# Probing for chiral $Z^{\prime}$ gauge boson through scattering measurement
experiments

Kento Asai<EMAIL_ADDRESS>Institute for Cosmic Ray Research (ICRR),
The University of Tokyo, Kashiwa, Chiba 277-8582, Japan  Arindam Das
<EMAIL_ADDRESS>Institute for the Advancement of Higher
Education, Hokkaido University, Sapporo 060-0817, Japan  Department of
Physics, Hokkaido University, Sapporo 060-0810, Japan  Jinmian Li
<EMAIL_ADDRESS>College of Physics, Sichuan University, Chengdu 610065, China
Takaaki Nomura<EMAIL_ADDRESS>College of Physics, Sichuan University,
Chengdu 610065, China  Osamu Seto<EMAIL_ADDRESS>Department
of Physics, Hokkaido University, Sapporo 060-0810, Japan

###### Abstract

Observation of tiny neutrino mass can not be explained within the framework of
Standard Model (SM) and it allows us to construct beyond the SM (BSM)
scenarios. We consider extra gauge extended scenarios which generate tiny
neutrino mass through seesaw mechanism. These scenarios are equipped with a
BSM neutral gauge boson called $Z^{\prime}$ to cancel gauge and mixed gauge-
gravity anomalies in the general $U(1)_{X}$ scenario which is a linear
combination of $U(1)_{Y}$ and $U(1)_{\rm B-L}$. In this case we find that left
and right handed fermions interact differently with $Z^{\prime}$. The
$Z^{\prime}$ gives rise to different processes involving neutrino-nucleon,
neutrino-electron, electron-nucleus and electron-muon scattering processes and
compared with FASER$\nu(2)$, SND$@$LHC, COHERENT, NA64, JSNS2 and MUonE
experiments, respectively. In addition to obtaining constraints on gauge
coupling with respect to $Z^{\prime}$ mass for different general $U(1)_{X}$
charges, we compare our results with those from proton, electron beam-dump
experiments. Recasting data from the long-lived and dark photon searches at
BaBaR, LHCb and CMS experiments, we estimate bounds on the gauge coupling and
the corresponding gauge boson masses. We derive limits on the general
$U(1)_{X}$ gauge coupling and $Z^{\prime}$ mass from the electron and muon
$g-2$ data for different $U(1)_{X}$ charges. We recast existing bounds from
different beam-dump experiments at Orsay, KEK, E137, CHARM, Nomad, $\nu-$cal,
E141, E774, NA64, KEK in our cases in addition to the prospective limits form
the future beam-dump scenarios at DUNE, FASER(2) and ILC. Finally we calculate
bounds recasting the data of the dilepton and dijet searches from the LEP
experiment for complementarity. Such parameter regions could be probed by
scattering, beam-dump and collider experiments in future.

††preprint: EPHOU-23-013

##  I Introduction

Tiny neutrino masses and flavor mixing are important experimental observations
[1] which allow us to think beyond the Standard Model(SM). In addition to
large scale cosmological data, galaxy rotation curve and bullet cluster data
indicate that nonluminous objects called dark matter candidate capture nearly
0.25 fraction of the energy budget of the Universe [2, 3] which further
indicate that extension of SM is certain. The origin of tiny neutrino mass can
be explained by the seesaw mechanism where SM is extended by SM-singlet
Majorana Right Handed Neutrinos(RHNs)[4, 5, 6, 7, 8]. In this case light
neutrino mass can be originated by the suppression of heavy mass scale of the
RHNs which introduces a lepton number violation of unit two which is a nice
realization of dimension five Weinberg operator [9].

This simple but interesting scenario at the tree level can also be observed if
SM is extended by a general U(1)X gauge group which is a linear combination of
$U(1)_{Y}$ and $U()1)_{\rm B-L}$. In this set-up three generations of SM-
singlet RHNs are introduced to cancel the gauge and mixed gauge-gravity
anomalies. In addition to the RHNs, there exists an SM-singlet scalar which
couples with the RHNs. After the general $U(1)_{X}$ symmetry is broken, the
scalar acquires vacuum expectation value(VEV) which lets the RHNs to acquire
Majorana mass. Followed by the electroweak symmetry breaking, a Dirac mass
term is generated from the Yukawa interaction between the SM lepton and Higgs
doublets along with the SM-singlet RHNs. These Majorana and Dirac masses get
involved in the seesaw mechanism to generate light neutrino mass and flavor
mixing [10]. All these couplings and interactions are protected by general
U(1)X gauge symmetry.

In general $U(1)_{X}$ extension of the SM, a neutral and beyond the SM (BSM)
gauge boson, commonly known as $Z^{\prime}$ is evolved which acquires mass
after general $U(1)_{X}$ symmetry breaking. In this scenario SM fermions could
be charged under the general $U(1)_{X}$ gauge group which takes a special
chiral nature after the anomaly cancellation where left and right handed SM
fermions have different general $U(1)_{X}$ charges affecting their interaction
with the $Z^{\prime}$ gauge boson. In this article we will study such a chiral
scenario to estimate the limits on the general $U(1)_{X}$ gauge coupling with
respect to the $Z^{\prime}$ mass for different general $U(1)_{X}$ charges and
compare with a variety of existing bounds for the $Z^{\prime}$ mass to be
probed. New physics contributions involving $Z^{\prime}$ scenarios are
considered to be very well motivated and being studied at low and high energy
experiments [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39] from various aspects.

We study the chiral nature of $Z^{\prime}$ which is a very interesting aspect
of the general $U(1)_{X}$ extension of the SM. We probe $Z^{\prime}$ mediated
neutrino-electron, neutrino-nucleon and electron-muon scattering process at
different experiments like FASER$\nu$ and FASER$\nu 2$ [40, 41, 42, 43, 44,
45, 46, 47], SND$@$LHC[48, 49, 50, 51], NA64 [52, 53, 54, 55, 56, 57] and
MuonE [58, 59]. In these cases, $Z^{\prime}$ appears in $t-$channel processes
where the interaction vertices depending on general $U(1)_{X}$ charges
manifest chiral nature which has not been studied before in the context of
these experiments. In this paper, we consider another general U(1)X extension
of the SM. In the previous case, we have three generations of RHNs which are
universal under the general U(1)X gauge group having same charge, however, we
consider another interesting aspect where first two generations of the RHNs
have general U(1)X charge $-4$ and that of the third generation RHN is $+5$.
In this case we involve two $SU(2)_{L}$ Higgs doublet, being differently
charged under the general U(1)X gauge group, where one of them couples only
with the RHNs and charged leptons making the scenario neutrinophilic. In
addition to the doublet scalars we introduce three SM-singlet scalars which
are differently charged under the general U(1)X gauge group. We call this
scenario as an alternative general $U(1)_{X}$ scenario. After the anomaly
cancellation we find that left and right handed SM fermions are differently
charged under the general U(1)X gauge group manifesting the chiral nature of
the model which will be reflected at time of their interactions with
$Z^{\prime}$ which we study from neutrino-electron, neutrino-nucleus,
electron-nucleus and electron-muon scatterings at different experiments like
FASER$\nu$, FASER$\nu 2$, SND$@$LHC, JSNS2, COHERENT, NA64 and MuonE
respectively. With a special anomaly free setting of the charges of the
charged fermions we find that, the general $U(1)_{X}$ charge assignment can
replicate this alternative scenario. Therefore we combine these two cases
together to study the limits on the general $U(1)_{X}$ gauge coupling and
$Z^{\prime}$ mass from different scattering experiments.

In order to compare our results with the existing bounds we obtain respective
bounds from different scattering and beam-dump experiments in chiral scenario.
To do that, we estimate bounds from electron/ positron beam-dump experiments
Orsay [60], NA64 [61], KEK [62], E141 [63], E137 [64] and E774 [65]
respectively. Studying the bounds from the neutrino-electron scattering
experiments from TEXONO [66, 67, 68], BOREXINO [69, 70, 71, 72, 73, 74] and
JSNS2 at J-PARC experiment [75], we compare our results for different U(1)X
charges. Depending on the choice of the $Z^{\prime}$ mass, we compare our
bounds for muon neutrino- and muon anti-neutrino-electron scattering from the
CHARM-II experiment [76, 77, 78, 79]. We study neutrino-nucleon scattering to
estimate bounds for different U(1)X charges from the COHERENT experiment [80,
80, 81, 82, 83, 84]. In addition to that we compare our results with the GEMMA
experiment studying neutrino magnetic moment [85, 26]. We compare the bounds
form the dark photon searches at the LHCb experiment [25, 14], dark photon
searches from the CMS experiment [86], visible and invisible decay of dark
photon from BaBar experiment [87, 88], respectively. We compare our results
with proton beam-dump experiments like Nomad [89], CHARM [90] and $\nu-$cal
[91, 92] experiments respectively. We compare our bounds with the dilepton and
dijet final states from the LEP-II study [93, 94, 95, 96]. Finally we estimate
bounds on general U(1)X coupling with respect to $M_{Z^{\prime}}$ from muon
and electron $g-2$ experiments to show the complementarity scenario.

Our paper is organized in the following way. We discuss the models in Sec. II.
We calculate constraints on the gauge coupling fro different U(1)X charges in
Sec. III. We discuss our results in Sec. IV and finally conclude the paper in
Sec. V.

##  II Model

A general U(1)X extension of the SM involves three generations of RHNs to
cancel the gauge and mixed gauge-gravity anomalies. As a result, we observe
that left and right handed charged fermions in the SM are differently charged
under $U(1)$ group. We write down the UV-complete models in the following:

###  II.1 Case-I

We consider a general U(1)X extension of the SM to investigate the chiral
scenario introducing three RHNs $(N_{R})$ and an SM-singlet scalar $(\Phi)$
field. Three generations of RHNs are introduced to cancel gauge and mixed
gauge-gravity anomalies. The corresponding field content is given in Tab. 1
where general U(1)X charges, being independent of generations, are written as
$\tilde{x}_{f}$ before anomaly cancellation and $f$ stands for the three
generations of quarks $(q_{L}^{\alpha},u_{R}^{\alpha},d_{R}^{\alpha})$ and
leptons $(\ell_{L}^{\alpha},e_{R}^{\alpha},N_{R}^{\alpha})$ respectively where
$\alpha(=1,~{}2,~{}3)$ is the generation index. The gauge and mixed gauge-
gravity anomaly cancellation conditions in terms of the general charges are
written below:

$\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm SU}(3)_{C}\right]^{2}$
$\displaystyle\ :$ $\displaystyle 2\tilde{x}_{q}-\tilde{x}_{u}-\tilde{x}_{d}$
$\displaystyle\ =\ 0~{},$ $\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm
SU}(2)_{L}\right]^{2}$ $\displaystyle\ :$ $\displaystyle
3\tilde{x}_{q}+\tilde{x}_{\ell}$ $\displaystyle\ =\ 0~{},$ $\displaystyle{\rm
U}(1)_{X}\otimes\left[{\rm U}(1)_{Y}\right]^{2}$ $\displaystyle\ :$
$\displaystyle\tilde{x}_{q}-8\tilde{x}_{u}-2\tilde{x}_{d}+3\tilde{x}_{\ell}-6\tilde{x}_{e}$
$\displaystyle\ =\ 0~{},$ $\displaystyle\left[{\rm
U}(1)_{X}\right]^{2}\otimes{\rm U}(1)_{Y}$ $\displaystyle\ :$
$\displaystyle{\tilde{x}_{q}}^{2}-{\tilde{2x}_{u}}^{2}+{\tilde{x}_{d}}^{2}-{\tilde{x}_{\ell}}^{2}+{\tilde{x}_{e}}^{2}$
$\displaystyle\ =\ 0~{},$ $\displaystyle\left[{\rm U}(1)_{X}\right]^{3}$
$\displaystyle\ :$
$\displaystyle{6\tilde{x}_{q}}^{3}-{3\tilde{x}_{u}}^{3}-{3\tilde{x}_{d}}^{3}+{2\tilde{x}_{\ell}}^{3}-{\tilde{x}_{\nu}}^{3}-{\tilde{x}_{e}}^{3}$
$\displaystyle\ =\ 0~{},$ $\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm
grav.}\right]^{2}$ $\displaystyle\ :$ $\displaystyle
6\tilde{x}_{q}-3\tilde{x}_{u}-3\tilde{x}_{d}+2\tilde{x}_{\ell}-\tilde{x}_{\nu}-\tilde{x}_{e}$
$\displaystyle\ =\ 0~{},$ (1)

respectively. The Yukawa interactions between the fermions and the scalars
$(H,\Phi)$ can be written following the $\mathcal{G}_{\rm SM}\otimes$
U$(1)_{X}$ gauge symmetry as

${\cal L}^{\rm
Yukawa}=-Y_{u}^{\alpha\beta}\overline{q_{L}^{\alpha}}Hu_{R}^{\beta}-Y_{d}^{\alpha\beta}\overline{q_{L}^{\alpha}}\tilde{H}d_{R}^{\beta}-Y_{e}^{\alpha\beta}\overline{\ell_{L}^{\alpha}}\tilde{H}e_{R}^{\beta}-Y_{\nu}^{\alpha\beta}\overline{\ell_{L}^{\alpha}}HN_{R}^{\beta}-Y_{N}^{\alpha}\Phi\overline{(N_{R}^{\alpha})^{c}}N_{R}^{\alpha}+{\rm
H.c.}~{},$ (2)

where H is the SM Higgs doublet and we transform it into $\tilde{H}$ following
$i\tau^{2}H^{*}$ where $\tau^{2}$ is the second Pauli matrix. Hence using Eq.
2 and following charge neutrality, we express the following relations between
the general U(1)X charges of the particles as

$\displaystyle-\frac{1}{2}x_{H}$ $\displaystyle=$
$\displaystyle-\tilde{x}_{q}+\tilde{x}_{u}\ =\ \tilde{x}_{q}-\tilde{x}_{d}\ =\
\tilde{x}_{\ell}-\tilde{x}_{e}=\ -\tilde{x}_{\ell}+\tilde{x}_{\nu},$
$\displaystyle 2x_{\Phi}$ $\displaystyle=$
$\displaystyle-2\tilde{x}_{\nu}~{}.$ (3)

The general U(1)X charges of the fermions can be obtained solving Eqs. (1) and
(3) which finally can be expressed using the scalar charges $x_{H}$ and
$x_{\Phi}$ respectively. Simply the anomaly free charge assignment of the
general U(1)X can be expressed in terms of a linear combination of two anomaly
free scenarios, namely, $U(1)_{Y}$ of the SM and B$-$L charges. Finally we
find that the left and right handed fremions under the general U(1)X scenario
have different charges and hence they interact differently with the neutral,
BSM gauge boson $Z^{\prime}$ in the model. We take $x_{\Phi}=1$ without the
loss of generality which corresponds to the $U(1)_{\rm B-L}$ and $U(1)_{R}$
scenarios with $x_{H}=0$ [97, 98, 99, 100, 101] and $x_{H}=2$, respectively.
$U(1)_{\rm B-L}$ is a vector like scenario where left and right handed
fermions of same type are equally charged under the $U(1)$ extension. In case
of $U(1)_{R}$ scenario we find that left handed fermions do not interact with
the $Z^{\prime}$.

| SU(3)C | SU(2)L | U(1)Y | U(1)X | $-2$ | $-1$ | $-0.5$ | $0$ | $0.5$ | $1$ | $2$
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  | $U(1)_{\rm{R}}$ |  |  | $U(1)_{\rm B-L}$ |  |  |
$q_{L}^{\alpha}$ | 3 | 2 | $\frac{1}{6}$ | $\tilde{x}_{q}$ | = | $\frac{1}{6}x_{H}+\frac{1}{3}x_{\Phi}$ | $0$ | $\frac{1}{6}$ | $\frac{1}{4}$ | $\frac{1}{3}$ | $\frac{5}{12}$ | $\frac{1}{2}$ | $\frac{1}{3}$
$u_{R}^{\alpha}$ | 3 | 1 | $\frac{2}{3}$ | $\tilde{x}_{u}$ | = | $\frac{2}{3}x_{H}+\frac{1}{3}x_{\Phi}$ | $-1$ | $-\frac{1}{3}$ | $0$ | $\frac{1}{3}$ | $\frac{1}{2}$ | $1$ | $\frac{5}{3}$
$d_{R}^{\alpha}$ | 3 | 1 | $-\frac{1}{3}$ | $\tilde{x}_{d}$ | = | $-\frac{1}{3}x_{H}+\frac{1}{3}x_{\Phi}$ | $1$ | $\frac{2}{3}$ | $\frac{1}{2}$ | $\frac{1}{3}$ | $\frac{1}{6}$ | $0$ | $-\frac{1}{3}$
$\ell_{L}^{\alpha}$ | 1 | 2 | $-\frac{1}{2}$ | $\tilde{x}_{\ell}$ | = | $-\frac{1}{2}x_{H}-x_{\Phi}$ | $0$ | $-\frac{1}{2}$ | $-\frac{3}{4}$ | $-1$ | $\frac{5}{4}$ | $-\frac{3}{2}$ | $-2$
$e_{R}^{\alpha}$ | 1 | 1 | $-1$ | $\tilde{x}_{e}$ | = | $-x_{H}-x_{\Phi}$ | $1$ | $0$ | $-\frac{1}{2}$ | $-1$ | $-\frac{3}{2}$ | $-2$ | $-3$
$N_{R}^{\alpha}$ | 1 | 1 | $0$ | $\tilde{x}_{\nu}$ | = | $-x_{\Phi}$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$
$H$ | 1 | 2 | $-\frac{1}{2}$ | $-\frac{1}{2}x_{H}$ | = | $-\frac{1}{2}x_{H}$ | $1$ | $\frac{1}{2}$ | $\frac{1}{4}$ | $0$ | $-\frac{1}{4}$ | $-\frac{1}{2}$ | $-1$
$\Phi$ | 1 | 1 | $0$ | $2x_{\Phi}$ | = | $2x_{\Phi}$ | $2$ | $2$ | $2$ | $2$ | $2$ | $2$ | $2$
Table 1:  Field content of general U(1)X extension of the SM in the minimal
form with the charges of the particles before and after anomaly cancellation
considering different benchmark values of $x_{H}$ setting $x_{\Phi}=1$. In
this charge assignment, $x_{H}=0$ and $-2$ are the $U(1)_{\rm B-L}$ and
$U(1)_{\rm{R}}$ scenarios. Among the chiral scenarios $U(1)_{\rm B-L}$ is a
vector like case.

Using the general form of the $U(1)$ charges of the charged fermions we notice
that for $x_{H}=-1$, the $U(1)$ charge of the right handed electron $(e_{R})$
becomes zero and as a result it has no direct interaction with $Z^{\prime}$
whereas other fermions will interact with $Z^{\prime}$ manifesting the chiral
nature of the model. In similar fashion we find that for $x_{H}=-0.5$, the
general U(1)X charge of the right handed up-type quark $u_{R}$ is zero
implying no direct interaction with $Z^{\prime}$ whereas other fermions will
have nonzero general U(1)X charges allowing direct interactions with
$Z^{\prime}$. Similar behavior could be observed when $x_{H}=1$ where general
U(1)X charge of right handed down type quark $(d_{R})$ is zero resulting into
no direct interaction with $Z^{\prime}$. Detail charge assignments for these
combinations of $x_{H}$ and $x_{\Phi}$ are given in Tab. 1. We consider two
more scenarios with $x_{H}=0.5$ and $2$ setting $x_{\Phi}=1$ where all the
charged fermions interact with $Z^{\prime}$ manifesting the chiral behavior of
the model.

The scalar sector of this scenario can be explored by introducing the
renormalizable potential of this model and that can be given by

$\displaystyle V\ =\
m_{H}^{2}(H^{\dagger}H)+\lambda_{H}(H^{\dagger}H)^{2}+m_{\Phi}^{2}(\Phi^{\dagger}\Phi)+\lambda_{\Phi}(\Phi^{\dagger}\Phi)^{2}+\lambda_{\rm
mix}(H^{\dagger}H)(\Phi^{\dagger}\Phi)~{},$ (4)

where $H$ and $\Phi$ can be separately approximated in the analysis of scalar
potential considering $\lambda_{\rm mix}$ to be very small. After the breaking
of general U(1)X and electroweak symmetries, the scalar fields develop vacuum
expectation values (VEVs) following

$\displaystyle\braket{H}\ =\ \frac{1}{\sqrt{2}}\begin{pmatrix}v+h\\\
0\end{pmatrix}~{},\quad{\rm and}\quad\braket{\Phi}\ =\
\frac{v_{\Phi}+\phi}{\sqrt{2}}~{},$ (5)

where $v=246$ GeV is marked as electroweak scale VEV at the potential minimum
and $v_{\Phi}$ is a free parameter. After general $U(1)_{X}$ symmetry is
broken, the mass of the BSM neutral gauge boson can be evolved setting
$x_{\Phi}=1$ as

$M_{Z^{\prime}}=2g_{X}v_{\Phi}~{},$ (6)

considering the limit $v_{\Phi}\gg v$. Here $g_{X}$ is the general U(1)X
coupling and the $Z^{\prime}$ mass is a free parameter.

###  II.2 Case-II

We consider another scenario where the SM is extended by a general U(1)X gauge
group with three generations of RHNs. The field content of the model is given
in Tab. 2. The general U$(1)$ charges of the charged fermion fields are same
for all generations, $\alpha(=1,2,3)$ where $\alpha$ is the generation index.
The general U(1)X charges of the fields are written as $\tilde{x}_{f}$ before
anomaly cancellation and $f$ stands for the three generations of quarks
$(q_{L}^{\alpha},u_{R}^{\alpha},d_{R}^{\alpha})$ and leptons
$(\ell_{L}^{\alpha},e_{R}^{\alpha})$ respectively. We introduce two $SU(2)$
doublet Higgs fields $(H_{1,2})$ where one is SM like $(H_{1})$ and the other
one is BSM $(H_{2})$ Higgs. Due to different general $U(1)_{X}$ charge
assignments $H_{1}$ does not couple with the BSM fermions. We introduce three
SM-singlet scalar fields $(\Phi_{1,2,3})$ which are differently charged under
general $U(1)_{X}$ gauge group. Due to general $U(1)_{X}$ gauge symmetry first
two generations of the RHNs have charge $-4$ each and the third generation RHN
has $+5$ charge under the general $U(1)_{X}$ gauge group being a unique choice
[102]. The RHNs with semi non-universal $U(1)$ charges in this model help to
cancel gauge and mixed gauge-gravity anomalies. As a result we call this model
an alternative general $U(1)_{X}$ scenario. Following the gauge and mixed
gauge-gravity anomaly cancellation conditions, we relate the general
$U(1)_{X}$ charges of the charged fermions as

| SU(3)C | SU(2)L | U(1)Y | U(1)X | $-2$ | $-1$ | $-0.5$ | $0$ | $0.5$ | $1$ | $2$
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  | U$(1)_{\rm{R}}$ |  |  | B$-$L |  |  |
$q_{L}^{\alpha}$ | 3 | 2 | $\frac{1}{6}$ | $\tilde{x}_{q}$ | = | $\frac{1}{6}x_{H}+\frac{1}{3}$ | $0$ | $\frac{1}{6}$ | $\frac{1}{4}$ | $\frac{1}{3}$ | $\frac{5}{12}$ | $\frac{1}{2}$ | $\frac{1}{3}$
$u_{R}^{\alpha}$ | 3 | 1 | $\frac{2}{3}$ | $\tilde{x}_{u}$ | = | $\frac{2}{3}x_{H}+\frac{1}{3}$ | $-1$ | $-\frac{1}{3}$ | $0$ | $\frac{1}{3}$ | $\frac{1}{2}$ | $1$ | $\frac{5}{3}$
$d_{R}^{\alpha}$ | 3 | 1 | $-\frac{1}{3}$ | $\tilde{x}_{d}$ | = | $-\frac{1}{3}x_{H}+\frac{1}{3}$ | $1$ | $\frac{2}{3}$ | $\frac{1}{2}$ | $\frac{1}{3}$ | $\frac{1}{6}$ | $0$ | $-\frac{1}{3}$
$\ell_{L}^{\alpha}$ | 1 | 2 | $-\frac{1}{2}$ | $\tilde{x}_{\ell}$ | = | $-\frac{1}{2}x_{H}-1$ | $0$ | $-\frac{1}{2}$ | $-\frac{3}{4}$ | $-1$ | $\frac{5}{4}$ | $-\frac{3}{2}$ | $-2$
$e_{R}^{\alpha}$ | 1 | 1 | $-1$ | $\tilde{x}_{e}$ | = | $-x_{H}-1$ | $1$ | $0$ | $-\frac{1}{2}$ | $-1$ | $-\frac{3}{2}$ | $-2$ | $-3$
$N_{R}^{1,2}$ | 1 | 1 | $0$ | $\tilde{x}_{\nu}$ | = | $-4$ | $-4$ | $-4$ | $-4$ | $-4$ | $-4$ | $-4$ | $-4$
$N_{R}^{3}$ | 1 | 1 | $0$ | $\tilde{x}_{\nu}^{\prime}$ | = | $5$ | $5$ | $5$ | $5$ | $5$ | $5$ | $5$ | $5$
$H_{1}$ | 1 | 2 | $-\frac{1}{2}$ | $\tilde{x}_{H_{1}}$ | = | $-\frac{x_{H}}{2}$ | $1$ | $\frac{1}{2}$ | $\frac{1}{4}$ | $0$ | $-\frac{1}{4}$ | $-\frac{1}{2}$ | $-1$
$H_{2}$ | 1 | 2 | $-\frac{1}{2}$ | $\tilde{x}_{H_{2}}$ | = | $-\frac{1}{2}x_{H}+3$ | $4$ | $\frac{7}{2}$ | $\frac{13}{2}$ | $3$ | $\frac{11}{4}$ | $\frac{5}{2}$ | $2$
$\Phi_{1}$ | 1 | 1 | $0$ | $\tilde{x}_{\Phi_{1}}$ | = | $+8$ | $+8$ | $+8$ | $+8$ | $+8$ | $+8$ | $+8$ | $+8$
$\Phi_{2}$ | 1 | 1 | $0$ | $\tilde{x}_{\Phi_{2}}$ | = | $-10$ | $-10$ | $-10$ | $-10$ | $-10$ | $-10$ | $-10$ | $-10$
$\Phi_{3}$ | 1 | 1 | $0$ | $\tilde{x}_{\Phi_{3}}$ | = | $-3$ | $-3$ | $-3$ | $-3$ | $-3$ | $-3$ | $-3$ | $-3$
Table 2:  Field content of general U(1)X extension of the SM in the minimal
form with the charges of the particles before and after anomaly cancellation
considering different benchmark values of $x_{H}$. Here, $x_{H}=0$ is an
alternative B$-$L case, which is purely vector like structure given as a
reference in this article.

$\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm SU}(3)_{C}\right]^{2}$
$\displaystyle\ :$ $\displaystyle 2\tilde{x}_{q}-\tilde{x}_{u}-\tilde{x}_{d}$
$\displaystyle\ =\ 0~{},$ $\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm
SU}(2)_{L}\right]^{2}$ $\displaystyle\ :$ $\displaystyle
3\tilde{x}_{q}+\tilde{x}_{\ell}$ $\displaystyle\ =\ 0~{},$ $\displaystyle{\rm
U}(1)_{X}\otimes\left[{\rm U}(1)_{Y}\right]^{2}$ $\displaystyle\ :$
$\displaystyle\tilde{x}_{q}-8\tilde{x}_{u}-2\tilde{x}_{d}+3\tilde{x_{\ell}}-6\tilde{x_{e}}$
$\displaystyle\ =\ 0~{},$ $\displaystyle\left[{\rm
U}(1)_{X}\right]^{2}\otimes{\rm U}(1)_{Y}$ $\displaystyle\ :$
$\displaystyle{\tilde{x}_{q}}^{2}-{2\tilde{x}_{u}}^{2}+{\tilde{x}_{d}}^{2}-{\tilde{x}_{\ell}}^{2}+{\tilde{x}_{e}}^{2}$
$\displaystyle\ =\ 0~{},$ $\displaystyle\left[{\rm U}(1)_{X}\right]^{3}$
$\displaystyle\ :$ $\displaystyle
3({6\tilde{x}_{q}}^{3}-{3\tilde{x}_{u}}^{3}-{3\tilde{x}_{d}}^{3}+{2\tilde{x}_{\ell}}^{3}-{\tilde{x}_{e}}^{3})-2\tilde{x}_{\nu}^{3}-\tilde{x}_{\nu}^{\prime^{3}}$
$\displaystyle\ =\ 0~{},$ $\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm
grav.}\right]^{2}$ $\displaystyle\ :$ $\displaystyle
3(6\tilde{x}_{q}-3\tilde{x}_{u}-3\tilde{x}_{d}+2\tilde{x}_{\ell}-\tilde{x}_{e})-2\tilde{x}_{\nu}-\tilde{x}_{\nu}^{\prime}$
$\displaystyle\ =\ 0~{},$ (7)

We find that due to general U(1)X charges the SM charged fermions interact
differently with $Z^{\prime}$ manifesting the chiral nature of the model. The
second Higgs doublet $H_{2}$ interacts with the SM lepton doublet
$(\ell^{\alpha}_{L})$ and first two generations of the RHNs $(N_{R}^{1,2})$
due to the general U(1)X symmetry. Hence the Dirac Yukawa mass term for
$N_{R}^{1,2}$ can be generated. On the other hand the corresponding Majorana
mass term for $N_{R}^{1,2}$ can be generated from the Dirac Yukawa coupling
with $\Phi_{1}$ followed by the general U(1)X symmetry breaking. The third
generation of the RHN, $N_{R}^{3}$ has no Dirac Yukawa coupling involving any
of the doublet Higgs fields being protected by the general U(1)X charge
assignments. Therefore it does not participate in the neutrino mass generation
mechanism at the tree level, however, it can have Dirac Yukawa interaction
with $\Phi_{2}$ which further generates a Majorana mass term for $N_{R}^{3}$
after the general U(1)X symmetry breaking. Finally we write the Yukawa
interaction among the BSM sector as

$\displaystyle-\mathcal{L}_{\rm int}$ $\displaystyle\ \supset\ $
$\displaystyle\sum_{\alpha=1}^{3}\sum_{\beta=1}^{2}Y_{1}^{\alpha\beta}\overline{\ell_{L}^{\alpha}}H_{2}N_{R}^{\beta}+\frac{1}{2}\sum_{\alpha=1}^{2}Y_{2}^{\alpha}\Phi_{1}\overline{(N_{R}^{\alpha})^{c}}N_{R}^{\alpha}+\frac{1}{2}Y_{3}\Phi_{2}\overline{(N_{R}^{3})^{c}}N_{R}^{3}+\rm{H.c.}~{},$
(8)

assuming $Y_{2}$ is diagonal without the loss of generality. As in the
previous case we can solve the gauge and mixed gauge-gravity anomalies to
estimate the charges of the SM particles in Tab. 2.

The scalar potential of this scenario can be given by

$\displaystyle V$ $\displaystyle\ =\ $ $\displaystyle
m_{H_{1}}^{2}(H_{1}^{\dagger}H_{1})+\lambda_{H_{1}}(H_{1}^{\dagger}H_{1})^{2}+m_{H_{2}}^{2}(H_{2}^{\dagger}H_{2})+\lambda_{H_{2}}(H_{2}^{\dagger}H_{2})^{2}$
(9)
$\displaystyle+m_{\Phi_{1}}^{2}(\Phi_{1}^{\dagger}\Phi_{1})+\lambda_{1}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+m_{\Phi_{2}}^{2}(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}$
$\displaystyle+m_{\Phi_{3}}^{2}(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{3}(\Phi_{3}^{\dagger}\Phi_{3})^{2}+(\mu\Phi_{3}(H_{1}^{\dagger}H_{2})+{\rm
H.c.})$
$\displaystyle+\lambda_{4}(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{2})+\lambda_{5}(H_{1}^{\dagger}H_{2})(H_{2}^{\dagger}H_{1})+\lambda_{6}(H_{1}^{\dagger}H_{1})(\Phi_{1}^{\dagger}\Phi_{1})$
$\displaystyle+\lambda_{7}(H_{1}^{\dagger}H_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{8}(H_{1}^{\dagger}H_{2})(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{9}(H_{2}^{\dagger}H_{2})(\Phi_{1}^{\dagger}\Phi_{1})$
$\displaystyle+\lambda_{10}(H_{1}^{\dagger}H_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{11}(H_{1}^{\dagger}H_{2})(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{12}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})$
$\displaystyle+\lambda_{13}(\Phi_{2}^{\dagger}\Phi_{2})(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{14}(\Phi_{3}^{\dagger}\Phi_{3})(\Phi_{1}^{\dagger}\Phi_{1})~{}.$

Choosing suitable parametrization for the scalar fields in this scenario to
develop their respective VEVs, we can write

$\displaystyle\braket{H_{1}}\ =\
\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}v_{h_{1}}\\\
0\end{array}\right)~{},\;\braket{H_{2}}\ =\
\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}v_{h_{2}}\\\
0\end{array}\right)~{},\braket{\Phi_{1}}\ =\
\frac{v_{1}}{\sqrt{2}}~{},\;\braket{\Phi_{2}}\ =\
\frac{v_{2}}{\sqrt{2}}~{},\;\braket{\Phi_{3}}\ =\
\frac{v_{3}}{\sqrt{2}}~{},~{}~{}~{}~{}$ (14)

with the condition, $\sqrt{v_{h_{1}}^{2}+v_{h_{2}}^{2}}=246~{}{\rm GeV}$. In
this alternative general U(1)X extension of the SM, we consider negligibly
small scalar quartic couplings among SM scalar doublets fields $H_{1,2}$ and
SM-singlet scalar fields $\Phi_{1,2,3}$. As a result this ensures higher order
mixing between the RHNs after the general U(1)X breaking to be very strongly
suppressed. In Eq. (9), we may consider $0<m_{\rm mix}^{2}=\mu
v_{3}/\sqrt{2}\ll m_{\Phi_{3}}^{2}$ which further leads to $v_{h_{2}}\sim
m_{\rm mix}^{2}v_{h_{1}}/m_{\Phi_{3}}^{2}\ll v_{h_{1}}$ [103].

Due to the presence of the general U(1)X gauge symmetry, the doublet scalar
sector $H_{1,2}$ and singlet scalar sector $\Phi_{1,2,3}$ interact only
through the coupling $\Phi_{3}(H_{1}^{\dagger}H_{2})+{\rm H.c.}$, however,
this coupling has no significant effect to determine the VEVs $(v_{1,2,3})$ of
the singlet scalar fields $(\Phi_{1,2,3})$ because there is already one
collider constraint present in the form of $v_{1}^{2}+v_{2}^{2}+v_{3}^{2}\gg
v_{h_{1}}^{2}+v_{h_{2}}^{2}$. Therefore we arrange the parameters of the
scalar potential in a way so that the VEVs of $\Phi_{1,2,3}$ will be almost
same following $\mu<v_{1}$ whereas $\Phi_{3}$ can be considered as a spurion
field which generate the mixing between $H_{1,2}$ in Eq. 9. Once $\Phi_{3}$
acquires VEV we get the mixing mass term between $H_{1,2}$ as $m_{\rm
mix}=\frac{\mu v_{3}}{\sqrt{2}}$ which resembles the potential of two Higgs
doublet model, however, due to the presence of the general U(1)X symmetry the
SM-singlet fields $\Phi_{1,2,3}$ do not mix. As a result there are two
existing physical Nambu-Goldstone (NG) bosons originating from the SM-singlet
scalars. Due to the tiny quartic couplings and gauge couplings, the SM-singlet
scalars become decoupled from the SM thermal bath in the early universe.
Additionally, we consider that the singlet scalars are heavier than the
neutral BSM gauge boson $Z^{\prime}$ preventing its decay into the NG bosons.
The breaking of general U(1) gauge symmetry helps $Z^{\prime}$ to acquire the
mass which copuld be given by

$\displaystyle
M_{Z^{\prime}}=g_{X}\sqrt{64v_{1}^{2}+100v_{2}^{2}+9v_{3}^{2}+\frac{1}{4}x_{H}^{2}v_{h_{1}}^{2}+\left(-\frac{1}{2}x_{H}+3\right)^{2}v_{h_{2}}^{2}}\simeq
g_{X}\sqrt{64v_{1}^{2}+100v_{2}^{2}+9v_{3}^{2}}~{}.$ (15)

which is a free parameter and the general U(1)X gauge coupling $g_{X}$ is also
a free parameter. Due to the general U(1)X gauge structure $H_{2}$ only
couples with $N_{R}^{1,2}$ making this case a neutrinophilic two Higgs Doublet
Model (2HDM) framework [103, 104, 105, 106, 107].

###  II.3 $Z^{\prime}$ interactions with the fermions

After the anomaly cancellation conditions are imposed, we notice that
$Z^{\prime}$ in the above models can interact with the left and right handed
SM fermions differently manifesting chiral nature of the models. Fixing
$x_{\Phi}=1$ in Case-I we find that the chiral nature is the same as the Case-
II. Therefore we write the interactions between the fermions with $Z^{\prime}$
in the following as

$\displaystyle\mathcal{L}_{\rm
int}=-g_{X}(\overline{f}\gamma_{\mu}q_{f_{L}}P_{L}f+\overline{f}\gamma_{\mu}q_{f_{R}}P_{R}f)Z_{\mu}^{\prime}~{},$
(16)

where $P_{L(R)}=(1\pm\gamma_{5})/2$, $q_{f_{L}}$ and $q_{f_{R}}$ are the
corresponding general U$(1)$ charges of the left handed $(f_{L})$ and right
handed $(f_{R})$ fermions. Hence we write vector coupling
$(c_{V}=\frac{q_{f_{L}}+q_{f_{R}}}{2})$ and axial vector coupling
$(c_{A}=\frac{q_{f_{L}}-q_{f_{R}}}{2})$ for the SM fermions following the
charge assignments of the Cases-I and II in Tab. 3 fixing $x_{\Phi}=1$.

| Vector coupling $(c_{V})$ | Axial-vector coupling $(c_{A})$
---|---|---
SM fermions | $x_{H}=-2$ | $-1$ | $-0.5$ | $0$ | $0.5$ | $1$ | $2$ | $x_{H}=-2$ | $-1$ | $-0.5$ | $0$ | $0.5$ | $1$ | $2$
Charged lepton $(\ell^{\alpha})$ | $-\frac{3}{4}x_{H}-1=\frac{1}{2}$ | $-\frac{1}{4}$ | $-\frac{5}{8}$ | $-1$ | $-\frac{11}{8}$ | $-\frac{7}{4}$ | $-\frac{5}{2}$ | $\frac{1}{4}x_{H}=-\frac{1}{2}$ | $-\frac{1}{4}$ | $-\frac{1}{8}$ | 0 | $\frac{1}{8}$ | $\frac{1}{4}$ | $\frac{1}{2}$
SM-like neutrino $(\nu^{\alpha}_{L})$ | $\frac{1}{4}x_{H}+\frac{1}{2}=0$ | $\frac{1}{4}$ | $\frac{3}{8}$ | $\frac{1}{2}$ | $\frac{5}{8}$ | $\frac{3}{4}$ | $1$ | $\frac{1}{4}x_{H}+\frac{1}{2}=0$ | $\frac{1}{4}$ | $\frac{3}{8}$ | $\frac{1}{2}$ | $\frac{5}{8}$ | $\frac{3}{4}$ | $1$
up-type quarks $(q_{u}^{\alpha})$ | $\frac{5}{12}x_{H}+\frac{1}{3}=-\frac{1}{2}$ | $-\frac{1}{12}$ | $\frac{1}{8}$ | $\frac{1}{3}$ | $\frac{13}{24}$ | $\frac{3}{4}$ | $\frac{7}{6}$ | $-\frac{1}{4}x_{H}=-\frac{1}{2}$ | $-\frac{1}{4}$ | $-\frac{1}{8}$ | 0 | $\frac{1}{8}$ | $\frac{1}{4}$ | $\frac{1}{2}$
down-type quarks $(q_{d}^{\alpha})$ | $-\frac{1}{12}x_{H}+\frac{1}{3}=-\frac{1}{2}$ | $-\frac{1}{4}$ | $-\frac{1}{8}$ | $\frac{1}{3}$ | $\frac{7}{24}$ | $\frac{1}{4}$ | $\frac{1}{6}$ | $-\frac{1}{4}x_{H}=-\frac{1}{2}$ | $-\frac{1}{4}$ | $-\frac{1}{8}$ | 0 | $\frac{1}{8}$ | $\frac{1}{4}$ | $\frac{1}{2}$
Table 3: Vector and axial vector couplings in general U(1)X scenarios for the
couplings between SM fermions and $Z^{\prime}$. In B$-$L case considering
$x_{H}=0$ and $x_{\Phi}=1$, the axial vector couplings for the charged
fermions vanish.

Hence, we notice that interactions between SM fermions and $Z^{\prime}$ are
chiral in nature in general U(1)X extension of the SM. The partial decay width
of $Z^{\prime}$ into different fermions can be calculated using Eq. (16) and
we write down the expression as

$\displaystyle\Gamma(Z^{\prime}\to\bar{f}f)$
$\displaystyle=N_{C}\frac{M_{Z^{\prime}}g_{X}^{2}}{24\pi}\left[\left(q_{f_{L}}^{2}+q_{f_{R}}^{2}\right)\left(1-\frac{m_{f}^{2}}{M_{Z^{\prime}}^{2}}\right)+6q_{f_{L}}q_{f_{R}}\frac{m_{f}^{2}}{M_{Z^{\prime}}^{2}}\right]~{},$
(17)

where $m_{f}$ is the mass of different SM fermions and $q_{L,R}$ are the
functions of $x_{H}$. Here, $N_{C}$ is the color factor being $1$ for the SM
leptons and $3$ for the SM quarks. The light neutrinos $(\nu_{L})$ are
considered to be massless due to their tiny mass and putting $q_{f_{R}}=0$ in
Eq. 17, we obtain the partial decay width of $Z^{\prime}$ into a pair of one
generation light neutrinos as

$\displaystyle\Gamma(Z^{\prime}\to\nu\nu)=\frac{M_{Z^{\prime}}g_{X}^{2}}{24\pi}q_{f_{L}}^{2}~{},$
(18)

were $q_{f_{L}}$ is a function of $x_{H}$. The $Z^{\prime}$ gauge boson can
decay into a pair of heavy Majorana neutrinos if $Z^{\prime}$ is heavier than
twice the mass of the heavy neutrinos. The corresponding partial decay width
into single generation of heavy neutrino pair can be written as

$\displaystyle\Gamma(Z^{\prime}\to
N_{R}^{\alpha}N_{R}^{\alpha})=\frac{m_{Z^{\prime}}g_{X}^{2}}{24\pi}q_{N_{R}}^{2}\left(1-4\frac{m_{N}^{2}}{M_{Z^{\prime}}^{2}}\right)^{\frac{3}{2}}~{},$
(19)

with $q_{N_{R}}$ is the general U(1)X charge of the heavy neutrinos which
could be found from Tabs. 1 and 2, respectively and $m_{N}$ is the mass of the
heavy neutrinos. If we consider that the RHNs are heavier than the half of the
$M_{Z^{\prime}}$ then the decay of $Z^{\prime}$ into a pair of RHN is
kinematically forbidden. We find that the U(1)X charges of the fermions in
Cases-II are same as those in Case-I with $x_{\Phi}=1$. As a result we can
utilize same bounds for both the cases.

###  II.4 Neutrino mass

The Yukawa interactions given in Eqs. 2 and 8 lead us to the generation of
neutrino mass mechanism. The general U(1)X symmetry breaking lead us to the
generation of the Majorana mass term for the three (first two) generations of
the RHNs in Case-I (II) where BSM scalar $\Phi_{(1)}$ is involved. The Dirac
mass term is generated after the electroweak symmetry breaking where SM Higgs
doublet $H_{1(2)}$ is involved in Case-I(II). The corresponding Dirac and
Majorana mass terms are written in Tab. 4.

Models | Majorana mass | Dirac mass
---|---|---
| $(m_{N})$ | $(m_{D})$
Case-I | $m_{N_{R}^{\alpha}}=\frac{Y_{N}^{\alpha}}{\sqrt{2}}v_{\Phi}$ | $m_{D}^{\alpha\beta}=\frac{Y_{\nu}}{\sqrt{2}}v_{1}$
Case-II | $m_{N_{R}^{1,2}}=\frac{Y_{2}^{1,2}}{\sqrt{2}}v_{1}$ | $m_{D}^{1,2}=\frac{Y_{1}^{1,2}}{\sqrt{2}}v_{h_{2}}$
Table 4: Dirac and Majorana masses in the neutrino sector. In Case-II we use
the collider constraints to set
$(v_{1}^{2}+v_{2}^{2}+v_{3}^{2})\gg(v_{h_{1}}^{2}+v_{h_{2}}^{2})$ and in this
case first two generations are participating in the neutrino mass generation
mechanism.

Hence finally the light neutrino mass is generated by the seesaw mechanism to
explain the origin of tiny neutrino mass term and flavor mixing. Following
Tab. 4 the generic formula for the neutrino mass matrix can be written as

$m_{\nu}=\begin{pmatrix}0&m_{D}\\\ m_{D}^{T}&m_{N}\end{pmatrix}.$ (20)

Diaginalizing the neutrino mass matrix we find the light neutrino mass
eigenvalues to be $`-m_{D}m_{N}^{-1}m_{D}^{T}$’. In Case-II $N_{R}^{1,2}$ will
generate the neutrino mass, on the other hand, at the tree level $N_{R}^{3}$
will not participate in the neutrino mass generation and it can be considered
as a potential DM candidate in some scenarios. The neutrino mass generation
mechanism and dark matter physics are not main motivations of this work,
however, we provide a simple outline in this paper for completeness because
these general U(1)X scenarios can generate neutrino mass at the tree level
from the so-called seesaw mechanism which is an important aspect for studying
such scenarios.

##  III Calculation of the constraints on the chiral gauge couplings

The chiral $Z^{\prime}$ gauge boson interacts with the SM fermions, and the
couplings depend on the $U(1)_{X}$ charge of the SM Higgs doublet. Therefore,
experiments for scattering measurements of the SM particles can search the
chiral $Z^{\prime}$ gauge boson by measuring deviations of the scattering
cross section from the SM value. In this section, we show the scattering cross
sections contributed by the chiral $Z^{\prime}$ gauge boson and methods to
calculate constraints on the $U(1)_{X}$ gauge coupling. In this work, we
consider FASER$\nu(2)$, SND$@$LHC, NA64 and MuonE as the experiments for
scattering measurement. In addition, we estimate the constraints from
$\nu-$electron, $\nu-$nucleon, electron and proton beam-dump, long-lived gauge
boson searches, respectively.

###  III.1 Scattering cross section contributed by chiral $Z^{\prime}$ at
FASER$\nu(2)$, SN$D$@LHC, NA64 and MuonE

In this subsection, we summarize scattering cross sections via chiral
$Z^{\prime}$ interactions at FASER$\nu(2)$, SND@LHC, NA64 and MuonE
experiments.

####  III.1.1 Prospects for FASER$\nu(2)$ and SND@LHC

The existence of a light $Z^{\prime}$ affects the neutral-current deep-
inelastic scattering at the LHC far-forward detectors. We study the
constraints from the FASER$\nu$, FASER$\nu$2 and SND@LHC experiments in this
subsection. The numbers of neutrinos that pass through the FASER$\nu$ and
SND@LHC detectors have been simulated in Refs. [43, 108]. It has been found
that the muon neutrino from pions and kaons decay is most abundant. The energy
distributions of which are shown in Fig. 1. The corresponding neutrino flux at
the FASER$\nu$2 detector can be obtained by rescaling the flux at the
FASER$\nu$, assuming the neutrino distributes uniformly on the detector
surface.

Figure 1: Number of muon neutrino pass through the FASER$\nu$ (left panel) and
SND@LHC (right panel) detectors. Thirty energy bins are defined uniformly on
the logarithmic scale in [10,$10^{4}$] GeV.

The MG5_aMC@NLO package is used to calculate the fixed target deep-inelastic
neutrino-proton scattering cross-section at the leading order. Assuming a
benchmark detector made of tungsten target, the nCTEQ15FullNuc_184_74 set
[109] as implemented in LHAPDF6 [110] is employed as the proton parton
distribution function. The partonic collision energy is taken as the
factorization and renormalization scales in our simulation. The neutrino-
proton scattering cross section $\sigma_{\nu p}$ is related to $g_{X}$,
$x_{H}$, $Z^{\prime}$ mass as well as the incoming neutrino energy $E_{\nu}$.
In Fig. 2, we present the relation between $E_{\nu}$ and $\sigma_{\nu
p}/E_{\nu}$ for a few sets of $x_{H}$ and $M_{Z^{\prime}}$. The $g_{X}$ is
fixed to the unity, since the change of $g_{X}$ can only lead to a total
rescaling. The $Z^{\prime}$ contribution is slightly increased with increasing
$x_{H}$ from $-1$ to 1. The dependence on the $M_{Z^{\prime}}$ is more
complicated due to the interference effects between the $Z$ and $Z^{\prime}$
bosons. The $Z^{\prime}$ contribution is negligible for $M_{Z^{\prime}}\gtrsim
100$ GeV, and the cross-section becomes identical to the SM one in this
region.

Figure 2: The neutrino-proton scattering cross section $\sigma_{\nu p}$
divided by the incoming neutrino energy $E_{\nu}$. The gauge coupling $g_{X}$
is set to the unity.

Given the neutrino-nucleus cross section ($\sigma_{\nu N}$, which is
$184\times\sigma_{\nu p}$), one can estimate the probability of a neutrino
interacting with the detector as

$\displaystyle P=\frac{\sigma_{\nu N}\times~{}\text{Number of
Nuclei}}{\text{Detector Area}}=\frac{\sigma_{\nu N}}{A}\frac{m_{\rm
det}}{m_{N}}~{},~{}$ (21)

where $N$ is the target nucleus, $A$ is the detector area, and $m_{N}$ is the
mass of the target nucleus. The relevant detector configurations are listed in
Tab. 5.

Detector | $A$ | $m_{\rm det}$ | Integrated Luminosity
---|---|---|---
FASER$\nu$ | 25 cm $\times$ 25 cm | 1.2 tons | 150 fb-1
FASER$\nu$2 | 50 cm $\times$ 50 cm | 10 tons | 3000 fb-1
SND@LHC | 39 cm $\times$ 39 cm | 0.8 tons | 150 fb-1
Table 5:  Detector configurations.

For given $g_{X}$, $x_{H}$ and $M_{Z^{\prime}}$, the number of interacting
neutrinos in each $E_{\nu}$ bin can be calculated by the products of the
number of neutrinos passing through the detector and the interaction
probability. For illustration, energy spectra for the interacting muon
neutrino at the FASER$\nu$, FASER$\nu$2 and SND@LHC detectors are shown in
Fig. 3. The $g_{X}$ is taken to be the unity. Three different values of
$x_{H}$ and six different values of $M_{Z^{\prime}}$ are used. The number of
interacting neutrinos is increased with increasing $x_{H}$ and decreasing
$M_{Z^{\prime}}$. Moreover, we can observe that the number at the FASER$\nu$
detector is around one order of magnitude larger than that at the SND@LHC
detector, while it is around two orders of magnitude below that at the
FASER$\nu$2 detector.

Figure 3: The muon neutrino spectra at the FASER$\nu$ (top panels),
FASER$\nu$2 (middle panels) and SND@LHC (bottom panels) detectors.

We follow a similar strategy as proposed in Ref. [32] to estimate the
sensitivity reach in the parameter space of $g_{X}$ and $M_{Z^{\prime}}$ for
our model. Only the total numbers of interacting neutrinos are used in
defining the $\chi^{2}$ measure (i.e. the shape of $E_{\nu}$ spectra are not
of concern):

$\displaystyle\chi^{2}=\min_{\alpha}\left[\frac{N^{\nu_{e}}_{\rm
BSM}-(1+\alpha)N^{\nu_{e}}_{\rm SM}}{N^{\nu_{e}}_{\rm
BSM}}+\frac{N^{\nu_{\mu}}_{\rm BSM}-(1+\alpha)N^{\nu_{\mu}}_{\rm
SM}}{N^{\nu_{\mu}}_{\rm BSM}}+\frac{N^{\nu_{\tau}}_{\rm
BSM}-(1+\alpha)N^{\nu_{\tau}}_{\rm SM}}{N^{\nu_{\tau}}_{\rm
BSM}}+(\frac{\alpha}{\sigma_{\rm norm}})^{2}\right]~{},$ (22)

where $N_{\rm BSM}$ and $N_{\rm SM}$ are the number of interacting neutrinos
of each flavor in our model and in the SM model. The systematic uncertainties
($\sigma_{\rm norm}$) in each neutrino flavor are assumed to be the same and
only one nuisance parameter $\alpha$ is used. The $\chi^{2}$ value is obtained
by minimizing over the $\alpha$. The 95% confidence level sensitivity reach
corresponds to $\chi^{2}=3.84$.

####  III.1.2 Constraints from the NA64

The fixed-target experiment NA64 at the CERN SPS [53, 54] aims to search for
the $Z^{\prime}$ that is produced through the bremsstrahlung process in the
high-energy electron beam colliding with heavy nuclei

$\displaystyle e^{-}Z\to e^{-}ZZ^{\prime};~{}~{}Z^{\prime}\to\nu\nu$ (23)

where the $Z^{\prime}$ is decaying invisibly. The partial width of each
$Z^{\prime}$ decay channel is calculated by the DARKCAST package [12, 14],
assuming that only vector interactions exist. The results are translated into
those in our chiral model by using the same method as we proposed in Ref.
[10].

The production cross section and the energy spectrum of the $Z^{\prime}$ is
simulated by the MG5_aMC@NLO package. Since the target nucleus is lead at the
NA64 experiment, the nCTEQ15_208_82 set in LHAPDF6 is employed as the proton
PDF. The total cross section is proportional to $g_{X}^{2}$. Taking
$g_{X}=0.2$ and incoming electron beam energy $E_{0}=100$ GeV, we present the
electron-proton scattering cross sections with respect to varying $x_{H}$ and
$M_{Z^{\prime}}$ in the left panel of Fig. 4. It can be observed that the
cross-section is increased with increasing $x_{H}$ and decreasing
$M_{Z^{\prime}}$. In the right panel of Fig. 4, the normalized bremsstrahlung
$Z^{\prime}$ spectra are shown. The shape of $Z^{\prime}$ spectrum is highly
dependent on the $Z^{\prime}$ mass while it is almost irrelevant to the
$x_{H}$ value. Some analytic discussions about the feature of the spectrum are
conducted in Ref. [111].

Figure 4: Left panel: the total cross section for the bremsstrahlung
$Z^{\prime}$ production. Right panel: normalized $E_{Z^{\prime}}/E_{0}$
distribution for different $Z^{\prime}$ masses. $E_{0}=100$ GeV is the
electron beam energy.

Having calculated the differential bremsstrahlung $Z^{\prime}$ production
cross section ($d\sigma_{Z^{\prime}}/dE_{Z^{\prime}}$) as well as the
$Z^{\prime}$ decay branching ratios for all of the channels, we are able to
calculate the number of signal events at the NA64 experiment as follows [52]

$\displaystyle
n_{Z^{\prime}}(g_{X},x_{H},M_{Z^{\prime}})=\int_{0.5E_{e}}^{E_{e}}C\frac{d\sigma_{Z^{\prime}}}{dE_{Z^{\prime}}}\left[\text{Br}(Z^{\prime}\to\nu\nu)+\sum_{l}\text{Br}(Z^{\prime}\to
l^{+}l^{-})\exp(-\frac{L_{\rm ECAL}+L_{\rm HCAL}}{L_{Z^{\prime}}})\right],$
(24)

where $C$ is related to the detector parameters. The
$L_{Z^{\prime}}=c\tau_{Z^{\prime}}E_{Z^{\prime}}/M_{Z^{\prime}}$, $L_{\rm
ECAL}$ and $L_{\rm HCAL}$ are the $Z^{\prime}$ decay length, electromagnetic
calorimeter length, and hadronic calorimeter length, respectively. The Ref.
[52] uses data with $3.22\times 10^{11}$ electrons on target collected during
2016–2021 runs at the NA64 experiment and obtains the 90% C.L. exclusion
limits for the $U(1)_{\rm B-L}$ model. The corresponding bounds for our chiral
model with different $x_{H}$ can be obtained by requiring the number of signal
events $n_{Z^{\prime}}(g_{X},x_{H},M_{Z^{\prime}})$ in Eq. 24 to be the same
as that in the B$-$L case.

####  III.1.3 Constraints from the MUonE

MUonE [112, 113] is an experiment at CERN aiming to measure elastic
scatterings between the 150 GeV $\mu^{+}$ beam and target being electrons in
beryllium atoms and determining a contribution of hadronic vacuum polarization
to the muon anomalous magnetic moment with a method [114]. The existence of
the chiral $Z^{\prime}$ changes the cross section of the elastic scattering
between positive muon and electron, and therefore, the chiral $Z^{\prime}$ can
be searched by estimating a deviation of the scattering cross section from the
SM prediction.

The elastic scattering cross section between the positive muon and electron is
given by

$\frac{d\sigma(\mu^{+}e^{-})}{dT}=\left.\frac{d\sigma(\mu^{+}e^{-})}{dT}\right|_{\rm
SM}\left.+\frac{d\sigma(\mu^{+}e^{-})}{dT}\right|_{Z^{\prime}}\left.+\frac{d\sigma(\mu^{+}e^{-})}{dT}\right|_{\rm
Int}~{},$ (25)

where $T$ denotes the electron recoil energy, and the first, second, and third
terms in the right-hand side (RHS) correspond to the contributions from the
only SM interaction, only $Z^{\prime}$ one and interference between the SM and
$Z^{\prime}$ ones. The purely SM contribution can be given by

$\displaystyle\left.\frac{d\sigma(\mu^{+}e^{-})}{dT}\right|_{\rm
SM}=\frac{\pi\alpha_{\rm
EM}^{2}}{(E_{\mu}^{2}-m_{\mu}^{2})m_{e}^{2}(T-m_{e})^{2}}\left\\{2E_{\mu}m_{e}(E_{\mu}-T+m_{e})-(T-m_{e})(2m_{e}^{2}+m_{\mu}^{2}-m_{e}T)\right\\}~{},$
(26)

where $\alpha_{\rm EM}$ denotes the fine structure constant, and $E_{\mu}$
stands for the energy of the positive muon. The contributions from
$Z^{\prime}$ exchanging diagram and interference between the SM and
$Z^{\prime}$ are estimated by

$\displaystyle\left.\frac{d\sigma(\mu^{+}e^{-})}{dT}\right|_{Z^{\prime}}=$
$\displaystyle\frac{g_{X}^{4}m_{e}}{128\pi
M_{Z^{\prime}}^{4}(E_{\mu}^{2}-m_{\mu}^{2})(M_{Z^{\prime}}^{2}+2m_{e}T-m_{e}^{2})^{2}}$
$\displaystyle\quad\times\left[(\tilde{x}_{e}+\tilde{x}_{\ell})^{4}\left\\{M_{Z^{\prime}}^{4}\left(2E_{\mu}(E_{\mu}-T+m_{e})+T^{2}-3m_{e}T-m_{\mu}^{2}+2m_{e}^{2}\right)\right\\}\right.$
$\displaystyle\qquad\quad+2(\tilde{x}_{e}+\tilde{x}_{\ell})^{3}(\tilde{x}_{e}-\tilde{x}_{\ell})\left\\{M_{Z^{\prime}}^{4}\left(2E_{\mu}^{2}-m_{e}T-m_{\mu}^{2}+m_{e}^{2}\right)\right\\}$
$\displaystyle\qquad\quad+2(\tilde{x}_{e}+\tilde{x}_{\ell})^{2}(\tilde{x}_{e}-\tilde{x}_{\ell})^{2}\left\\{M_{Z^{\prime}}^{4}\left(2E_{\mu}(E_{\mu}+T-m_{e})-(T-m_{e})^{2}\right)\right.$
$\displaystyle\hskip
142.26378pt\left.+2m_{e}m_{\mu}^{2}M_{Z^{\prime}}^{2}(T-m_{e})+2m_{e}^{2}m_{\mu}^{2}(T-m_{e})^{2}\right\\}$
$\displaystyle\qquad\quad+2(\tilde{x}_{e}+\tilde{x}_{\ell})(\tilde{x}_{e}-\tilde{x}_{\ell})^{3}\left\\{M_{Z^{\prime}}^{4}\left(2E_{\mu}^{2}+m_{e}T+m_{\mu}^{2}-m_{e}^{2}\right)+4m_{e}m_{\mu}^{2}M_{Z^{\prime}}^{2}(T-m_{e})\right.$
$\displaystyle\hskip
142.26378pt\left.+4m_{e}^{2}m_{\mu}^{2}(T-m_{e})^{2}\right\\}$
$\displaystyle\qquad\quad+(\tilde{x}_{e}-\tilde{x}_{\ell})^{4}\left\\{M_{Z^{\prime}}^{4}\left(2E_{\mu}(E_{\mu}-T+m_{e})+T^{2}-m_{e}T+m_{\mu}^{2}\right)+4m_{e}m_{\mu}^{2}M_{Z^{\prime}}^{2}(T-m_{e})\right.$
$\displaystyle\hskip
142.26378pt\left.\left.+4m_{e}^{2}m_{\mu}^{2}(T-m_{e})^{2}\right\\}\right]~{},$
(27) $\displaystyle\left.\frac{d\sigma(\mu^{+}e^{-})}{dT}\right|_{\rm Int}=$
$\displaystyle\frac{\alpha_{\rm
EM}g_{X}^{2}}{8(E_{\mu}^{2}-m_{\mu}^{2})(T-m_{e})(M_{Z^{\prime}}^{2}+2m_{e}T-m_{e}^{2})}$
$\displaystyle\quad\times\left[(\tilde{x}_{e}+\tilde{x}_{\ell})^{2}\left\\{2E_{\mu}\left(E_{\mu}-T+m_{e}\right)+T^{2}-3m_{e}T-m_{\mu}^{2}+2m_{e}^{2}\right\\}\right.$
$\displaystyle\qquad\quad\left.+(\tilde{x}_{e}+\tilde{x}_{\ell})(\tilde{x}_{e}-\tilde{x}_{\ell})\left(2E_{\mu}^{2}-m_{e}T-m_{\mu}^{2}+m_{e}^{2}\right)+(\tilde{x}_{e}-\tilde{x}_{\ell})^{2}\left(T-m_{e}\right)\left(2E_{\mu}-T+m_{e}\right)\right].~{}~{}$
(28)

The number of elastic scattering signals in $i$-th bin of the electron recoil
energy ($T_{i}<T<T_{i}+\Delta T$) is evaluated by [34]

$\displaystyle N_{i}=\mathcal{L}\int_{T_{i}}^{T_{i}+\Delta
T}\mathrm{d}T\frac{\mathrm{d}\sigma(\mu^{+}e^{-})}{\mathrm{d}T}\Theta(T)\Theta\left(T_{\rm
max}(E_{\mu})-T\right)~{},$ (29)

where $\mathcal{L}$ is the integrated luminosity, and $\mathcal{L}=150$ fb-1
for MUonE. In Eq. (29)$,\Theta(T)$ is the Heaviside step function, and $T_{\rm
max}(E_{\mu})$ stands for the maximal value of the electron recoil energy
determined by

$\displaystyle T_{\rm
max}(E_{\mu})=\frac{2m_{e}(E_{\mu}^{2}-m_{\mu}^{2})}{2E_{\mu}m_{e}+m_{e}^{2}+m_{\mu}^{2}}~{}.$
(30)

In Fig. 5, the number of events contributed only by the SM particles and the
deviation of the number of events in chiral $Z^{\prime}$ model from that in
the SM are shown.

Figure 5:  Top-left panel : the distribution of the number of events
contributed only by the SM particles in MUonE. Top-right, bottom-left, and
bottom-right panels : the distributions of the deviation of the number of
events in MUonE from that contributed only by the SM particles : $\delta
N_{i}/N_{i}^{\rm SM}\equiv(N_{i}^{{\rm SM}+Z^{\prime}}-N_{i}^{\rm
SM})/N_{i}^{\rm SM}$ for $(M_{Z^{\prime}}/{\rm
GeV},g_{X})=(10^{-3},10^{-3}),(10^{-1},10^{-3}),(10,10^{-3})$, respectively.

In this paper, we evaluate the sensitivity of MUonE to the chiral $Z^{\prime}$
gauge boson by the $\chi^{2}$-function, following Ref. [34]. The
$\chi^{2}$-function is calculated by

$\displaystyle\chi^{2}=\sum_{i=1}^{100}\frac{(N_{i}-N_{i}^{\rm
SM})^{2}}{\sigma_{{\rm stat},i}^{2}+\sigma_{{\rm sys},i}^{2}}~{},$ (31)

where $\sigma_{{\rm stat},i}=\sqrt{N_{i}}$ stands for the statistical
uncertainty, and $\sigma_{{\rm sys},i}=10^{-5}N_{i}$ stands for the systematic
uncertainty at the level of 10 ppm [112]. The 95% confidence level sensitivity
is obtained by solving $\chi^{2}=3.84$.

###  III.2 Constraints from proton/electron beam dump experiments

For constraints from proton beam dump experiments NOMAD and CHARM, we obtain
bound curves by rescaling the bounds of U(1)B-L case given in ref. [11]. The
upper bound on $\\{m_{Z^{\prime}},g_{X}\\}$ plane is approximately derived
applying scaling [12, 28]

$\tau_{Z^{\prime}}(g_{B-L}^{\rm max})\sim\tau_{Z^{\prime}}(g_{X}^{\rm
max},x_{H},x_{\Phi})~{},$ (32)

where $g_{B-L}$ denotes the gauge coupling in the U(1)B-L case, and
$\tau_{Z^{\prime}}$ is the lifetime of the $Z^{\prime}$. The lower bound is
also scaled by applying

$g_{X}^{\rm low}\sim g_{B-L}^{\rm low}\sqrt{\frac{{\rm BR}(M\to
Z^{\prime}_{B-L}\gamma)~{}{\rm BR}(Z^{\prime}_{B-L}\to
e^{+}e^{-})\tilde{\tau}_{Z^{\prime}}}{{\rm BR}(M\to Z^{\prime}\gamma)~{}{\rm
BR}(Z^{\prime}\to e^{+}e^{-})\tilde{\tau}_{Z^{\prime}_{B-L}}}}~{},$ (33)

where $\tilde{\tau}$ is lifetime with gauge coupling being unity, and
$Z^{\prime}$ is produced via meson decay with $M=\pi^{0}$ for MONAD and
$M=\eta$ for CHARM dominantly. Here, meson decay branching ratio is estimated
using the method given in ref. [12].

For proton beam dump experiment $\nu$-cal, the bremsstrahlung process
dominantly produces $Z^{\prime}$ boson. In this work we use the excluded
region given in ref. [10] on $\\{M_{Z^{\prime}},g_{X}\\}$ space where we take
into account chiral structure of the $Z^{\prime}$ interactions in estimating
the $Z^{\prime}$ production cross section.

For electron beam dump constraints from from Orsay and KEK, The bound curves
are obtained by rescaling the bounds of U(1)B-L case as for the proton beam
dump one. We derive the constraint on the upper region on
$\\{M_{Z^{\prime}},g_{X}\\}$ plane approximately by scaling with Eq. (32),
which is the same as the proton beam dump case. On the other hand, the
constraint on the lower region is estimated by [28]

$g_{X}^{\rm low}\sim g_{B-L}^{\rm low}\sqrt{\frac{2{\rm
BR}(Z^{\prime}_{B-L}\to
e^{+}e^{-})\tilde{\tau}_{Z^{\prime}}}{(5x_{H}^{2}/4+3x_{H}x_{\Phi}+2x_{\Phi}^{2}){\rm
BR}(Z^{\prime}\to e^{+}e^{-})\tilde{\tau}_{Z^{\prime}_{B-L}}}}~{},$ (34)

where $Z^{\prime}$ is considered to be produced via bremsstrahlung process.

For electron beam dump constraints from E137 and E141 experiments, we use the
results in ref. [10] for excluded region on $\\{M_{Z^{\prime}},g_{X}\\}$ in
which chiral structure of the $Z^{\prime}$ interactions is taken into account
in estimating the $Z^{\prime}$ production cross section via bremsstrahlung
process.

###  III.3 Constraints from electron-(anti)neutrino scattering in neutrino
experiments

Here we discuss constraints on $\\{M_{Z^{\prime}},g_{X}\\}$ from
electron-(anti)neutrino scattering processes that are tested by neutrino
experiments: BOREXINO, TEXONO, GEMMA, CHARM-II and the J-PARC Sterile Neutrino
Search at the J-PARC Spallation Neutron Source(JSNS$2$). To obtain the
constraints we estimate the electron-(anti)neutrino scattering cross sections
under the existence of $Z^{\prime}$ interactions. The differential cross
section can be expressed as

$\frac{d\sigma(\nu e)}{dT}=\left.\frac{d\sigma(\nu e)}{dT}\right|_{\rm
SM}+\left.\frac{d\sigma(\nu
e)}{dT}\right|_{Z^{\prime}}+\left.\frac{d\sigma(\nu e)}{dT}\right|_{\rm Int}$
(35)

where $T$ denotes the electron recoil energy and the first, second and third
terms in the RHS correspond to the contributions from the only SM
interactions, the only $Z^{\prime}$ interactions and interference between the
SM and $Z^{\prime}$ interactions. The terms in RHS are given as follows [28].
The purely SM contribution can be written by

$\displaystyle\left.\frac{d\sigma(\nu e)}{dT}\right|_{\rm SM}$
$\displaystyle=\frac{2G_{F}^{2}m_{e}}{\pi
E_{\nu}^{2}}\left(a^{2}_{1}E^{2}_{\nu}+a_{2}^{2}(E_{\nu}-T)^{2}-a_{1}a_{2}m_{e}T\right),$
(36)

where $E_{\nu}$ is the energy of initial neutrino. Here $a_{1}$ and $a_{2}$
are given by

$\displaystyle a_{1}$ $\displaystyle=\left\\{\sin^{2}\theta_{W}+\frac{1}{2},\
\sin^{2}\theta_{W},\ \sin^{2}\theta_{W}-\frac{1}{2},\
\sin^{2}\theta_{W}\right\\}\ {\rm for}\
\\{\nu_{e}e,\bar{\nu}_{e}e,\nu_{\beta}e,\bar{\nu}_{\beta}e\\},$ (37)
$\displaystyle a_{2}$ $\displaystyle=\left\\{\sin^{2}\theta_{W},\
\sin^{2}\theta_{W}+\frac{1}{2},\ \sin^{2}\theta_{W},\
\sin^{2}\theta_{W}-\frac{1}{2}\right\\}\ {\rm for}\
\\{\nu_{e}e,\bar{\nu}_{e}e,\nu_{\beta}e,\bar{\nu}_{\beta}e\\},$ (38)

where $\beta=\\{\mu,\tau\\}$. The contribution from $Z^{\prime}$ exchanging
diagram is estimated by

$\displaystyle\left.\frac{d\sigma(\overset{(-)}{\nu}_{\alpha}e)}{dT}\right|_{Z^{\prime}}$
$\displaystyle=\frac{g_{X}^{4}(\tilde{x}_{\ell})^{2}m_{e}}{4\pi
E_{\nu}^{2}(2m_{e}T+M^{2}_{Z^{\prime}})}[(2E^{2}_{\nu}-2E_{\nu}T+T^{2})(b^{2}_{1}+b^{2}_{2})\pm
2b_{1}b_{2}(2E_{\nu}-T)T-m_{e}T(b^{2}_{1}-b_{2}^{2})],$ (39)

where $b_{1}=\frac{\tilde{x}_{\ell}+\tilde{x}_{e}}{2}$ and
$b_{2}=\frac{\tilde{x}_{\ell}-\tilde{x}_{e}}{2}$ with $\tilde{x}_{\ell,e}$
from Table. 1, and the negative sign of $\pm$ is for the process of anti
neutrino. The contributions from interference between the SM and $Z^{\prime}$
are also written, depending on the process, as follows:

$\displaystyle\left.\frac{d\sigma(\nu_{e}e)}{dT}\right|_{\rm int}=$
$\displaystyle\frac{G_{F}g_{X}^{2}\tilde{x}_{\ell}m_{e}}{\sqrt{2}\pi
E^{2}_{\nu}(2m_{e}T+M^{2}_{Z^{\prime}})}[2E_{\nu}^{2}(b_{1}+b_{2})+(2E^{2}_{\nu}-2E_{\nu}T+T^{2})(b_{1}c_{1}+b_{2}c_{2})]$
$\displaystyle+T(2E_{\nu}-T)(b_{1}c_{2}+b_{2}c_{1})-m_{e}T(b_{1}-b_{2}+b_{1}c_{1}-b_{2}c_{2})],$
(40) $\displaystyle\left.\frac{d\sigma(\bar{\nu}_{e}e)}{dT}\right|_{\rm int}=$
$\displaystyle\frac{G_{F}g_{X}^{2}\tilde{x}_{\ell}m_{e}}{\sqrt{2}\pi
E^{2}_{\nu}(2m_{e}T+M^{2}_{Z^{\prime}})}[2(E_{\nu}-T)^{2}(b_{1}+b_{2})+(2E^{2}_{\nu}-2E_{\nu}T+T^{2})(b_{1}c_{1}+b_{2}c_{2})]$
$\displaystyle-T(2E_{\nu}-T)(b_{1}c_{2}+b_{2}c_{1})-m_{e}T(b_{1}-b_{2}+b_{1}c_{1}-b_{2}c_{2})],$
(41)
$\displaystyle\left.\frac{d\sigma(\overset{(-)}{\nu}_{\beta}e)}{dT}\right|_{\rm
int}=$ $\displaystyle\frac{G_{F}g_{X}^{2}\tilde{x}_{\ell}m_{e}}{\sqrt{2}\pi
E^{2}_{\nu}(2m_{e}T+M^{2}_{Z^{\prime}})}[(2E_{\nu}^{2}-2E_{\nu}T-T^{2})2(b_{1}c_{1}+b_{2}c_{1})\pm
T(2E_{\nu}-T)(b_{1}c_{2}+b_{2}c_{1})]$ $\displaystyle-
m_{e}T(b_{1}c_{1}-b_{2}c_{2})],$ (42)

where $c_{1}=-1/2+2\sin^{2}\theta_{W}$ and $c_{2}=-1/2$. We then estimate the
differential cross sections and derive the constraints for each experiment in
the following way.

BOREXINO: The cross section of $\nu_{e}$-$e$ scattering process is estimated
by the experiment where $\langle E_{\nu}\rangle=862$ keV and
$T\simeq[270,665]$ keV for ${}^{7}B_{e}$ solar neutrino. We require the cross
section with $Z^{\prime}$ interaction should not be more than $8\%$ above that
of the SM prediction [70] to obtain the constraint on
$\\{M_{Z^{\prime}},g_{X}\\}$.

TEXONO: $\bar{\nu}_{e}$-$e$ scattering process is measured by the experiment
using 187 kg of CsI(Tl) scintillating crystal array with 29882/7369 kg-day of
reactor ON/OFF data with electron recoil energy of $T\simeq[3,8]$ MeV. The
$\chi^{2}$ value is estimated as

$\chi^{2}=\sum_{\rm bin}\frac{(R_{\rm data}-R_{\rm th})^{2}}{\Delta R^{2}},$
(43)

where $R_{\rm data}$ and $R_{\rm th}$ are the event ratios measured by the
experiment and predicted by the cross section in Eq. (35), and $\Delta R$ is
the experimental uncertainty, for each recoil energy bin taken from data in
ref. [66]. Here we also applied anti neutrino flux in the reference. The
constraint on $\\{M_{Z^{\prime}},g_{X}\\}$ plane is then obtained by
$\chi^{2}$ analysis with $90\%$ C.L.

GEMMA: $\bar{\nu}_{e}$-$e$ scattering is observed with 1.5 kg HPGe detector
where energy of neutrino is $\langle E_{\nu}\rangle\sim 1$-$2$ MeV and flux is
$2.7\times 10^{13}$ cm-2s-1. The $\chi^{2}$ value is estimated applying the
formula Eq. (43) for the data given in ref. [115] with 13000 ON-hours and 3000
OFF-hours, and we derive the upper limit curve on $\\{M_{Z^{\prime}},g_{X}\\}$
with $90\%$ C.L.

CHARM-II: $\nu_{\mu}(\bar{\nu}_{\mu})$-electron scattering is observed where
$2677\pm 82$ and $2752\pm 88$ events are respectively obtained for $\nu_{\mu}$
and $\bar{\nu}_{\mu}$ cases. The mean neutrino energies for $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ are respectively $\langle E_{\nu_{\mu}}\rangle=23.7$ GeV and
$\langle E_{\bar{\nu}_{\mu}}\rangle=19.1$ GeV, and the range of observed
recoil energy is 3-24 GeV. The $\chi^{2}$ value is estimated using the formula
Eq. (43) for the data given in ref. [78, 79], and we derive the upper curve on
$\\{M_{Z^{\prime}},g_{X}\\}$ with $90\%$ C.L.

JSNS$2$: In the experiment, 3 GeV proton collides with mercury target
producing pions giving neutrino beams. We consider $\nu_{e}$-$e$ and
$\bar{\nu}_{\mu}$-$e$ scattering processes to obtain the constraint. We
estimate number of scattering events applying $3.8\times 10^{22}$ protons on
target per year, 17 tons of a gadolinium(Gd)-loaded liquid-scintillator
detector (LS) detector and neutrino fluxes given in ref. [75]. The
corresponding $\chi^{2}$ is estimated by

$\chi^{2}=\underset{\alpha}{\rm min}\left[\frac{(N_{\rm th}-(1+\alpha)N_{\rm
SM})^{2}}{N_{\rm th}}+\left(\frac{\alpha}{\sigma_{\rm
norm}}\right)^{2}\right],$ (44)

where $N_{\rm th}$ and $N_{\rm SM}$ are the expected number of events in our
models and in the SM for 1 year, $\sigma_{\rm norm}$ is the systematic
uncertainty in the neutrino flux normalization, and $\alpha$ is nuisance
parameter. Here $\sigma_{\rm norm}$ is assumed to be $5\%$ as the reference
value. Then the future sensitivity on $\\{M_{Z^{\prime}},g_{X}\\}$ plane is
estimated by requiring $\chi^{2}$ value to be less than that of $90\%$ C.L.

###  III.4 Limits from coherent neutrino-nucleus scattering

An upper limit curve on $\\{M_{Z^{\prime}},g_{X}\\}$ is also obtained from
coherent elastic neutrino-nucleus scattering (CE$\nu$NS) that is measured by
COHERENT experiment with CsI and Ar targets [116, 83]. Here we derive the
curve by rescaling the limit curve for $U(1)_{B-L}$ case given in ref. [82] by
comparing number of events in $U(1)_{B-L}$ and other cases. We estimate the
number of events at COHERENT experiment adopting formulas in the reference, as
discussed below.

Firstly, the differential cross section for CE$\nu$NS process is estimated by
[117, 118]

$\frac{d\sigma_{\nu-N}}{dT}(E,T)=\frac{G_{F}^{2}M}{\pi}\left(1-\frac{MT}{2E^{2}}\right)Q_{{\rm
SM}+Z^{\prime}}^{2},$ (45)

where $E$ is the initial neutrino energy, $T$ is the recoil energy, $M$ is the
mass of target nucleus and $Q_{{\rm SM}+Z^{\prime}}$ is the factor coming from
interactions including SM and $Z^{\prime}$ gauge bosons. In our models
$Q_{{\rm SM}+Z^{\prime}}$ is given by

$Q_{{\rm
SM}+Z^{\prime}}=\left(g^{p}_{V}(\nu_{\ell})+2\epsilon^{uV}_{\ell\ell}+\epsilon^{dV}_{\ell\ell}\right)ZF_{Z}(|{\bf
q}^{2}|)+\left(g^{n}_{V}(\nu_{\ell})+\epsilon^{uV}_{\ell\ell}+2\epsilon^{dV}_{\ell\ell}\right)NF_{N}(|{\bf
q}^{2}|),$ (46)

where $g^{p(n)}_{V}$ is the neutrino-proton(neutron) coupling in the SM,
$Z(N)$ is the number of proton(neutron) in the target nucleus, and
$F_{Z(N)}(|{\bf q}^{2}|)$ is the from factors of the proton(neutron) for the
target nucleus. The $\epsilon^{qV}_{\ell\ell}$ is effective coupling
explicitly given by

$\epsilon^{qV}_{\ell\ell}=\frac{g_{X}^{2}\tilde{x}_{\ell}\tilde{x}_{q}}{\sqrt{2}G_{F}({\bf
q}^{2}+M^{2}_{Z^{\prime}})}.$ (47)

We adopt the values of $g_{V}^{p}(\nu_{e})=0.0401$, $g_{V}^{p}=0.0318$ and
$g_{V}^{n}=-0.5094$ for the neutrino-proton(neutron) coupling in the SM [119,
120]. The Helm parametrization [121] is applied for the form factors
$F_{Z(N)}(|{\bf q}^{2}|)$ using proton rms radii $\\{R_{p}({\rm
Cs}),R_{p}({\rm I}),R_{p}({\rm Ar})\\}=\\{4.804,4.749,3.448\\}$ [fm] and
neutron rms radii $\\{R_{n}({\rm Cs}),R_{n}({\rm I}),R_{n}({\rm
Ar})\\}=\\{5.01,4.94,3.55\\}$ [fm] [122, 123, 124].

Then we adopt the neutrino fluxes for the CE$\nu$NS event rate in the
experiment which depend on the neutrino fluxes produced from the Spallation
Neutron Source (SNS) at the Oak Ridge National Laboratories. They are given by
[116, 83]

$\displaystyle\frac{dN_{\nu_{\mu}}}{dE}$
$\displaystyle=\eta\delta\left(E-\frac{m_{\pi}^{2}-m_{\mu}^{2}}{2m_{\pi}}\right),$
(48) $\displaystyle\frac{dN_{\nu_{\bar{\mu}}}}{dE}$
$\displaystyle=\eta\frac{64E^{2}}{m_{\mu}^{3}}\left(\frac{3}{4}-\frac{E}{m_{\mu}}\right),$
(49) $\displaystyle\frac{dN_{\nu_{e}}}{dE}$
$\displaystyle=\eta\frac{192E^{2}}{m^{3}_{\mu}}\left(\frac{1}{2}-\frac{E}{m_{\mu}}\right),$
(50)

where $\eta=rN_{\rm POT}/(4\pi L^{2})$ with $L$, $N_{\rm POT}$ and $r$ being
respectively the distance between the source and the detector, the number of
proton-on-target (POT), and the number of neutrinos per flavor that are
produced for each POT. For these values, we use $r=9\times 10^{-2}$, $N_{\rm
POT}=13.7\times 10^{22}$ and $L=27.5$ m for Ar detector, and $r=0.08$, $N_{\rm
POT}=17.6\times 10^{22}$ and $L=19.5$ m for CsI detector, respectively.

Finally the theoretical number of events for each energy bin in the experiment
is derived from

$N_{i}=N(\mathcal{N})\int^{T_{i+1}}_{T_{i}}dTA(T)\int_{E_{\rm min}}^{E_{\rm
max}}dE\sum_{\nu=\nu_{e},\nu_{\mu},\nu_{\bar{\mu}}}\frac{dN_{\nu}}{dE}\frac{d\sigma_{\nu-N}}{dT}(E,T),$
(51)

where $i$ corresponds to each recoil energy bin, $E_{\rm
min(max)}=\sqrt{MT/2}(m_{\mu}/2)$, and $A(T)$ is the energy-dependent
reconstruction efficiency. We estimate upper limit of the coupling $g_{X}$ for
each $Z^{\prime}$ mass for different models by rescaling that of $U(1)_{B-L}$
case in ref. [82] by comparing the number of events for upper limit of $g_{X}$
in $U(1)_{B-L}$ with the number of events in each model.

###  III.5 Limits from collider experiments: LEP-II, CMS, LHCb and BarBar

Here we briefly summarize our estimation of limit on
$\\{M_{Z^{\prime}},g_{X}\\}$ from results of several collider experiments:

(i) The limit from LEP-II: We estimate upper limit curve on
$\\{M_{Z^{\prime}},g_{X}\\}$ from the results of LEP-II [125, 96] that measure
$e^{+}e^{-}\to\bar{f}f$ scattering cross sections at the $Z$ peak with $f$
being the SM fermions. The scattering cross sections are estimated including
$Z^{\prime}$ exchanging diagram in addition to the SM processes. Then the
value of cross section is compared with the observed value. As a result, we
consider constraints coming from the cross section of $e^{+}e^{-}\to q\bar{q}$
process giving hadronic final state whose value is $\sigma=41.544\pm 0.037$
nb, and $e^{+}e^{-}\to\ell^{+}\ell^{-}$ process with
$R_{\ell}=\Gamma_{had}/\Gamma_{\ell}=20.768\pm 0.024$. We thus obtain upper
limit of $g_{X}$ as function of $m_{Z^{\prime}}$ when we require the total
cross section is within the 90% C.L. of the observed value.

(ii) The limit from dark photon search at the LHC experiments: The bounds on
$\\{M_{Z^{\prime}},g_{X}\\}$ are also obtained from the results of CMS [86]
and LHCb [126] experiments searching for dark photon $A^{\prime}$ which decays
into $\mu^{+}\mu^{-}$ pair. They provide us the bounds on the mass of dark
photon $m_{A^{\prime}}$ and kinetic mixing parameter $\epsilon$. The upper
limit on $g_{X}$ as a function of $M_{Z^{\prime}}$ can be estimated in use of
following re-scaling

$g_{X}^{\rm max}(M_{Z^{\prime}}=m_{A^{\prime}})=\epsilon^{\rm
max}(m_{A^{\prime}})e\sqrt{\frac{\sigma(pp\to
A^{\prime})BR(A^{\prime}\to\mu^{+}\mu^{-})}{\sigma(pp\to
Z^{\prime})BR(Z^{\prime}\to\mu^{+}\mu^{-})}},$ (52)

where $\sigma(pp\to A^{\prime}(Z^{\prime}))$ is dark photon($Z^{\prime}$)
production cross section estimated by CalcHEP3.5 [127] implementing relevant
interactions, and $\epsilon^{\rm max}(m_{A^{\prime}})$ is the upper limit of
kinetic mixing parameter as a function of dark photon mass.

(iii) The limit from BaBar experiment At the BaBar experiment, dark photon is
searched for via the process $e^{+}e^{-}\to A^{\prime}\gamma$ [87, 88]. They
consider visible $A^{\prime}$ which decays into
$\\{e^{+}e^{-},\mu^{+}\mu^{-},\text{light mesons}\\}$ final states and
invisible $A^{\prime}$ which decays into invisible final sate such as
neutrinos. To estimate the bounds on $g_{X}$, we rescale the upper limit of
gauge coupling in $U(1)_{B-L}$ case given in ref. [11] as function of
$Z^{\prime}$ mass. The rescaling formula for visible $Z^{\prime}$ decay mode
is

$g_{X}^{\rm max}(M_{Z^{\prime}})=g_{B-L}^{\rm
max}(M_{Z^{\prime}})\sqrt{\frac{\sigma(e^{+}e^{-}\to\gamma
Z^{\prime}_{B-L})BR(Z^{\prime}_{B-L}\to\text{visible
states})}{\sigma(e^{+}e^{-}\to\gamma Z^{\prime})BR(Z^{\prime}\to\text{visible
states})}},$ (53)

where $Z^{\prime}_{B-L}$ indicate $Z^{\prime}$ boson in case of $U(1)_{B-L}$.
The rescaling formula for invisible $Z^{\prime}$ decay mode is

$g_{X}^{\rm max}(M_{Z^{\prime}})=g_{B-L}^{\rm
max}(M_{Z^{\prime}})\sqrt{\frac{\sigma(e^{+}e^{-}\to\gamma
Z^{\prime}_{B-L})BR(Z^{\prime}_{B-L}\to\bar{\nu}\nu)}{\sigma(e^{+}e^{-}\to\gamma
Z^{\prime})BR(Z^{\prime}\to\bar{\nu}\nu)}},$ (54)

where all neutrino modes are taken into account.

###  III.6 Limits from electron and muon $(g-2)$

The $Z^{\prime}$ boson contributes to electron and muon $(g-2)$, $\Delta
a_{\mu,e}$, at one loop level. Calculating one-loop diagram, we obtain the
formula as [128]

$\Delta
a_{\ell}=\frac{g_{X}^{2}}{8\pi^{2}}\frac{m_{\ell}^{2}}{M_{Z^{\prime}}^{2}}\int^{1}_{0}dx\frac{(\tilde{x}_{\ell}+\tilde{x}_{e})^{2}2x^{2}(1-x)+(\tilde{x}_{\ell}-\tilde{x}_{e})^{2}\left(2x(1-x)(x-4)-4x^{3}\frac{m_{\ell}^{2}}{M_{Z^{\prime}}^{2}}\right)}{(1-x)\left(1-x\frac{m_{\ell}^{2}}{M_{Z^{\prime}}^{2}}\right)+x\frac{m_{\ell}^{2}}{M_{Z^{\prime}}^{2}}},$
(55)

where $\ell=\\{e,\mu\\}$. As a reference we derive the region that accommodate
experimental values of electron and muon $(g-2)$ in the models. For electron
$(g-2)$, the experimentally obtained range of

$\displaystyle\Delta a_{e}(^{133}\text{Cs})$ $\displaystyle=-(8.8\pm
3.6)\times
10^{-13}~{}~{}\text{\cite[cite]{[\@@bibref{Number}{Parker:2018vye}{}{}]}}\,,$
$\displaystyle\Delta a_{e}(^{87}\text{Rb})$ $\displaystyle=(4.8\pm 3.0)\times
10^{-13}~{}~{}\text{\cite[cite]{[\@@bibref{Number}{Morel:2020dww}{}{}]}}\,,$
(56)

where the corresponding deviation from the SM are $2.4\sigma$ and $1.6\sigma$
respectively. We consider $\Delta a_{e}$ constraint on $Z^{\prime}$
interaction to satisfy either range of $\Delta a_{e}$ depending on the charge
assignment. For muon $(g-2)$ we apply the experimentally obtained range of
[131]

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

that corresponds to 4.2$\sigma$ deviation from the SM prediction. We show the
parameter region that satisfies the $\Delta a_{\mu}$ range in the models.

##  IV Results and Discussions

Figure 6: Limits on $g_{X}-M_{Z^{\prime}}$ plane for $x_{H}=-2$ (upper panel,
$U(1)_{R}$ case) and $x_{H}=-1$ (lower panel) taking $x_{\Phi}=1$ considering
$10^{-3}$ GeV $\leq M_{Z^{\prime}}\leq 150$ GeV. For $x_{H}=-2$ we show the
region sensitive to the MuonE experiment. Recasting the data we compare the
parameter region with those we estimated from LEP, Dark photon searches at
BaBaR, LHCb and CMS (CMS Dark) and different beam-dump experiments at Orsay,
KEK, E137, CHARM, Nomad, $\nu-$cal, E141, E774, NA64, KEK and involving
prospective bounds from FASER(2), DUNE and ILC-BD, respectively. For
$x_{H}=-1$ we show the regions sensitive to FASER$\nu$, FASER$\nu$2,
SND$@$LHC, NA64$(eN)$ and JSNS2 experiments respectively. We compare our
results recasting limits from LEP, CHARM-II, GEMMA, BOREXINO, COHERENT,
TEXONO, Dark photon search at BaBaR (vis and invis), LHCb and CMS (CMS Dark)
and different beam-dump experiments at Orsay, KEK, E137, CHARM, Nomad,
$\nu-$cal, E141, E774, NA64, KEK and involving prospective bounds from
FASER(2), DUNE and ILC, respectively. In this case MUonE bound can not be
calculated because $e_{R}$ does not interact with $Z^{\prime}$.

We calculate the exclusion and future sensitivity regions for the chiral
$Z^{\prime}$ gauge boson from neutrino-nucleon scattering measurements
(FASER$\nu$(2), SND@LHC), missing energy search (NA64), muon-electron
scattering measurements (MUonE), proton and electron beam dump experiments
(NOMAD, CHARM), electron-neutrino scattering measurements (BOREXINO, TEXONO,
JSNS2), coherent neutrino-nucleus scattering (COHERENT), and collider
experiments (LEP-II, CMS, LHCb, BarBar). In Figs. 6-9, we show the exclusion
and future sensitivity regions for $x_{H}=-2,-1,-0.5,0,0.5,1$, and $2$
considering $x_{\Phi}=1$ where the horizontal and vertical axes are
$Z^{\prime}$ mass within the range $10^{-3}$ GeV $\leq M_{Z^{\prime}}\leq 150$
GeV and gauge coupling constant $g_{X}$, respectively. The shaded regions have
already been excluded, and on the other hand, the curves without shaded
regions are sensitivity curves by future experiments. Along this line we
compare the sensitivity lines from Supernova (only in case of $x_{H}=-2$,
bounds for other $x_{H}$ can be found in [10]) and beam-dump experiments at
FASER(2), DUNE and ILC (ILC-BD) from [10], respectively. The shaded regions in
the figures are ruled out by respective experiments.

We show the prospective sensitivity in the $g_{X}-M_{Z^{\prime}}$ plane
obtained from the elastic $\mu^{+}-e^{-}$ scattering at the MUonE experiment
in the upper panel of Fig. 6 for $x_{H}=-2$ where left handed fermions do not
interact with $Z^{\prime}$. Bounds obtained from the MUonE experiment vary
between $6\times 10^{-4}\leq g_{X}\leq 0.4$ for $10^{-3}$ GeV $\leq
M_{Z^{\prime}}\leq 150$ GeV. It is found that MUonE bounds for
$M_{Z^{\prime}}\leq 0.03$ GeV are rules out by beam-dump searches from NA64
and E774 respectively whereas bounds for $M_{Z^{\prime}}\geq 0.175$ GeV are
ruled out by estimated limits obtained from the dark photon searches at BaBaR,
LHCb, CMS (CMS Dark), respectively except for a narrow window from the LHCb
experiment around $M_{Z^{\prime}}\simeq 1$ GeV. Limits obtained from
electronic magnetic moment for Cesium-133 is stronger than the limits obtained
from $\mu-e$ scattering at MUonE for $0.0174$ GeV $\leq M_{Z^{\prime}}\leq
0.0894$ GeV with $1.4\times 10^{-4}\leq g_{X}\leq 7.16\times 10^{-4}$.
Constrains obtained from electron $g-2$ using Cesium-133 is stronger than the
MUonE results for $0.02$ GeV $\leq M_{Z^{\prime}}\leq 0.1$ GeV. Following the
analysis given in [10] we find that prospective sensitivity from the beam-dump
experiments at FASER(2) can reach up to $5\times 10^{-3}$ at
$M_{Z^{\prime}}=10^{-3}$ GeV whereas ILC-BD reaches up to $2.5\times 10^{-3}$
at that $M_{Z^{\prime}}$, however, these bounds are ruled out by the results
from different beam-dump searches from NA64, E774 and E141 whereas prospective
reach from FASER(2) and ILC-BD can be improved for $M_{Z^{\prime}}\simeq 0.3$
GeV. Following the analysis in [10] we find that Supernova bounds can be
slightly stronger than the $\nu-$cal bounds around $M_{Z^{\prime}}\simeq 0.1$
GeV. Within $10^{-3}$ GeV $\leq M_{Z^{\prime}}\leq 0.18$ GeV the prospective
bounds from DUNE (prospective DUNE sensitivity plots can be found in [10] for
$0.1$ GeV $\leq M_{Z^{\prime}}\leq 3$ GeV) is weaker than $\nu-$cal. Recasting
BaBaR(vis) data for dark photon searches we obtain a stringent bound for
$M_{Z^{\prime}}\leq 0.175$ GeV. Recasting the dark photon searches at the LHCb
and CMS experiments, we find that stringent bounds can be calculated for
$0.175$ GeV $\leq M_{Z^{\prime}}\leq 70$ GeV and $1$ GeV $\leq
M_{Z^{\prime}}\leq 8$ GeV respectively except for small windows in these
ranges where $M_{Z^{\prime}}\simeq 10$ GeV, stringent bounds can be obtained
from the visible final state in BaBaR experiment. For $M_{Z^{\prime}}\leq
0.19$ GeV we compare our results recasting the limits from different beam-dump
experiments including Nomad, CHARM, KEK, E774, E141, E137 and $\nu-$cal
experiments with the prospective limits obtained from FASER(2), DUNE and ILC-
BD. Furthermore dilepton and dijet searches also rules out the bounds for
$M_{Z^{\prime}}\geq 70$ GeV. We also show that dilepton and dijet limits when
$M_{Z^{\prime}}$ is at the $Z-$pole giving a bound about $3\times 10^{-3}$
when $x_{H}=-2$. Due to the structure of the $U(1)_{R}$ scenario where left
handed fermions do not interact with $Z^{\prime}$, we find no other constrains
in this scenario from experiments like $\nu-$electron, $\nu-$nucleon, etc
where we can not explore the coupling between $\nu_{L}$ and $Z^{\prime}$.

Limits for $x_{H}=-1$ are shown in the lower panel of Fig. 6. For this charge
there is no coupling between $e_{R}$ and $Z^{\prime}$ resulting no direct
bound from $\mu^{+}-e^{-}$ scattering at the MUonE experiment. However, for
this value of $x_{H}$, we obtain prospective limits from FASER$\nu(2)$,
SND$@$LHC, $\nu-$nucleus scattering at NA64 (NA64$(eN)$) which we compare
recasting the bounds from the existing scattering experiments like GEMMA,
BOREXINO, CHARM-II, TEXONO and COHERENT. Where bounds obtained from the
TEXNONO and COHERENT experiments are stronger than the other bounds obtained
from the existing scattering experiments for the $Z^{\prime}$ roughly within
$0.03$ GeV $\leq M_{Z^{\prime}}\leq 0.1$ GeV. The prospective bounds on the
$U(1)_{X}$ gauge coupling varies between $7\times 10^{-6}\leq g_{X}\leq 1$
from the electron-nucleon scattering process in the NA64$(eN)$ experiment for
$10^{-3}$ GeV $\leq M_{Z^{\prime}}\leq 10$ GeV. However, the bound is
comparable with the bounds obtained from the TEXONO and COHERENT experiments
for the $Z^{\prime}$ mass within $0.03$ GeV $\leq M_{Z^{\prime}}\leq 0.1$ GeV.
Whereas bounds obtained from NA64$(eN)$ experiment is weak compared to the
beam-dump experiments for $M_{Z^{\prime}}\leq 0.028$ GeV. Here we recast
existing results from different beam-dump experiments like Orsay, KEK, E137,
CHARM, Nomad, $\nu-$cal, E141, E774, NA64, KEK, respectively for $x_{H}=-1$ to
show complementarity with the other experiments. Prospective bounds on the
$U(1)_{X}$ gauge coupling obtained from the beam-dump experiments like
FASER(2) and ILC-BD could be stringent compared to the estimated bounds from
the existing beam-dump experiments for $M_{Z^{\prime}}\geq 0.2$ GeV.
Interestingly the NA64$(eN)$ line crosses ILC-BD and FASER2 lines at this mass
point. Prospective limits obtained performing the beam-dump study at DUNE we
obtain that those limits are weaker than the limits obtained recasting the
existing results from the $\nu-$cal experiment. Estimated prospective bounds
obtained from $\nu-$nucleon scattering at FASER$\nu$, SND$@$LHC vary between
$6.69\times 10^{-4}\leq g_{X}\leq 0.29$ and those in case of FASER$\nu 2$ vary
between $2.8\times 10^{-4}\leq g_{X}\leq 0.14$ for $0.01$ GeV $\leq
M_{Z^{\prime}}\leq 150$ GeV, respectively showing stronger prospective bounds
for $91.2$ GeV $\leq M_{Z^{\prime}}\leq 150$ GeV. We recast dilepton and dijet
results from LEP which provide most stringent limit as $g_{X}\simeq 5\times
10^{-3}$ at the $Z-$pole. We recast recent results of dark photon searches at
the LHCb and CMS providing constrains for $0.2$ GeV $\leq M_{Z^{\prime}}\leq
70$ GeV and $1$ GeV $\leq M_{Z^{\prime}}\leq 8$ GeV respectively. Constrains
on $g_{X}$ from LHCb vary between $3.12\times 10^{-5}\leq g_{X}\leq 1.09\times
10^{-3}$ and those from CMS Dark vary between $4.0\times 10^{-4}\leq g_{X}\leq
1.1\times 10^{-3}$ leaving some narrow windows where BaBaR(vis) provides
strongest bound around $M_{Z^{\prime}}\simeq 10$ GeV as $g_{X}\simeq 7\times
10^{-4}$. Studying $\nu-$electron scattering at JSNS2 we find that a
prospective sensitivity for $g_{X}$ could reach as low as $2.78\times 10^{-5}$
and as high as $3\times 10^{-3}$ for $M_{Z^{\prime}}=0.022$ GeV and $0.22$ GeV
respectively which could be probed in future. JSNS2 provides stronger bound
for $M_{Z^{\prime}}$ above the $Z-$pole. Recasting the data from the different
beam-dump experiments we find that prospective bounds obtained from JSNS2
could be weaker than some of these bounds for $M_{Z^{\prime}}\leq 0.025$
GeV.In addition to that we estimate bounds on the $U(1)_{X}$ coupling
analyzing the electron $g-2$ data which belongs to the shaded region being
ruled out by the scattering, dark photon search and beam-dump experiments,
respectively.

Figure 7: Limits on $g_{X}-M_{Z^{\prime}}$ plane for $x_{H}=-0.5$ (upper
panel) and $x_{H}=0$ (lower panel, B-L case) taking $x_{\Phi}=1$ considering
$10^{-3}$ GeV $\leq M_{Z^{\prime}}\leq 150$ GeV showing the regions sensitive
to FASER$\nu$, FASER$\nu$2, SND$@$LHC, NA64$(eN)$ and JSNS2 experiments.
Recasting the existing results in our case, we compare parameter regions
obtained from LEP, dark photon searches at BaBaR(vis and invis), LHCb and
CMS(CMS Dark), scattering experiments CHARM-II, GEMMA, BOREXINO, COHERENT,
TEXONO and different beam-dump experiments from Orsay, KEK, E137, CHARM,
Nomad, $\nu-$cal, E141, E774, NA64, KEK involving prospective bounds from
FASER(2), DUNE and ILC(ILC-BD), respectively.

Limits for $x_{H}=-0.5$ are shown in the upper panel of Fig. 7. In this case
$u_{R}$ does not interact with $Z^{\prime}$. We obtain that the prospective
sensitivity on $g_{X}$ estimating $\mu^{-}-e^{+}$ scattering process involving
$Z^{\prime}$ contribution at MUonE experiment. Estimated limits are weaker
than the projected sensitivities obtained from the $\nu-$nucleon scattering at
the experiments like FASER$\nu$ and SND$@$LHC except for $M_{Z^{\prime}}\leq
0.73$ GeV. Studying $\nu-$nucleon scattering for the FASER$\nu 2$ experiment
we find that the prospective sensitivities could be stronger than the
projections of FASER$\nu$, SND$@$LHC and MUonE experiments respectively. The
prospective limit obtained from the FASER$\nu 2$ experiment could reach at
$g_{X}\simeq 0.15$ for $M_{Z^{\prime}}$ going beyond the $Z-$pole. At the
$Z-$pole we recast dilepton and dijet bounds from LEP experiment and find that
the exclusion limit on the coupling could reach at $g_{X}\simeq 3\times
10^{-3}$. Limits calculated recasting the COHERENT $\nu-$nucleus scattering
are obtained to be strong for $0.066$ GeV $\leq M_{Z^{\prime}}\leq 0.213$ GeV
where coupling varies between $9.0\times 10^{-5}\leq g_{X}\leq 2.36\times
10^{-4}$. We find that COHERENT limits are weaker than the limits obtained by
the $e-$nucleus scattering at the NA64 experiment (NA64$(eN)$) for
$M_{Z^{\prime}}\leq 0.07$ GeV while NA64$(eN)$ weaker than the limits from
COHERENT, TEXONO and BOREXINO experiments beyond this $M_{Z^{\prime}}$.
Recasting the Dark photon searches at the experiments like BaBRaR, LHCb and
CMS we find that stringent limits come from LHCb within $0.21$ GeV $\leq
M_{Z^{\prime}}\leq 70$ GeV whereas those from BaBaR(vis) could have severe
bounds around $M_{Z^{\prime}}\simeq 10$ GeV and CMS Dark could have severe
bound around $M_{Z^{\prime}}\simeq 1$ GeV and $3$ GeV, respectively. We obtain
prospective sensitivity from the JSNS2 experiment studying $\nu-$electron
scattering where the most stringent bound comes within the range $0.024$ GeV
$\leq M_{Z^{\prime}}\leq 0.215$ GeV where the $U(1)_{X}$ gauge coupling varies
between $2.46\times 10^{-5}\leq g_{X}\leq 1.97\times 10^{-4}$. JSNS2 could
also produces a stringent prospective bound on $g_{X}$ for $M_{Z^{\prime}}$
beyond the $Z-$pole which is close to the prospective limit from FASER$\nu 2$
for $M_{Z^{\prime}}\simeq 150$ GeV. The prospective sensitivity line from
JSNS2 crosses over the bounds obtained from the prospective limits obtained
from the beam-dump scenarios like FASER at $M_{Z^{\prime}}=0.0287$ GeV whereas
FASER2 and ILC-BD lines at $M_{Z^{\prime}}=0.0326$ GeV and the corresponding
couplings are $2.8\times 10^{-5}$ and $3.21\times 10^{-5}$, respectively.
These values could be verified in the near future. Recasting the data obtained
from the existing results of E774, E137, E141, NA64, Orsay, KEK, Nomad and
$\nu-$cal we find that prospective bounds obtained from DUNE for
$M_{Z^{\prime}}\leq 0.15$ GeV are weaker than the bounds obtained from
recasting the data of the $\nu-$cal experiment. We estimated bounds on $g_{X}$
from the electron and muon $g-2$ data. However, the limits are weaker compared
to different existing and prospective scattering and dark photon experiments
mentioned for this charge. Similar behavior is seen for the sensitivity line
obtained after recasting the data from GEMMA experiment which is weak compared
to existing beam-dump, scattering and dark photon searches, respectively.

We estimate limits on $g_{X}$ for $x_{H}=0$ depending on $M_{Z^{\prime}}$ in
the lower panel of Fig. 7. We mention that $x_{H}=0$ is the well known B$-$L
scenario. We estimate prospective limits at experiments like MUonE,
FASER$\nu(2)$ and SND$@$LHC and find that these sensitivities belong to the
shaded region below the $Z-$pole. However, above $Z-$pole the prospective
limits from FASER$\nu 2$ becomes stringent for $M_{Z^{\prime}}\leq 150$ GeV,
the range of $Z^{\prime}$ mass under consideration. Recasting the dilepton and
dijet search results from LEP we find the strongest bound at $Z-pole$ to be
$0.0018$. Recasting available data, we find that limits obtained from GEMMA,
CHARM-II, COHERENT also belong to the shaded region being weak compared to the
bounds from different scattering experiments involving TEXONO and BOREXINO,
beam-dump experiments involving E774, E131, E141, NA64, Orsay, KEK, Nomad and
$\nu-$cal, CHARM and dark photon search experiments involving BaBaR(visible
and invisible modes), LHCb and CMS after we recast the available data from
these experiments. We calculate prospective limits from $e-$nucleon scattering
at the NA64 experiment which are slightly stronger than existing bounds
obtained from the TEXONO experiment for $0.02$ GeV $\leq M_{Z^{\prime}}\leq
0.164$ GeV where limits on the $U(1)_{X}$ coupling can reach up to $2.0\times
10^{-5}\leq g_{X}\leq 7.2\times 10^{-4}$. We show bounds obtained from the
dark photon search experiments at BaBaR(visible and invisible), LHCb and CMS
and found that LHCb provides a strong bound on the $U(1)_{X}$ gauge coupling
for $0.21$ GeV $\leq M_{Z^{\prime}}\leq 70$ GeV whereas for narrow windows
around $M_{Z^{\prime}}\simeq 10$ GeV where limits obtained from BaBaR are
stringent and limits on $g_{X}$ are stringent around $M_{Z^{\prime}}\simeq 1$
GeV and 3 GeV from the CMS Dark, respectively. Prospective bounds from the
JSNS2 experiment could be stronger than the other scattering experiments for
$0.028$ GeV $\leq M_{Z^{\prime}}\leq 0.21$ GeV where constrains on the
$U(1)_{X}$ coupling constant could be as strong as $1.7\times 10^{-5}\leq
g_{X}\leq 1.3\times 10^{-4}$. Starting from beyond the $Z-$pole, prospective
search reach from JSNS2 could be around $g_{X}=0.02$ for $M_{Z^{\prime}}\leq
150$ GeV. We compare our results with different beam-dump scenarios involving
electron and proton in the context of E774, E131, E141, NA64, Orsay, KEK,
Nomad, $\nu-$cal, CHARM experiments from which we could estimate stronger
bounds on the $g_{X}-M_{Z^{\prime}}$ plane along with the prospective search
reaches for the beam-dump sceanrios at DUNE, FASER(2) and ILC-BD. Doing that
we find that prospective reach from JSNS2 could intersect the prospective
beam-dump lines from FASER(2) and ILC-BD at
$\\{M_{Z^{\prime}},g_{X}\\}=\\{0.025~{}\rm GeV,1.55\times
10^{-5}\\},\\{0.032~{}\rm GeV,1.92\times 10^{-5}\\},\\{0.035~{}\rm
GeV,2.13\times 10^{-5}\\}$, respectively which could be probed in the near
future. Prospective limits obtained from the DUNE are weaker than the
$\nu-$cal bounds for $M_{Z^{\prime}}\leq 0.08$ GeV.

Figure 8: Limits on $g_{X}-M_{Z^{\prime}}$ plane for $x_{H}=0.5$ (upper panel)
and $x_{H}=1$(lower panel) taking $x_{\Phi}=1$ considering $10^{-3}$ GeV $\leq
M_{Z^{\prime}}\leq 150$ GeV showing the regions sensitive to FASER$\nu$,
FASER$\nu$2, SND$@$LHC, NA64$(eN)$ and JSNS2 experiments. Recasting the
existing results in our case, we compare parameter regions obtained from
scattering experiments at LEP, CHARM-II, GEMMA, BOREXINO, COHERENT, TEXONO,
Dark photon searches at BaBaR(vis and invis), LHCb and CMS(CMS Dark)
experiments and different beam-dump experiments at Orsay, KEK, E137, CHARM,
Nomad, $\nu-$cal, E141, E774, NA64, KEK involving prospective bounds from
FASER(2), DUNE and ILC(ILC-BD), respectively.

The limits on $g_{X}-M_{Z^{\prime}}$ plane for $x_{H}=0.5$ has been shown in
the upper panel of Fig. 8 where all the fermions interact with $Z^{\prime}$,
however, left and right handed interactions with $Z^{\prime}$ are different.
We estimated prospective limits from the $\mu^{+}-e^{-}$ scattering at the
MUonE experiment and compare it with the bounds recasting the data from other
scattering experiment involving GEMMA, TEXONO, BOREXONO, CHARM-II and
COHERENT. We find that bounds from TEXONO rules out the other results. We also
estimate the bounds from the prospective search reaches at the experiments
involving SND$@$LHC, FASER$\nu(2)$, however, FASER$\nu 2$ could provide a
stringent limit beyond the $Z-$pole up to $M_{Z^{\prime}}\leq 150$ GeV where
constrains on $g_{X}$ could reach around 0.1. Using the dilepton and dijet
searches from LEP we recast those data in our scenario to obtain a strong
limit on $M_{Z^{\prime}}$ at the $Z-$pole as $g_{X}=0.0012$. We find that
electron-nucleon scattering in NA64$(eN)$ gives a strong bound on the
$U(1)_{X}$ coupling as $2.22\times 10^{-5}\leq g_{X}\leq 1.17\times 10^{-4}$
for the $Z^{\prime}$ within $0.028$ GeV $\leq M_{Z^{\prime}}\leq 0.14$ GeV
which is slightly stronger than the bounds obtained from TEXONO and COHERENT
within that range of $M_{Z^{\prime}}$. We calculate bounds on $g_{X}$
recasting the data obtained from the dark photon searches at BaBaR, LHCb and
CMS, respectively. Hence we find that LHCb provides a stringent bound for the
$Z^{\prime}$ mass within $0.21$ GeV $\leq M_{Z^{\prime}}\leq 70$ GeV where
limits on $g_{X}$ vary between $10^{-5}\leq g_{X}\leq 2\times 10^{-4}$,
however, around $M_{Z^{\prime}}\simeq 10$ GeV BaBaR provides stringent
constrain whereas around $M_{Z^{\prime}}\simeq 1$ GeV and $3$ GeV CMS Dark
provides stringent constrains on the $U(1)_{X}$ gauge coupling. We study
$\nu-e$ scattering for JSNS2 experiment. JSNS2 shows prospective strongest
bounds compared to other scattering experiments and dark photon searches
within $0.031$ GeV $\leq M_{Z^{\prime}}\leq 0.21$ GeV where limits on the
coupling vary between $1.32\times 10^{-5}\leq g_{X}\leq 9.1\times 10^{-5}$.
Following the JSNS2 line we find that it may provide a strong bound on $g_{X}$
beyond the $Z-$pole which might reach up to $g_{X}\simeq 0.06$ for
$M_{Z^{\prime}}\leq 150$ GeV. Limits on $g_{X}$ calculating electron and muon
$g-2$ scenarios are found to be weak compared to the existing scattering,
beam-dump and dark photon search experiments. Recasting the data from the
beam-dump experiments involving NA64, E141, E137, CHARM, Nomad, $\nu-$cal,
KEK, we find the limits on $g_{X}-M_{Z^{\prime}}$ plane shown by the shaded
areas. We also show the prospective sensitivities from FASER(2), DUNE and ILC-
BD where sensitivity from DUNE is weaker compared to $\nu-$cal for
$M_{Z^{\prime}}\leq 0.08$ GeV. The prospective sensitivity line from JSNS2
crosses the prospective FASER, FASER2 and ILC-BD lines at
$\\{M_{Z^{\prime}},g_{X}\\}=\\{0.0305\rm{GeV},1.29\times
10^{-5}\\},~{}\\{0.035\rm{GeV},1.54\times
10^{-5}\\},~{}\\{0.0374\rm{GeV},1.65\times 10^{-5}\\}$, respectively. These
limits could be probed by the scattering and beam-dump experiments in future.

In the lower panel of Fig. 8 we show the constrains on the general $U(1)_{X}$
coupling for different $M_{Z^{\prime}}$ using $x_{H}=1$ where $d_{R}$ does not
interact with $Z^{\prime}$. We estimate prospective limits from
$\mu^{+}-e^{-}$ scattering at the MUonE experimet and compare it with the
bounds recasting the results from CHARM-II, TEXONO, BOREXINO, GEMMA, and
COHERENT experiments, respectively. We find that comparing with all these
limits MUonE is weak staying in the shaded region for $10^{-3}$ GeV $\leq
M_{Z^{\prime}}\leq 150$ GeV. Prospective sensitivity obtained from the
electron-nucleon scattering at the NA64 experiment are denoted by NA64$(eN)$
line. This provides a strongest prospective limit within $0.0257$ GeV $\leq
M_{Z^{\prime}}\leq 0.0085$ GeV where $U(1)_{X}$ coupling could reach down to
$1.423\times 10^{-5}\leq g_{X}\leq 10^{-5}$. Recasting the TEXONO data we find
that in this context strong constrains come for $0.0254$ GeV $\leq
M_{Z^{\prime}}\leq 0.072$ GeV where limits on the gauge coupling vary within
$1.72\times 10^{-5}\leq g_{X}\leq 4.64\times 10^{-5}$. We find that bounds
obtained from TEXONO can be stronger than the limits obtained from recasting
the dilepton and dijet searches at the LEP experiment beyond the $Z-$ pole but
$M_{Z^{\prime}}\leq 150$ GeV where limits on the gauge coupling could vary
between $0.031\leq g_{X}\leq 0.051$. In addition, we obtain that LEP bounds at
$Z-$pole could reach at $g_{X}\simeq 0.0019$ form the dilepton and dijet
searches. FASER$\nu 2$ provides a prospective sensitivity beyond the $Z-$pole
and below 150 GeV where limits could vary between $0.03\leq g_{X}\leq 0.05$.
The fermions in the $U(1)_{X}$ scenario under consideration interact equally
with $Z^{\prime}$ making a generation independent nature of the model
irrespective of $x_{H}$ which affects bounds from $g-2$ analysis. This is true
for any value of $x_{H}$. We find that limits obtained from the dark photon
searches at LHCb shows strongest bounds on $U(1)_{X}$ gauge coupling within
$8.41\times 10^{-6}\leq g_{X}\leq 1.637\times 10^{-4}$ for $0.21$ GeV $\leq
M_{Z^{\prime}}\leq 70$ GeV. We find that dark photon search at CMS provides
strongest bounds on the gauge coupling for $M_{Z^{\prime}}$ around 1 GeVa nd 3
GeV respectively whereas same scenario appears from the BaBaR experiment for
$M_{Z^{\prime}}$ around 10 GeV. Prospective limits estimated in the context of
JSNS2 experiment are found to reach at $1.85\times 10^{-4}\leq g_{X}\leq
1.95\times 10^{-4}$ for $0.517$ GeV $\leq M_{Z^{\prime}}\leq 0.56$ GeV and
beyond $Z-$pole, the bounds are comparable with the prospective ones from
FASER$\nu 2$. Prospective limits from JSNS2 crosses the prospective
sensitivity lines at the beam-dump experiments involving FASER, FASER2 and
ILC-BD at $\\{M_{Z^{\prime}},g_{X}\\}=\\{0.033\rm{GeV},1.05\times
10^{-5}\\},~{}\\{0.036\rm{GeV},1.21\times
10^{-5}\\},~{}\\{0.0375\rm{GeV},1.30\times 10^{-5}\\}$, respectively which
could be probed in future. We compare these bounds recasting the data from
different beam-dump experiments like E141, NA64, KEK, Orsay, CHARM, E137 and
$\nu-$cal which are shown by different shaded regions. We find that
prospective bounds obtained from DUNE are weaker than $\nu-$cal for
$M_{Z^{\prime}}\leq 0.06$ GeV.

Figure 9: Limits on $g_{X}-M_{Z^{\prime}}$ plane for $x_{H}=2$ taking
$x_{\Phi}=1$ considering $10^{-3}$ GeV $\leq M_{Z^{\prime}}\leq 150$ GeV
showing the regions sensitive to FASER$\nu$, FASER$\nu$2, SND$@$LHC,
NA64$(eN)$ and JSNS2 experiments. Recasting the existing results in our case,
we compare parameter regions obtained from the scattering experiments at LEP,
CHARM-II, GEMMA, BOREXINO, COHERENT, TEXONO, dark photon searches at BaBaR(vis
and invis), LHCb and CMS(CMS Dark) and different beam-dump experiments at
Orsay, KEK, E137, CHARM, Nomad, $\nu-$cal, E141, E774, NA64, KEK and involving
prospective bounds from FASER(2), DUNE and ILC(ILC-BD) respectively.

We show the bounds from different experiments for $x_{H}=2$ in Fig. 9. This
charge is another example where all fermions interact with $Z^{\prime}$,
however, their left and right handed counterparts interact differently with
$Z^{\prime}$. Being influenced by the charge assignment we find that
prospective bounds on $g_{X}$ for different $M_{Z^{\prime}}$ obtained form the
MUonE experiment are weaker while compared to the bounds after recasting the
data from TEXONO, BOREXINO, CHARM-II and COHERENT experiments. We estimate
prospective bounds on $g_{X}$ for different prospective experiments like
SND$@$LHC and FASER$\nu(2)$, however, prospective searches from FASER$\nu 2$
are strong only beyond the $Z-$pole and up to $M_{Z^{\prime}}\leq 150$GeV. In
this mass range the limits on $g_{X}$ vary between $0.014\leq g_{X}\leq
0.025$. Recasting the dilepton and dijet searches from the LEP experiment we
find that limit on $g_{X}$ at the $Z-$pole could be as stringent as $6.8\times
10^{-4}$. We find that electron-nucleon scattering in NA64 experiment shown by
the line NA64$(eN)$ provides a strong bound on the gauge coupling around
$1.1\times 10^{-5}\leq g_{X}\leq 1.1\times 10^{-4}$ for $0.0261$ GeV $\leq
M_{Z^{\prime}}\leq 0.21$ GeV. Studying $\nu-$electron scattering in context of
JSNS2 experiment we estimate the prospective bounds on $g_{X}$ with respect to
$M_{Z^{\prime}}$. The strongest future limits on $g_{X}$ can be estimated for
$Z^{\prime}$ within $0.032$ GeV $\leq M_{Z^{\prime}}\leq 0.212$ GeV as
$7.735\times 10^{-6}\leq g_{X}\leq 5.13\times 10^{-5}$ which crosses
respective future sensitivity lines obtained from the beam-dump experiments
like FASER, FASER2 and ILC-BD at
$\\{M_{Z^{\prime}},g_{X}\\}=\\{0.033~{}\rm{GeV},7.855\times
10^{-6}\\},~{}\\{0.0371~{}\rm{GeV},8.57\times
10^{-6}\\},~{}\\{0.0386~{}\rm{GeV},9.07\times 10^{-6}\\}$ which could be
probed in future. We compare our results for $x_{H}=2$ recasting the bounds
obtained from the existing results from the beam-dump experiments like E141,
NA64, KEK, Orsay, CHARM, E137 and $\nu-$cal respectively. The excluded regions
are shown by different shades. We find that prospective bounds obtained from
the beam-dump scenario at DUNE for $M_{Z^{\prime}}\leq 0.04$ GeV are weaker
than those obtained from $\nu-$cal experiment recasting the existing data.
Dark photon searches from LHCb provides stronger limit for $0.21$ GeV $\leq
M_{Z^{\prime}}\leq 70$ GeV and beyond $Z-$pole up to $M_{Z^{\prime}}=150$ GeV.
We find that LHCb limits below $Z-$pole vary within $3.5\times 10^{-6}\leq
g_{X}\leq 10^{-4}$. Stringent limits can be obtained from the dark photon
search experiments at BaBaR and CMS Dark around $M_{Z^{\prime}}\simeq 10$ GeV
and $M_{Z^{\prime}}\simeq 1$ GeV and $3$ GeV within narrow windows from LHCb
experiment. Finally we comment that limits obtained from muon and electron
$g-2$ studies are weaker than the scattering and beam-dump experiments due to
the generation independent nature of the fermionic couplings with
$Z^{\prime}$.

##  V Conclusions

In this paper we consider chiral scenarios where $Z^{\prime}$ interacts with
the left and right handed fermions differently. We obtain that depending on
$U(1)_{X}$ charges the interactions of the femions with $Z^{\prime}$ get
modified by manifesting chiral nature of the scenarios under consideration.
Such interactions affect $Z^{\prime}$ mediated neutrino-electron, electron-
nucleon, electron-muon scattering processes which could be probed at the
experiments like FASER$\nu(2)$, SND$@$LHC, NA64$(eN)$, MUonE, JSNS2, dark
photon searches at BaBaR, LHCb and CMS experiments respectively. Further we
compare our results with dilepton, dijet searches from LEP, neutrino-nucleus
coherent scattering at COHERENT experiment, electron-neutrino scattering
experiments like BOREXINO, TEXONO, GEMMA and CHARM-II respectively. We compare
our results studying visible and invisible final states at the BaBaR
experiment. Finally we show complementarity with different beam-dump
experiments like $\nu-$cal, E137, E141, NA64, E774, Orsay, CHARM, KEK, Nomad
and future experiments like FASER, FASER2 and ILC-BD. Analyzing different
interactions we find that some prospective bounds at NA64$(eN)$, FASER$\nu 2$,
JSNS2 could be probed in future. Some of the existing experimental limits from
LEP, TEXONO, BaBaR(visible), dark photon searches at LHCb and CMS show
stringent upper limits on $g_{X}$ for the respective $Z^{\prime}$ mass. JSNS2
bounds crosses the future sensitivities estimated from FASER, FASER2 and ILC-
BD which could also be verified in future, however, their cross-overs depend
on $x_{H}$ which could be checked after the realistic experimental results
will be available. Depending on $U(1)_{X}$ charge, we find that beam-dump
experiments like $\nu-$cal, E137, E141, NA64, E774, Orsay, CHARM, KEK, Nomad
rules out the values of $U(1)_{X}$ coupling between $10^{-6}\leq g_{X}\leq
0.01$ depending on the $Z^{\prime}$ mass for $M_{Z^{\prime}}\leq 0.08$ GeV.
Within the mass range of $Z^{\prime}$, we find that DUNE will provide a weaker
bound from the beam-dump scenario. We find that weaker limits obtained
analyzing the $g-2$ data because in our model three generations of the
fermions are equally coupled with $Z^{\prime}$. We point out that limits from
SND$@$LHC, FASER$\nu$ are weak compared to the other scattering experiments
below the $Z-$pole. Finally, from our analysis it has been found that
scattering experiment could probe lighter $Z^{\prime}$ between $0.02$ GeV
$\leq M_{Z^{\prime}}\leq 0.2$ GeV which could be simultaneously probed by
proposed beam-dump experiments involving FASER, FASER2 and ILC-BD. On the
other hand heavier $Z^{\prime}$ above the $Z-$pole but $M_{Z^{\prime}}\leq
150$ GeV which could be probed by high energy colliders experiments in future.
Hence we conclude that studying $Z^{\prime}$ mediated interactions in addition
to the SM processes, limits on general $U(1)_{X}$ couplings could be
interesting to probe $\mathcal{G}_{\rm SM}\otimes U(1)_{X}$ scenario in
future.

## Acknowledgments

We thank Takashi Shimomura and Yuichi Uesaka for useful advice. This work is
supported by JSPS KAKENHI Grant Numbers JP21K20365 and JP23K13097 [KA],
JP19K03860, JP19K03865 and JP23K03402 [OS], by the Fundamental Research Funds
for the Central Universities [TN, JL], by the Natural Science Foundation of
Sichuan Province under grant No. 2023NSFSC1329 and the National Natural
Science Foundation of China under grant No. 11905149 [JL].

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