# Revisiting for maximal flavor violating $Z^{\prime}_{e\mu}$ and its
phenomenology constraints

Jia Liu<EMAIL_ADDRESS>School of Physics and State Key Laboratory of
Nuclear Physics and Technology, Peking University, Beijing 100871, China
Center for High Energy Physics, Peking University, Beijing 100871, China
Muyuan Song<EMAIL_ADDRESS>Center for High Energy Physics, Peking
University, Beijing 100871, China  School of Physics and State Key Laboratory
of Nuclear Physics and Technology, Peking University, Beijing 100871, China
Haohao Zhang<EMAIL_ADDRESS>Center for High Energy Physics, Peking
University, Beijing 100871, China

###### Abstract

Lepton flavor violation (LFV), observed conclusively in neutrino oscillations,
remains a pivotal area of investigation due to its absence in the Standard
Model (SM). Beyond the Standard Model (BSM) physics explores charged lepton
flavor violation (CLFV), particularly through new particle candidates such as
the $Z^{\prime}$. This article focuses on maximal LFV interactions facilitated
by the $Z^{\prime}$ boson, specifically targeting its off-diagonal
interactions with the first and second generations of charged and neutral
leptons. In our ultraviolet (UV) model for the origin of the $Z^{\prime}$,
inspired by the work of [R.Foot et al., Phys.Rev. D50 (1994) 4571-4580], we
utilize the discrete $Z_{2}$ symmetry to investigate the maximal LFV mediated
by the $Z^{\prime}$ between the muon ($\mu$) and electron ($e$) arising from
the additional scalars. This symmetry prohibits flavor-conserving interactions
between $Z^{\prime}$ and $\mu^{+}\mu^{-},\,e^{+}e^{-}$. In conjunction with
collider, $(g-2)_{\mu},(g-2)_{e}$, inverse $\mu$ decay and LFV decay
constraints, we provide forecasts for anticipated limits derived from
processes such as $\nu_{\mu}N\to\nu_{e}\mu^{+}e^{-}N$ in neutrino trident
experiments like the DUNE experiment. These limits highlight the prospective
scope and significance of LFV investigations within these experimental
frameworks. From the range of mass between 0.01 GeV and 10 GeV, the most
stringent limit arise from $\mathcal{B}(\mu\to e+X+\gamma)$ for
$M_{Z^{\prime}}<m_{\mu}$ while neutrino trident processes constrain
effectively as $M_{Z^{\prime}}\to 10$ GeV.

###### Contents

  1. I Introduction
  2. II Maximal LFV $Z^{\prime}$ model
    1. II.1 An ultraviolet (UV) model with maximal LFV $Z^{\prime}$
  3. III LFV limits
    1. III.1 $(g-2)_{\mu}$ and $(g-2)_{e}$
    2. III.2 Collider constraints
    3. III.3 Muon decay limits
    4. III.4 Inverse $\mu$ decay
    5. III.5 Muonium-anti Muonium oscillation ($\rm{Mu}-\overline{\rm{Mu}}$)
    6. III.6 Neutrino trident production with maximal LFV $Z^{\prime}$
  4. IV Results
  5. V Conclusions
  6. A Scalar potential with neutral scalars
  7. B Kinematic variables for the neutrino trident process.

##  I Introduction

Lepton Flavor Violation (LFV) holds considerable significance in particle
physics, offering insights into fundamental interactions beyond the Standard
Model (SM). The theoretical motivation for searching for LFV processes stems
from the possibility that new symmetries might suppress non-conservative
leptonic flavor interactions, rendering these phenomena less apparent
Feinberg:1959ui ; Feinberg:1961zz ; Cirigliano:2005ck ; Calibbi:2017uvl .
Investigating LFV in novel particle interactions continues to be a dynamic and
compelling area of research, despite the yet unrealized expectations, such as
searches for axion-like particles (ALPs) Bjorkeroth:2018dzu ; Cornella:2019uxs
; Calibbi:2020jvd ; Cheung:2021mol ; Calibbi:2024rcm ; Davoudiasl:2024vje .
Moreover, LFV studies involving right-handed neutrinos with non-zero mass in
SUSY Hisano:1995cp ; Abada:2014kba , little Higgs models Blanke:2007db ;
Han:2011aq , seesaw (with EFT) Ardu:2023yyw , SMEFT Altmannshofer:2023tsa ;
Altmannshofer:2022fvz , B-LSSM Huo:2024puy  and dark matter Toma:2013zsa ;
Tapender:2024ktc  have spurred investigations into various lepton flavor
violating interactions. In contrast, exploring smaller masses around the
electroweak scale has prompted investigations both with and without new gauge
symmetries, as highlighted in recent studies such as xiaogang:1991a ;
xiaogang:1991b ; Foot:1994vd ; Iguro:2020rby ; AtzoriCorona:2022moj ;
Crivellin:2023sig ; Espinosa-Gomez:2023xrq ; Eguren:2024oov ; marin:2024LFV ,
even offering potential explanations for discrepancies observed in quantities
like $(g-2)_{\mu}$ Cheng:2021okr ; Kang:2019vng ; Eijima:2023yiw . There has
been significant emphasis on new gauge interactions involving $U(1)^{\prime}$,
with researchers actively searching for flavor-conserving interactions between
the $Z^{\prime}$ boson and leptons, from theoretical studies to experimental
investigations Buras:2021btx ; ATLAS:2018sky ; Araki:2022xqp . Special
attention has been given to maximal flavor violation, where off-diagonal
couplings are considered dominant, as highlighted in references Foot:1994vd ;
CDF:2008zud ; Foldenauer:2016rpi ; Kriewald:2022erk ; ding2024study 111Indeed,
maximal flavor violation has been explored extensively in the quark sector,
even in the absence of maximal flavor violation in the lepton sector Bar-
Shalom:2007xeu .. For experiment searches, neutrino experiments have
undertaken several LFV searches, as labeled by references such as CHARM-
II:1990dvf ; CCFR:1991lpl ; NuTeV:1999wlw . In neutrino experiments, the new
gauge interaction undergoes testing through a process known as neutrino
trident production Czyz:1964zz ; Fujikawa:1971nx , where an initial neutrino
collides with stable nuclei, resulting in the production of a pair of charged
leptons. Previous searches have primarily focused on pairs of $\mu$ with
opposing charges CHARM-II:1990dvf ; CCFR:1991lpl ; NuTeV:1999wlw , extending
to potential combinations with LHC energy regimes Francener:2024wul ;
Altmannshofer:2024hqd  and exploring new physics effects from atmospheric
neutrinos Ge:2017poy . Nevertheless, maximal mixing between $\mu$ and $e$
flavors could produce a clear signal without background interference. Such
distinct flavor signatures considerably simplify the analysis compared to
previous searches. Moreover, proposed explorations on DUNE intend to
investigate final states featuring muon-electron pairs within SM
Ballett:2018uuc , thereby augmenting sensitivity to lepton flavor-violating
interactions. From a theoretical standpoint, Ref. Foot:1994vd , emphasized
maximal LFV involving the $Z^{\prime}\bar{\mu}\tau$ interaction, suggesting
equivalence in the effects on the Yukawa couplings $H\bar{\mu}\mu$ and
$H\bar{\tau}\tau$. However, recent Higgs measurements contradict this
assumption. On the other hand, the recent Higgs decay measurement still
survives the scenario where $H\bar{\mu}\mu,H\bar{e}e$ couplings to be
identical in the presence of LFV $Z^{\prime}\mu^{\pm}e^{\mp}$ within similar
models. Therefore, exploring $\mu$ and $e$ flavor interactions proves crucial
for advancing LFV investigations from both experimental and theoretical
perspectives.

In this study, we explore the implications of neutrino trident production
limits on maximal LFV involving $Z^{\prime}$ in the DUNE experiment. We
demonstrate that the neutrino trident process sets stringent constraints on
LFV interactions between $\mu$ and $e$ compared to collider experiments (LEP
and Belle-II) and other LFV interaction limits, particularly in the mass range
$m_{\mu}<M_{Z^{\prime}}<10$ GeV. In the lower mass regime, the most
restrictive limit originates from the inclusive search for the
$\mathcal{B}(\mu\to e+\gamma+X)$ process when $M_{Z^{\prime}}<m_{\mu}$.

The paper is structured as follows: Section II clarifies the simplified
$Z^{\prime}$ model, emphasizing its dominant new off-diagonal gauge
interaction and discussing its implications for charged and neutral leptons.
Additionally, we introduce a UV model in Section II that comprises three
scalar doublets and an additional singlet, where the singlet is pivotal in
providing mass for the maximal LFV $Z^{\prime}$. Section III addresses the LFV
constraints on $Z^{\prime}$, incorporating findings from collider experiments,
LFV $\mu$ decays, neutrino-charged lepton scattering (inverse $\mu$ decay),
and the neutrino trident process. Additionally, we analyze the implications of
LFV couplings for $(g-2)_{\mu,e}$. In Section IV, we depict exclusion limit
plots derived from the couplings between $Z^{\prime}$ and leptons. Finally,
Section V concludes with a discussion on future prospects.

##  II Maximal LFV $Z^{\prime}$ model

In models incorporating an additional $Z^{\prime}$ boson, maintaining a global
anomaly-free symmetry is essential within the SM framework. Previous studies
of such models have concentrated on interactions between the $Z^{\prime}$
boson and leptons, highlighting scenarios where a discrete symmetry prevents
diagonal couplings Foot:1994vd . This precaution is crucial to prevent mixing
between the $A$ and $Z^{\prime}$ bosons, as described by transformations
within the kinetic Lagrangian,

$\displaystyle\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\kappa
F_{\mu\nu}F^{{}^{\prime}\mu\nu}-\frac{1}{4}F^{\prime}_{\mu\nu}F^{{}^{\prime}\mu\nu},$
(1)

where $B_{\mu}\to B_{\mu},Z^{\prime}_{\mu}\to-Z^{\prime}_{\mu}$. Under the
$Z_{2}$ discrete symmetry, the above mixing between $A$ and $Z^{\prime}$ is
forbidden.

To satisfy the anomaly-free requirements, we assign distinct $U(1)^{\prime}$
charges to different generations of leptons. Specifically, we consider a
scenario where these $U(1)^{\prime}$ charge assignments apply to the first and
second generations while the third generation remains uncharged under
$U(1)^{\prime}$. Alternatively, opposite charges could be assigned to any
other pair of generations, such as $\mu/\tau$ or $e/\tau$. In such cases, the
flavor states under $SU(3)\times SU(2)\times U(1)_{Y}\times U(1)^{\prime}$ are
structured as described in Ref. Foot:1994vd ,

$\displaystyle\ell_{1L}$ $\displaystyle=$
$\displaystyle\frac{(e+\mu)_{L}}{\sqrt{2}},\ell_{2L}=\frac{(e-\mu)_{L}}{\sqrt{2}},e_{1R}=\frac{(e+\mu)_{R}}{\sqrt{2}},e_{2R}=\frac{(e-\mu)_{R}}{\sqrt{2}},$
$\displaystyle\ell_{1L}$ $\displaystyle\thicksim$
$\displaystyle(\mathbf{1},\mathbf{2})(-1,2a),\ell_{2L}\thicksim(\mathbf{1},\mathbf{2})(-1,-2a),$
$\displaystyle e_{1R}$ $\displaystyle\thicksim$
$\displaystyle(\mathbf{1},\mathbf{1})(-2,2a),e_{2R}\thicksim(\mathbf{1},\mathbf{1})(-2,-2a),$
(2)

where $a$ is a parameter for the $U(1)^{\prime}$ charge, $\ell_{1L,2L}$ and
$e_{1R,2R}$ are the flavor states, while $e$ and $\mu$ are the eigenstates in
the mass basis. With the discrete symmetry ($\ell_{1L}\to\ell_{2L},e_{1R}\to
e_{2R},Z^{\prime}_{\mu}\to-Z^{\prime}_{\mu}$) in place, the interaction
between $Z^{\prime}$ and flavor states would demonstrate the maximal mixing
feature for lepton mass states:

$\displaystyle\mathcal{L}^{Z^{\prime}}_{\rm{NC}}$ $\displaystyle=$
$\displaystyle
g_{e\mu}\,Z^{\prime}_{\mu}(\bar{\mu}\gamma^{\mu}e+\bar{e}\gamma^{\mu}\mu),$
(3)

and neutrino-$Z^{\prime}$ interaction,

$\displaystyle\mathcal{L}^{Z^{\prime}}_{\rm{NC}}$ $\displaystyle=$
$\displaystyle g_{e\mu}\,Z^{\prime}_{\mu}(\bar{\nu}_{\mu
L}\gamma^{\mu}\nu_{eL}+\bar{\nu}_{eL}\gamma^{\mu}\nu_{\mu L}),$ (4)

where the $Z^{\prime}$ does not couple to the diagonal term in the mass basis.

Figure 1: The decay width of $Z^{\prime}$ correlated with charged and neutral
leptons with $g_{e\mu}=10^{-2}(\text{solid}),10^{-4}(\text{dashed})$. The red
lines label the $\Gamma(Z^{\prime}\to e^{\pm}\mu^{\mp})$ and the black lines
label the $\Gamma(Z^{\prime}\to\nu_{e(\mu)}\bar{\nu}_{\mu(e)})$.

Next, the decay width of the $Z^{\prime}$ can be straightforwardly calculated
from the above interaction, considering the limit where $m_{e}\rightarrow 0$,

$\displaystyle\Gamma(Z^{\prime}\to e^{\pm}\mu^{\mp})$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{e\mu}(M^{2}_{Z^{\prime}}-m^{2}_{\mu})^{2}(2M^{2}_{Z^{\prime}}+m^{2}_{\mu})}{24\pi
M_{Z^{\prime}}^{5}},$ (5)
$\displaystyle\Gamma(Z^{\prime}\to\nu_{e(\mu)}\bar{\nu}_{\mu(e)})$
$\displaystyle=$ $\displaystyle\frac{g^{2}_{e\mu}M_{Z^{\prime}}}{24\pi},$ (6)

Fig. 1 depicts the decay widths of $Z^{\prime}$ across a mass range from 0.01
GeV to 10 GeV, with $g_{e\mu}=10^{-2}$ (solid line) and $10^{-4}$ (dashed
line). It becomes apparent that the two decay channels become comparable when
$M_{Z^{\prime}}$ is significantly larger than $m_{\mu}+m_{e}$. Furthermore,
for large $M_{Z^{\prime}}$, the decay width into neutrinos is half that of the
charged lepton decay,

$\displaystyle
2\,\mathcal{B}(Z^{\prime}\to\nu_{e(\mu)}\bar{\nu}_{\mu(e)})=\mathcal{B}(Z^{\prime}\to
e^{\pm}\mu^{\mp}).$ (7)

because the neutrinos only have left-handed spinors.

###  II.1 An ultraviolet (UV) model with maximal LFV $Z^{\prime}$

The preceding section examines a simplified LFV $Z^{\prime}$ and explores its
new gauge interaction. This prompts the inquiry into how such a $Z^{\prime}$
could be generated within a UV framework. In our UV model, we establish the
existence of a model comprising three scalar doublets and an additional real
singlet, with specific charge assignments outlined in Tab. 1 as extended by
Ref. Foot:1994vd .

Group | $\ell_{1L}$ | $\ell_{2L}$ | $e_{1R}$ | $e_{2R}$ | $\phi_{1}$ | $\phi_{2}$ | $\phi_{3}$ | $\phi_{4}$
---|---|---|---|---|---|---|---|---
$SU(3)_{c}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
$SU(2)_{L}$ | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1
$U(1)_{Y}$ | -1 | -1 | -2 | -2 | 1 | 1 | 1 | 0
$U(1)^{\prime}$ | 2a | -2a | 2a | -2a | -4a | 4a | 0 | $1$
Table 1: Particle contents and charges in the $U(1)^{\prime}$ model with
$\ell_{1L,2L},e_{1R,2R}$ and four scalars $\phi_{1,2,3,4}$.

The electromagnetic neutral components of three scalar doublets and one
singlet are given in the following,

$\displaystyle\phi_{1}$ $\displaystyle\supset$ $\displaystyle
v_{1}+\left(\rho_{1}+i\eta_{1}\right)/\sqrt{2},\,\phi_{2}\supset
v_{2}+\left(\rho_{2}+i\eta_{2}\right)/\sqrt{2},\,$ $\displaystyle\phi_{3}$
$\displaystyle\supset$ $\displaystyle
v_{3}+\left(\rho_{3}+i\eta_{3}\right)/\sqrt{2},\,\phi_{4}=v_{4}+\rho_{4}/\sqrt{2},$
(8)

where we will propose the discrete symmetry $1\leftrightarrow 2$
($\ell_{1L}\to\ell_{2L},e_{1R}\to e_{2R},\phi_{1}\to\phi_{2}$), thereby
enforcing $v_{0}\equiv v_{1}=v_{2}$. In contrast to Ref. Foot:1994vd , we
introduced an additional singlet scalar $\phi_{4}$, which is charged solely
under $U(1)^{\prime}$. The possible scalar potential is given in Appendix A.

On the other side, the different generation of leptons have exactly opposite
charges under $U(1)^{\prime}$ in Table 1 , which ensures the anomaly free
condition. Therefore, the Yukawa interaction for leptons under the above
charge assignments would be expressed as follows Foot:1994vd ,

$\displaystyle\mathcal{L}^{\ell}_{\text{Yukawa}}$ $\displaystyle=$
$\displaystyle\mathcal{Y}_{1}\bar{\ell}_{1L}e_{1R}\phi_{3}+\mathcal{Y}_{1}\bar{\ell}_{2L}e_{2R}\phi_{3}+\mathcal{Y}_{3}\bar{\ell}_{3L}e_{3R}\phi_{3}$
(9) $\displaystyle+$
$\displaystyle\mathcal{Y}_{12}(\bar{\ell}_{1L}e_{2R}\phi_{2}+\bar{\ell}_{2L}e_{1R}\phi_{1})+\rm{h.c},$

where the Yukawa sector obeys the discrete symmetry $1\leftrightarrow 2$. The
scalar $\phi_{4}$ does not participate in Yukawa interactions due to its
specific charge assignment. Instead, it contributes solely to the mass
generation of the $Z^{\prime}$ boson through the process of symmetry breaking.
As detailed in Appendix A, the mass eigenstate $H$ emerges from the
diagonalization of mass matrices, particularly when the flavor state
$\rho_{3}$ aligns with the physical state $H$, facilitated by setting
$\lambda_{4}\to 0$. The SM-like Higgs ($H$) exhibits Yukawa couplings with
leptons as follows,

$\displaystyle\mathcal{L}^{\ell}_{H}=\frac{m_{\tau}}{\sqrt{2}v_{3}}\bar{\tau}\tau
H+\frac{m_{\mu}+m_{e}}{2\sqrt{2}v_{3}}(\bar{\mu}\mu+\bar{e}e)H,$ (10)

where the off-diagonal Yukawa terms vanish. This cancellation occurs because
the terms $(\bar{\ell}_{1L}e_{2R})$ and $(\bar{\ell}_{2L}e_{1R})$ eliminate
the off-diagonal contributions:

$\displaystyle\bar{\ell}_{1L}e_{2R}\phi_{2}+\bar{\ell}_{2L}e_{1R}\phi_{1}+\text{h.c}$
(11) $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2}}(\bar{e}_{L}e_{R}\mathcal{Y}_{12}+\bar{e}_{L}\mu_{R}\mathcal{Y}_{12}-\bar{\mu}_{L}e_{R}\mathcal{Y}_{12}-\bar{\mu}_{L}\mu_{R}\mathcal{Y}_{12})\rho_{1}$
$\displaystyle+$
$\displaystyle\frac{1}{2\sqrt{2}}(\bar{e}_{L}e_{R}\mathcal{Y}_{12}-\bar{e}_{L}\mu_{R}\mathcal{Y}_{12}+\bar{\mu}_{L}e_{R}\mathcal{Y}_{12}-\bar{\mu}_{L}\mu_{R}\mathcal{Y}_{12})\rho_{2}$
$\displaystyle+$
$\displaystyle\frac{\sqrt{2}}{2}(\bar{e}_{L}e_{R}\mathcal{Y}_{1}+\bar{\mu}_{L}\mu_{R}\mathcal{Y}_{1})\rho_{3}+\text{h.c},$

Applying the linear transformation from $\rho_{1},\rho_{2}$ to the neutral
states $S_{1},S_{2}$,

$\displaystyle
S_{1}=\frac{\rho_{1}+\rho_{2}}{\sqrt{2}},S_{2}=\frac{\rho_{1}-\rho_{2}}{\sqrt{2}},$
(12)

results in the elimination of off-diagonal Yukawa-type interactions involving
$S_{1}$. All off-diagonal interactions are transferred to the additional
scalar $S_{2}$, which we note could be significantly heavier due to large
number of parameters from scalar potential and thus not of primary interest in
this context. The Yukawa couplings from the Lagrangian above display a
distinctive feature wherein the couplings for $\mu$ and $e$ are identical. In
Ref. Foot:1994vd , the model presents a unique aspect where the Yukawa
couplings between $H-\bar{\mu}\mu$ and $H-\bar{\tau}\tau$ are equal. However,
the measurement of Higgs decay to muon and tau leptons have already ruled out
this possibility ATLAS:2022vkf ; CMS:2022dwd . Hence, our attention is
directed exclusively towards the mixing between $\mu$ and $e$, given the
current relatively low precision in measurements of $H-\bar{\mu}\mu$ and
$H-\bar{e}e$. Latest measurements of the branching ratios for
$\mathcal{B}(H\to\mu^{+}\mu^{-})<2.6\times 10^{-4}$ by ATLAS ATLAS:2022vkf
and $\mathcal{B}(H\to e^{+}e^{-})<3\times 10^{-4}$ by CMS CMS:2022urr  suggest
that setting the Yukawa couplings for both electrons and muons to
$\mathcal{Y}_{e}\sim\mathcal{Y}_{\mu}<\mathcal{O}(10^{-4})$ would not impose
stringent constraints on the model.

Conversely, following the transformation in eq. (12), the CP-even squared mass
matrices result in $S_{1},\rho_{3}$ being non-diagonal when $\lambda_{4}\neq
0$, indicating that $S_{1},\rho_{3}$ are not mass eigenstates. Instead, after
diagonalization (as shown in eqs. (50 and 57)), the mass eigenstates are
$S^{\prime}_{1},H$ respectively. The transformation from flavor states
($\rho_{1},\rho_{2},\rho_{3},\rho_{4}$) to mass states
($S^{\prime}_{1},S_{2},H,S_{4}$) leads to the new SM Yukawa interaction
between $H$ and leptons, represented as follows (where $S_{2}$ and $S_{4}$ are
already the mass eigenstates):

$\displaystyle\mathcal{L}^{\ell}_{H}$ $\displaystyle=$
$\displaystyle\frac{m_{\tau}}{\sqrt{2}v_{3}}\bar{\tau}\tau
H+\left[\frac{m_{e}+m_{\mu}}{2\sqrt{2}v_{3}}+\frac{\lambda_{4}^{2}}{(\lambda_{1}+\lambda_{4})^{2}}\frac{v_{0}}{v_{3}}\frac{m_{\mu}-m_{e}}{2\sqrt{2}v_{3}}\right]\bar{\mu}\mu
H$ (14)
$\displaystyle+\left[\frac{m_{e}+m_{\mu}}{2\sqrt{2}v_{3}}+\frac{\lambda_{4}^{2}}{(\lambda_{1}+\lambda_{4})^{2}}\frac{v_{0}}{v_{3}}\frac{m_{e}-m_{\mu}}{2\sqrt{2}v_{3}}\right]\bar{e}eH,$

Once again, the off-diagonal interaction terms vanish for the same reasons as
previously mentioned. It is noteworthy that the Yukawa couplings between $H$
and $\mu,e$ are no longer identical. However, directly converting the formal
expression back to the SM $\mu$ and $e$ Yukawa couplings would require setting
$v_{0}=v_{3}$, which would affect the Yukawa couplings for other fermions,
such as the $\tau$ Yukawa coupling. Instead, simplifying this scenario
involves setting $\lambda_{4}=0$ to reduce one parameter of the model. In this
case, the Yukawa couplings for $\mu$ and $e$ become nearly equivalent and
revert to the formal condition in Eq. (10).

From the $U(1)^{\prime}$ covariant derivative
($D_{\mu}=\partial_{\mu}+ig^{\prime}Q_{U(1)^{\prime}}Z^{{}^{\prime}}_{\mu}$),
the gauge interaction between $Z^{\prime}$ and lepton flavor states can be
easily expressed as follows,

$\displaystyle\mathcal{L}^{Z^{\prime}}_{\text{gauge}}=g_{e\mu}(\bar{\ell}_{1L}\gamma^{\mu}\ell_{1L}-\bar{\ell}_{2L}\gamma^{\mu}\ell_{2L}+\bar{e}_{1R}\gamma^{\mu}e_{1R}-\bar{e}_{2R}\gamma^{\mu}e_{2R})\,,$
(15)

where $g_{e\mu}=g^{\prime}\,2a$ is the product of charge and coupling under
the $U(1)^{\prime}$. According to the $U(1)^{\prime}$ charge assignments,
$\phi_{3}$ would not couple with $U(1)^{\prime}$ gauge boson and $\phi_{4}$
only connects with $Z^{\prime}$. After electroweak symmetry breaking
associated with $U(1)^{\prime}$, the masses of the SM $Z$ boson and
$Z^{\prime}$ are determined from the covariant derivative,

$\displaystyle|D_{\mu}\phi_{1}|^{2}+|D_{\mu}\phi_{2}|^{2}+|D_{\mu}\phi_{3}|^{2}$
$\displaystyle\supset$ $\displaystyle\frac{1}{2}M^{2}_{Z}Z^{\mu}Z_{\mu},$
$\displaystyle|D_{\mu}\phi_{1}|^{2}+|D_{\mu}\phi_{2}|^{2}+|D_{\mu}\phi_{4}|^{2}$
$\displaystyle\supset$
$\displaystyle\frac{1}{2}M^{2}_{Z^{\prime}}Z^{\prime\mu}Z^{\prime}_{\mu},$
$\displaystyle
M^{2}_{Z}=\frac{1}{4}(g^{2}+g^{2}_{Y})(v^{2}_{3}+2\,v^{2}_{0}),\,\,M^{2}_{Z^{\prime}}$
$\displaystyle\sim$ $\displaystyle g^{2}_{e\mu}(2\,v^{2}_{0}+v^{2}_{4})\sim
g^{2}_{e\mu}\,v^{2}_{Z^{\prime}}.$ (16)

where $v_{0}<v_{3}$. By taking $(v^{2}_{3}+2v^{2}_{0})\approx
v^{2}_{\text{SM}}$, this ensures that $M_{Z}$ remains unchanged while allowing
$M_{Z^{\prime}}$ to be a free parameter through varying the value of
$v_{Z^{\prime}}$. In comparison to Ref. Foot:1994vd , their model features
$M_{Z^{\prime}}$ strongly constrained by $M_{Z}$, resulting in no available
region due to both Higgs-Yukawa and $M_{Z}$ constraints.

##  III LFV limits

In this section, our attention turns to the phenomenology of low mass
$Z^{\prime}$, where we conduct a thorough analysis to constrain $g_{e\mu}$ and
$M_{Z^{\prime}}$. Initially, we consider the anomalous magnetic moment ($g-2$)
of the muon/electron and the constraints provided by electron-positron
colliders such as the Belle-II experiment and LEP measurements. Then, we
consider the LFV decays ($\mu$ decays) since the majority interaction between
$Z^{\prime}$ with leptons are $\mu$ and $e$. Thirdly, inverse $\mu$ decay and
Muonium-anti Muonium oscillation are taking into account. Finally, the
neutrino trident process would be discussed and we will present the difference
between SM process and distinctive signal from the new $Z^{\prime}$
contribution.

###  III.1 $(g-2)_{\mu}$ and $(g-2)_{e}$

According to the window observable theory Ce:2022kxy ; Colangelo:2022vok  and
the SM prediction Acaroglu:2023cza , the current disagreement concerning the
$\Delta a_{\mu}$ designations has emerged as follows Muong-2:2023cdq ;
Armando:2023zwz ; Acaroglu:2023cza ; Borsanyi:2020mff :

$\displaystyle\Delta a^{\rm{window}}_{\mu}=(1.81\pm 0.47)\times 10^{-9},$ (17)

On the other hand, the electron magnetic dipole moment Giudice:2012ms  with
its experiment measurement through Rb in 2020 Morel:2020dww ; Fan:2022eto ,
the discrepancy with SM prediction can be expressed as follows:

$\displaystyle\Delta a^{\text{Rb}}_{e}=34(16)\times 10^{-14},$ (18)

whereas the Cs atoms measurement method search has obtained a lower bound
Parker:2018vye ,

$\displaystyle\Delta a^{\text{Cs}}_{e}=-102(26)\times 10^{-14},$ (19)

$\displaystyle\gamma$$\displaystyle Z^{{}^{\prime}}$$\displaystyle
e^{-}$$\displaystyle
e^{-}$$\displaystyle\mu^{-}$$\displaystyle\mu^{-}$$\displaystyle\gamma$$\displaystyle
Z^{{}^{\prime}}$$\displaystyle e^{-}$$\displaystyle
e^{-}$$\displaystyle\mu^{-}$$\displaystyle\mu^{-}$ Figure 2: The
representative Feynman diagrams for $(g-2)_{e/\mu}$ contribution via the LFV
$Z^{\prime}$ interaction.

The LFV coupling ($g_{ij}$) would thus contribute to ($g-2$) with the Fig. 2
and its contribution calculated through package-X Patel:2015tea  via the
following expression,

$\displaystyle\Delta a_{e}$ $\displaystyle=$
$\displaystyle\frac{g_{e\mu}^{2}m_{e}m_{\mu}\left(-m_{\mu}^{6}+4M_{Z^{\prime}}^{6}-3m_{\mu}^{3}M_{Z^{\prime}}^{4}-6m_{\mu}^{3}M_{Z^{\prime}}^{4}\log\left(\frac{M_{Z^{\prime}}^{2}}{m_{\mu}^{2}}\right)\right)}{16\pi^{2}M_{Z^{\prime}}^{2}(M_{Z^{\prime}}^{2}-m_{\mu}^{2})^{3}},$
(20) $\displaystyle\Delta a_{\mu}$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{e\mu}}{16\pi^{2}}\left(-\frac{(2M^{2}_{Z^{\prime}}-m^{2}_{\mu})^{2}}{m^{2}_{\mu}M^{2}_{Z^{\prime}}}-\frac{2M^{2}_{Z^{\prime}}(3m^{2}_{\mu}-2M^{2}_{Z^{\prime}})\log\left(\frac{M^{2}_{Z^{\prime}}}{|M^{2}_{Z^{\prime}}-m^{2}_{\mu}|}\right)}{m^{4}_{\mu}}\right).$
(21)

We apply the 2$\sigma$ bounds based on the recent measurements of eqs. (17 and
18) to separately illustrate $\Delta a_{\mu}$ and $\Delta a_{e}$.

###  III.2 Collider constraints

For an electron-positron collider, particularly the Belle-II experiment, the
search for LFV $Z^{\prime}$ bosons is linked with $e\mu+\not{E}$ final states.
We use their constraints on the product of the cross-section
$\sigma(e^{+}e^{-}\to\mu^{\pm}e^{\pm}+\not{E})$ (see Fig. 3) and the detector
efficiency $\epsilon$ to set the upper bound Belle-II:2019qfb . We used the
product ($\epsilon\times\sigma$) to constrain the coupling in the mass region
up to approximately 8 GeV. The numerical cross-section for the $Z^{\prime}$ is
directly proportional to $g^{2}_{e\mu}$ and can be computed by solving the
relation between the existing product limit and their selection of
$M_{Z^{\prime}}$ values.

$\displaystyle e^{-}$$\displaystyle e^{+}$$\displaystyle
e^{+}/\mu^{-}$$\displaystyle\gamma/Z$$\displaystyle
Z^{\prime}$$\displaystyle\overline{\nu}$${}_{e}(\overline{\nu}_{\mu})$$\displaystyle\nu$${}_{\mu}(\nu_{e})$$\displaystyle\mu^{-}/e^{+}$
Figure 3: Typical Feynman diagrams for $e^{+}e^{-}\to
e^{\pm}\mu^{\mp}+\not{E}$.

In the process $e^{+}e^{-}\to\mu^{+}\mu^{-}$ DELPHI:2000ztm , the LFV
interaction occurs via a $t$-channel exchange mediated by $M_{Z^{\prime}}$. As
reported in the aforementioned reference, LEP conducted measurements of this
process at $\sqrt{s}=189$ GeV. The background from the $s$-channel in the SM
(mediated by $H$ bosons) is suppressed by $m_{\mu}m_{e}/M^{2}_{H}$ due to
small Yukawa couplings, resulting in significant interference contributions
between new physics and the SM background associated only with s-channel
($\gamma$ and Z) diagrams
($\mathcal{M}_{\text{total}}=\mathcal{M}_{Z^{\prime}(t)}+\mathcal{M}_{\gamma(s)}+\mathcal{M}_{Z(s)}$)
as depicted in Fig. 4.

$\displaystyle e^{-}$$\displaystyle
e^{+}$$\displaystyle\mu^{+}$$\displaystyle\mu^{-}$$\displaystyle
e^{-}$$\displaystyle
e^{+}$$\displaystyle\mu^{+}$$\displaystyle\mu^{-}$$\displaystyle\gamma/Z$$\displaystyle
Z^{\prime}$ Figure 4: Typical Feynman diagrams for
$e^{+}e^{-}\to\mu^{+}\mu^{-}$. Left panel: SM contribution. Right panel:
$Z^{\prime}$ contribution.

The cross-section of the process expands the terms to $\mathcal{O}(s^{-1})$ as
follows:

$\displaystyle\sigma(e^{+}e^{-}\to\mu^{+}\mu^{-})=\frac{e^{4}\left(1+24s_{w}^{4}\right)}{768\pi
s_{w}^{4}\left(1-s_{w}^{2}\right)^{2}s}+\frac{e^{2}g^{2}_{e\mu}\left(1+4s_{w}^{2}\right)\left(2\log(s/M_{Z^{\prime}}^{2})-3\right)}{64\pi
s_{w}^{2}\left(1-s_{w}^{2}\right)s}+\frac{g^{4}_{e\mu}}{4\pi
M_{Z^{\prime}}^{2}}.$ (22)

The squared amplitude of the new physics process is proportional to
$1/(t-M_{Z^{\prime}}^{2})^{2}$, where the variable $t$ depends on the
scattering angle $\theta$. Upon integrating over $\theta$, the cross-section
becomes proportional to the $1/(t-M_{Z^{\prime}}^{2})$ term. Consequently,
this factor reaches its maximum value of $1/M^{2}_{Z^{\prime}}$ when $t$
approaches zero. Furthermore, the contribution from the term
$q^{\mu}q^{\nu}/M_{Z^{\prime}}^{2}$ is proportional to
$m_{\mu}^{4}/(M_{Z^{\prime}}^{4}s)$, which is negligible compared to the
$1/M^{2}_{Z^{\prime}}$ contribution associated with the term $g^{\mu\nu}$ in
the region of interest for $M_{Z^{\prime}}$. In summary, the factor
$1/M^{2}_{Z^{\prime}}$ in the aforementioned equation arises from the
t-channel in the context of new physics.

We conducted an analysis of the LEP cross-sections DELPHI:2000ztm  at
$\sqrt{s}=189\,$GeV, corresponding to an integrated luminosity of
$155.21\,\text{pb}^{-1}$. This analysis encompasses the range of $\cos\theta$
from $-0.97$ to $0.97$, partitioned into 10 bins, in order to establish
constraints on the $Z^{\prime}$ boson.This enables us to perform a $\chi^{2}$
test based on the differential cross-section of the process (denoted as
$\text{N}^{j}_{\text{signal}}=\int d\sigma/d\cos\theta\times\mathcal{L}$),

$\displaystyle\chi^{2}=\sum^{10}_{j=1}\frac{\left(\text{N}^{j}_{\text{experiment}}-(\text{N}^{j}_{\text{background}}+\text{N}^{j}_{\text{signal}})\right)^{2}}{(\sigma^{j}_{\text{error}})^{2}}.$
(23)

where $\text{N}^{i}_{\text{experiment}}$ represents the number of events
measured by the LEP, and $\text{N}^{i}_{\text{background}}$ represents the
difference between the theoretical predictions and the measured events in each
bin. The two independent parameters, $g_{e\mu}$ and $M_{Z^{\prime}}$, can then
be constrained using the 10 bins with $\chi^{2}\leq 5.991$ (95% CL).

$\displaystyle\mu^{-}$$\displaystyle e^{-}$$\displaystyle
Z^{{}^{\prime}}$$\displaystyle\gamma$$\displaystyle\mu^{-}$$\displaystyle
e^{-}$$\displaystyle
Z^{{}^{\prime}}$$\displaystyle\gamma$$\displaystyle\mu^{-}$$\displaystyle
e^{-}$ Figure 5: The representative Feynman diagrams for 3-body decay of the
muon which $Z^{\prime}$ is lighter than $m_{\mu}$.

###  III.3 Muon decay limits

The occurrence of LFV decays, characterized as $\ell_{i}\to\ell_{j}+X$,
imposes constraints on the underlying interactions. In the context of muon
decays, such as $\mu\to e\gamma$, $\mu\to 3e$ and $\mu\to e+X$ (A summarized
overview is provided in Renga:2019mpg ), these processes define significant
boundaries for new physics. Notably, the $\mu\to e\gamma$ decay has been
scrutinized extensively at the MEG experiment MEG:2011naj ; MEG:2013oxv ;
MEG:2016leq , while $\mu\to 3e$ has been explored by various experiments
SINDRUM:1987nra ; Mu3e:2020gyw ; Perrevoort:2023qhn . Additionally, $\mu\to
e+X$ processes have been investigated by the TWIST collaboration TWIST:2014ymv
. The absence of kinetic mixing, such as $Z-Z^{\prime}$ or $A-Z^{\prime}$
interactions, precludes the occurrence of $\mu\to 3e$ due to the lack of
diagonal coupling vertices between $Z^{\prime}$ and charged leptons.
Consequently, the constraints on LFV decays primarily focus on two-body
processes, where the final decay channel involves $\mu\to eZ^{\prime}$ and
subsequent $Z^{\prime}$ decays exclusively to neutrinos with a branching ratio
of 100%. The TWIST experiment utilized muon stopped events to establish an
upper limit for $\mathcal{B}(\mu\to e+X)$ of approximately
$\mathcal{O}(10^{-5})$ within the mass region of $X$, ranging from 13 MeV to
80 MeV TWIST:2014ymv . In addition, the PSI experiment with a HPGe detector,
testing masses ranging from 100 MeV to 150 MeV, obtained an upper limit of
$\mathcal{B}(\mu^{+}\to e^{+}+X)<5.7\times 10^{-4}$ Bilger:1998rp . Therefore,
we will employ the current limit established by the TWIST collaboration to
assess the constraint on the $Z^{\prime}$.

The decay width of $\mu\to e+Z^{\prime}$ can be expressed as follows under the
approximation of neglecting $m_{e}$,

$\displaystyle\Gamma(\mu\to
e+Z^{\prime})=\frac{g_{e\mu}^{2}(m^{6}_{\mu}-3m^{2}_{\mu}M^{4}_{Z^{\prime}}+2M^{6}_{Z^{\prime}})}{16\pi
m^{3}_{\mu}M^{2}_{Z^{\prime}}},$ (24)

with its branching ratio $\left(\tau_{\mu}=2.2\times
10^{-6}s\sim\Gamma_{\mu}=3\times 10^{-19}\text{GeV}\right)$ Workman:2022ynf ,

$\displaystyle\mathcal{B}(\mu\to e+Z^{\prime})=\Gamma(\mu\to
e+Z^{\prime})/\Gamma_{\mu},$ (25)

However, as reported by Ref. Renga:2019mpg , the Crystal Box collaboration
Goldman:1987hy  has showcased the capability to examine the inclusive $\mu\to
e+\gamma+X$ decay process with $\mu\to e+\gamma$. They have established an
upper bound for the branching ratio $\mathcal{B}(\mu\to e+\gamma+X)<1.3\times
10^{-9}$ Goldman:1987hy . The three-body decay, $\mu\to e+\gamma+X$, can be
expressed as follows,

$\displaystyle\Gamma(\mu\to
e+\gamma+Z^{\prime})\simeq\frac{\alpha_{\text{EM}}g_{e\mu}^{2}m_{\mu}^{5}}{128\pi^{2}E_{\gamma}^{2}M_{Z^{\prime}}^{2}}.$
(26)

where $E_{\gamma}$ correspond to minimum energy cut to prevent infrared
divergence (typical energy is picked for 38 MeV in Goldman:1987hy ). For
$M_{Z^{\prime}}$ reduces to $m_{e}$, the decay process is predominantly
governed by $m_{\mu}$, leading to tighter constraints on its coupling.
Conversely, as the mass of $Z^{\prime}$ approaches $m_{\mu}$, the situation is
reversed. This is due to the enhancement of $m^{3}_{\mu}/M^{2}_{Z^{\prime}}$
(arising from the longitudinal polarization mode of $Z^{\prime}$), which has
been extensively discussed for many channels based on the Goldstone Boson
Equivalence Theorem (e.g., $\tau\to\mu Z^{\prime}$, $t\to bW^{+}$
Heeck:2016xkh ; CHANOWITZ1985379 ; Peskin:2017emn ; Hill:2023dym ). Meanwhile,
an additional $\alpha_{\text{EM}}$ factor and the three body phase space
factor further suppresses the new physics contribution compared to the $\mu\to
e+Z^{\prime}$ process. Thus, the three-body channel would then be the most
stringent limit compares with the two-body channel.

###  III.4 Inverse $\mu$ decay

$\displaystyle\nu_{\mu}$$\displaystyle e^{-}$$\displaystyle
W^{+}$$\displaystyle\mu^{-}$$\displaystyle\nu_{e}$$\displaystyle\overline{\nu}_{\mu}$$\displaystyle
e^{-}$$\displaystyle
Z^{\prime}$$\displaystyle\mu^{-}$$\displaystyle\overline{\nu}_{e}$ Figure 6:
Feynman diagrams for inverse $\mu$ decay with the SM electroweak interaction
and the LFV $Z^{\prime}$ interaction.

In Ref. Amsterdam-CERN-Hamburg-Moscow-Rome:1980qbb ; NuTeV:2001bgq , CHARM and
NuTeV collaborations tested the inverse $\mu$ decay
$\nu_{\mu}e^{-}\to\mu^{-}\nu_{e}$ and
$\bar{\nu}_{\mu}e^{-}\to\mu^{-}\bar{\nu}_{e}$, established the upper limit for
the ratio of the process as Fig. 6 (where there is no SM contribution for the
second channel),

$\displaystyle R_{\nu
e}=\frac{\sigma(\bar{\nu}_{\mu}e^{-}\to\mu^{-}\bar{\nu}_{e})}{\sigma(\nu_{\mu}e^{-}\to\mu^{-}\nu_{e})}<\bigg{\\{}\begin{array}[]{c}0.09\,(90\%\,\text{C.L.)~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Amsterdam-CERN-Hamburg-Moscow-
Rome:1980qbb}{\@@citephrase{(}}{\@@citephrase{)}}} }\\\
\,0.017\,(90\%\,\text{C.L.)~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{NuTeV:2001bgq}{\@@citephrase{(}}{\@@citephrase{)}}}
}\end{array},$ (29)

In fact, there is an additional $u$-channel diagram from $Z^{\prime}$
contribution supply for the $\nu_{\mu}e^{-}\to\mu^{-}\nu_{e}$ despite the SM
($W$) diagram. However, compared to the SM contribution, the $Z^{\prime}$
contribution can be neglected. Therefore, the resulting ratio can be
simplified as follows:

$\displaystyle\frac{\sigma(\bar{\nu}_{\mu}e^{-}\to\mu^{-}\bar{\nu}_{e})}{\sigma(\nu_{\mu}e^{-}\to\mu^{-}\nu_{e})}\simeq\frac{g^{4}_{e\mu}}{2\,G_{F}^{2}\,M^{2}_{Z^{\prime}}s},(M_{Z^{\prime}}<\sqrt{s}),$
(30)
$\displaystyle\frac{\sigma(\bar{\nu}_{\mu}e^{-}\to\mu^{-}\bar{\nu}_{e})}{\sigma(\nu_{\mu}e^{-}\to\mu^{-}\nu_{e})}\simeq\frac{g^{4}_{e\mu}}{3\,G_{F}^{2}\,M^{4}_{Z^{\prime}}},(M_{Z^{\prime}}>\sqrt{s}).$
(31)

where $\sqrt{s}\approx\sqrt{2E_{\nu_{\mu}}m_{e}}$. In this study, the limit
was established by the newest NuTeV search.

###  III.5 Muonium-anti Muonium oscillation ($\rm{Mu}-\overline{\rm{Mu}}$)

One specific observable, Muonium (Mu), forming a bound state of
$\mu^{+}e^{-}$, would be relevant for exploring new physics, as it could
induce Muonium-anti Muonium transitions due to the mediation of the
$Z^{\prime}$ between two bound states (e.g., $\mu^{+}+e^{-}\to\mu^{-}+e^{+}$
in Fig. 7.) Kriewald:2022erk ; Fukuyama:2021iyw .

$\displaystyle e^{-}$$\displaystyle e^{+}$$\displaystyle
Z^{\prime}$$\displaystyle\mu^{-}$$\displaystyle\mu^{+}$$\displaystyle
e^{-}$$\displaystyle\mu^{+}$$\displaystyle
e^{+}$$\displaystyle\mu^{-}$$\displaystyle Z^{\prime}$ Figure 7: Feynman
diagrams for muonium (Mu) bound state.

This transition probability has been tested by the PSI experiment
Willmann:1998gd , and its expression could be labeled as follows (only vector
current and left/right-handed vertices are the same result in the constraint
only proportional to $g^{2}_{e\mu}/M^{2}_{Z^{\prime}}$),

$\displaystyle P$ $\displaystyle=$ $\displaystyle\frac{2.57\times
10^{-5}}{\text{G}^{2}_{f}}\left(|c_{0,0}|^{2}|-G_{3}+G_{123}|^{2}+|c_{1,0}|^{2}|G_{3}+G_{123}|^{2}\right)<8.3\times
10^{-11}$ (32)

with parameters,

$\displaystyle G_{1}$ $\displaystyle=$ $\displaystyle
G_{2}=\sqrt{2}\frac{g^{2}_{e\mu}}{8M^{2}_{Z^{\prime}}},\,G_{3}=\sqrt{2}\frac{2g^{2}_{e\mu}}{8M^{2}_{Z^{\prime}}},G_{123}=\frac{G_{1}+G_{2}-1/2\,G_{3}}{\sqrt{1+X^{2}}}.$
(33)

In the above equations $|c_{0,0}|^{2}=0.32$ and $|c_{1,0}|^{2}=0.18$ represent
the population states of the Mu Fukuyama:2021iyw  and $X=6.31\times
B/\rm{Tesla}$, with $B=0.1\,\,\rm{Tesla}$ Willmann:1998gd . By using the above
transition probability, we can set limits on the $Z^{\prime}$ couplings.

###  III.6 Neutrino trident production with maximal LFV $Z^{\prime}$

Neutrino trident production Fujikawa:1971nx ; Czyz:1964zz ; Koike:1971tu ;
Koike:1971vg ; Brown:1972vne , where a neutrino collides with the electric
field of a nucleus and produces a pair of charged leptons, is a rare process
involve neutral and charged lepton electroweak interactions. This phenomenon
has been investigated by several experimental groups, as listed in Table 2. It
is worth noting that the proposed SHiP experiment Pastore:2762117 ;
DiCrescenzo:2023czg  will be sensitive to LFV in future searches.
Additionally, these processes probe new types of interactions mediated by
particles beyond the SM, such as the LFV $Z^{\prime}$ boson. Previous
experiments have concentrated on utilizing produced $\mu^{+}\mu^{-}$ pairs to
constrain the couplings of $Z^{\prime}$ with leptons. These searches involving
$Z^{\prime}$ consider both off-diagonal couplings between different flavors
and diagonal couplings with the same flavor. The main challenge lies in
distinguishing the new physics signal from significant SM backgrounds, where
final states could include $e^{+}e^{-}\nu,\mu^{+}\mu^{-}\nu$, or
$e^{+}\mu^{-}\nu$, as shown in the lower figures of Fig. 8. However, the
incoming $\nu_{\mu}$ beam also provides sensitivity to $\mu^{+}e^{-}\nu$, a
final state not produced by SM. For instance, the DUNE experiment
Altmannshofer:2019zhy  distinguishes between incoming $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ fluxes, facilitating the isolation of final lepton states.
Hence, it offers a clear signal for searches involving the
$Z^{\prime}\mu^{+}e^{-}$ vertex, as shown in the upper two figures of Fig. 8.

Experiment |  $E^{\rm{mean}}_{\nu_{\mu}}$ (GeV) | Target
---|---|---
CHARM-II CHARM-II:1990dvf ; Conrad:1997ne  | $\sim 24$ | silicon (Z=14,A=28)
CCFR Sakumoto:1990py ; CCFR:1991lpl ; King:1991gs ; Spentzouris:1998pf  | 160 | iron (Z=26,A=56)
NuTeV NuTeV:1999wlw ; NuTeV:2001bgq  | 140 | iron (Z=26,A=56)
DUNE Magill:2016hgc ; Ballett:2018uuc ; Altmannshofer:2019zhy  |  $\thicksim$ 40 | argon (Z=18,A=40)
SHiP(proposed) Pastore:2762117 ; DiCrescenzo:2023czg  | $\thicksim 40$ | Tungsten (Z=74,A=184)
Table 2:  Previous and proposed experiments on neutrino tridents along with
their respective setup configurations.  $\displaystyle\nu_{\mu}$$\displaystyle
e^{-}$$\displaystyle
Z^{\prime}$$\displaystyle\mu^{+}$$\displaystyle\nu_{e}$$\displaystyle
e^{-}$$\displaystyle N$$\displaystyle
N$$\displaystyle\nu_{\mu}$$\displaystyle\mu^{+}$$\displaystyle
Z^{\prime}$$\displaystyle
e^{-}$$\displaystyle\nu_{e}$$\displaystyle\mu^{+}$$\displaystyle
N$$\displaystyle
N$$\displaystyle\gamma$$\displaystyle\gamma$$\displaystyle\nu_{\mu}$$\displaystyle\mu^{+}/e^{+}$$\displaystyle
Z$$\displaystyle\mu^{-}/e^{-}$$\displaystyle\nu_{\mu}$$\displaystyle
N$$\displaystyle N$$\displaystyle\nu_{\mu}$$\displaystyle W^{+}$$\displaystyle
N$$\displaystyle
N$$\displaystyle\gamma$$\displaystyle\gamma$$\displaystyle\mu^{+}/e^{+}$$\displaystyle\mu^{-}$$\displaystyle\nu_{\mu}/\nu_{e}$$\displaystyle\mu^{+}/e^{+}$$\displaystyle\mu^{+}/e^{+}$
Figure 8: Top panels: Feynman diagrams illustrating the neutrino trident
process ($\nu_{\mu}+N$) featuring off-diagonal lepton flavor violation (LFV)
interactions. Lower panels: The muon neutrino trident process mediated by the
SM electroweak interactions through $W/Z$ exchange. In the latter, photon
couplings with negatively charged leptons diagrams are omitted.

The amplitude for the distinctive clean signal channel
($\nu_{\mu}(k_{1})+\gamma(q)\to\nu_{e}(k_{2})+\mu^{+}(p_{1})+e^{-}(p_{2})$)
based on EPA method vonWeizsacker:1934nji ; Williams:1934ad  will be expressed
as follows222The EPA method is valid for dimuon channels when the neutrino
energy exceeds 20 GeV Ballett:2018uuc . For mixed flavor states, its shows a
factor of two enhancement against the full 2-to-4 calculations. To account for
this, we adjust the cross-section enhancement applying a factor of one-half
for large neutrino energy ($E_{\nu}>20$ GeV). Belusevic:1987cw ;
Vysotsky:2002ix ; Altmannshofer:2014pba ,

$\displaystyle\mathcal{M}_{1}$ $\displaystyle=$ $\displaystyle
g_{e\mu}\,e\,\bar{u}(p2)\bigg{(}\epsilon^{\mu}\gamma_{\mu}.\frac{g^{\alpha\beta}-\frac{k^{\alpha}k^{\beta}}{M_{Z^{\prime}}^{2}}}{k^{2}-M^{2}_{Z^{\prime}}}\frac{\gamma.(p_{1}-q)+m_{\mu}}{(p_{1}-q)^{2}-m^{2}_{\mu}}\gamma^{\alpha}$
$\displaystyle+$
$\displaystyle\epsilon^{\mu}\gamma_{\mu}.\frac{g^{\alpha\beta}-\frac{k^{\alpha}k^{\beta}}{M_{Z^{\prime}}^{2}}}{k^{2}-M^{2}_{Z^{\prime}}}\frac{\gamma.(p_{2}-q)+m_{e}}{(p_{2}-q)^{2}-m^{2}_{e}}\gamma^{\alpha}\bigg{)}v(p_{1}),$
$\displaystyle\mathcal{M}_{2}$ $\displaystyle=$ $\displaystyle
g_{e\mu}\bar{u}(k_{2})\gamma^{\beta}u(k_{1}),\,$ (35) $\displaystyle
d\sigma^{\rm{Z^{\prime}}}_{\nu_{\mu}\gamma}$ $\displaystyle\approx$
$\displaystyle\frac{1}{2s}\frac{1}{2}\frac{dt}{2s}\frac{dl}{2\pi}v\frac{d\Omega}{4\pi}|\mathcal{M}_{\nu_{\mu}\gamma\to\nu_{e}\mu^{+}e^{-}}|^{2},$
(36)

In the above expression $k=k_{1}-k_{2}$, and the velocity $v$ is defined as
$v=\frac{l-m^{2}_{\mu}}{l+m^{2}_{\mu}}$ where $m_{e}\to 0$. Here, $l$
represents the invariant momentum squared of the $\mu$ and $e$, given by
$l=(P_{\mu^{+}}+P_{e^{-}})^{2}$. The resulting amplitude is formulated to
exclude terms involving $m_{\mu}$, as they are deemed negligible compared to
other contributions. Subsequently, we reformulate the momentum products using
new Mandelstam variables Vysotsky:2002ix  (refer to the Appendix B for
details). After integrating out $t$ and $l$, we obtain the following
expression when $m_{\mu}<M_{Z^{\prime}}<\sqrt{s}$,

$\displaystyle\sigma^{Z^{\prime}}_{\nu_{\mu}\gamma}\simeq\frac{1}{M_{Z^{\prime}}^{2}}\frac{g_{e\mu}^{4}\alpha_{\text{EM}}}{2\pi^{2}}\log\left(\frac{M_{Z^{\prime}}^{2}}{m_{\mu}^{2}}\right),$
(37)

while for $M_{Z^{\prime}}<m_{\mu}<\sqrt{s}$, the cross-section can be
simplified as follows,

$\displaystyle\sigma^{Z^{\prime}}_{\nu_{\mu}\gamma}\simeq\frac{1}{m_{\mu}^{2}}\frac{2\,g_{e\mu}^{4}\alpha_{\text{EM}}}{5\,\pi^{2}}\log\left(\frac{m_{\mu}^{2}}{M_{Z^{\prime}}^{2}}\right),$
(38)

Thus, the total cross-section for the neutrino trident production with
distinctive $Z^{\prime}$ signals can be expressed as follows,

$\displaystyle\sigma(\nu_{\mu}+\rm{N}\to\nu_{e}+\mu^{+}+e^{-}+\rm{N})$
$\displaystyle\approx$
$\displaystyle\frac{Z^{2}\alpha_{\text{EM}}}{\pi}\int^{s_{\text{Max}}}_{s_{\text{Min}}}\frac{1}{s}ds\,\int^{q^{2}_{\text{Max}}}_{q^{2}_{\text{Min}}}dq^{2}\,\frac{1}{q^{2}}\,F^{2}(A,q^{2})\,\,\sigma^{\rm{Z^{\prime}}}_{\nu_{\mu}\gamma},$
(39) $\displaystyle q^{2}_{\text{Max}}$ $\displaystyle\thicksim$
$\displaystyle\left(\frac{m_{\pi}}{A^{1/3}}\right)^{2},\,q^{2}_{\text{Min}}=\left(\frac{s}{2E_{\nu}}\right)^{2},\,$
$\displaystyle s_{\text{Max}}$ $\displaystyle=$ $\displaystyle
2E_{\nu}\,q_{\text{Max}},\,s_{\text{Min}}=m^{2}_{\mu},$ (40)

where $Z,A$ are the relevant atomic number and atomic mass of the material in
the target and we adopted the electromagnetic nucleus form factor
$F(A,q^{2})=\text{exp}\\{-\frac{\left(1.3\text{fm}\,A^{1/3}\right)^{2}q^{2}}{10}\\}$
as in Ref. Altmannshofer:2014pba . We used the full
$\sigma^{Z^{\prime}}_{\nu\gamma}$ results to evaluate the expected
$Z^{\prime}$ signals. Since the process is essentially background free, the
number expected events would be proposed statistically three events at 95%
C.L. The number of coherent events will be obtained by the following form
Altmannshofer:2019zhy ,

$\displaystyle
N_{\text{trident}}=F_{\nu_{\mu}}\sigma(\nu_{\mu}N\to\nu_{e}\mu^{+}e^{-}N)\frac{M_{\text{detector}}}{M_{\text{target
atom}}}N_{\text{POT}}.$ (41)

where $F_{\nu_{\mu}}$ is the integrated $\nu_{\mu}$ flux correspond to the
incoming $\nu_{\mu}$ energy beam, and $N_{\text{POT}}$ is the number of
protons on target. The expected events are assumed to have 100% efficiency,
with searches covering 86% of the mixing flavor states Ballett:2018uuc ,
resulting in negligible differences in the outcomes. For the DUNE experiment,
the values are given by $F_{\nu_{\mu}}=1.04\times 10^{-3}$ m-2 POT-1 and
$N_{\text{POT}}=1.1\times 10^{21}$ POT per year. The ratio
$M_{\text{detector}}/M_{\text{target atom}}$ is correlated with DUNE’s argon
liquid detector (LArTPC) mass (147 tonnes) and atomic mass number (A = 40). In
this study, our focus lies on the DUNE experiment in general, with the
anticipation that estimations for other experiments would be similar.

Figure 9: The exclusion limit in the ($M_{Z^{\prime}}$, $g_{e\mu}$) parameter
space under various constraints. The blue region denotes $\Delta
a_{e}(2\sigma)$. The green band indicates the $\Delta a_{\mu}(2\sigma)$. The
exclusion of Mu-$\overline{\rm{Mu}}$ oscillation is represented by the brown
region Willmann:1998gd , while the black line depicts the exclusion limits for
DUNE with mean $\nu_{\mu}$ energy of 40 GeV Altmannshofer:2019zhy . The top
red region is exclusion from the LEP ($e^{+}e^{-}\to\mu^{+}\mu^{-}$) at
$\sqrt{s}=189$ GeV DELPHI:2000ztm  and the orange region is Belle-II exclusion
from ($e^{+}e^{-}\to\mu^{\pm}e^{\mp}+\not{E}$) Belle-II:2019qfb . The inverse
$\mu$ decay mode from the NuTeV search, providing the ratio $R_{\nu e}$ (eq.
(29)), is labeled in magenta region NuTeV:2001bgq . The dark green line labels
the exclusion region from $\mathcal{B}(\mu\to e+X)$ TWIST:2014ymv . The most
stringent constraints for $g_{e\mu}$ below $m_{\mu}$ originate from
$\mathcal{B}(\mu\to e+\gamma+X)$ (red) Goldman:1987hy .

##  IV Results

To verify the bounds for $g_{e\mu}$ and $M_{Z^{\prime}}$, we adopted the
Lagrangian generated by the Feynrules Alloul:2013bka  and using the Madgraph5
Alwall:2011uj  to calculate the LFV three-body decay and cross section results
to compare with the measurements. The decay widths of $Z^{\prime}$ and
analytical evaluations are obtained by using the FeynArt Hahn:2000kx  and
FeynCalc MERTIG1991345 ; Shtabovenko:2016sxi ; Shtabovenko:2020gxv . Since our
focus is within the mass range of $0.01$ GeV to $10$ GeV, we do not consider
the constraints from BBN and cosmology.

The constraints of $g_{e\mu}$ versus $M_{Z^{\prime}}$ are shown in Fig. 9. For
the collider approach, the LEP measurement of
$\sigma(e^{+}e^{-}\to\mu^{+}\mu^{-})$ DELPHI:2000ztm  has labeled the top red
region, while the orange region is excluded by Belle-II
($e^{+}e^{-}\to\mu^{\pm}e^{\mp}+\not{E}$) Belle-II:2019qfb . For the
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ process, the exclusion bound remains flat for
$M_{Z^{\prime}}>m_{\mu}$, primarily due to the dominant effect of the $1/s$
term. However, as $M_{Z^{\prime}}$ decreases further, the enhancement of
$1/M^{2}_{Z^{\prime}}$ dominates, results in a reduction of the cross-section,
leading to tighter limits as $M_{Z^{\prime}}$ approaches $m_{e}$. This
tightening effect is attributed to the $1/(t-M^{2}_{Z^{\prime}})$ from new
physics diagram.

The magenta and brown areas represent exclusion regions for inverse $\mu$
decay ratio $R_{\nu e}$, defined by equation eq. (29) and
Mu-$\overline{\rm{Mu}}$ oscillation Willmann:1998gd , respectively. From the
simplified analytical formulas eqs. (30 and 31), the constraints are
proportional to $g^{4}_{e\mu}/M^{2}_{Z^{\prime}}$ with an additional $1/s$
suppression for $M_{Z^{\prime}}<\sqrt{s}$ and to
$g^{4}_{e\mu}/M^{4}_{Z^{\prime}}$ for $M_{Z^{\prime}}>\sqrt{s}$. Meanwhile,
the limit from the brown region is proportional to
$g^{2}_{e\mu}/M^{2}_{Z^{\prime}}$, leading to a linear enhancement of the
bounds as $M_{Z^{\prime}}\to m_{e}$.

The green band shows the 2$\sigma$ allowed range of $\Delta a_{\mu}$. We see
that the $Z^{\prime}$ itself with $e-\mu$ couplings can only resolve the
discrepancy $\Delta{a_{\mu}}$ within small mass region
($0.04\,\text{GeV}\,<M_{Z^{\prime}}<0.08\,\text{GeV}$) while the experimental
constraints have already ruled out this scenario.

In the context of the neutrino trident process, the DUNE experiment estimates
the significance with a solid black line for a mean neutrino energy of
$E_{\nu_{\mu}}=40$ GeV. The exclusion region from trident events is slightly
more stringent than that from $\Delta a_{e}$, as indicated by the plot. From
eqs. (37 and 38), the cross-sections exhibit dependencies proportional to
$g^{4}_{e\mu}\log\left(1/M^{2}_{Z^{\prime}}\right)$ and
$g^{4}_{e\mu}/M^{2}_{Z^{\prime}}\log\left(M^{2}_{Z^{\prime}}\right)$,
respectively. As $M_{Z^{\prime}}$ increases from below $m_{\mu}$ to around 10
GeV, the constraint curve transitions smoothly, reflecting the enhanced cross-
section when $M_{Z^{\prime}}<m_{\mu}$ due to the logarithmic term
$\log\left(1/M^{2}_{Z^{\prime}}\right)$, which strengthens the coupling limit
$g_{e\mu}$. Conversely, the trident cross-section is suppressed as
$M_{Z^{\prime}}$ exceeds $m_{\mu}$ owing to the inverse-square dependence
$1/M_{Z^{\prime}}^{2}$, resulting in a weaker constraint on the coupling.

Based on the plot, it is apparent that the most stringent constraint arises
from the $\mathcal{B}(\mu\to e+\gamma+Z^{\prime})$ process when
$M_{Z^{\prime}}<m_{\mu}$. This limitation stems from the $M_{Z^{\prime}}$
enhancement in the denominator, an extra factor of $\alpha_{\text{EM}}$, and
the suppression of the three-body phase space, which includes an additional
factor related to the density of states when integrating the final momenta of
the particles, $\prod\limits^{3}_{i=1}\frac{d^{3}\vec{p}_{i}}{(2\pi)^{3}}$.
Therefore, considering these exclusion limits, we anticipate that the neutrino
trident experiment will capitalize on its background-free signal, thereby
offering a promising avenue to explore maximal LFV $Z^{\prime}$.

##  V Conclusions

In this work, we investigated a novel $U(1)^{\prime}$ gauge symmetry, focusing
on the $Z^{\prime}$ boson that couples with $e$ and $\mu$ through a specific
charge configuration. We propose a UV model featuring four scalars—three
doublets and one singlet—that contribute to the $M_{Z^{\prime}}$ after
symmetry breaking, without altering the SM $Z$ boson mass. The model employs a
discrete symmetry $\ell_{1L}\to\ell_{2L},\,e_{1R}\to
e_{2R},Z^{\prime}_{\mu}\to-Z^{\prime}_{\mu}$, leading to maximally off-
diagonal interactions between $Z^{\prime}$ and leptons, manifesting
exclusively as lepton flavor-violating (LFV) processes. A notable aspect
indicates that the Yukawa coupling between $\mu/e$ and the SM Higgs remains
identical. Given recent measurements of the Higgs’ leptonic decay branching
ratios, we anticipate that future precision measurements will be sensitive to
this model.

Next, we analyzed the existing experiments aimed at constraining the LFV
coupling $g_{e\mu}$ of $Z^{\prime}$. Specifically, the discrepancy between the
SM prediction and the measured magnetic moment of the muon, $\Delta a_{\mu}$,
cannot be explained by the $Z^{\prime}$, as the parameter space where
$Z^{\prime}$ could account for this discrepancy has already been ruled out by
other experimental measurements. We considered several LFV search limits,
including those from Belle-II, LEP, LFV decays, inverse muon decay, Muonium-
anti Muonium oscillation, and the newly proposed neutrino trident process
($\nu_{\mu}+\text{N}\to\nu_{e}+\mu^{+}+e^{-}+\text{N}$).

Neutrino trident processes are measured from the DUNE and some other similar
experiments to test the LFV interaction. Most searches have focused on muon
pair final states. However, the LFV interaction involve the maximal flavor
violation can contribute a distinctive signal $\mu^{+}e^{-}$ from $\nu_{\mu}$
beams without any SM background as the SM does not produce $\mu^{+}e^{-}$ in
the final state via $\nu_{\mu}$. We employed the neutrino trident production
method to investigate the $Z^{\prime}\mu^{+}e^{-}$ vertex, utilizing its
framework to incorporate the amplitude of a distinct, background-free signal
through the DUNE experiment. In the appendix, we listed the kinematic
variables for the 2-to-3 scattering process
$\nu_{\mu}+\gamma\to\nu_{e}+\mu^{+}+e^{-}$ and calculated the final cross-
section for an initial $\nu_{\mu}$ collides with the nucleus. The resulting
constraint on $Z^{\prime}$ is stricter compared to those from colliders and
LFV decays when $M_{Z^{\prime}}>m_{\mu}$, with the limit forming a quadratic
curve. For $M_{Z^{\prime}}<m_{\mu}$, the primary constraint comes from
$\mathcal{B}(\mu\to e+\gamma+Z^{\prime})$ due to the enhancement by the factor
$m^{3}_{\mu}/M^{2}_{Z^{\prime}}$ and the suppression both from the three-body
phase space factor and QED coupling $\alpha_{\text{EM}}$.

##  Appendix A Scalar potential with neutral scalars

The scalar potential can be expressed as follows, incorporating the discrete
symmetry ($\phi_{1}\to\phi_{2}$) Foot:1994vd ,

$\displaystyle V_{\phi_{1},\phi_{2},\phi_{3},\phi_{4}}$ $\displaystyle=$
$\displaystyle\lambda_{1}(\phi^{\dagger}_{3}\phi_{3}-v^{2}_{3})^{2}+\lambda_{2}(\phi^{\dagger}_{1}\phi_{1}+\phi^{\dagger}_{2}\phi_{2}-2\,v^{2}_{0})^{2}$
(42) $\displaystyle+$
$\displaystyle\lambda_{3}(\phi^{\dagger}_{1}\phi_{1}-\phi^{\dagger}_{2}\phi_{2})^{2}+\lambda_{4}(\phi^{\dagger}_{1}\phi_{1}+\phi^{\dagger}_{2}\phi_{2}+\phi^{\dagger}_{3}\phi_{3}-v^{2}_{3}-2\,v^{2}_{0})^{2}$
$\displaystyle+$
$\displaystyle\lambda_{5}[\phi^{\dagger}_{1}\phi_{1}\phi^{\dagger}_{2}\phi_{2}-\phi^{\dagger}_{1}\phi_{2}\phi^{\dagger}_{2}\phi_{1}]$
$\displaystyle+$
$\displaystyle\lambda_{6}[\phi^{\dagger}_{3}\phi_{3}(\phi^{\dagger}_{1}\phi_{1}+\phi^{\dagger}_{2}\phi_{2})-\phi^{\dagger}_{3}\phi_{1}\phi^{\dagger}_{1}\phi_{3}-\phi^{\dagger}_{3}\phi_{2}\phi^{\dagger}_{2}\phi_{3}]$
$\displaystyle+$
$\displaystyle\lambda_{7}[\phi^{\dagger}_{3}\phi_{3}(\phi^{\dagger}_{1}\phi_{1}+\phi^{\dagger}_{2}\phi_{2})-\phi^{\dagger}_{3}\phi_{1}\phi^{\dagger}_{3}\phi_{2}-\phi^{\dagger}_{1}\phi_{3}\phi^{\dagger}_{2}\phi_{3}]$
$\displaystyle+$
$\displaystyle\frac{1}{2}\,\mu^{2}_{4}\phi^{2}_{4}+\lambda_{44}\phi^{4}_{4},$

Based on the aforementioned reference and the alteration of our set up, one
can redefine the CP-even scalar components,

$\displaystyle
S_{1}=\frac{\rho_{1}+\rho_{2}}{\sqrt{2}},S_{2}=\frac{\rho_{1}-\rho_{2}}{\sqrt{2}},$
(43)

The CP even squared mass matrices can be expressed as follows ($v_{0}\equiv
v_{1}=v_{2}$),

$\displaystyle M^{2}_{S_{1}}$ $\displaystyle=$ $\displaystyle
8v^{2}_{0}\lambda_{2},$ (44) $\displaystyle M^{2}_{S_{2}}$ $\displaystyle=$
$\displaystyle 8\lambda_{3}v^{2}_{0}+2v^{2}_{3}\lambda_{7},$ (45)
$\displaystyle M^{2}_{H}$ $\displaystyle=$ $\displaystyle
4v_{3}^{2}\lambda_{1},$ (46) $\displaystyle M^{2}_{S_{4}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\mu^{2}_{4}+8v^{2}_{4}\lambda_{44}),$ (47)

The above $M_{H}$ would be the SM Higgs mass, with the flavor states and mass
states being identical ($\rho_{1},\rho_{2},\rho_{3},\rho_{4}\to
S_{1},S_{2},H,S_{4}$) when we assume $\lambda_{4}\to 0$. If $\lambda_{4}\neq
0$, one could have non-diagonalized mass matrices,

$\displaystyle
M^{2}_{S_{1},\rho_{3}}=\bigg{(}\begin{array}[]{cc}8v_{0}^{2}(\lambda_{2}+\lambda_{4})&4\sqrt{2}v_{0}v_{3}\lambda_{4}\\\
4\sqrt{2}v_{0}v_{3}\lambda_{4}&4v^{2}_{3}(\lambda_{1}+\lambda_{4})\end{array}\bigg{)},$
(50)

On the other hand, $M^{2}_{S_{2}}$ and $M^{2}_{S_{4}}$ are automatically
diagonalized as shown in eqs. (45 and 47), with $S_{2},S_{4}$ representing the
mass states. The states ($S_{1}$ and $\rho_{3}$) are no longer directly
equivalent to their mass eigenstates. Instead, the mass eigenstates
($S^{\prime}_{1}$ and $H$) are introduced, and their relationship can be
expressed through the following 2-by-2 mixing matrix, which rotates from the
flavor states $S_{1}$ and $\rho_{3}$ to the mass states $S^{\prime}_{1}$ and
$H$),

$\displaystyle\left(\begin{array}[]{cc}S_{1}\\\
\rho_{3}\end{array}\right)=\left(\begin{array}[]{cc}\frac{\sqrt{2}v_{0}v_{3}\lambda_{4}}{v^{2}_{3}\lambda_{1}-2v^{2}_{0}\lambda_{2}}&1\\\
1&\frac{\sqrt{2}v_{0}v_{3}\lambda_{4}}{-v^{3}_{3}\lambda_{1}+2v^{2}_{0}\lambda_{2}}\end{array}\right)\,\left(\begin{array}[]{c}H\\\
S^{\prime}_{1}\end{array}\right)+\mathcal{O}(\lambda^{2}_{4}),$ (57)

Thus, the flavor states ($\rho_{1},\rho_{2},\rho_{3},\rho_{4}$) transform into
the physical mass states ($S^{\prime}_{1},S_{2},H,S_{4}$) resembles the SM-
like Higgs ($H$).

In this context, we exclusively focus on the CP-even scalars due to their
correlation with SM-like scalar Yukawa couplings. The CP-odd and charged
components are not pertinent to our current discussion. The scalar potential
introduces numerous parameters, resulting in a vast parameter space that could
potentially include additional scalar states, which we do not investigate
here.

##  Appendix B Kinematic variables for the neutrino trident process.

The kinematic variables that transform the 4-momentum product into Mandelstam
variables in the 2-to-3 neutrino trident production
($\nu_{\mu}(k_{1})+\gamma(q)\to\nu_{e}(k_{2})+\mu^{+}(p_{1})+e^{-}(p_{2})$)
are expressed as follows:

$\displaystyle k_{1}+q$ $\displaystyle=$ $\displaystyle k_{2}+p_{1}+p_{2},$
(58) $\displaystyle k_{1}^{2}$ $\displaystyle=$ $\displaystyle
k_{2}^{2}=0,\,k=k_{1}-k_{2},$ (59) $\displaystyle\kappa_{1}$ $\displaystyle=$
$\displaystyle m_{\mu}^{2}-p_{1}^{2},\,\kappa_{2}=-p_{2}^{2},$ (60)
$\displaystyle s$ $\displaystyle=$
$\displaystyle(k_{1}+q)^{2},\,l=(p_{1}+p_{2})^{2},\,t=2k.q=\kappa_{1}+\kappa_{2}=l-k^{2}-q^{2},$
(61) $\displaystyle\kappa_{1}$ $\displaystyle\to$
$\displaystyle\frac{t}{2}\left(1+\frac{m^{2}_{\mu}}{l}+x(1-\frac{m^{2}_{\mu}}{l})\right),$
(62) $\displaystyle\kappa_{2}$ $\displaystyle\to$
$\displaystyle\frac{t}{2}\left(1-\frac{m^{2}_{\mu}}{l}-x(1-\frac{m^{2}_{\mu}}{l})\right),$
(63) $\displaystyle\kappa_{3}$ $\displaystyle\to$ $\displaystyle l-t,$ (64)
$\displaystyle k_{1}.q$ $\displaystyle\to$
$\displaystyle\frac{s}{2},\,k_{2}.q\to\frac{s-t}{2},\,p_{1}.q\to\frac{\kappa_{1}}{2},\,p_{2}.q\to\frac{\kappa_{2}}{2},$
(65) $\displaystyle k_{1}.p_{1}$ $\displaystyle\to$
$\displaystyle\frac{l-\kappa_{1}}{2}+\frac{s-l}{4}\left(1+\frac{m^{2}_{\mu}}{l}+x(1-\frac{m^{2}_{\mu}}{l})\right),$
(66) $\displaystyle k_{1}.p_{2}$ $\displaystyle\to$
$\displaystyle\frac{l-\kappa_{2}}{2}+\frac{s-l}{4}\left(1-\frac{m^{2}_{\mu}}{l}-x(1-\frac{m^{2}_{\mu}}{l})\right),$
(67) $\displaystyle k_{2}.p_{1}$ $\displaystyle\to$
$\displaystyle\frac{s-l}{4}\left(1+\frac{m^{2}_{\mu}}{l}+x(1-\frac{m^{2}_{\mu}}{l})\right),$
(68) $\displaystyle k_{2}.p_{2}$ $\displaystyle\to$
$\displaystyle\frac{s-l}{4}\left(1-\frac{m^{2}_{\mu}}{l}-x(1-\frac{m^{2}_{\mu}}{l})\right),$
(69) $\displaystyle p_{1}.p_{2}$ $\displaystyle\to$
$\displaystyle\frac{l}{2}-m^{2}_{\mu},\,k_{1}.k_{2}\to-\frac{\kappa_{3}}{2}.$
(70)

We employed the notations introduced in Ref. Vysotsky:2002ix , where
$x=v\cos\theta_{qp_{1}}$, with $v$ representing the velocity of the muon in
the center-of-mass (COM) frame of the $\mu^{+}(p_{1})e^{-}(p_{2})$ pair, and
we take $q^{2}\to 0$. In these expressions, the electron mass is neglected as
it remains the lightest particle regardless of the value of $M_{Z^{\prime}}$.

###### Acknowledgements.

We thank Jinhui Guo and Yuxuan He for their discussion on the early stage of
the work. The work of JL is supported by Natural Science Foundation of China
under grant No. 12075005 and 12235001.

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