Electron scattering

Neutrino-electron scattering is a viable alternative to collider searches for
physics beyond the SM Feng:2022inv . In this section, we discuss the neutrino-
electron scattering in the SM and with the presence of a dark photon.

###  III.1 Standard Model cross section

Table 1: Valued for $a$ and $b$ in the expression of Eq. (4), where $\alpha=\mu,\leavevmode\nobreak\ \tau$ Process | $a$ |  $b$
---|---|---
$\nu_{e}e^{-}\rightarrow\nu_{e}e^{-}$ |  $\sin^{2}\theta_{w}+\frac{1}{2}$ |  $\sin^{2}\theta_{w}$
$\bar{\nu}_{e}e^{-}\rightarrow\bar{\nu}_{e}e^{-}$ |  $\sin^{2}\theta_{w}$ |  $\sin^{2}\theta_{w}+\frac{1}{2}$
$\nu_{\alpha}e^{-}\rightarrow\nu_{\alpha}e^{-}$ |  $\sin^{2}\theta_{w}-\frac{1}{2}$ |  $\sin^{2}\theta_{w}$
$\bar{\nu}_{\alpha}e^{-}\rightarrow\bar{\nu}_{\alpha}e^{-}$ |  $\sin^{2}\theta_{w}$ |  $\sin^{2}\theta_{w}-\frac{1}{2}$

In SM, the $\nu_{e}$-$e$ scattering takes place via the charged and neutral
current interactions. However, the $\nu_{\mu}$-$e$ and $\nu_{\tau}$-$e$
scatterings are mediated only by the neutral $Z$ boson. The differential
cross-section in the rest frame of the electron can be expressed as
Bilmis:2015lja ; Chakraborty:2021apc

$\big{[}\frac{d\sigma}{dT}(\nu e^{-}\rightarrow\nu
e^{-})\big{]}_{SM}=\frac{2G^{2}_{\rm F}m_{\rm e}}{\pi
E^{2}_{\rm\nu}}(a^{2}E^{2}_{\rm\nu}+b^{2}(E_{\rm\nu}-T)^{2}-abm_{\rm e}T)\;,$
(4)

where $G_{\rm F}$ denotes the Fermi constant, $T$ indicates the recoil energy
of the electron, $E_{\rm\nu}$ is the energy of the incoming neutrino, and
$m_{\rm e}$ is the mass of electron. The values for $a$ and $b$ depend on the
neutrino flavor, which are given in Table. 1. The contributing Feynman
diagrams of the SM neutrino-electron scattering in the SM are shown in Fig.
1(a).

The SM production of $\nu_{e,\mu,\tau}e^{-}\to\nu_{e,\mu,\tau}e^{-}$ is the
irreducible background to the scattering via the dark photon. On the other
hand, a reducible background is the charged-current scattering with the
nucleon $\nu_{e}N\to e^{-}N^{\prime}$. In the signal process, the incoming
neutrino does not scatter with the nucleon, and therefore the nucleon remains
intact. On the other hand, the charged-current scattering with the nucleon
will break the nucleon and thus distinguishable from the signal process.
Nevertheless, the hadronic part in this charged current interaction of $\nu$
with the nucleon may not be fully identified with a 100% efficiency.
Therefore, whether this reducible background can be completely removed depends
on the experimental setup and resolution. This is beyond the scope of this
work. To a large extent we can include this uncertainty into the systematic
uncertainties. We will show the results with systematic uncertainties of 5%
and 20%, the latter should be large enough to account for the uncertainty of
this reducible background.

###  III.2 Dark photon contribution to the cross-section

The $U(1)_{B-L}$ gauge boson has vectorial couplings to SM fermions, but the
$U(1)^{\prime}$ gauge boson has different couplings to left- and right-handed
fermions. In the following, we focus on the $U(1)_{B-L}$. Later, we extend the
calculation to the $U(1)^{\prime}$.

We obtain the differential cross section for neutrino-electron scattering as a
function of the recoil energy. The effective renormalizable Lagrangian given
in Eq. (3), where the SM photon and dark photon mixed via a kinetic term. This
mixing has been studied widely in literature Essig:2013lka ; Essig:2010gu ;
Essig:2009nc . The $U(1)_{B-L}$ gauging induces a new coupling $g_{B-L}$ in
addition to its mass $M_{A^{\prime}}$ and $\epsilon$. The parameter space of
the current dark photon model contains a set of three variables
($M_{A^{\prime}},\epsilon,g_{B-L}$). In this study we focus on the dark model
with two parameters, $M_{A^{\prime}}$ and $g_{B-L}$. Figure 1(b) shows the
Feynman diagram of neutrino-electron scattering mediated by the dark photon.

Figure 1:  (a) Contributing Feynman diagrams for anti-electron-neutrino and
electron scattering via the $W$ and $Z$ boson exchanges. Note that muon- and
tau-neutrinos (including anti-muon- and anti-tau-neutrinos) scatter with
electron only via the $Z$ boson exchange. (b) The Feynman diagram showing the
scattering of (anti-)neutrino with electron via the $t$-channel dark photon
$A^{\prime}$ exchange.

The scattering cross section from the pure contribution of $A^{\prime}$ is
expressed as follows Bilmis:2015lja .

$\big{[}\frac{d\sigma}{dT}(\nu e^{-}\rightarrow\nu
e^{-})\big{]}_{DP}=\frac{g^{4}_{B-L}m_{e}}{4\pi E^{2}_{\nu}(M^{2}_{\rm
A^{\prime}}+2m_{e}T)^{2}}(2E_{\nu}^{2}+T^{2}-2TE_{\nu}-m_{e}T)$ (5)

where the cross section is flavor blind. On the other hand, the interference
term in the differential cross-section depends on the neutrino flavor and
whether it is the neutrino or anti-neutrino. The interference terms are given
by Bilmis:2015lja .,

$\frac{d\sigma_{\rm
INT}(\nu_{e}e^{-})}{dT}=\frac{g^{2}_{B-L}G_{F}m_{e}}{2\sqrt{2}\pi
E^{2}_{\nu}(M^{2}_{\rm A^{\prime}}+2m_{e}T)}(2E_{\nu}^{2}-m_{e}T+\beta)$ (6)
$\frac{d\sigma_{\rm
INT}(\bar{\nu}_{e}e^{-})}{dT}=\frac{g^{2}_{B-L}G_{F}m_{e}}{2\sqrt{2}\pi
E^{2}_{\nu}(M^{2}_{\rm
A^{\prime}}+2m_{e}T)}(2E_{\nu}^{2}+2T^{2}-T(4E_{\nu}+m_{e})+\beta)$ (7)
$\frac{d\sigma_{\rm
INT}(\nu_{\alpha}e^{-})}{dT}=\frac{g^{2}_{B-L}G_{F}m_{e}}{2\sqrt{2}\pi
E^{2}_{\nu}(M^{2}_{\rm A^{\prime}}+2m_{e}T)}(-2E_{\nu}^{2}+m_{e}T+\beta)$ (8)

$\frac{d\sigma_{\rm
INT}(\bar{\nu}_{\alpha}e^{-})}{dT}=\frac{g^{2}_{B-L}G_{F}m_{e}}{2\sqrt{2}\pi
E^{2}_{\nu}(M^{2}_{\rm
A^{\prime}}+2m_{e}T)}(-2E_{\nu}^{2}-2T^{2}-T(4E_{\nu}+m_{e})+\beta)$ (9)

where $\beta$ is given by

$\beta=\sin^{2}\theta_{w}(8E^{2}_{\nu}-8E_{\nu}T-4m_{e}T+4T^{2}).$

Here $\alpha$ in $\nu_{\alpha}$ denotes either $\mu$ or $\tau$. The
$\nu_{\mu}$-$e^{-}$ and $\nu_{\tau}$-$e^{-}$ scatterings can occur through the
SM $Z$ and the dark photon $A^{\prime}$ mediated processes, while the
$\nu_{e}$-$e^{-}$ scattering is mediated by the SM $Z,W$ bosons and dark
photon $A^{\prime}$. The pure contribution of the dark photon cross section is
proportional to the fourth power of $g_{B-L}$ while the interference term is
proportional to the second power of $g_{B-L}$. The total cross-section
$\sigma_{T}$ is then given by
$\sigma_{T}=\sigma_{DP}+\sigma_{INT}+\sigma_{SM}$ .

##  IV Sensitivity reach on parameter space of the Dark Photon model at the
Forward physics Experiments

Proton-proton collisions at the Large Hadron Collider (LHC) produce many
particles along the beam axis. Forward Physics Facilities (FPF) Feng:2022inv ;
Anchordoqui:2021ghd ; FASER:2019aik ; FASER:2021ljd ; FASER:2021cpr ;
FASER:2020gpr ; FASER:2018bac ; Kling:2021gos ; FASER:2019dxq ; Ismail:2020yqc
, situated several hundred meters away from the ATLAS interaction point and
protected by concrete and rock, will host a wide variety of experiments
probing the SM processes and also searching for new physics beyond the
standard model (BSM) Jodlowski:2020vhr ; Feng:2022inv ; Feng:2017vli ;
Kling:2018wct ; Feng:2018pew ; Deppisch:2019kvs ; Cheung:2021tmx ;
Bakhti:2020szu ; Jho:2020jfz ; Okada:2020cue ; Bahraminasr:2020ssz ;
Kelly:2020pcy ; Falkowski:2021bkq ; FASER:2018eoc ; Cottin:2021lzz ;
Ismail:2021dyp ; Ansarifard:2021elw ; Cheung:2022oji ; Batell:2021blf ;
Batell:2021aja ; Batell:2021snh . Each experiment in the FPF is uniquely
optimized for its own physics goals.

In this section, we study the neutrino-electron scattering mediated by the
$t$-channel dark photon exchange at the running FASER$\nu$ and the proposed
FASER$\nu 2$, FLArE (Forward Liquid Argon Experiment), and SND$@$LHC, where
$\rm FASER\nu 2$ and FLArE are parts of the proposed Forward Physics Facility
(FPF). FPF also contains other detectors called the FASER2, Advanced SND, and
FORMOSA. The dark photon sensitivity at FASER2 was discussed in the literature
Feng:2022inv ; Bauer:2018onh ; FASER:2018eoc .

The FASER$\nu$ and SND@LHC detectors are made of tungsten with masses of 1.1
tons and 830 kg respectively. FASER$\nu$ is made of 1000 layers of emulsion
films interleaved with tungsten plates of thickness of 1 mm. The size of
FASER$\nu$ is $25\,{\rm cm}\times 30\,{\rm cm}\times 1.1\,{\rm m}$ talk-2022 ;
talk-2022(2) . That of SND@LHC is $39\,{\rm cm}\times 39\,{\rm cm}$
talk-2022(2) . The upgraded version of FASER$\nu$ – FASER$\nu 2$ will have a
size $50\,{\rm cm}\times 50\,{\rm cm}\times 5\,{\rm m}$ and will weigh 10
tons. FLArE is composed of 10 ton-scale liquid argon time projection chamber
(LArTPC) of the type being employed in several modern neutrino experiments.
The size of the FLArE detector is $1\,{\rm m}\times 1\,{\rm m}\times 7\,{\rm
m}$ and will weigh 10 tons.

We are interested in the neutrino-electron scattering at the aforementioned
detectors. It is well known that huge number of hadrons, such as pions, kaons
and other hadrons, are produced along the beam direction. These hadrons will
decay during the flight, thus producing a lot of neutrinos of all three
flavors at very high energy up to a few TeVKling:2021gos . It was shown
FASER:2020gpr  that muon neutrinos are mostly produced from charged-pion
decays, electron neutrinos from hyperon, kaon and $D$-meson decays, and tau
neutrinos from $D_{s}$ meson decays. With average energies ranging from 600
GeV to 1 TeV, the spectra of the three neutrino flavors cover a broad energy
range.

The uncertainties in the coming neutrino flux were studied in Ref.
Kling:2021gos . The muon-neutrino flux has less than 10% uncertainty up to 1
TeV, the electron-neutrino flux has less than 10% uncertainty up to 0.8 TeV,
and the tau-neutrino flux has less than 10% uncertainty up to $0.3\sim 0.4$
TeV. Afterwards, the uncertainties are within a factor of 2. Such
uncertainties will propagate to the event rates predictions of order a few
percent up to 20%. Such uncertainties are included in the systematic
uncertainties.

###  IV.1  Dark photon interactions at FASER$\nu$ and FASER$\nu 2$

In this subsection we discuss the neutrino-electron $\nu e^{-}\rightarrow\nu
e^{-}$ scattering at FASER$\nu$ and FASER$\nu 2$ mediated by a new neutral
$U(1)_{B-L}$ gauge field $A^{\prime}$. The cross-section of the fixed target
neutrino-electron scattering is given in Sec. III. We consider the dark photon
of mass $M^{\prime}_{A}$ ranging from $10^{-6}$ GeV to 10 GeV with an incoming
neutrino beam energy $E_{\nu}$ from 10 GeV to 10 TeV. The cross section is
calculated by setting the value of $g_{B-L}=10^{-2}$. We use the neutrino
fluxes and energy spectra obtained in FASER:2019dxq ; Kling:2021gos  to study
the neutrinos that pass through FASER$\nu$ and FASER$\nu 2$. We estimate the
90% C.L. sensitivity reach on the parameter space of the dark photon model. We
show that the best sensitivity can be achieved in the small $M_{A^{\prime}}$
region.

####  IV.1.1 $\rm FASER\nu$

The FASER$\nu$ neutrino detector is located in front of the FASER main
detector FASER:2019dxq ; FASER:2018bac , which is aligned along the collision
axis to maximize the neutrino interaction. FASER$\nu$FASER:2019dxq  is a
$25\,{\rm cm}\times 30\,{\rm cm}\times 1.1\,{\rm m}$ emulsion detector,
consisting of 770 layers of emulsion films interleaved with 1-mm-thick
tungsten plates with mass 1.2 tons talk-2022  .The features of high density
and shorten radiation length of tungsten are advocated as the material for the
detector, because such features help keeping the detector small in size and
localizing electromagnetic showers in a small volume.

Figure 2:  Total number of active neutrinos produced at FASER$\nu$ from
neutrino-electron scattering
($\nu_{\alpha}e^{-}\rightarrow\nu_{\alpha}e^{-}$), where $\alpha=e,\mu,\tau$.
Contributions from neutrinos and anti-neutrinos are included.

The expected numbers of neutrino events from the three flavors of active
neutrinos with different dark photon masses $M_{A^{\prime}}$ in FASER$\nu$ are
shown in Fig. 2. The solid lines represent the number of neutrino events as a
function of dark photon mass $M_{A^{\prime}}$ while the dashed lines represent
the SM event rates. The number of events associated with dark photon overlaps
with the SM value as the dark photon mass $M_{A^{\prime}}$ increases. The
highest number of events comes in with the muon neutrino $\nu_{\mu}$
scattering with electron($\nu_{\mu}e^{-}\rightarrow\nu_{\mu}e^{-}$), while the
lowest one is from the tau neutrino $\nu_{\tau}$ scattering
($\nu_{\tau}e^{-}\rightarrow\nu_{\tau}e^{-}$), and that from the electron
neutrino $\nu_{e}$ scattering is in between. Total number of events follows
the same behavior as the scattering cross section. We can summarize the
results in Fig. 2 as follows,
$N_{\nu_{\tau}}(g_{B-L},M_{A^{\prime}})<N_{\nu_{e}}(g_{B-L},M_{A^{\prime}})<N_{\nu_{\mu}}(g_{B-L},M_{A^{\prime}})$.

To estimate the sensitivity reach in the parameter space ($g_{B-L}$ vs
$M_{A^{\prime}}$) of the dark photon model, we first calculate the predicted
number of events $N_{BSM}$ for the dark photon model and the SM number of
events $N_{SM}$, and treat the statistical error as $\sqrt{N_{BSM}}$ and
systematic uncertainty $\sigma$ as a fraction ($\sigma$= 20%, 5%) of the
normalization of the SM predictions. We then define the measure of $\chi^{2}$
as a function of ($g_{B-L}$) and a nuisance parameter $\alpha$ as follows
Cheung:2021tmx ; Ballett:2019xoj

$\displaystyle\chi^{2}(g_{B-L},\alpha)$ $\displaystyle=$
$\displaystyle\min_{\alpha}\Biggr{[}\frac{(N^{\nu_{e}}_{BSM}-(1+\alpha)N^{\nu_{e}}_{SM})^{2}}{N^{\nu_{e}}_{BSM}}+\frac{(N^{\nu_{\mu}}_{BSM}-(1+\alpha)N^{\nu_{\mu}}_{SM})^{2}}{N^{\nu_{\mu}}_{BSM}}$
(10)
$\displaystyle+\frac{(N^{\nu_{\tau}}_{BSM}-(1+\alpha)N^{\nu_{\tau}}_{SM})^{2}}{N^{\nu_{\tau}}_{BSM}}+\left(\frac{\alpha}{\sigma}\right)^{2}\Biggr{]}\;,$

where $N_{BSM}=N_{DP}+N_{INT}+N_{SM}$ and the minimization is over the
nuisance parameter $\alpha$. Here $N_{DP}$ is the number of events from the
dark photon mediated process only, $N_{INT}$ is the interference term. Here we
have treated the systematic uncertainties in each neutrino flavor to be the
same and use only one nuisance parameter $\alpha$. Physics-wise the systematic
uncertainties come from theoretical calculations, the flux of neutrinos from
the ATLAS IP, detector response, etc.

Figure 3:  Left Panel: Sensitivity reach on the coupling $g_{B-L}$ versus the
dark photon mass $M_{A^{\prime}}$ achieved at FASER$\nu$. Systematic
uncertainties $\sigma$ = 5%, 20% and without systematic uncertainties are
shown. Right Panel: sensitivity reach on the coupling $g^{\alpha}_{B-L}$ for
each neutrino flavor $\alpha=e,\mu,\tau$ versus the dark photon mass
$M_{A^{\prime}}$ achieved at FASER$\nu$ (without systematic uncertainties) are
shown.

We show in the left panel of Fig. 3 the 90% C.L. sensitivity reach
(corresponding to $\chi^{2}=2.71$) of the coupling strength $g_{B-L}$ versus
the dark photon mass $M_{A^{\prime}}$ at FASER$\nu$. The higher the systematic
uncertainty the weaker the limit on $g_{B-L}$ will be. Nevertheless, the
differences among $\sigma=5\%,20\%$ and without systematic uncertainties are
relatively small. The sensitivity reach on $g_{B-L}$ is the best at very small
$M_{A^{\prime}}$, around $2\times 10^{-6}$ at $M_{A^{\prime}}=10^{-6}$ GeV and
reduces to about $2\times 10^{-2}$ at $M_{A^{\prime}}=10$ GeV.

FASER$\nu$ is primarily designed for the purpose of identifying the flavors of
neutrinos FASER:2019dxq . The expected sensitivity for each flavor at
FASER$\nu$ is shown in the right panel of Fig. 3. We only show the curves with
no systematic uncertainty included. The curves can be compared to the
corresponding one “Without Systematic” of the left panel of Fig. 3. For the
evaluation of $\chi^{2}$, we consider the special case of Eq. 10 with
$\alpha=0$ and it reduces to

$\chi^{2}(g^{\alpha}_{B-L})=[\frac{(N^{\nu_{\alpha}}_{BSM}-N^{\nu_{\alpha}}_{SM})^{2}}{N^{\nu_{\alpha}}_{BSM}}]$
(11)

where $\alpha=e,\mu,\tau$. The green and orange curves of Fig. 3 depict the
sensitivity reach of $g^{\tau}_{B-L}$ and $g^{e}_{B-L}$ ($\chi^{2}=2.71$)
versus $M_{A^{\prime}}$, respectively, while the blue curve represents the
sensitivity reach of $g^{\mu}_{B-L}$ versus $M_{A^{\prime}}$. It is clear from
the right and left panel of Fig. 3 that the overall sensitivity reach of
$g_{B-L}$ is dominated by $g^{\mu}_{B-L}$. The sensitivity reach by the
neutrino flavors is in the order of $\nu_{\tau}<\nu_{e}<\nu_{\mu}$.

####  IV.1.2 FASER$\nu 2$

FASER$\nu 2$ is a much more advanced version of FASER$\nu$Anchordoqui:2021ghd
. FASER$\nu 2$ detector is currently proposed to be composed of 3330 emulsion
layers interleaved with 2 mm thick tungsten plates. It will also accommodate a
veto detector and an interface detector to the FASER2 spectrometer, with one
detector in the middle of the emulsion modules and the other detector
downstream of the emulsion modules to make the global analysis and muon charge
measurement possible. Both the emulsion modules and interface detectors will
be put in a cooling system. The total volume of the tungsten target is
$50\,{\rm cm}\times 50\,{\rm cm}\times 5\,{\rm m}$, and the mass is 10 tons.
The detector length, including the emulsion films and interface detectors,
will be about 8 m.

Figure 4:  Total number of active neutrinos produced at FASER$\nu 2$ from
neutrino-electron scattering
($\nu_{\alpha}e^{-}\rightarrow\nu_{\alpha}e^{-}$), where $\alpha=e,\mu,\tau$.
Contributions from neutrinos and anti-neutrinos are included.

In this subsection we calculate the sensitivity reach on the parameter space
of the dark photon model at FASER$\nu 2$. We follow the same procedures as the
last subsection for the estimation of the total number of neutrino events at
FASER$\nu 2$. The total numbers of neutrino events at FASER$\nu 2$ are shown
in Fig. 4, which follows the same notation as Fig. 2. Comparing with
FASER$\nu$, FASER$\nu 2$ could collect a larger number of neutrino events.
There are sizable increments in both the BSM neutrino events and SM neutrino
events. In the heavy $M_{A^{\prime}}$ mass regime ($M_{A^{\prime}}\gtrsim 5$
GeV), the $N_{\nu_{\mu}}$ curve overlaps with the $N_{\nu_{e}}$ curve. This
behavior arises from the large interference effect associated with the muon
neutrinos. Among the three neutrino flavors, the tau neutrino is the least in
event rates.

Figure 5:  Left Panel: Sensitivity reach on the coupling $g_{B-L}$ versus the
dark photon mass $M_{A^{\prime}}$ achieved at FASER$\nu 2$. Systematic
uncertainties $\sigma$ = 5%, 20% and without systematic uncertainties are
shown. Right Panel: sensitivity reach on the coupling $g^{\alpha}_{B-L}$ for
each neutrino flavor $\alpha=e,\mu,\tau$ versus the dark photon mass
$M_{A^{\prime}}$ achieved at FASER$\nu 2$ (without systematic uncertainties)
are shown.

Sensitivity reach at FASER$\nu 2$ is carried out by a similar $\chi^{2}$
analysis in Eq. (10). We estimate the 90% C.L. sensitivity reach
(corresponding to $\chi^{2}=2.71$) of the coupling strength $g_{B-L}$ versus
dark photon mass $M_{A^{\prime}}$ at FASER$\nu 2$, as shown in Fig. 5. The
dark photon sensitivity reach at FASER$\nu 2$ is shown in the left panel of
Fig. 5, in which the curves with $\sigma=5\%$ and and $20\%$ are overlapping
with each other. FASER$\nu 2$ has better sensitivity compared to FASER$\nu$.
In the very small dark photon mass region,
$M_{A^{\prime}}=10^{-6}\rm\leavevmode\nobreak\ GeV$, FASER$\nu 2$ can reach as
low as $10^{-6}$ while FASER$\nu$ only reaches around $2\times 10^{-6}$. The
sensitivity devalues with the increment of dark photon mass.

The expected sensitivity reach of FASER$\nu 2$ for each neutrino flavor is
given in the right panel of Fig. 5. The effect of interference is clearly
visible from the cross-over of $g^{e}_{B-L}$ curve over the $g^{\mu}_{B-L}$
curve. In the higher $M_{A^{\prime}}$ mass region, the $g^{e}_{B-L}$ curve has
better sensitivity compared with the $g^{\mu}_{B-L}$ curve. The total
sensitivity curve of $g_{B-L}$ in the left panel of Fig. 5 resembles the
$g^{\mu}_{B-L}$ curve of the right panel in the low $M_{A^{\prime}}$ region
and switches to $g^{e}_{B-L}$ in the heavier $M_{A^{\prime}}$ region.

###  IV.2  Dark photon interactions at SND@LHC

Figure 6:  Total number of active neutrinos produced at SND@LHC from neutrino-
electron scattering ($\nu_{\alpha}e^{-}\rightarrow\nu_{\alpha}e^{-}$), where
$\alpha=e,\mu,\tau$. Contributions from neutrinos and anti-neutrinos are
included.

SND@LHC is a compact detector of neutrinos at the LHC and it consists of a
neutrino target followed downstream by a device for detecting muons, which are
produced when neutrinos interact with the target. It is especially
complementary to the FASER$\nu$. Targets are made from tungsten plates
interlaced with emulsion films and electronic tracking devices. Emulsion films
display the tracks of particles produced in neutrino interactions, while
electronic tracking devices provide the time stamps for these tracks. The
tracking devices also measure the energy of the neutrinos along with the muon
detector.

The predicted number of neutrino events mediated by the dark photon of mass
ranging from $M_{A^{\prime}}\sim 10^{-6}-10\rm\leavevmode\nobreak\ GeV$ is
shown in Fig. 6. Comparing with FASER$\nu$ and FASER$\nu 2$, SND@LHC has fewer
neutrino events. We summarize the predicted numbers of neutrino events for
these three forward-physics experiments as follows, $\rm N^{\rm
SND@LHC}_{BSM}(g_{B-L},M_{A^{\prime}})<N^{FASER\nu}_{BSM}(g_{B-L},M_{A^{\prime}})<N^{FASER\nu
2}_{BSM}(g_{B-L},M_{A^{\prime}})$. The difference in the event rate arises
from the neutrino flux at each detector volume.

Figure 7:  Left Panel: Sensitivity reach on the coupling $g_{B-L}$ versus the
dark photon mass $M_{A^{\prime}}$ achieved at SND@LHC. Systematic
uncertainties $\sigma$ = 5%, 20% and without systematic uncertainties are
shown. Right Panel: sensitivity reach on the coupling $g^{\alpha}_{B-L}$ for
each neutrino flavor $\alpha=e,\mu,\tau$ versus the dark photon mass
$M_{A^{\prime}}$ achieved at SND@LHC (without systematic uncertainties) are
shown.

A similar $\chi^{2}$ analysis (Eq. 10) is carried out at SND@LHC to assess
dark photon sensitivity. Accordingly, a 90% C.L. sensitivity reach is
estimated (corresponding to $\chi^{2}=2.71$) for the coupling strength
$g_{B-L}$ versus the dark photon mass $M_{A^{\prime}}$ at the SND@LHC, as
shown in the left panel of Fig. 7. Here we have assumed a benchmark detector
made of tungsten with dimensions $39\,{\rm cm}\times 39\,{\rm cm}$ at the 14
TeV LHC with an integrated luminosity of $L=150\leavevmode\nobreak\ fb^{-1}$.
The higher the systematic uncertainty the weaker the limit on $g_{B-L}$ will
be. Nevertheless, the differences among $\sigma=5\%,20\%$ and without
systematic uncertainties are relatively small. The sensitivity reach on
$g_{B-L}$ is the best at very small $M_{A^{\prime}}$ as low as $4\times
10^{-6}$ at $M_{A^{\prime}}=10^{-6}$ GeV and devalues to about $8\times
10^{-2}$ at $M_{A^{\prime}}=10$ GeV. The reach of $g^{\alpha}_{B-L}$ for each
neutrino flavor as a function of dark photon mass $M_{A^{\prime}}$ is shown in
the right panel of Fig. 7.

Figure 8:  Total number of active neutrinos produced at FLArE (10 tons) from
neutrino-electron scattering
($\nu_{\alpha}e^{-}\rightarrow\nu_{\alpha}e^{-}$), where $\alpha=e,\mu,\tau$.
Contributions from neutrinos and anti-neutrinos are included.

###  IV.3  Dark photon interactions at FLArE (10 tons)

As a part of the suite of detectors for the FPF, the liquid argon time
projection chamber (LArTPC) is being considered. This type of detector offers
the capability to determine particle identification, track angle, and kinetic
energy over a wide range of energies. Dark matter (DM) direct detection
searches Batell:2021blf  and neutrino experiments have successfully used
liquid argon as an active detector. Detecting neutrino energy
Anchordoqui:2021ghd  and searching for dark matter are therefore well suited
to be put together to utilize such a liquid-argon detector. There is an
imperative motivation to detect and measure TeV-scale neutrino events from a
laboratory-generated and well-characterized source. Moreover, the LHC could be
the only source of high-energy intense tau neutrinos on Earth. Here we have
assumed a benchmark detector made of tungsten with dimensions $1{\rm m}\times
1\,{\rm m}\times 8\,{\rm m}$ at the 14 TeV LHC with an integrated luminosity
of $L=3000\leavevmode\nobreak\ fb^{-1}$.

The expected number of neutrino events at FLArE(10 tons) is presented in Fig.
8. Comparing with the aforementioned forward physics experiments, the expected
number of neutrino events at the FLArE as a function of dark photon mass
$M_{A^{\prime}}$ and coupling $g_{B-L}$ are in the following ordering: $\rm
N^{\rm
SND@LHC}_{BSM}(g_{B-L},M_{A^{\prime}})<N^{FASER\nu}_{BSM}(g_{B-L},M_{A^{\prime}})<N^{FLArE}_{BSM}(g_{B-L},M_{A^{\prime}})<N^{FASER\nu
2}_{BSM}(g_{B-L},M_{A^{\prime}})$. This ordering will reflect the bound on the
dark photon coupling.

Figure 9:  Left Panel: Sensitivity reach on the coupling $g_{B-L}$ versus the
dark photon mass $M_{A^{\prime}}$ achieved at FLArE (10 tons). Systematic
uncertainties $\sigma$ = 5%, 20% and without systematic uncertainties are
shown. Right Panel: sensitivity reach on the coupling $g^{\alpha}_{B-L}$ for
each neutrino flavor $\alpha=e,\mu,\tau$ versus the dark photon mass
$M_{A^{\prime}}$ achieved at FLArE(10 tons) (without systematic uncertainties)
are shown.

.

The bound on the dark-photon coupling is obtained through the $\chi^{2}$
defined in Eq. (10). We obtain the 90% C.L. dark photon sensitivity at FLArE
(10 tons), as shown in the left panel of the Fig. 9, which shows the coupling
$g_{B-L}$ versus the dark photon mass $M_{A^{\prime}}$ with systematic
uncertainties $\sigma$ = 5%, 20% and without systematic uncertainties. The
right panel of Fig. 9 obtained by Eq. (11) shows the sensitivity reach on the
coupling $g^{\alpha}_{B-L}$ for each neutrino flavor $\alpha=e,\mu,\tau$
versus the dark photon mass $M_{A^{\prime}}$ achieved at FLArE (10 tons)
(without systematic uncertainties). The effect of interference is visible from
the right panel of the Fig. 9, in which the $g^{e}_{B-L}$ curve overlaps with
the $g^{\mu}_{B-L}$ curve for $M_{A^{\prime}}\gtrsim 0.1$ GeV.

##  V Complementarity of FASER$\nu/2\nu$, SND@LHC and FLArE Results

We have obtained the future sensitivity reach in the parameter space
($g_{B-L},M_{A^{\prime}}$) of the dark photon model at FASER$\nu$,
FASER$2\nu$, SND@LHC, and FLArE(10 tons). In this section, we compare these
forward physics experiments with existing neutrino-electron scattering
experiments. While comparing the Forward Physics Experiments to one another,
FASER$\nu 2$ has the best sensitivity among the four experiments while SND@LHC
is the least sensitive one. For the case of a very small dark photon mass
$M_{A^{\prime}}=10^{-6}\rm\leavevmode\nobreak\ GeV$ the sensitivity reach of
$g_{B-L}$ is around $4\times 10^{-6}$ at SND@LHC, $2\times 10^{-6}$ at
FASER$\nu$, $1.6\times 10^{-6}$ at FLArE(10 tons), and $10^{-6}$ at FASER$\nu
2$. FASER$\nu 2$ has the best sensitivity for the whole range of dark photon
mass considered in this work.

Figure 10:  Future sensitivity reach of FASER$\nu$ (military green),
FASER$2\nu$ (blue), SND@LHC (violet), and FLArE (10 tons)(red) at 90%
C.L.(without systematic uncertainties) on the gauge coupling $g_{B-L}$ of the
$U_{B-L}(1)$ group as the function of dark photon mass $M_{A^{\prime}}$. Other
existing constraints at 90% C.L. shown include (i) $\bar{\nu}_{e}-e$
scattering measurement at Gemma (dashed magenta curve), (ii) NPCGe (dashed
orange curve), (iii) HPGe (dashed blue curve), (iv) elastic scattering of
neutrinos using a liquid scintillator at Borexino (dashed black curve), (v)
CsI (dashed red curve) measurement of $\bar{\nu}_{e}-e^{-}$, (vi) LSND (dashed
light green curve) measurement of $\nu_{e}-e^{-}$, (vii) CHARM-
II-$\nu_{\mu}-e$ (dashed light blue), (viii) CHARM-II-$\bar{\nu}_{\mu}-e$
(dashed light yellow), and (ix) NA64 experiment at the CERN SPS(unbroken
$U(1)_{\rm B-L}$)(dashed brown). The light blue region indicates the solar
constraints (luminosity analysis in the conversion of plasmons in the Sun),
the gray region represents the constraint derived from energy loss due to dark
photons in globular clusters, and the orange region indicates the constraint
due to supernova cooling. These colored regions are taken from Ref.
Bilmis:2015lja ; Harnik:2012ni .

The GEMMA Collaboration Beda:2009kx  measured the $\bar{\nu}_{e}-e$ scattering
cross section, which can be used to put a bound on neutrino magnetic moments.
The Gemma (dashed magenta curve) in Fig. 10 has the best sensitivity among all
the considered forward Physics experiments at the low dark photon mass region.
The Gemma sensitivity gets weakened with increment of dark photon mass, after
$M_{A^{\prime}}>0.01\leavevmode\nobreak\ \rm GeV$ there is a considerable
reduction in the Gemma sensitivity to the dark photon coupling. In the heavier
mass region the forward physics experiments estimated in this work in general
have better sensitivity.

The TEXONO-NPCGeChen:2014dsa  estimated the constraints on millicharged
neutrinos via analysis of data from atomic ionizations. NPCGe (dashed orange
curve) also has better sensitivity in the low dark photon mass region, and the
sensitivity gets weakened with increases in dark photon mass. The sensitivity
curve of NPCGe follows the similar pattern of Gemma. The TEXONO-HPGe
TEXONO:2002pra  estimated the limit on the electron-neutrino magnetic moment.
FASER$\nu 2$ (blue curve) has better sensitivity than that of HPGe (dashed
blue) in the very small dark photon mass $M_{A^{\prime}}\sim
10^{-6}\leavevmode\nobreak\ \rm GeV$. However, in the intermediate mass region
$\rm 10^{-6}\leavevmode\nobreak\ GeV<M_{A^{\prime}}<0.01\leavevmode\nobreak\
GeV$, HPGe has better sensitivity than all the forward physics experiments
considered in this work.

The Borexino Collaboration Bellini:2011rx  measured the spectrum of the
$\rm{}^{7}Be$ solar neutrino (with 862 keV energy) via the elastic scattering
of neutrinos using a liquid scintillator. The sensitivity curve of Borexino
(dashed black curve) also followed the similar pattern of TEXONO. However,
both FLArE (red curve) and FASER$\nu 2$ (blue) have better sensitivity at
$M_{A^{\prime}}=10^{-6}\leavevmode\nobreak\ \rm GeV$ compared with Borexino.

Figure 11:  Future sensitivity reach of FASER$\nu$ (orange), FASER$2\nu$
(red), SND@LHC (blue), and FLArE (10 tons)(green) at 90% C.L.(without
systematic uncertainties) on the kinetic mixing parameter $\epsilon$ as a
function of dark photon mass $M_{A^{\prime}}$. The colored regions are the
same as in Fig. 10 Bilmis:2015lja ; Harnik:2012ni

TEXONO-CsI TEXONO:2009knm  measured the $\bar{\nu}_{e}-e^{-}$ scattering.
FASER$\nu$ (dark green), FASER$\nu 2$ (blue), and FLArE (red curve) all have
better sensitivity in the very low $M_{A^{\prime}}$ mass region
($M_{A^{\prime}}<2\times 10^{-6}\leavevmode\nobreak\ \rm GeV$). However, in
the heavier dark photon mass region ($M_{A^{\prime}}<1\rm\leavevmode\nobreak\
GeV$) CsI has better sensitivity than any of the forward physics experiments.
LSND LSND:2001akn  also followed the similar pattern as CsI. However
SND@LHC(violet curve) has better sensitivity compared with LSND (dashed light
green) in the low $M_{A^{\prime}}$ mass region. In all the above, except CsI,
the forward physics experiments have better sensitivity in the heavy dark
photon mass region. CHARM II CHARM-II:1993phx  also measured the neutrino-
electron scattering in the lower dark photon mass region. The forward physics
experiments have better sensitivity compared to CHARM II (CHARM-II
$\nu_{\mu}-e$ (dashed light blue) and CHARM-II $\bar{\nu}_{\mu}-e$ (dashed
light yellow)). A search for a new $Z^{\prime}$ gauge boson associated with
(un)broken $B-L$ symmetry in the keV- GeV mass range is carried out for the
first time using the missing-energy technique in the NA64 experiment (dashed
brown- unbroken $U(1)_{B-L}$) at the CERN SPSAndreev:2022hxz . In the Low dark
photon mass $M_{A^{\prime}}\sim 10^{-6}$, the FPF experiments (exclude
SND@LHC) have better sensitivity compared to NA64e.

The bounds on the kinetic mixing parameter $\epsilon$ as a function of dark
photon mass $M_{A^{\prime}}$ are shown in Fig. 11. The sensitivity reach on
the dark photon kinetic mixing $\epsilon$ follows the same pattern as
$g_{B-L}$ coupling. While comparing the bounds on $\epsilon$ achieved in the
forward physics experiments with existing neutrino-electron scattering
experiments, the forward physics experiments have better bounds in the very
low dark photon mass region $(M_{A^{\prime}}\sim
10^{-6}\rm\leavevmode\nobreak\ GeV)$ and in the heavier dark photon mass
region $M_{A^{\prime}}>0.1\rm\leavevmode\nobreak\ GeV$. For the case of a very
small dark photon mass $M_{A^{\prime}}=10^{-6}\rm\leavevmode\nobreak\ GeV$ the
sensitivity reach of $\epsilon$ is around $2\times 10^{-4}$ for SND@LHC,
$1.3\times 10^{-4}$ for FASER$\nu$, $1.2\times 10^{-4}$ for FLArE(10 tons),
and $8\times 10^{-5}$ for FASER$\nu 2$. FASER$\nu 2$ has the best sensitivity
for all dark photon mass range considered in this work. The sensitivity on
$\epsilon$ gets weakened with increment of dark photon mass.

Sensitivities of the coupling $g_{B-L}$ or the kinetic mixing $\epsilon$ of
the dark photon have been evaluated for the FASER and/or FASER2, as we have
mentioned before. Here we make comparisons with those in literature
Feng:2022inv ; Bauer:2018onh ; FASER:2018eoc . Note that the sensitivities
obtained at FASER/FASER2 are based on the dark photon(s) produced mainly from
the decays of hadrons, such as $\pi,\eta,K$, while those obtained in this work
are based on the neutrino-electron scattering in FASER$\nu$. In Ref.
Feng:2022inv ; Bauer:2018onh , sensitivities of the coupling $g_{B-L}$ at
FASER (FASER2) can be down to $3\times 10^{-6}-0.7\times 10^{-6}$ ($5\times
10^{-7}-4\times 10^{-8}$) for $M_{A^{\prime}}=2\times 10^{-3}-0.1$ GeV
($M_{A^{\prime}}=2\times 10^{-3}-1$ GeV). In Ref. FASER:2018eoc , $g_{B-L}$
can be probed down to $8\times 10^{-7}-2\times 10^{-7}$ for
$M_{A^{\prime}}=3\times 10^{-6}-0.3$ GeV. Another model Little String Theory
was studied in Ref. Anchordoqui:2022kuw  that the coupling $g_{X}$ can be
probed down to almost $10^{-8}$ for $M_{X}=0.2-1.3$ GeV. It is clear that the
study in this work showed that the sensitivities are not as good as those
listed above. Nevertheless, the FPF experiments studied here showed advantage
for $M_{A^{\prime}}>1$ GeV.

Note that for $M_{A^{\prime}}=10^{-6}-10^{-1}$ GeV there are more stringent
constraints from stella and supernova cooling Dent:2012mx ; Kazanas:2014mca .
We add the constraint due to coupling in the SunRedondo:2008aa ;
Raffelt:1988rx ; An:2013yfc  (light blue), globular clustersRedondo:2008aa ;
Raffelt:1988rx ; An:2013yfc ; Jaeckel:2010ni  (gray), and SN1987ADent:2012mx ;
Kazanas:2014mca  (orange) to the summary figures in Fig. 10 and 11 for
$g_{B-L}$ and $\epsilon$, respectively. For $M_{A^{\prime}}=10^{-6}-3\times
10^{-4}$ GeV, the whole range of $g_{B-L},\,\epsilon=10^{-8}-0.1$ is excluded.
The SN1987A constraint ruled out $g_{B-L}<0.5\times 10^{-7}$ and
$\epsilon<10^{-7}$ for $M_{A^{\prime}}=10^{-4}-10^{-1}$ GeV.

##  VI Conclusions

Forward Physics Facilities (FPF) can host a number of experiments that make
use of the unique opportunities of neutrino beams in the energy range of a few
hundred GeV to TeV in exploring physics beyond the SM. We have investigated
sensitivity reach on a couple of dark photon models at a number of
experiments, including FASER$\nu$, FASER$\nu 2$, SND@LHC, and FLArE(10 tons).
We employed two models: (i) $U(1)_{B-L}$ and (ii) a hidden $U(1)^{\prime}$
that mixes with the SM gauge fields via the kinetic mixing (vector portal).
The presence of such a $U(1)_{B-L}$ gauge field or a dark photon gives rise to
a detectable signal in neutrino-electron scattering. We have studied neutrino-
electron scattering mediated via $t$-channel exchange of the $U(1)_{B-L}$
gauge field at the FPF experiments such as FASER$\nu$, FASER$\nu 2$, SND@LHC
and FLArE (10 tons), and calculated the expected sensitivity reach on the
parameter space of the $U(1)_{B-L}$ model. We investigated the advantage of
FPF in a wide mass range of the $U(1)_{B-L}$ gauge field and to determine the
flavor dependence of the coupling between neutrino and this new gauge boson.
We found that both FASER$\nu$ and SND@LHC are more sensitive to
$g_{B-L}^{\mu}$. On the other hand, FASER$\nu 2$ is more sensitive to
$g_{B-L}^{\mu}$ in the lower dark photon mass region, but shifts to
$g_{B-L}^{e}$ in the heavier dark photon mass region due to the interference
effect. A similar pattern happens in FLArE, however, in the heavier dark
photon mass region both $g_{B-L}^{\mu}$ and $g_{B-L}^{e}$ receive similar
sensitivity.

On the other hand, the sensitivities on the kinetic mixing parameter
$\epsilon$ are calculated as a function of dark photon mass $M_{A^{\prime}}$.
The bound of kinetic mixing also follows the same pattern of the gauge
coupling $g_{B-L}$.

We have covered a wide mass range of dark photon $10^{-6}\leavevmode\nobreak\
\rm GeV\leq M_{A^{\prime}}\leq 10\leavevmode\nobreak\ GeV$ in our study. Among
all the proposed forward physics experiments, FASER$\nu 2$ has the best
sensitivity to the $U(1)_{B-L}$ and the dark photon model, whereas the SND@LHC
has the least. The sensitivity curves for all the experiments follow a similar
pattern, in which the sensitivity gets weakened with the increment of dark
photon mass. We also compared the 90% C.L. sensitivity reach on the parameter
space of the models at the FPF experiments with the other existing bounds from
the experiments such as TEXONO, GEMMA, BOREXINO, LSND, CHARM II, NA64e.

In summary, the FPF experiments considered in this work can achieve better
sensitivities at higher $M_{A^{\prime}}$ mass region because of the higher
energy range of the neutrino flux coming from the ATLAS IP.

Before closing we would like to address an interesting idea of testing the
dark photon model at ICECUBE. To our knowledge the dark photon studies at
ICECUBE Ardid:2017lry  were based on the decay of dark photons into charged
leptons or pions, followed by cascade decays into neutrinos, e.g., $\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}$ and $\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}$. On the
other hand, using neutrino-electron or neutrino-nucleon scattering via dark
photon, one can use the existing ICECUBE neutrino flux measurements
IceCube:2021uhz ; IceCube:2020acn  to constrain the dark photon model
parameters. The measurement of muon-neutrino flux is based on induced muon
tracks. Electron- and tau-neutrino fluxes are determined by electromagnetic
and/or hadronic cascade caused by the electron or low-energy tau neutrinos.
The presence of the dark photon $A^{\prime}$ would modify the scattering of
$\nu e^{-}\to\nu e^{-}$, which would then affect the electron-neutrino flux
measurement. Since the energy range of the neutrino at ICECUBE is much higher
than that of FASER$\nu(2)$, we expect ICECUBE could be more sensitive to
higher mass region of $M_{A^{\prime}}>1$ GeV. We will explore this possibility
in the future.

## Acknowledgement

This work was supported in parts by the Ministry of Science and Technology
with the grant number MoST-110-2112-M-007-017-MY3.

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