# Probing general $U(1)^{\prime}$ models with non-universal lepton charges at
FASER/FASER2, COHERENT and long-baseline oscillation experiments

Tobias Felkl<EMAIL_ADDRESS>Sydney Consortium for Particle Physics and
Cosmology,
School of Physics, The University of New South Wales, Sydney, New South Wales
2052, Australia  Tong Li<EMAIL_ADDRESS>School of Physics, Nankai
University, Tianjin 300071, China  Jiajun Liao<EMAIL_ADDRESS>School of Physics, Sun Yat-Sen University, Guangzhou 510275, China  Michael A.
Schmidt<EMAIL_ADDRESS>Sydney Consortium for Particle Physics and
Cosmology,
School of Physics, The University of New South Wales, Sydney, New South Wales
2052, Australia

###### Abstract

The general anomaly-free $U(1)^{\prime}$ models allow non-universal lepton
charges. We explore the sensitivities of FASER/FASER2, COHERENT and DUNE/T2HK
precision experiments to the new gauge boson $Z^{\prime}$ and the new CP-even
scalar $\phi$. With non-universal lepton charges, distinctive reaches at
FASER/FASER2 emerge in the regime of low $m_{Z^{\prime}}$ and small gauge
coupling $g_{BL}$ for different $U(1)^{\prime}$ charge setups. The COHERENT
experiment and the future long-baseline experiments DUNE/T2HK also provide
complementary probes to the available parameter space. For
$m_{\phi}<2m_{Z^{\prime}}$, the search for the scalar $\phi$ at FASER/FASER2
is sensitive to the mixing angle between the scalar singlet and the SM Higgs.
In the case of $m_{\phi}>2m_{Z^{\prime}}$, the kinematically allowed decay
$\phi\to Z^{\prime}Z^{\prime}$ changes the lifetime and decay rates of the
scalar $\phi$. The sensitivity reach highly depends on the $Z^{\prime}$ mass
and the gauge coupling $g_{BL}$.

††preprint: CPPC-2023-03

###### Contents

  1. I Introduction
  2. II The general $U(1)^{\prime}$ models with non-universal leptonic charges
  3. III Sensitivity of FASER and neutrino experiments to $Z^{\prime}$ in general $U(1)^{\prime}$ models
    1. III.1 The sensitivity of FASER/FASER2 to the gauge boson $Z^{\prime}$
    2. III.2 The search reach of COHERENT and neutrino oscillation for $Z^{\prime}$
      1. III.2.1 COHERENT
      2. III.2.2 Neutrino oscillation experiments
  4. IV Sensitivity of FASER to scalar $\phi$ and increased $Z^{\prime}$ production
    1. IV.1 Sensitivity of FASER to scalar $\phi$
    2. IV.2 Additional $Z^{\prime}$ production from $\phi\to Z^{\prime}Z^{\prime}$
  5. V Conclusions
  6. A Texture zeroes and lepton mixing

##  I Introduction

Neutrino oscillations have been observed by various neutrino experiments in
the last two decades. Their explanation requires non-vanishing neutrino masses
which cannot be accounted for by the Standard Model (SM). It thus provides a
strong motivation for new physics beyond the SM associated with neutrinos. The
most economical mechanism for generating neutrino masses is the seesaw
mechanism Minkowski (1977); Yanagida (1979); Gell-Mann  _et al._ (1979);
Glashow (1980); Mohapatra and Senjanovic (1980); Shrock (1981); Schechter and
Valle (1980), in which one introduces at least two right-handed neutrinos.
Right-handed neutrinos also allow the introduction of an extra Abelian gauge
symmetry $U(1)^{\prime}$ by choosing their charges to cancel gauge anomalies
Carlson (1987). The simplest and best-studied anomaly-free formulation for a
gauged $U(1)^{\prime}$ would be the $U(1)_{B-L}$ extension Davidson (1979);
Mohapatra and Marshak (1980); Marshak and Mohapatra (1980); Wetterich (1981);
Masiero  _et al._ (1982); Mohapatra and Senjanovic (1983); Buchmuller  _et
al._ (1991), where $B$ and $L$ stand for baryon number and lepton number,
respectively. In this model, the $U(1)^{\prime}$ has no $U(1)_{Y}$ component
and the charges are the same for three generations of SM fermions, i.e.,
$B=1/3$ for SM quarks and $L=-1$ for charged leptons and neutrinos.

This is however only one special case. The anomaly conditions allow non-
universal $U(1)^{\prime}$ charges for different generations of the SM leptons.
The non-universal gauge couplings lead to a non-trivial flavor structure for
the fermion masses and may in particular lead to texture zeros in the neutrino
Yukawa couplings and the Majorana mass matrix Araki  _et al._ (2012); Liao
_et al._ (2013); Bhatia  _et al._ (2017); Bonilla  _et al._ (2018); Babu  _et
al._ (2017); Ellis  _et al._ (2018); Alonso  _et al._ (2017); Bhatia  _et al._
(2021). Phenomenologically, the general $U(1)^{\prime}$ models with non-
universal lepton charges can result in distinguishable searches in high-
intensity experiments and may evade the severe constraints by electron beam-
dump experiments Abdallah  _et al._ (2020); Bauer  _et al._ (2021); Kling
(2020); de la Vega  _et al._ (2021); Coy and Xu (2021). Moreover, the new
scalar field spontaneously breaking the $U(1)^{\prime}$ symmetry is not
necessarily very heavy and can be naturally below 1 TeV or even much lighter
Iso  _et al._ (2009); Okada and Orikasa (2012); Dev  _et al._ (2021). The new
light scalar in the same model can also be probed through the mixing with the
SM Higgs boson.

The newly predicted particles in $U(1)^{\prime}$ models with very weak
couplings would become long-lived and can be searched for at the Forward
Physics Facility (FPF) such as the far-forward experiments at the Large Hadron
Collider (LHC). FASER (ForwArd Search ExpeRiment) is an on-going experiment in
the far-forward region of ATLAS and located about 480 m from the ATLAS
interaction point Feng  _et al._ (2018a); Ariga  _et al._ (2019). The detector
will be upgraded to FASER2 in the high luminosity LHC (HL-LHC) era. The long-
lived particles (LLPs) can be produced by proton-beam bremsstrahlung and the
rare decays of mesons, and then decay in the detector. The searches for LLPs
such as a dark photon, a dark Higgs as well as the vector boson in
$U(1)_{B-L}$ model have been simulated at FASER/FASER2 Feng  _et al._ (2018b);
Ariga  _et al._ (2019); Boiarska  _et al._ (2020); Araki  _et al._ (2021); Li
_et al._ (2022a); Asai  _et al._ (2022). We expect that the new gauge boson in
$U(1)^{\prime}$ models with non-universal gauge couplings would lead to
distinctive search reaches.

The $U(1)^{\prime}$ models also introduce non-standard interactions (NSIs)
between neutrinos and other SM fermions via neutral current interactions after
integrating out the new gauge boson. Coherent elastic neutrino-nucleus
scattering (CE$\nu$NS) Freedman (1974) and precision neutrino oscillation
experiments can probe the new physics in such NSIs. The CE$\nu$NS process was
first observed by the COHERENT experiment in the Spallation Neutron Source
(SNS) at the Oak Ridge National Laboratory Akimov  _et al._ (2017). The
neutrinos measured at COHERENT are produced by the decays of stopped pions and
muons through $\pi^{+}\to\mu^{+}\nu_{\mu}$ and $\mu^{+}\to
e^{+}\nu_{e}\bar{\nu}_{\mu}$. Thus, the COHERENT results would be sensitive to
the $U(1)^{\prime}$ models with gauge couplings to electron or muon neutrinos
Han  _et al._ (2019); Flores  _et al._ (2020); Cadeddu  _et al._ (2021); de la
Vega  _et al._ (2021); Cheung  _et al._ (2021); Atzori Corona  _et al._
(2022). Moreover, due to the change of neutrino propagation in matter in terms
of neutral current NSIs, the long-baseline neutrino oscillation experiments
such as DUNE are also sensitive to probe the NSIs in the $U(1)^{\prime}$
models Han  _et al._ (2019); Flores  _et al._ (2020); Chatterjee  _et al._
(2021); Asai  _et al._ (2022).

In this work, we study general $U(1)^{\prime}$ models with non-universal
lepton charges. A new scalar singlet is introduced with non-zero
$U(1)^{\prime}$ charge. After the scalar acquires a vacuum expectation value,
the $U(1)^{\prime}$ symmetry is broken and the masses of the neutral gauge
boson $Z^{\prime}$ and of Majorana neutrinos are generated. The scalar is
mixed with the SM Higgs doublet and there are two CP-even physical Higgses,
i.e., the SM-like $h$ and $\phi$. We then consider the sensitivities of above
precision experiments to the new particles in three scenarios Dev  _et al._
(2021):

  * •

$Z^{\prime}$ production and decay with non-universal lepton charges ;

  * •

$\phi$ production and decay for $m_{\phi}<2m_{Z^{\prime}}$ or
$m_{\phi}>2m_{Z^{\prime}}$ with $\phi\to Z^{\prime}Z^{\prime}$ open ;

  * •

$Z^{\prime}$ production when $m_{\phi}>2m_{Z^{\prime}}$ with $\phi\to
Z^{\prime}Z^{\prime}$ open .

In the first scenario, the non-universal lepton charges affect the lifetime
and decay rates of the gauge boson $Z^{\prime}$. We expect distinctive reaches
in the regime of low $m_{Z^{\prime}}$ and small gauge couplings for different
$U(1)^{\prime}$ charge setups. The dimensionless coupling parameters in the
NSIs are scaled as $\epsilon\sim g^{2}_{BL}v^{2}/m_{Z^{\prime}}^{2}$ where
$v\approx 246$ GeV is the electroweak vacuum expectation value and $g_{BL}$ is
the $U(1)^{\prime}$ gauge coupling. The COHERENT experiment and the neutrino
oscillation experiments also provide complementary reach with a linear
dependence on $g_{BL}^{2}/m_{Z^{\prime}}^{2}$. In the second scenario with
$m_{\phi}<2m_{Z^{\prime}}$, analogous to the dark Higgs case Feng  _et al._
(2018b), the search reach of $\phi$ is sensitive to the mixing angle with the
SM Higgs boson. However, if $m_{\phi}>2m_{Z^{\prime}}$, the kinematically
allowed decay $\phi\to Z^{\prime}Z^{\prime}$ changes the lifetime and decay
rates of the scalar $\phi$. The probe of $\phi$ depends on additional
parameters which are the $Z^{\prime}$ mass and the gauge coupling $g_{BL}$.
Moreover, with $\phi\to Z^{\prime}Z^{\prime}$ open in the third scenario,
additional production modes of the $Z^{\prime}$ appear.

This paper is organized as follows. In Sec. II we introduce the general
$U(1)^{\prime}$ models and discuss their details which are essential for the
phenomenological studies below. In Sec. III we study the sensitivities of
FASER/FASER2 and neutrino experiments to a light $Z^{\prime}$ in general
$U(1)^{\prime}$ models with non-universal lepton charges. Then, we investigate
the search for $\phi$ and the improved $Z^{\prime}$ search at FASER/FASER2 in
Sec. IV. Finally, in Sec. V we draw our main conclusions.

##  II The general $U(1)^{\prime}$ models with non-universal leptonic charges

Particles | $SU(3)_{C}$ | $SU(2)_{L}$ | $U(1)_{Y}$ | $U(1)^{\prime}$
---|---|---|---|---
$L_{iL}$ | 1 | 2 | $-1/2$ | $Q_{i}^{\prime}$
$e_{iR}$ | 1 | 1 | $-1$ | $Q_{i}^{\prime}$
$\nu_{iR}$ | 1 | 1 | $-1$ | $Q_{\nu i}^{\prime}$
$Q_{iL}$ | 3 | 2 | $1/6$ | $1/3$
$u_{iR}$ | 3 | 1 | $2/3$ | $1/3$
$d_{iR}$ | 3 | 1 | $-1/3$ | $1/3$
$H$ | 1 | 2 | $1/2$ | $0$
$\Phi_{m}$ | 1 | 1 | 0 | $Q_{\Phi,m}^{\prime}$

Table 1: Particles content and the charges of the model, where $i$ is the
fermion generation index, $H$ is the SM Higgs doublet and $\Phi_{m}$ denotes
the additional scalar singlets.

We consider general $U(1)^{\prime}$ models based on non-universal leptonic
charges as given in Table 1. We allow the charges of the right-handed (RH)
neutrinos to be different from that of the SM leptons in contrast to Ref.
Kownacki  _et al._ (2017) in which they were assumed to be equal. The
universal quark charges of $1/3$ are taken to avoid the constraints from quark
flavor-changing neutral current processes. As also shown in Ref. Kownacki  _et
al._ (2017), the $[SU(3)_{c}]^{2}U(1)^{\prime}$ and
$U(1)_{Y}[U(1)^{\prime}]^{2}$ anomaly conditions and the usual SM anomaly
conditions are automatically satisfied. The other anomaly conditions amount to

$\displaystyle[U(1)^{\prime}]^{3}$
$\displaystyle:\sum_{i}\left(2[Q_{i}^{\prime}]^{3}-[Q_{i}^{\prime}]^{3}\right)-\sum_{i}[Q_{\nu
i}^{\prime}]^{3}\;,$ (1) $\displaystyle[SU(2)_{L}]^{2}U(1)^{\prime}$
$\displaystyle:\frac{1}{2}\left(3+\sum_{i}Q_{i}^{\prime}\right)\;,$ (2)
$\displaystyle[U(1)_{Y}]^{2}U(1)^{\prime}$
$\displaystyle:3\left[6\left(\frac{1}{6}\right)^{2}-3\left(\frac{2}{3}\right)^{2}-3\left(-\frac{1}{3}\right)^{2}\right]\frac{1}{3}+\left[2\left(-\frac{1}{2}\right)^{2}-(-1)^{2}\right]\sum_{i}Q_{i}^{\prime}\;,$
(3) $\displaystyle[\mathrm{gravity}]^{2}U(1)^{\prime}$
$\displaystyle:\sum_{i}(2Q_{i}^{\prime}-Q_{i}^{\prime})-\sum_{i}Q_{\nu i}\;,$
(4)

where we have taken into account the fact that the SM quarks do not contribute
to the $[U(1)^{\prime}]^{3}$ and the $[\mathrm{gravity}]^{2}U(1)^{\prime}$
anomalies due to their vectorial couplings. The above anomaly conditions can
be resolved with

$\displaystyle\sum_{i}Q_{i}^{\prime}$ $\displaystyle=-3\;,$
$\displaystyle\sum_{i}Q^{\prime}_{\nu i}$
$\displaystyle=\sum_{i}Q_{i}^{\prime}\;,$ $\displaystyle\sum_{i}Q_{\nu
i}^{\prime 3}$ $\displaystyle=\sum_{i}Q_{i}^{\prime 3}\;.$ (5)

As a result, there are only three independent leptonic charges and at least
one of them is from the RH neutrinos. Without loss of generality, we utilize
the first two equalities to determine the charges of $Q_{\tau}^{\prime}$ and
$Q_{\nu 3}^{\prime}$ which are
$Q_{\tau}^{\prime}=-3-Q_{e}^{\prime}-Q_{\mu}^{\prime}$ and $Q_{\nu
3}^{\prime}=-3-Q_{\nu 1}^{\prime}-Q_{\nu 2}^{\prime}$. We further use the last
equality to eliminate one of the RH neutrino charges $Q_{\nu 1}^{\prime}$ and
$Q_{\nu 2}^{\prime}$. Thus, the free leptonic charges include
$Q_{e}^{\prime}$, $Q_{\mu}^{\prime}$ as well as one independent charge for the
RH neutrino. Note that the charges do not have to be quantized and can take
arbitrary real values.

The kinetic terms for the Abelian sector are given by

$\displaystyle\mathcal{L}_{gauge}$
$\displaystyle=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{4}F^{\prime\mu\nu}F^{\prime}_{\mu\nu}-\frac{\epsilon}{2}F^{\mu\nu}F^{\prime}_{\mu\nu}\;,$
(6)

with $F^{\mu\nu}=\partial^{\mu}B^{\nu}-\partial^{\nu}B^{\mu}$ and
$F^{\prime\mu\nu}=\partial^{\mu}B^{\prime\nu}-\partial^{\nu}B^{\prime\mu}$
being the field strength tensors. Here $B^{\mu}$ and $B^{\prime\mu}$ are the
gauge fields for $U(1)_{Y}$ and $U(1)^{\prime}$, respectively. Although the
kinetic mixing term can be generated via charged fermion loops, it is
generally negligible compared to direct couplings. Since the mixing between
the Abelian gauge bosons has to be very small, we work in the case with
$\epsilon=0$. The kinetic terms for the matter fields are

$\displaystyle\mathcal{L}_{kinetic}$ $\displaystyle=$ $\displaystyle
i\overline{Q}_{L}\gamma^{\mu}D_{\mu}Q_{L}+i\overline{u}_{R}\gamma^{\mu}D_{\mu}u_{R}+i\overline{d}_{R}\gamma^{\mu}D_{\mu}d_{R}$
(7)
$\displaystyle{}+i\overline{L}_{L}\gamma^{\mu}D_{\mu}L_{L}+i\overline{e}_{R}\gamma^{\mu}D_{\mu}e_{R}+i\overline{\nu}_{R}\gamma^{\mu}D_{\mu}\nu_{R}\;,$

where the covariant derivative for the $U(1)^{\prime}$ is

$\displaystyle
D_{\mu}f_{L,R}=\partial_{\mu}f_{L,R}+ig_{BL}Q^{\prime}_{f}B^{\prime}_{\mu}f_{L,R}\;,$
(8)

with $f$ being the SM fermions. The $Z^{\prime}$ gauge boson directly couples
to quarks and leptons proportional to their charges $Q^{\prime}_{f}$ which
enter in the covariant derivative
$D_{\mu}=\partial_{\mu}+ig_{BL}Q^{\prime}_{f}B^{\prime}_{\mu}$. In the
following analysis we choose the basis where the charged lepton and up-type
quark mass matrices are diagonal.

The most general renormalizable Lagrangian for the neutrino sector reads

$\displaystyle-\mathcal{L}_{Y}^{\nu}$ $\displaystyle=$ $\displaystyle
y_{ij}^{D}\overline{l_{i}}_{L}\tilde{H}\nu_{Rj}+\frac{1}{2}M_{ij}\overline{\nu_{Ri}^{C}}\nu_{Rj}+\frac{1}{2}y_{ij,m}^{M}\overline{\nu_{Ri}^{C}}\nu_{Rj}\Phi_{m}+h.c.\;,$
(9)

where $\tilde{H}=i\sigma_{2}H^{\ast}$ and the scalar singlet fields $\Phi_{m}$
carry $U(1)^{\prime}$ charges $Q_{\Phi,m}^{\prime}$. The Dirac and RH Majorana
mass matrices are

$\displaystyle M_{D}$ $\displaystyle=\frac{y^{D}v_{0}}{\sqrt{2}}\;,$
$\displaystyle M_{R}$ $\displaystyle=M+\frac{y^{M}_{m}v_{m}}{\sqrt{2}}\;,$
(10)

where the scalar fields get their vacuum expectation values as $\left\langle
H\right\rangle^{T}=(0,v_{0}/\sqrt{2})$ and
$\left\langle\Phi_{m}\right\rangle=v_{m}/\sqrt{2}$. The light neutrino mass
matrix is then given by the usual seesaw expression

$\displaystyle m_{\nu}=-M_{D}M_{R}^{-1}M_{D}^{T}\;.$ (11)

The observation of two non-zero neutrino masses further restricts the leptonic
charges. The Dirac neutrino mass matrix $M_{D}$ and the RH neutrino Majorana
mass matrix $M_{R}$ are forced to have at least rank $2$. The condition on the
rank of the Dirac neutrino mass matrix enforces $Q_{\nu
i}^{\prime}=Q_{i}^{\prime}$ without loss of generality, taking into account
the fact that the RH neutrinos can always be reordered. Thus, this condition
further removes the degree of freedom of the RH neutrino charge and there are
at most two independent charges $Q_{e,\mu}^{\prime}$. For the RH neutrino mass
matrix, a rank 2 matrix requires at least two independent non-zero entries. As
the charges of the scalars $Q_{\Phi,m}^{\prime}$ are not constrained, it is
always possible to generate a contribution from the Yukawa interaction by
choosing $Q_{\Phi,m}^{\prime}$ appropriately without imposing additional
constraints on the charge assignment. For the direct mass term, the condition
$Q_{e}^{\prime}=0$ ($Q_{\mu}^{\prime}=0$)
[$Q_{e}^{\prime}+Q_{\mu}^{\prime}=-3$] leads to a non-zero $(1,1)$ ($(2,2)$)
[$(3,3)$] entry. Similarly, the conditions $Q_{e}^{\prime}+Q_{\mu}^{\prime}=0$
($Q_{e}^{\prime}=-3$) [$Q_{\mu}^{\prime}=-3$] individually induce non-zero
off-diagonal terms in the $(1,2)$ ($(2,3)$) [$(1,3)$] entry.

For illustration, we take a model based on
$U(1)_{B-3L_{e}+Q^{\prime}L_{\mu}-Q^{\prime}L_{\tau}}$ with $Q^{\prime}\neq 3$
as well as one scalar singlet $\Phi$ with charge $Q^{\prime}_{\Phi}$ and
$\left\langle\Phi\right\rangle=v_{1}/\sqrt{2}$. This charge assignment leads
to a diagonal neutrino Yukawa matrix $y^{D}$ with arbitrary entries. The
combination of the Majorana mass matrix and the Yukawa couplings matrix
$y^{M}$ for $Q_{\Phi}^{\prime}=-2Q^{\prime}_{e}$ results in an effective
Majorana mass matrix 111Although two non-zero Majorana mass entries with a
diagonal Dirac neutrino mass matrix can explain neutrino masses, the observed
leptonic mixing requires more non-zero entries. It can be straightforwardly
generated by introducing multiple scalars $\Phi_{m}$. See Appendix A for a
detailed discussion of neutrino Yukawa textures.

$\displaystyle M+\frac{v_{1}}{\sqrt{2}}y^{M}$
$\displaystyle=\begin{pmatrix}\frac{v_{1}}{\sqrt{2}}y^{M}_{11}&0&0\\\
.&0&M_{23}\\\ .&.&0\\\ \end{pmatrix}\qquad\Rightarrow\qquad
m_{\nu}=-\frac{v_{0}^{2}}{2}\begin{pmatrix}\frac{(y^{D}_{11})^{2}\sqrt{2}}{y^{M}_{11}v_{1}}&0&0\\\
.&0&\frac{y^{D}_{22}y^{D}_{33}}{M_{23}}\\\ .&.&0\\\ \end{pmatrix}\;.$ (12)

This texture agrees with $L_{\mu}-L_{\tau}$ symmetry. Note that the charge
$Q^{\prime}$ is not quantized and can be arbitrarily chosen. It is
straightforward to construct similar models with a free charge which is not
quantized. Motivated by the possibility to have free charges, in the next
section we will investigate the phenomenology of a gauged $U(1)^{\prime}$
symmetry with arbitrary charges $Q_{e,\mu}^{\prime}$ in a simple model with
one scalar $\Phi$. Even if there is no free charge, the analysis in the next
section can also be generally used for models with fixed charges. For
instance, the minimal $U(1)_{B-L}$ model Carlson (1987) with universal
$U(1)^{\prime}$ charges, the $U(1)_{B-3L_{e}-L_{\mu}+L_{\tau}}$,
$U(1)_{B-L_{e}-3L_{\mu}+L_{\tau}}$, or $U(1)_{B-3L_{\mu}}$ model are motivated
by the introduction of vanishing textures in the neutrino mass matrix Araki
_et al._ (2012); Liao  _et al._ (2013); Bhatia  _et al._ (2017, 2021) and can
have implications for non-standard neutrino interactions Han  _et al._ (2019);
Bauer  _et al._ (2021).

The general Lagrangian for the electroweak Higgs doublet $H$ and the scalar
field $\Phi$ with $U(1)^{\prime}$ charge $Q_{\Phi}^{\prime}$ is given by

$\displaystyle\mathcal{L}_{scalar}$ $\displaystyle=$
$\displaystyle(D_{\mu}H)^{\dagger}(D^{\mu}H)+(D_{\mu}\Phi)^{\dagger}(D^{\mu}\Phi)-V(H,\Phi)\;.$
(13)

Once getting the vacuum expectation value $\langle\Phi\rangle\to
v_{1}/\sqrt{2}$, the extra scalar breaks the new $U(1)^{\prime}$ gauge
symmetry with
$D_{\mu}\Phi=\partial_{\mu}\Phi+ig_{BL}Q^{\prime}_{\Phi}B^{\prime}_{\mu}\Phi$.
The phenomenologically relevant interactions of the scalar $\Phi$ are governed
by the scalar potential

$\displaystyle V(H,\Phi)$ $\displaystyle=$ $\displaystyle
m_{H}^{2}H^{\dagger}H+\lambda_{H}(H^{\dagger}H)^{2}+m_{\Phi}^{2}\Phi^{\dagger}\Phi+\lambda_{\Phi}(\Phi^{\dagger}\Phi)^{2}+\lambda_{1}(H^{\dagger}H)(\Phi^{\dagger}\Phi)$
(14)

which is parameterized by five real parameters,
$m_{H}^{2},\lambda_{H},m_{\Phi}^{2},\lambda_{\Phi}$ and the Higgs portal
coupling $\lambda_{1}$. The electroweak symmetry and the $U(1)^{\prime}$
symmetry are spontaneously broken. We parameterize the SM Higgs doublet $H$
and the singlet $\Phi$ as
$H^{T}=(H^{+},H^{0})=(H^{+},(v_{0}+h^{0}+iG^{0})/\sqrt{2})$ and
$\Phi=(v_{1}+\phi_{1}+i\chi_{1})/\sqrt{2}$ with the vacuum expectation values

$\displaystyle v_{0}^{2}$
$\displaystyle=\frac{4m_{H}^{2}\lambda_{\Phi}-2\lambda_{1}m_{\Phi}^{2}}{\lambda_{1}^{2}-4\lambda_{H}\lambda_{\Phi}}\;,$
$\displaystyle v_{1}^{2}$
$\displaystyle=\frac{4m_{\Phi}^{2}\lambda_{H}-2\lambda_{1}m_{H}^{2}}{\lambda_{1}^{2}-4\lambda_{H}\lambda_{\Phi}}\;.$
(15)

After the spontaneous symmetry breaking, the Goldstone boson $\chi_{1}$ is
eaten by the neutral gauge boson $Z^{\prime}$ and the $Z^{\prime}$ acquires a
mass $m_{Z^{\prime}}=g_{BL}Q_{\Phi}^{\prime}v_{1}$. The masses of the two CP-
even scalars are described by

$\displaystyle\mathcal{M}^{2}=\begin{pmatrix}2\lambda_{H}v_{0}^{2}&\lambda_{1}v_{0}v_{1}\\\
\lambda_{1}v_{0}v_{1}&2\lambda_{\Phi}v_{1}^{2}\\\ \end{pmatrix}\;.$ (16)

The physical Higgs mass eigenstates are defined as

$\displaystyle\begin{pmatrix}h\\\ \phi\\\
\end{pmatrix}=\begin{pmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\\\ \end{pmatrix}\begin{pmatrix}h^{0}\\\ \phi_{1}\\\
\end{pmatrix}\qquad\text{with}\qquad\tan
2\theta=\frac{\lambda_{1}v_{0}v_{1}}{\lambda_{H}v_{0}^{2}-\lambda_{\Phi}v_{1}^{2}}\;.$
(17)

We thus parameterize the new physics in terms of the $Z^{\prime}$ mass
$m_{Z^{\prime}}$, the gauge coupling $g_{BL}$, the two CP-even scalar masses
$m_{h}$ and $m_{\phi}$, and the mixing angle $\theta$. The three dimensionless
parameters are

$\displaystyle\lambda_{H}$ $\displaystyle=$
$\displaystyle(c_{\theta}^{2}m_{h}^{2}+s_{\theta}^{2}m_{\phi}^{2})/(2v_{0}^{2})\;,$
$\displaystyle\lambda_{\Phi}$ $\displaystyle=$
$\displaystyle(s_{\theta}^{2}m_{h}^{2}+c_{\theta}^{2}m_{\phi}^{2})/(2v_{1}^{2})\;,$
$\displaystyle\lambda_{1}$ $\displaystyle=$ $\displaystyle
c_{\theta}s_{\theta}(m_{h}^{2}-m_{\phi}^{2})/(v_{0}v_{1})\;,$ (18)

where $s_{\theta}=\sin\theta$ and $c_{\theta}=\cos\theta$. The two vacuum
expectation values $v_{0}$ and $v_{1}$ are given in terms of the Fermi
constant $v_{0}=(\sqrt{2}G_{F})^{-1/2}$ and the ratio
$v_{1}=m_{Z^{\prime}}/(g_{BL}Q_{\Phi}^{\prime})$. If $m_{\phi}<m_{h}/2$, the
SM-like Higgs boson can decay into a pair of light CP-even scalars $\phi$. The
scalar $\phi$ couples to the SM particle through the small mixing angle
$\theta$ and can be long-lived. The Higgs invisible width is given by

$\displaystyle\Gamma(h\to\phi\phi)=\frac{\lambda_{h\phi\phi}^{2}}{8\pi
m_{h}}\sqrt{1-\frac{4m_{\phi}^{2}}{m_{h}^{2}}}\;,\qquad\lambda_{h\phi\phi}=\frac{s_{2\theta}}{4v_{0}v_{1}}(m_{h}^{2}+2m_{\phi}^{2})(v_{0}c_{\theta}+v_{1}s_{\theta})\;.$
(19)

The constraint on $\lambda_{h\phi\phi}$ comes from the invisible decay of
Higgs boson Sirunyan  _et al._ (2019); Aaboud  _et al._ (2019); Aad  _et al._
(2021); Tumasyan  _et al._ (2022) and the latest limit on the corresponding
decay branching fraction is 0.18 at the 95% CL Tumasyan  _et al._ (2022).

As outlined above in the phenomenological analysis we focus on the
$U(1)^{\prime}$ models which consist of three right-handed neutrinos, the
$Z^{\prime}$ gauge boson and a (light) scalar field $\Phi$ which breaks the
$U(1)^{\prime}$ gauge symmetry spontaneously. While we assume universal
charges for the quarks $Q_{1,2,3}^{\prime}=1/3$, we employ non-universal
charges $Q_{i}^{\prime}$ for the lepton fields $L_{Li}$, $e_{Ri}$ and
$\nu_{Ri}$. The kinetic mixing is neglected in the analysis because its
contribution is negligible, and thus the couplings of the $Z^{\prime}$ gauge
boson are entirely fixed by the charges of the SM fermions. We also assume
decoupled right-handed neutrinos for simplicity, so the gauge boson
$Z^{\prime}$ does not decay into them. An additional partial decay width
$\Gamma(\phi\to\nu_{R}\nu_{R})$ decreases the lifetime $\tau_{\phi}$ and leads
to a suppression of the visible branching ratio by a factor $\Gamma_{\rm
visible}/(\Gamma(\phi\to\mathrm{SM})+\Gamma(\phi\to\nu_{R}\nu_{R}))$. See
Refs. Ariga  _et al._ (2019); Berryman  _et al._ (2020); Boiarska  _et al._
(2021) for recent works on the search for heavy neutral leptons at FASER, DUNE
or CHARM.

##  III Sensitivity of FASER and neutrino experiments to $Z^{\prime}$ in
general $U(1)^{\prime}$ models

###  III.1 The sensitivity of FASER/FASER2 to the gauge boson $Z^{\prime}$

For the gauge boson $Z^{\prime}$, as seen in the last section, both of its
mass and its couplings to the SM matter particles are determined by the gauge
coupling $g_{BL}$. A very weak coupling $g_{BL}$ can induce a light and long-
lived $Z^{\prime}$ which can be searched for in far-forward experiments such
as FASER/FASER2.

The major production mechanisms of $Z^{\prime}$ bosons at FASER include
proton-beam bremsstrahlung as well as the decays of light mesons Feng  _et
al._ (2018a). The $Z^{\prime}$ production cross sections and meson decay
widths can be easily obtained from those of the dark photon $A^{\prime}$ with
$\varepsilon$ being the $\gamma-A^{\prime}$ kinetic mixing, as presented in
the darkcast package Ilten  _et al._ (2018). The approximate ratio of the
bremsstrahlung cross sections is

$\displaystyle\frac{\sigma(pZ\to pZZ^{\prime})}{\sigma(pZ\to
pZA^{\prime})}\approx\frac{g_{BL}^{2}(2Q^{\prime}_{u}+Q^{\prime}_{d})^{2}}{(\varepsilon
e)^{2}}\;,$ (20)

where $Q^{\prime}_{u}=Q^{\prime}_{d}=1/3$ in our setup. The general decay
width ratios of $\pi^{0}$ and $\eta$ are then given by

$\displaystyle{\Gamma(\pi^{0}\to Z^{\prime}\gamma)\over\Gamma(\pi^{0}\to
A^{\prime}\gamma)}$ $\displaystyle=$
$\displaystyle\Big{(}{g_{BL}\over\varepsilon
e}\Big{)}^{2}{|(Q^{\prime}_{u}-Q^{\prime}_{d}){\rm
BW}_{\rho}+3(Q^{\prime}_{u}+Q^{\prime}_{d}){\rm BW}_{\omega}|^{2}\over|{\rm
BW}_{\rho}+{\rm BW}_{\omega}|^{2}}\;,$ (21) $\displaystyle{\Gamma(\eta\to
Z^{\prime}\gamma)\over\Gamma(\eta\to A^{\prime}\gamma)}$ $\displaystyle=$
$\displaystyle\Big{(}{g_{BL}\over\varepsilon
e}\Big{)}^{2}{|9(Q^{\prime}_{u}-Q^{\prime}_{d}){\rm
BW}_{\rho}+3(Q^{\prime}_{u}+Q^{\prime}_{d}){\rm
BW}_{\omega}+6Q^{\prime}_{s}{\rm BW}_{\phi}|^{2}\over|9{\rm BW}_{\rho}+{\rm
BW}_{\omega}-2{\rm BW}_{\phi}|^{2}}\;,$ (22)

where ${\rm BW}_{V}$ denotes the Breit-Wigner form factors Tulin (2014); Ilten
_et al._ (2018) for vector mesons $V=\rho,\omega,\phi$ in the framework of
vector meson dominance (VMD) Fujiwara  _et al._ (1985). In FASER detectors,
the gauge boson can decay into all kinematically accessible states. The
$Z^{\prime}$ with a mass smaller than $2m_{\tau}$ mainly decays into
neutrinos, $e^{+}e^{-}$, $\mu^{+}\mu^{-}$ and light hadrons. The partial width
of the $Z^{\prime}$ decay into SM final states is given by

$\displaystyle\Gamma(Z^{\prime}\to
f\bar{f})={C_{f}(g_{BL}Q^{\prime}_{f})^{2}\over
12\pi}m_{Z^{\prime}}\left(1+{2m_{f}^{2}\over
m_{Z^{\prime}}^{2}}\right)\sqrt{1-{4m_{f}^{2}\over m_{Z^{\prime}}^{2}}}\;,$
(23)

where the color factor is $C_{f}=1$ for charged leptons, 3 for quarks, and
$1/2$ for active neutrinos. Here we assume the right-handed neutrinos are
heavy enough to decouple. The VMD approach was also used to estimate the decay
width for specific hadronic final states as stated in Ref. Ilten  _et al._
(2018).

Figure 1: The decay branching ratios of $Z^{\prime}$ bosons for $U(1)_{B-L}$
(left) and $U(1)_{B-3L_{e}-L_{\mu}+L_{\tau}}$ (right) models.

We use the darkcast package Ilten  _et al._ (2018) to calculate the lifetime
and branching fractions of the $Z^{\prime}$ in general $U(1)^{\prime}$ models
as shown in Fig. 1 for illustration. Apart from the two peaks from vector
meson resonances, where hadronic final states dominate, the $Z^{\prime}$ boson
dominantly decays to leptons. One can see that the different charge
$Q^{\prime}_{e}$ would change the dominant decay channel and the branching
fractions of viable decay products significantly. These results are then
passed to the FORESEE package Kling and Trojanowski (2021). In the FORESEE
package, we modify the production rates of the dark photon to obtain the
forward $Z^{\prime}$ flux and specify the allowed channels of $Z^{\prime}$
decays to be those into $e^{+}e^{-}$, $\mu^{+}\mu^{-}$ and hadrons. FORESEE
then provides the number of signal events passing the selection criteria in
FASER detectors. The sensitivity is then estimated by requiring at least three
signal events and negligible SM backgrounds. The projected gauge boson
sensitivity reaches are derived using LHC Run 3 with 150 fb-1 for FASER, and
HL-LHC with 3 ab-1 for FASER2.

For illustration we first consider the four benchmark $U(1)_{B-L}$,
$U(1)_{B-3L_{\mu}}$, $U(1)_{B-3L_{e}-L_{\mu}+L_{\tau}}$ and
$U(1)_{B-L_{e}-3L_{\mu}+L_{\tau}}$ models Araki  _et al._ (2012); Liao  _et
al._ (2013); Bhatia  _et al._ (2017); Han  _et al._ (2019); Bauer  _et al._
(2021); Bhatia  _et al._ (2021). The first one is the minimal $U(1)_{B-L}$
model with universal $U(1)^{\prime}$ charge $1/3$ for SM quarks and $-1$ for
leptons Carlson (1987). The other three models with non-universal lepton
charges are motivated by the introduction of vanishing textures in the
neutrino mass matrix Araki  _et al._ (2012); Liao  _et al._ (2013); Bhatia
_et al._ (2017, 2021) and have implications for non-standard neutrino
interactions Han  _et al._ (2019); Bauer  _et al._ (2021). The gauge boson
$Z^{\prime}$ in the second model only couples to the muon with $-3$ charge,
and the third and fourth models have different electron and muon charges. In
Fig. 2, we show the sensitivity of FASER and FASER2 to the $Z^{\prime}$ in
these four models. We reproduce the projected future sensitivities of
FASER/FASER2 to the gauge boson $Z^{\prime}$ in the minimal $U(1)_{B-L}$ model
Ariga  _et al._ (2019). FASER can probe the gauge coupling $g_{BL}$ down to
$10^{-6}$ and $m_{Z^{\prime}}$ below 0.1 GeV. FASER2 extends the sensitivity
to $m_{Z^{\prime}}\sim 1$ GeV and $g_{BL}\sim 10^{-7}$ for the majority of the
$m_{Z^{\prime}}$ region. Since it is a detector at a high-energy collider, it
turns out that FASER/FASER2 is able to probe larger $g_{BL}$ than the DUNE
near detector Dev  _et al._ (2021) and heavier $m_{Z^{\prime}}$ compared to
the beam-dump experiments Bauer  _et al._ (2018). Other experiments, such as
NA64e Andreev  _et al._ (2022), CHARM-II Vilain  _et al._ (1993, 1994), BaBar
Lees  _et al._ (2014) and TEXONO Deniz  _et al._ (2010), can constrain the
region of larger $g_{BL}\gtrsim 10^{-5}$. The less stringent constraints can
be found in Refs. Bauer  _et al._ (2018); Dev  _et al._ (2021).

Figure 2: The sensitivity of FASER (solid red) and FASER2 (solid orange) to
the $Z^{\prime}$ boson in $U(1)_{B-L}$, $U(1)_{B-3L_{\mu}}$,
$U(1)_{B-3L_{e}-L_{\mu}+L_{\tau}}$ and $U(1)_{B-L_{e}-3L_{\mu}+L_{\tau}}$
models. The other existing or prospective bounds for the minimal $U(1)_{B-L}$
model and $U(1)_{B-3L_{\mu}}$ are mainly taken from Ref. Bauer  _et al._
(2018); Dev  _et al._ (2021) and Bauer  _et al._ (2021), respectively. For
$U(1)_{B-L}$, the constraints are from existing beam-dump searches Bauer  _et
al._ (2018) (purple), recent NA64e Andreev  _et al._ (2022) (silver), BaBar
Lees  _et al._ (2014) (blue), CHARM-II Vilain  _et al._ (1993, 1994) (green),
TEXONO Deniz  _et al._ (2010) (salmon), Supernova 1987A (SN1987A) Knapen  _et
al._ (2017) (cyan) and the DUNE Near Detector Dev  _et al._ (2021) (black).
For $U(1)_{B-3L_{\mu}}$, the constraints come from NA62 Cortina Gil  _et al._
(2019) (black), BaBar Lees  _et al._ (2016) (blue), LHCb Aaij  _et al._ (2020)
(purple), NA64$\mu$ Bauer  _et al._ (2021) (silver), NuCal Blümlein and
Brunner (2014) (cyan) and future SHiP Anelli  _et al._ (2015) (green). The
bounds from COHERENT (dashed brown) and from the simulation of future DUNE and
T2HK Han  _et al._ (2019) (dashed magenta) are also shown.

For the $U(1)_{B-3L_{\mu}}$ model without electron charge, the gauge boson
$Z^{\prime}$ can also couple to the electron through the vacuum polarization
diagram of gauge bosons Coy and Xu (2021). However, due to the loop
suppression, the coupling is proportional to $e^{2}/(4\pi)^{2}$ and has no
significant impact on the search for light $Z^{\prime}$ bosons. As a result,
FASER/FASER2 has no sensitivity for $m_{Z^{\prime}}<2m_{\mu}$ in this model.
Other beam-dump experiments and fixed target experiments are not sensitive to
this region either. Only the search for missing energy in pion decay at NA62
is able to constrain $30\lesssim m_{Z^{\prime}}\lesssim 130$ MeV. Above the
dimuon threshold, FASER2 can reach isolated regions for $10^{-8}\lesssim
g_{BL}\lesssim 10^{-6}$ in which there is overlap with the reach of NuCal
Merkel  _et al._ (2014) and the projection of SHiP Anelli  _et al._ (2015).
Collider experiments such as BaBar Lees  _et al._ (2016) and LHCb Aaij  _et
al._ (2020) exclude the region of $g_{BL}\gtrsim 10^{-4}$. The missing energy
search at future NA64$\mu$ Banerjee  _et al._ (2019) provides a stronger bound
for $g_{BL}\gtrsim 5\times 10^{-6}$. For other constraints that are less
stringent or beyond the masses of interest here we refer the reader to Ref.
Bauer  _et al._ (2021).

For the two non-universal models with electron $U(1)^{\prime}$ charge, there
is a blank region above $m_{Z^{\prime}}\simeq 0.3$ GeV. As seen from Fig. 1 in
Ref. Ilten  _et al._ (2018), for $m_{Z^{\prime}}$ above 0.3 GeV the
$\pi^{+}\pi^{-}$ channel is open and makes the $Z^{\prime}$ lifetime shorter.
Thus, fewer events are detected although the $\mu^{+}\mu^{-}$ decay is
present. The lower panels of Fig. 2 show that the smaller decay rate of
$\pi^{+}\pi^{-}$ also makes it difficult to detect the hadrons compared to the
$U(1)_{B-L}$ model. Moreover, for the $U(1)_{B-3L_{e}-L_{\mu}+L_{\tau}}$ model
with larger $U(1)^{\prime}$ charge for electron, FASER/FASER2 extends the
sensitivity to lower $g_{BL}$. The neutrino experiment TEXONO measured the
elastic $\bar{\nu}_{e}+e^{-}$ scattering and its bound applies to the cases
with charge $Q^{\prime}_{e}=-1$, such as the
$U(1)_{B-L_{e}-3L_{\mu}+L_{\tau}}$ model. As relying on interference effects
of the $Z^{\prime}$ with the SM, the bounds from other neutrino experiments
are not straightforward to be rescaled for other models and require further
work. Similarly, the beam-dump experiments measuring a very displaced vertex
are also difficult to apply for these models. We leave their study for future
work.

Figure 3: FASER/FASER2’s sensitivity and existing constraints from COHERENT
(blue-shaded region) as function of $Q^{\prime}_{e}$ and $Q^{\prime}_{\mu}$
for different benchmark values of $m_{Z^{\prime}}$ and $g_{BL}$. The grey
regions with dashed (dotted) lines as boundaries show the parameter space
which will be probed by DUNE+T2HK via $\epsilon_{\tau\tau}-\epsilon_{\mu\mu}$
($\epsilon_{ee}-\epsilon_{\mu\mu}$), as quoted in Table 2.

Figure 4: FASER/FASER2’s sensitivity and constraints as function of $g_{BL}$
and $Q_{e}^{\prime}$ for different benchmark values of $m_{Z^{\prime}}$ and
$Q_{\mu}^{\prime}$. The grey regions with dashed lines as boundaries show the
future sensitivity of DUNE+T2HK to $g_{BL}$ as a function of $Q_{e}^{\prime}$
from the constraint on $\epsilon_{\tau\tau}-\epsilon_{\mu\mu}$. The constraint
from $\epsilon_{ee}-\epsilon_{\mu\mu}$ is weaker. The blue region shows the
current constraint from COHERENT.

Next we vary two free leptonic charges $Q^{\prime}_{e}$ and $Q^{\prime}_{\mu}$
and show the sensitivity in the plane of these two charges in Fig. 3. FASER
(FASER2) is sensitive to the regions inside the red (orange) contours. In the
upper two panels we fix $m_{Z^{\prime}}=0.01$ GeV and $g_{BL}=10^{-5}$ or
$10^{-6}$. One can see that, as expected, FASER is sensitive to a broader
region of $U(1)^{\prime}$ charge for larger $g_{BL}$.
$|Q^{\prime}_{e}|>0.1~{}(1.5)$ can be reached when
$g_{BL}=10^{-5}~{}(10^{-6})$. For positive $Q^{\prime}_{e}$, FASER can probe
the region of $Q^{\prime}_{\mu}<1.5$ and the detector is sensitive to larger
$Q^{\prime}_{\mu}$ for negative $Q^{\prime}_{e}$ because the tau lepton is
assigned smaller $|Q^{\prime}_{\tau}|$. FASER2 can reach the whole region of
$|Q^{\prime}_{e}|>0.1$ and has no dependence on $Q^{\prime}_{\mu}$. In the
lower two panels we fix $g_{BL}=10^{-6}$ and $m_{Z^{\prime}}=0.1$ GeV or 0.25
GeV. FASER has no sensitivity for these larger $m_{Z^{\prime}}$ values. FASER2
can cover a larger region for smaller $m_{Z^{\prime}}$ and only leaves an
island of sensitivity in the plane of these two charges for
$m_{Z^{\prime}}=0.25$ GeV.

Finally, in Fig. 4 we show the constraints as function of $g_{BL}$ and
$Q^{\prime}_{e}$ for different benchmark values of $m_{Z^{\prime}}$ and
$Q^{\prime}_{\mu}$. For small $Z^{\prime}$ mass of 0.01 GeV, FASER (FASER2)
can place a constraint on $g_{BL}$ down to $10^{-6}~{}(10^{-7})$ as long as
$|Q^{\prime}_{e}|>0.1$. For a large value of $m_{Z^{\prime}}=0.25$ GeV,
however, only FASER2 can cover a narrow region of $10^{-7}<g_{BL}<10^{-6}$.

###  III.2 The search reach of COHERENT and neutrino oscillation for
$Z^{\prime}$

While FASER and FASER2 are far-forward detectors which are sensitive to long-
lived particles and thus small gauge couplings $g_{BL}$, COHERENT and neutrino
oscillation experiments probe larger gauge couplings $g_{BL}$. Thus COHERENT
and neutrino oscillation experiments provide complementary sensitivity to
FASER and other far-forward detectors.

####  III.2.1 COHERENT

In order to account for the latest data release on the measurement of coherent
elastic neutrino-nucleus scattering (CE$\nu$NS) in CsI by COHERENT Akimov  _et
al._ (2022), we employ the Poissonian least-squares function

$\chi^{2}=2\sum_{i=2}^{9}\sum_{j=1}^{11}\left[N^{\text{th}}_{ij}-N^{\text{exp}}_{ij}+N^{\text{exp}}_{ij}\ln\frac{N^{\text{exp}}_{ij}}{N^{\text{th}}_{ij}}\right]+\left(\frac{\alpha_{0}}{\sigma_{0}}\right)^{2}+\left(\frac{\alpha_{3}}{\sigma_{3}}\right)^{2}$
(24)

with

$N^{\text{th}}_{ij}=(1+\alpha_{0})N^{\text{CE$\nu$NS}}_{ij}(\alpha_{4},\alpha_{6},\alpha_{7})+(1+\alpha_{3})N^{\text{SSB}}_{ij}$
(25)

as well as $\sigma_{0}=0.1145$ and $\sigma_{3}=0.021$. For more information on
how to obtain the bin-wise signal prediction $N^{\text{CE$\nu$NS}}_{ij}$ from
integrating over the relevant differential cross sections and flux
distributions, we refer the reader to Ref. De Romeri  _et al._ (2022),
including the references therein, that our analysis is closely aligned with.
Several systematic uncertainties are captured via the nuisance parameters
$\alpha_{i}$, the numerical values of which are unknown a priori and thus must
be fitted from the data. The above indicates that we only include the beam-
uncorrelated steady-state background (SSB) which constitutes over 98% of the
background as is reported in Akimov  _et al._ (2022). We also follow their
distribution into energy bins, expressed in photoelectrons, with the endpoints

$E\sim(8,12,16,20,24,32,40,48,60)~{}\text{PE}$ (26)

and recoil time bins with the endpoints

$t_{\text{rec}}\sim(0,0.125,0.25,0.375,0.5,0.625,0.75,0.875,1,2,4,6)~{}\mu\text{s}.$
(27)

We do not include the lowest-energy bin $E\sim(0,8)~{}\text{PE}$ in the
analysis, because the used detector efficiency function is negative in that
region. The contribution from the $Z^{\prime}$ boson to CE$\nu$NS is induced
via a modification of the effective weak charge of the nucleus (see for
instance Ref. Atzori Corona  _et al._ (2022)),

$\displaystyle Q_{\nu,k}$
$\displaystyle=Q^{\text{SM}}_{i,k}+Q^{Z^{\prime}}_{i,k}=g_{p,i}Z_{0,k}+g_{n,i}N_{0,k}+\frac{g_{BL}^{2}Q^{\prime}_{i}\left(Z_{0,k}+N_{0,k}\right)}{\sqrt{2}G_{F}\left(|\mathbf{q}|^{2}+m_{Z^{\prime}}^{2}\right)}$
(28)

with the three neutrino species
$\nu_{i}\in\nu_{e},\nu_{\mu},\overline{\nu}_{\mu}$ and the two nuclei
$k=\text{Cs},\text{I}$, and $\mathbf{q}$ denotes the 3-momentum of the virtual
$Z^{\prime}$ boson. We have explicitly checked that the contributions from
neutrino-electron scattering are entirely negligible in the considered mass
range $m_{Z^{\prime}}\gtrsim 0.01$ GeV. The contour lines visible in Figs. 2,
3 and 4 arise from bounding the profile
$\Delta\chi^{2}\equiv\chi^{2}-\chi^{2}_{\text{min}}$ from above, where for
each (fixed) set of model parameters
($Q^{\prime}_{e},Q^{\prime}_{\mu},g_{BL},m_{Z^{\prime}}$) we choose the
nuisance parameters $\alpha_{i}$ such that $\chi^{2}$ is minimised, and
$\chi^{2}_{\text{min}}$ is the (global) minimum value taken when both the
model parameters and the nuisance parameters are optimised simultaneously. The
confidence level is set to 90%. As can be seen from Eq. (28), there is a
degeneracy in the model parameters since we are not sensitive to the gauge
coupling and charges individually, but only to the product
$g_{BL}^{2}Q_{i}^{\prime}$.

For the $Z^{\prime}$ mass and gauge coupling fixed, we find that the following
two non-SM-like cases are generally in good agreement with the COHERENT data:
(i) $Q_{\mu}^{\prime}>0$ for $|Q_{\mu}^{\prime}|$ large, and (ii)
$Q_{e}^{\prime}<0$ for $|Q_{e}^{\prime}|$ large and, if $Q_{\mu}^{\prime}<0$,
$|Q_{\mu}^{\prime}|$ not too large, as can be seen in the bottom plot of Fig.
3. We interpret this as an indication that besides SM-like cases, there is
also a preference for negative corrections to $Q_{i,k}^{\text{SM}}$. In the
$U(1)_{B-L}$ model, for a range of $Z^{\prime}$ masses this can be realised if
the $g_{BL}$ gauge coupling becomes large enough. Indeed, the lower tip of a
non-excluded region for $g_{BL}\gtrsim 10^{-3}$ is visible the top-left plot
of Fig. 2. This is in agreement with Fig. 4 which demonstrates that for the
benchmark case $Q_{\mu}^{\prime}=-1$ and a moderately large
$|Q_{e}^{\prime}|$, a specific region of larger $g_{BL}$ can exist that is
less constrained than $g_{BL}$ smaller. The $U(1)_{B-3L_{e}-L_{\mu}+L_{\tau}}$
model represents a more extreme case to which COHERENT does not provide any
sensitivity and thus there is no constraint at all. Lastly, we comment that
the slope of the lower bound of the excluded region for the $U(1)_{B-L}$,
$U(1)_{B-3L_{\mu}}$ and $U(1)_{B-L_{e}-3L_{\mu}+L_{\tau}}$ models approaches a
constant value for larger $Z^{\prime}$ masses, consistent with the expectation
that the contribution to CE$\nu$NS will approximately scale like
$g_{BL}^{2}/m_{Z^{\prime}}^{2}$ in that regime.

####  III.2.2 Neutrino oscillation experiments

The Hamiltonian for neutrino propagation in matter in the presence of a
$U(1)^{\prime}$ model becomes

$\displaystyle H={1\over 2E}U\left(\begin{array}[]{ccc}0&0&0\\\ 0&\delta
m_{21}^{2}&0\\\ 0&0&\delta m_{31}^{2}\\\
\end{array}\right)U^{\dagger}+V_{CC}\left(\begin{array}[]{ccc}1+\epsilon_{ee}&\epsilon_{e\mu}&\epsilon_{e\tau}\\\
\epsilon_{e\mu}^{\ast}&\epsilon_{\mu\mu}&\epsilon_{\mu\tau}\\\
\epsilon_{e\tau}^{\ast}&\epsilon_{\mu\tau}^{\ast}&\epsilon_{\tau\tau}\\\
\end{array}\right)\;,$ (35)

where $U$ is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix,
$V_{CC}=\sqrt{2}G_{F}N_{e}$ is the standard matter potential, and
$\epsilon_{\alpha\beta}=\sum_{f}\epsilon^{f}_{\alpha\beta}N_{f}/N_{e}$ with
$N_{f}$ being the number density of fermions $f=u,d$ and $e$. The quark number
densities satisfy

$\displaystyle N_{u}=2N_{p}+N_{n}\;,~{}~{}N_{d}=N_{p}+2N_{n}\;,$ (36)

where $N_{p}$ and $N_{n}$ are proton and neutron number densities,
respectively, and the matter neutrality implies $N_{p}=N_{e}$. The diagonal
parameters of non-standard neutrino interactions (NSI) are given by Han  _et
al._ (2019)

$\displaystyle\epsilon_{\alpha\alpha}^{f}={g_{BL}^{2}Q^{\prime}_{\alpha}Q^{\prime}_{f}\over\sqrt{2}G_{F}m_{Z^{\prime}}^{2}}\;,$
(37)

where $\alpha$ denotes charged lepton flavor. Note that there are no off-
diagonal terms, since the theory is defined in the basis where the charged
lepton masses are diagonal. As a result, in our general $U(1)^{\prime}$
models, the diagonal components of the matter part in the Hamiltonian are

$\displaystyle\epsilon_{\alpha\alpha}={g_{BL}^{2}\over\sqrt{2}G_{F}m_{Z^{\prime}}^{2}}\Big{[}Q^{\prime}_{\alpha}Q^{\prime}_{e}+Q^{\prime}_{\alpha}Q^{\prime}_{q}\Big{(}3+3{N_{n}\over
N_{e}}\Big{)}\Big{]}\;,$ (38)

where $N_{n}/N_{e}\approx 1.051$ in Earth matter.

In Table 2, we show the $2\sigma$ allowed ranges for the diagonal NSI
parameters from the simulation of future long-baseline experiments DUNE and
T2HK Liao  _et al._ (2017); Han  _et al._ (2019). The neutrino oscillation
data constrain differences between two diagonal $\epsilon$ parameters. Thus,
there is no bound on the $U(1)_{B-L}$ model from neutrino oscillation, but
only on those with non-universal lepton charges as shown in Fig. 2. Note that
due to the presence of the generalized mass ordering degeneracy Coloma and
Schwetz (2016), the allowed region near $\epsilon_{ee}-\epsilon_{\mu\mu}=-2$
cannot be ruled out by neutrino oscillation experiments alone. In the
literature studying the current neutrino oscillation data Esteban  _et al._
(2018); Coloma  _et al._ (2023), there exist bounds on only electron NSI
couplings $\epsilon^{e}_{\alpha\alpha}$, only quark couplings
$\epsilon^{q}_{\alpha\alpha}$, or components $\epsilon_{\alpha\alpha}$ from a
global analysis including CE$\nu$NS data within an effective theory and thus
only applicable for $m_{Z^{\prime}}>50$ MeV. We thus do not consider the
constraints from the current neutrino oscillation data here.

As seen in Fig. 2, DUNE+T2HK will be able to probe lower $g_{BL}$ than
COHERENT for $m_{Z^{\prime}}<30$ MeV in $U(1)_{B-3L_{\mu}}$ and
$U(1)_{B-L_{e}-3L_{\mu}+L_{\tau}}$ models. For the
$U(1)_{B-3L_{e}-L_{\mu}+L_{\tau}}$ model in which COHERENT loses sensitivity,
they can also probe the regime of large $g_{BL}$ and small $m_{Z^{\prime}}$.
In Figs. 3 and 4, the grey regions show the future sensitivity of DUNE+T2HK.
They provide complementary projected bounds compared to FASER/FASER2 and
COHERENT. It turns out that the $\epsilon_{\tau\tau}-\epsilon_{\mu\mu}$
quantity can place stronger bounds. The two “peaks” losing sensitivity from
left to right in Fig. 4 correspond to $\epsilon_{\alpha\alpha}=0$ and
$\epsilon_{ee}-\epsilon_{\mu\mu}=\epsilon_{\tau\tau}-\epsilon_{\mu\mu}=0$,
respectively.

| DUNE+T2HK
---|---
$\epsilon_{ee}-\epsilon_{\mu\mu}$ | $[-2.50,-1.66]\oplus[-0.443,+0.394]$
$\epsilon_{\tau\tau}-\epsilon_{\mu\mu}$ | $[-0.105,+0.105]$
Table 2: The $2\sigma$ allowed ranges for the diagonal NSI parameters from the
simulation of future DUNE and T2HK Liao  _et al._ (2017); Han  _et al._
(2019).

##  IV Sensitivity of FASER to scalar $\phi$ and increased $Z^{\prime}$
production

###  IV.1 Sensitivity of FASER to scalar $\phi$

The interactions of the scalar $\phi$ can be obtained via the mixing with the
SM Higgs boson. It can also become a light and weakly interacting particle
governed by the small mixing angle $\theta$ Winkler (2019). Below we will
explore the sensitivity of FASER/FASER2 to light scalars $\phi$.

The scalar $\phi$ has flavour-changing interactions via its mixing with the SM
Higgs boson. For down-type quarks they are described by the effective
Lagrangian Willey and Yu (1982); Leutwyler and Shifman (1990)222See
Kachanovich  _et al._ (2020) for a calculation of the effective vertex of the
second Higgs in $R_{\xi}$ gauge.

$\mathcal{L}_{\rm
FCNC}=\frac{-\phi}{v_{0}}\sin\theta\sum_{\alpha,\beta}C_{\alpha\beta}\left(m_{d_{\beta}}\,\bar{d}_{\alpha}P_{R}d_{\beta}\,\right)+\mathrm{h.c.}\qquad\mathrm{with}\quad
C_{\alpha\beta}\approx\frac{3m_{t}^{2}V_{t\alpha}^{\ast}V_{t\beta}}{16\pi^{2}v_{0}^{2}}\;.$
(39)

Normally, the light scalar $\phi$ is mainly produced in the flavor changing
2-body rare decay of $B$ mesons $b\to X_{s}\phi$. This process includes all
b-flavored hadrons and all strange-flavored decay products. The corresponding
branching fraction is given by Feng  _et al._ (2018b)

$\displaystyle{\rm BR}(b\to X_{s}\phi)=5.7\times\Big{(}1-{m_{\phi}^{2}\over
m_{b}^{2}}\Big{)}^{2}\times\sin^{2}\theta\;.$ (40)

Moreover, the Kaons are expected to be produced more copiously than $B$ mesons
at the LHC. The light scalar $\phi$ can also be produced from Kaon decays
$K^{\pm}\to\pi^{\pm}\phi$ and $K_{L(S)}\to\pi^{0}\phi$ Leutwyler and Shifman
(1990); Feng  _et al._ (2018b); Dev  _et al._ (2020); Batell  _et al._ (2021);
Dev  _et al._ (2021); Li  _et al._ (2022b). The amplitude of the Kaon decay
into the light scalar receives contributions from the effective couplings of
the scalar to the gluon field strength tensor which are proportional to the
coupling constants $\gamma_{1},\gamma_{2}$ and the effective FCNC coupling of
the scalar in Eq. (39) Leutwyler and Shifman (1990). It takes the form

$\displaystyle\mathcal{M}={-\sin\theta\over v_{0}}\Big{[}\gamma_{1}{7\over
18}(m_{K}^{2}-m_{\phi}^{2}+m_{\pi}^{2})-\gamma_{2}{7\over
9}m_{K}^{2}+\frac{1}{2}C_{ds}m_{s}{m_{K}^{2}-m_{\pi}^{2}\over
m_{s}-m_{d}}f_{0}^{K\pi}\Big{]}\;,$ (41)

where $\gamma_{1}\approx 3.1\times 10^{-7}$, $\gamma_{2}\approx 0$ Leutwyler
and Shifman (1990) and the form factor $f_{0}^{K\pi}=0.96$ Boiarska  _et al._
(2019). After only taking the dominant third term in the amplitude, the
branching fractions of Kaon decays become

$\displaystyle{\rm BR}(K^{\pm}\to\pi^{\pm}\phi)$
$\displaystyle=\frac{\lambda^{1/2}(m_{K}^{2},m_{\pi}^{2},m_{\phi}^{2})}{16\pi
m_{K}^{3}\Gamma_{K^{\pm}}}|\mathcal{M}|^{2}\approx 1.7\times
10^{-3}\lambda^{1/2}\left(1,\frac{m_{\pi}^{2}}{m_{K}^{2}},\frac{m_{\phi}^{2}}{m_{K}^{2}}\right)\sin^{2}\theta\;,$
(42) $\displaystyle{\rm BR}(K_{L}\to\pi^{0}\phi)$
$\displaystyle=\frac{\lambda^{1/2}(m_{K}^{2},m_{\pi}^{2},m_{\phi}^{2})}{16\pi
m_{K}^{3}\Gamma_{K_{L}}}(\mathrm{Re}\mathcal{M})^{2}\approx 6.1\times
10^{-3}\lambda^{1/2}\left(1,\frac{m_{\pi}^{2}}{m_{K}^{2}},\frac{m_{\phi}^{2}}{m_{K}^{2}}\right)\sin^{2}\theta\;,$
(43) $\displaystyle{\rm BR}(K_{S}\to\pi^{0}\phi)$
$\displaystyle=\frac{\lambda^{1/2}(m_{K}^{2},m_{\pi}^{2},m_{\phi}^{2})}{16\pi
m_{K}^{3}\Gamma_{K_{S}}}(\mathrm{Im}\mathcal{M})^{2}\approx 2.0\times
10^{-6}\lambda^{1/2}\left(1,\frac{m_{\pi}^{2}}{m_{K}^{2}},\frac{m_{\phi}^{2}}{m_{K}^{2}}\right)\sin^{2}\theta$
(44)

with the Källén function $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+ac+bc)$. The
Kaon decays contribute to the production of lighter $\phi$ which is more
severely constrained from other experiments. We include this production mode
for completeness. The branching ratio for Higgs decay $h\to\phi\phi$ is
BR($h\to\phi\phi)\approx
0.19\,(500\,\mathrm{GeV}Q_{\Phi}^{\prime}g_{BL}/m_{Z^{\prime}})^{2}(\sin\theta/0.1)^{2}$
and thus strongly constrained by searches for the $Z^{\prime}$ gauge boson in
most of the available parameter space, see e.g. Kling (2020) for a compilation
of many existing constraints. Thus, for simplicity, we neglect the production
of the light scalar $\phi$ from SM Higgs decay.

The decays of light $\phi$ into SM matters actually include
$\phi\to\ell^{+}\ell^{-}$ ($\ell=e,\mu$), $\gamma\gamma$ and $\pi\pi$ for
$m_{\phi}<1$ GeV Dev  _et al._ (2021). The partial decay widths are then given
by

$\displaystyle\Gamma(\phi\to\ell^{+}\ell^{-})$ $\displaystyle=$
$\displaystyle{G_{F}m_{\phi}m_{\ell}^{2}\sin^{2}\theta\over
4\sqrt{2}\pi}\Big{(}1-{4m_{\ell}^{2}\over m_{\phi}^{2}}\Big{)}^{3/2}\;,$ (45)
$\displaystyle\Gamma(\phi\to\pi^{+}\pi^{-})$ $\displaystyle=$ $\displaystyle
2\Gamma(\phi\to\pi^{0}\pi^{0})={G_{F}m_{\phi}^{3}\sin^{2}\theta\over
8\sqrt{2}\pi}\Big{(}1-{4m_{\pi}^{2}\over
m_{\phi}^{2}}\Big{)}^{1/2}|G(m_{\phi}^{2})|^{2}\;,$ (46)
$\displaystyle\Gamma(\phi\to\gamma\gamma)$ $\displaystyle=$
$\displaystyle{G_{F}\alpha^{2}m_{\phi}^{3}\sin^{2}\theta\over
128\sqrt{2}\pi^{3}}\Big{|}\sum_{f}C_{f}Q_{f}^{2}A_{1/2}(\tau_{f})+A_{1}(\tau_{W})\Big{|}^{2}\;,$
(47)

where $G(m_{\phi}^{2})$ is a dimensionless transition amplitude Donoghue  _et
al._ (1990), $A_{1/2}(\tau_{f})$ and $A_{1}(\tau_{W})$ with
$\tau_{X}=m_{\phi}^{2}/4m_{X}^{2}$ are the standard fermion and $W$ loop
functions for the SM Higgs decay Djouadi (2008), as summarized in the Appendix
of Ref. Dev  _et al._ (2021). When $m_{\phi}\gtrsim 1$ GeV, the $KK$ decay
mode is allowed. Note that hadronic scalar decays have a large theory
uncertainty. See Winkler (2019); Gorbunov  _et al._ (2023) for two recent
independent calculations for the hadronic decays. As hadronic scalar decays
dominate the total decay width above the pion threshold, the scalar lifetime
has a similar uncertainty. In our numerical calculation below, we adopt the
default inputs of the dark Higgs model in FORESEE for the scalar decay into SM
matter fields. Moreover, when $m_{\phi}>2m_{Z^{\prime}}$, the scalar can also
decay into a pair of $Z^{\prime}$ boson from the covariant derivative term
$(D_{\mu}\Phi)^{\dagger}(D^{\mu}\Phi)$. The corresponding decay width is
determined by the gauge coupling $g_{BL}$ as follows 333We find that the decay
width is larger by a factor 4 compared to Eq. (2.8) in Ref. Dev  _et al._
(2021).

$\displaystyle\Gamma(\phi\to
Z^{\prime}Z^{\prime})={g_{BL}^{2}(Q^{\prime}_{\Phi})^{2}m_{\phi}^{3}\over
32\pi m_{Z^{\prime}}^{2}}\Big{(}1-{4m_{Z^{\prime}}^{2}\over
m_{\phi}^{2}}\Big{)}^{1/2}\Big{(}1-{4m_{Z^{\prime}}^{2}\over
m_{\phi}^{2}}+{12m_{Z^{\prime}}^{4}\over m_{\phi}^{4}}\Big{)}\;.$ (48)

Thus, there are two additional parameters $m_{Z^{\prime}}$ and $g_{BL}$ in
this case. In Fig. 5 we show the total decay length of $\phi$ (left) and the
branching ratios (right) of $\phi$ decay into $\ell^{+}\ell^{-}$ ($\ell=e$ and
$\mu$) with or without the decay $\phi\to Z^{\prime}Z^{\prime}$. Without
$\phi\to Z^{\prime}Z^{\prime}$, the $e^{+}e^{-}$ channel is dominant for
$m_{\phi}<0.2$ GeV before the $\mu^{+}\mu^{-}$ channel is kinematically
allowed. Given $\phi\to Z^{\prime}Z^{\prime}$ is allowed with
$m_{\phi}=3m_{Z^{\prime}}$ and $g_{BL}=10^{-6}$, in the region of
$m_{\phi}<2m_{\mu}$, the total decay length and BR$(\phi\to e^{+}e^{-})$ are
both suppressed by four orders of magnitude. When $g_{BL}$ becomes large
enough, $\phi\to Z^{\prime}Z^{\prime}$ dominates over all decay channels into
SM products and the decay length decreases linearly with $m_{\phi}$.

Figure 5: The decay length (left) and decay branching ratios of $\phi$ into
$e^{+}e^{-}$ and $\mu^{+}\mu^{-}$ (right), for $m_{\phi}<2m_{Z^{\prime}}$ with
$|\sin\theta|=10^{-2}$ (black) and $m_{\phi}=3m_{Z^{\prime}}$ with
$g_{BL}=10^{-6}$ (red) or $g_{BL}=10^{-4}$ (blue).

To evaluate the sensitivity of FASER/FASER2 to the scalar $\phi$, we consider
two cases: $m_{\phi}<2m_{Z^{\prime}}$ (case A) and $m_{\phi}>2m_{Z^{\prime}}$
(case B). The FORESEE package is then modified to take into account the
production of $\phi$ from both $B$ and $K$ meson decay and the two-dimensional
parameterization of $\sin\theta$ and $m_{\phi}$ for the
$m_{\phi}>2m_{Z^{\prime}}$ case. In Fig. 6 we show the sensitivity of
FASER/FASER2 to the mixing $\sin\theta$ for the scalar $\phi$ in the plane of
$\sin\theta$ vs. $m_{\phi}$. FASER2 yields the most sensitive constraint for
$m_{\phi}\gtrsim 0.4$ GeV as the viable decay products such as
$\mu^{+}\mu^{-}$ or $\pi\pi$ are dominant once kinematically allowed. For the
case of $m_{\phi}<2m_{Z^{\prime}}$, FASER and FASER2 can respectively reach
$\sin\theta\sim 6\times 10^{-4}$ and $5\times 10^{-6}$ at most. For lower
$m_{\phi}$, the constraint becomes weaker to the level of $\sin\theta\sim
10^{-2}$ and $10^{-3}$ as only the $\phi\to e^{+}e^{-}$ channel dominates. For
the case B with $m_{\phi}>2m_{Z^{\prime}}$, we fix $m_{\phi}=3m_{Z^{\prime}}$
and $g_{BL}=10^{-8}$ or $g_{BL}=10^{-6}$ for illustration. As seen from the
lower panels of Fig. 6, the projected constraints change dramatically as
$g_{BL}$ increases. For $m_{\phi}<2m_{\mu}$, as $g_{BL}$ increases, the decay
length decreases and a lower range of $\sin\theta$ can be constrained. Given
$m_{\phi}=0.1$ GeV and $g_{BL}=10^{-6}$, a mixing $\sin\theta$ as small as
$10^{-5}$ can be reached.

Figure 6: The sensitivity of FASER and FASER2 to the mixing $\sin\theta$ of
the scalar $\phi$ in case A ($m_{\phi}<2m_{Z^{\prime}}$) and case B
($m_{\phi}=3m_{Z^{\prime}}$, $g_{BL}=10^{-8}$ or $g_{BL}=10^{-6}$). The other
existing bounds for the $m_{\phi}<2m_{Z^{\prime}}$ case are taken from Refs.
Kling and Trojanowski (2021) and Dev  _et al._ (2021), including NA62 Ruggiero
(2020) (light green), CHARM Winkler (2019) (green), MicroBooNE Abratenko  _et
al._ (2021) (blue), BNL-E949 Artamonov  _et al._ (2009) (cyan), LSND Foroughi-
Abari and Ritz (2020) (salmon), LHCb Aaij  _et al._ (2015, 2017) (purple) and
DUNE near detector Dev  _et al._ (2021) (black).

###  IV.2 Additional $Z^{\prime}$ production from $\phi\to
Z^{\prime}Z^{\prime}$

When $m_{\phi}>2m_{Z^{\prime}}$, there exists an additional $Z^{\prime}$
production mode through $B\to K\phi$ or $K\to\pi\phi$ followed by the decay
$\phi\to Z^{\prime}Z^{\prime}$. The decays $b\to s\phi(\to
Z^{\prime}Z^{\prime})$ and $s\to d\phi(\to Z^{\prime}Z^{\prime})$ can be
described by the FCNC Lagrangian in Eq. (39) and the $\phi
Z^{\prime}Z^{\prime}$ coupling Bird  _et al._ (2004); Altmannshofer  _et al._
(2009); Feng  _et al._ (2018b)

$\displaystyle\mathcal{L}_{\phi
Z^{\prime}Z^{\prime}}=Q^{\prime}_{\Phi}g_{BL}m_{Z^{\prime}}Z^{\prime\mu}Z^{\prime}_{\mu}\phi\;.$
(49)

We estimate the contribution from $B$ meson decays by considering the
differential partial decay width for the inclusive decay $b\to
sZ^{\prime}Z^{\prime}$

$\displaystyle{d\Gamma_{b\to sZ^{\prime}Z^{\prime}}\over dq^{2}}$
$\displaystyle=$
$\displaystyle\frac{\sin^{2}\theta\,m_{b}^{3}m_{Z^{\prime}}^{2}Q_{\Phi}^{\prime
2}g_{BL}^{2}}{256\pi^{3}v_{0}^{2}}\left|\frac{3m_{t}^{2}V_{ts}^{\ast}V_{tb}}{16\pi^{2}v_{0}^{2}}\right|^{2}$
(50)
$\displaystyle\times\frac{1}{(q^{2}-m_{\phi}^{2})^{2}+\Gamma_{\phi}^{2}m_{\phi}^{2}}\Big{(}1-{4m_{Z^{\prime}}^{2}\over
q^{2}}\Big{)}^{1/2}\Big{(}1-{q^{2}\over
m_{b}^{2}}\Big{)}^{2}\Big{[}2+{(q^{2}-2m_{Z^{\prime}}^{2})^{2}\over
4m_{Z^{\prime}}^{4}}\Big{]}\;,$

where $q$ is the sum of the two $Z^{\prime}$ bosons’ four-momenta. The total
decay width $\Gamma_{\phi}$ depends on all parameters $m_{\phi}$,
$\sin\theta$, $m_{Z^{\prime}}$ and $g_{BL}$ in this case, as mentioned in the
above subsection and thus the (differential) branching ratio does not simply
scale with $g_{BL}^{2}$, but also depends on the total decay width
$\Gamma_{\phi}$. Similarly, the expression for $K\to\pi Z^{\prime}Z^{\prime}$
is

$\displaystyle\frac{d\Gamma_{K\to\pi Z^{\prime}Z^{\prime}}}{dq^{2}}=$
$\displaystyle\frac{\sin^{2}\theta\,m_{s}^{2}m_{Z^{\prime}}^{2}Q_{\Phi}^{\prime
2}g_{BL}^{2}}{256\pi^{3}m_{K}v_{0}^{2}}\left(\frac{m_{K}^{2}-m_{\pi}^{2}}{m_{s}-m_{d}}\right)^{2}\left|\frac{3m_{t}^{2}V_{td}^{\ast}V_{ts}}{16\pi^{2}v_{0}^{2}}\right|^{2}$
(51)
$\displaystyle\times\frac{\lambda^{1/2}(1,m_{\pi}^{2}/m_{K}^{2},q^{2}/m_{K}^{2})|f_{0}^{K\pi}(q^{2})|^{2}}{(q^{2}-m_{\phi}^{2})^{2}+m_{\phi}^{2}\Gamma_{\phi}^{2}}\left(1-\frac{4m_{Z^{\prime}}^{2}}{q^{2}}\right)^{1/2}\left[2+\frac{(q^{2}-2m_{Z^{\prime}}^{2})^{2}}{4m_{Z^{\prime}}^{2}}\right]$

and the $K\to\pi$ form factor is given by $f_{0}^{K\pi}(q^{2})=0.96$ Boiarska
_et al._ (2019). We implemented the 3-body decay in FORESEE. To efficiently
translate the result from the reference coupling $g_{BL}=1$ and to avoid
generating events for each coupling, we rescale the result with the change in
the total cross section. The 3-body decays turn out to be subdominant compared
to 2-body decays for the benchmark scenarios which we considered and thus the
resulting sensitivity regions of FASER/FASER2 remain unchanged.

##  V Conclusions

We study the general $U(1)^{\prime}$ models with non-universal lepton charges.
A scalar singlet is introduced with non-zero $U(1)^{\prime}$ charge to
spontaneously break the $U(1)^{\prime}$ gauge symmetry. This model with small
masses and couplings can be tested in precision experiments such as
FASER/FASER2, COHERENT and the long-baseline neutrino oscillation experiments
DUNE and T2HK. We explore the sensitivities of above precision experiments to
the new gauge boson $Z^{\prime}$ and the new CP-even scalar $\phi$. Our main
findings are the following:

  * •

With non-universal lepton charges, distinctive reaches emerge in the regime of
low $m_{Z^{\prime}}$ and small gauge coupling for different $U(1)^{\prime}$
charge setups. The COHERENT experiment and the future long-baseline neutrino
oscillation experiments DUNE and T2HK also provide complementary
sensitivities.

  * •

For $m_{\phi}<2m_{Z^{\prime}}$, the search for the scalar $\phi$ at
FASER/FASER2 is sensitive to the mixing angle between the scalar singlet and
the SM Higgs, which is analogous to the dark Higgs search.

  * •

In the case of $m_{\phi}>2m_{Z^{\prime}}$, the kinematically allowed decay
$\phi\to Z^{\prime}Z^{\prime}$ changes the lifetime and decay rates of the
scalar $\phi$. The probe reach highly depends the $Z^{\prime}$ mass and the
gauge coupling $g_{BL}$ as additional parameters. With $\phi\to
Z^{\prime}Z^{\prime}$ open, an additional production mode of $Z^{\prime}$
appears which is however subdominant to other production modes.

###### Acknowledgements.

We would like to thank Yi Cai and Yongchao Zhang for discussion. T.L. is
supported by the National Natural Science Foundation of China (Grant No.
11975129, 12035008) and “the Fundamental Research Funds for the Central
Universities”, Nankai University (Grant No. 63196013). J.L. is supported by
the National Natural Science Foundation of China (Grant No. 11905299,
12275368) and Guangdong Basic and Applied Basic Research Foundation (Grant No.
2020A1515011479). T.F. and M.S. acknowledge support by the Australian Research
Council Discovery Project DP200101470. This research includes computations
using the computational cluster Katana supported by Research Technology
Services at UNSW Sydney.

##  Appendix A Texture zeroes and lepton mixing

We first assume one non-zero charge for leptons, for instance
$Q^{\prime}_{e}=-3$, and then one has the following Yukawa matrix textures

$\displaystyle y^{D}$ $\displaystyle=\begin{pmatrix}*&0&0\\\ 0&*&*\\\ 0&*&*\\\
\end{pmatrix}\;,$ $\displaystyle M$ $\displaystyle=\begin{pmatrix}0&0&0\\\
0&*&*\\\ 0&*&*\\\ \end{pmatrix}\;,$ (52) $\displaystyle
y^{M}(Q^{\prime}_{\Phi}=3)$ $\displaystyle=\begin{pmatrix}0&*&*\\\ *&0&0\\\
*&0&0\\\ \end{pmatrix}\;,$ $\displaystyle y^{M}(Q^{\prime}_{\Phi}=6)$
$\displaystyle=\begin{pmatrix}*&0&0\\\ 0&0&0\\\ 0&0&0\\\ \end{pmatrix}\;,$
(53)

with the following ranks

$\displaystyle\mathrm{rank}[y^{D}]=3\;,~{}\mathrm{rank}[M]=2\;,~{}\mathrm{rank}[y^{M}(Q^{\prime}_{\Phi}=3)]=2\;,~{}\mathrm{rank}[y^{M}(Q^{\prime}_{\Phi}=6)]=1\;,$
$\displaystyle\mathrm{rank}[M+y^{M}(Q^{\prime}_{\Phi}=3)]=3\;,~{}\mathrm{rank}[M+y^{M}(Q^{\prime}_{\Phi}=6)]=3\;.$
(54)

In this case the assignment of $Q^{\prime}_{\Phi}=3$ leads to the neutrino
mass matrix without zeros, but $Q^{\prime}_{\Phi}=6$ allows two zero entries.
Next we take two non-zero charges, for instance
$Q^{\prime}_{e}+Q^{\prime}_{\mu}=-3$ with $Q^{\prime}_{e}\neq 0$ and
$Q^{\prime}_{\mu}\neq 0$. For the choice of
$Q^{\prime}_{e}=Q^{\prime}_{\mu}=-3/2$, we can have the Yukawa matrices

$\displaystyle y^{D}$ $\displaystyle=\begin{pmatrix}*&*&0\\\ *&*&0\\\ 0&0&*\\\
\end{pmatrix}\;,$ $\displaystyle M$ $\displaystyle=\begin{pmatrix}0&0&0\\\
0&0&0\\\ 0&0&*\\\ \end{pmatrix}\;,$ (55) $\displaystyle
y^{M}(Q^{\prime}_{\Phi}=3)$ $\displaystyle=\begin{pmatrix}*&*&0\\\ *&*&0\\\
0&0&0\\\ \end{pmatrix}\;,$ $\displaystyle y^{M}(Q^{\prime}_{\Phi}=3/2)$
$\displaystyle=\begin{pmatrix}0&0&*\\\ 0&0&*\\\ *&*&0\\\ \end{pmatrix}\;,$
(56)

with the following ranks

$\displaystyle\mathrm{rank}[y^{D}]=3\;,~{}\mathrm{rank}[M]=1\;,~{}\mathrm{rank}[y^{M}(Q^{\prime}_{\Phi}=3)]=2\;,~{}\mathrm{rank}[y^{M}(Q^{\prime}_{\Phi}=3/2)]=2\;,$
$\displaystyle\mathrm{rank}[M+y^{M}(Q^{\prime}_{\Phi}=3)]=3\;,~{}\mathrm{rank}[M+y^{M}(Q^{\prime}_{\Phi}=3/2)]=2\;.$
(57)

One can see that, without introducing additional scalars, the assignment of
$Q^{\prime}_{\Phi}=3/2$ cannot give rise to three massive light neutrinos in
this case. The choice of $Q^{\prime}_{\Phi}=3$ can generate a full rank mass
matrix, but there are still two zeros in the neutrino mass matrix. By
including two scalar fields $\Phi,\Phi^{\prime}$ with $Q^{\prime}_{\Phi}=3$
and $Q^{\prime}_{\Phi^{\prime}}=3/2$, one can generate a full rank mass matrix
and remove the zeros in the neutrino mass matrix. For the case of
$Q^{\prime}_{e}+Q^{\prime}_{\mu}=-3$ and $Q^{\prime}_{e}\neq
Q^{\prime}_{\mu}$, we have

$\displaystyle y^{D}$ $\displaystyle=\begin{pmatrix}*&0&0\\\ 0&*&0\\\ 0&0&*\\\
\end{pmatrix}\;,$ $\displaystyle M$ $\displaystyle=\begin{pmatrix}0&0&0\\\
0&0&0\\\ 0&0&*\\\ \end{pmatrix}\;,$ $\displaystyle y^{M}(Q^{\prime}_{\Phi}=3)$
$\displaystyle=\begin{pmatrix}0&*&0\\\ *&0&0\\\ 0&0&0\\\ \end{pmatrix}\;,$
(58)

with the ranks as

$\displaystyle\mathrm{rank}[y^{D}]=3\;,~{}\mathrm{rank}[M]=1\;,~{}\mathrm{rank}[y^{M}(Q^{\prime}_{\Phi}=3)]=2\;,~{}\mathrm{rank}[M+y^{M}(Q^{\prime}_{\Phi}=3)]=3\;.$
(59)

It turns out that there are two zero entries in the neutrino mass matrix.
Other assignments of $Q^{\prime}_{\Phi}$ give at most one non-zero entry in
$y^{M}$ and thus there must be two zeros in neutrino mass matrix. For the case
with three non-zero charges such as $Q^{\prime}_{e}=Q^{\prime}_{\mu}\neq
Q^{\prime}_{\tau}$, one has the following matrices

$\displaystyle y^{D}$ $\displaystyle=\begin{pmatrix}*&*&0\\\ *&*&0\\\ 0&0&*\\\
\end{pmatrix}\;,$ $\displaystyle M$ $\displaystyle=\begin{pmatrix}0&0&0\\\
0&0&0\\\ 0&0&0\\\ \end{pmatrix}\;,$ (60) $\displaystyle
y^{M}(Q^{\prime}_{\Phi}=-(Q^{\prime}_{e}+Q^{\prime}_{\tau}))$
$\displaystyle=\begin{pmatrix}0&0&*\\\ 0&0&*\\\ *&*&0\\\ \end{pmatrix}\;,$
$\displaystyle y^{M}(Q^{\prime}_{\Phi}=-2Q^{\prime}_{e})$
$\displaystyle=\begin{pmatrix}*&*&0\\\ *&*&0\\\ 0&0&0\\\ \end{pmatrix}\;,$
(61)

with the ranks as

$\displaystyle\mathrm{rank}[y^{D}]=3\;,~{}\mathrm{rank}[M]=0\;,~{}\mathrm{rank}[y^{M}(Q^{\prime}_{\Phi}=-(Q^{\prime}_{e}+Q^{\prime}_{\tau}))]=\mathrm{rank}[y^{M}(Q^{\prime}_{\Phi}=-2Q^{\prime}_{e})]=2\;,$
$\displaystyle\mathrm{rank}[M+y^{M}(Q^{\prime}_{\Phi}=-(Q^{\prime}_{e}+Q^{\prime}_{\tau}))]=\mathrm{rank}[M+y^{M}(Q^{\prime}_{\Phi}=-2Q^{\prime}_{e})]=2\;.$
(62)

We find that this case cannot generate three massive neutrinos and one needs
additional scalar fields to give a full rank mass matrix. For the most general
case with $Q^{\prime}_{e}\neq Q^{\prime}_{\mu}\neq Q^{\prime}_{\tau}$, such as
$Q^{\prime}_{e}=-3$ and $Q^{\prime}_{\mu}=-Q^{\prime}_{\tau}\neq 3$ in
particular, we have the following Yukawa matrices

$\displaystyle y^{D}$ $\displaystyle=\begin{pmatrix}*&0&0\\\ 0&*&0\\\ 0&0&*\\\
\end{pmatrix}\;,$ $\displaystyle M$ $\displaystyle=\begin{pmatrix}0&0&0\\\
0&0&*\\\ 0&*&0\\\ \end{pmatrix}\;,$ (63) $\displaystyle
y^{M}(Q^{\prime}_{\Phi}=-2Q^{\prime}_{e})$
$\displaystyle=\begin{pmatrix}*&0&0\\\ 0&0&0\\\ 0&0&0\\\ \end{pmatrix}\;,$
$\displaystyle y^{M}(Q^{\prime}_{\Phi}=-(Q^{\prime}_{e}+Q^{\prime}_{\mu}))$
$\displaystyle=\begin{pmatrix}0&*&0\\\ *&0&0\\\ 0&0&0\\\ \end{pmatrix}\;,$
(64) $\displaystyle y^{M}(Q^{\prime}_{\Phi}=-2Q^{\prime}_{\mu})$
$\displaystyle=\begin{pmatrix}0&0&*\\\ 0&*&0\\\ *&0&0\\\ \end{pmatrix}\;,$
$\displaystyle y^{M}(Q^{\prime}_{\Phi}=2Q^{\prime}_{\mu}))$
$\displaystyle=\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&0&*\\\ \end{pmatrix}\;,$
(65)

For the most general case with $Q^{\prime}_{e}\neq Q^{\prime}_{\mu}\neq
Q^{\prime}_{\tau}$, we have the Yukawa matrices

$\displaystyle y^{D}$ $\displaystyle=\begin{pmatrix}*&0&0\\\ 0&*&0\\\ 0&0&*\\\
\end{pmatrix}\;,$ $\displaystyle M$ $\displaystyle=\begin{pmatrix}0&0&0\\\
0&0&0\\\ 0&0&0\\\ \end{pmatrix}\;,$ (66) $\displaystyle
y^{M}(Q^{\prime}_{\Phi}=-(Q^{\prime}_{e}+Q^{\prime}_{\tau}))$
$\displaystyle=\begin{pmatrix}0&0&*\\\ 0&0&0\\\ *&0&0\\\ \end{pmatrix}\;,$
$\displaystyle y^{M}(Q^{\prime}_{\Phi}=-2Q^{\prime}_{e})$
$\displaystyle=\begin{pmatrix}*&0&0\\\ 0&0&0\\\ 0&0&0\\\ \end{pmatrix}\;,$
(67)

with the ranks as

$\displaystyle\mathrm{rank}[y^{D}]=3\;,~{}\mathrm{rank}[M]=0\;,~{}\mathrm{rank}[y^{M}(Q^{\prime}_{\Phi}=-(Q^{\prime}_{e}+Q^{\prime}_{\tau}))]=2\;,~{}\mathrm{rank}[y^{M}(Q^{\prime}_{\Phi}=-2Q^{\prime}_{e})]=1\;,$
$\displaystyle\mathrm{rank}[M+y^{M}(Q^{\prime}_{\Phi}=-(Q^{\prime}_{e}+Q^{\prime}_{\tau}))]=2\;,~{}\mathrm{rank}[M+y^{M}(Q^{\prime}_{\Phi}=-2Q^{\prime}_{e})]=1\;.$
(68)

Again, the additional scalar fields are needed to give a full rank mass
matrix.

It turns out that adding one scalar singlet can lead to three massive light
neutrinos and no zeros in the neutrino mass matrix in the case with one non-
zero lepton charge. In other non-universal models, additional scalar fields
are needed to generate a full rank mass matrix and remove the zero entries in
the neutrino mass matrix. Alternatively, we could generate the missing entries
in the Dirac neutrino Yukawa matrix. The cases with one or two independent
zeros were also discussed in the literature, where either one or two scalar
singlets are needed. To summarize, it is quite arbitrary to include the scalar
singlets breaking the non-universal gauge symmetry to give the allowed PMNS
matrix.

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