###### Abstract

We present an analysis of the lepton-flavor violating decay of the Higgs boson
mediated by an ultralight gauge boson, $\chi$. Our analysis matches a model
generating the lepton flavor-violating interaction
$\bar{\ell}_{i}\ell_{j}\chi$ at tree level with an effective field theory,
safeguarding the $\chi$-boson mass as it tends to zero. By utilizing the upper
bounds on $H\to\ell_{i}\bar{\ell}_{j}$ decays from CMS and ATLAS
Collaborations, we establish an indirect upper limit on the nonstandard decay
$H\to\ell_{i}\bar{\ell}_{j}\chi$. The analysis encompasses various observables
such as the lepton energy spectrum, Dalitz plot distribution, and Lepton
Charge and Forward-Backward Asymmetries.

Lepton Flavor-Violating Higgs Decays Mediated by Ultralight Gauge Boson

Marcela<EMAIL_ADDRESS>R.
Gaitán$\,{}^{1}$, R. Martinez$\,{}^{2}$

1 Departamento de Física, FES-Cuautitlán, UNAM, C.P. 54770, Estado de México,
México.

2 Departamento de Física, Universidad Nacional de Colombia, K. 45 No. 26-85,
Bogotá, Colombia.

##  1 Introduction

The discovery of the Higgs boson in 2012 by the ATLAS and CMS Collaborations
at the Large Hadron Collider (LHC), with a mass of approximately $125~{}{\rm
GeV}$ [1, 2, 3], marked a significant milestone in our understanding of the
Standard Model (SM). Since then, precise measurements of its properties,
including decay branching fractions, have consistently aligned with the
expectations of the SM [4, 5, 6, 7, 8, 9, 10, 11]. However, amidst this
conformity lies an intriguing frontier: exploring decay channels that deviate
from the SM predictions. While extensive efforts are underway to measure the
decay rates within SM-favored channels precisely, there is a corresponding
need for commensurate attention to be directed toward those subdominant or
absent channels within the SM. Especially intriguing are scenarios in which
new physics phenomena may emerge, potentially surpassing the expectations set
forth by the SM.

The SM assumption that left-handed neutrinos are massless implies conserving
the lepton family number. However, neutrino oscillation experiments have
unequivocally demonstrated that lepton flavor conservation is not a
fundamental symmetry of nature [12, 13]. In the minimal extension of the SM
where neutrinos have nonvanishing masses, charged Lepton Flavor Violation
(cLFV) is made possible through neutrino oscillation. Even so, cLFV processes
are heavily suppressed by the GIM mechanism, making them unobservable in
current experiments. Nevertheless, the potential detection of cLFV would
provide an enticing avenue for exploring physics Beyond the Standard Model
(BSM).

Lepton-flavor violating (LFV) decays $H\rightarrow\mu e$111In this manuscript,
the notation $H\to\mu e$ refers to $H\to\mu\bar{e}$ or $H\to\bar{\mu}e$. The
same convention applies to the other cLFV modes., $H\rightarrow\tau e$, or
$H\rightarrow\tau\mu$ are strictly forbidden within the confines of the SM
framework and effectively suppressed in its minimal extension [14]. However,
they find theoretical justification in BSM theories, where LFV interaction can
arise through off-diagonal LFV Yukawa couplings, which facilitate the
interaction of the Higgs boson with leptons of differing flavors. These
processes are primarily induced via virtual boson exchange mechanisms [15, 16,
17, 18, 19], often occurring within extended Higgs sectors, or involving
several scalar doublets [20, 21, 22, 23, 24, 25, 26]. LHC Collaborations have
undertaken experimental efforts to probe Higgs decays [27]. Specifically, the
ATLAS and CMS Collaborations have conducted extensive measurements for the SM
decay modes $H\rightarrow\tau\bar{\tau}$ and $H\rightarrow\mu\bar{\mu}$,
alongside establishing upper bounds for $H\rightarrow e\bar{e}$ and all
potentially cLFV channels of $H\rightarrow\ell_{i}\bar{\ell}_{j}$ decay. The
outcomes of these searches are detailed in Table 1.

SM modes | BR |  | cLFV modes | BR ($95\%$CL) |
---|---|---|---|---|---
$H\to\tau\bar{\tau}$ | $(6.0^{+0.8}_{-0.7})\%$ | [4] | $H\to\tau\mu$ | $\leq 1.5\times 10^{-3}$ | [28]
$H\to\mu\bar{\mu}$ | $(2.6\pm 1.3)\times 10^{-4}$ | [4] | $H\to\tau e$ | $\leq 2.2\times 10^{-3}$ | [28]
$H\to e\bar{e}$ |  $\leq 3.6\times 10^{-4}$ ($95\%$CL) | [29] | $H\to\mu e$ | $\leq 6.1\times 10^{-5}$ | [29]
Table 1: Experimental measured and upper bounds on the branching fraction of
$H\to\ell_{i}\bar{\ell}_{j}$.

This paper presents the phenomenological implications of cLFV Higgs decays,
$H\to\ell_{i}\bar{\ell}_{j}$, mediated by an ultralight gauge boson, $\chi$,
where the cLFV interaction with this boson arising at tree level.
Additionally, we delve into the scenario where $\chi$ is on-shell establishing
an upper bound for these decays. This paper is structured as follows: Section
2 provides an overview of the model inducing the cLFV interactions and its
correspondence with an effective Lagrangian. Section 3 discusses the
analytical outcomes of cLFV Higgs decays mediated by $\chi$, accompanied by
phenomenological analyses of these processes. We analyze scenarios with the
gauge boson off-shell in Section 3.1 and on-shell in Section 3.2. Finally,
Section 4 presents our conclusions.

##  2 Effective Lagrangian description

In the letter [30], one of the authors proposes two models enabling LFV
interactions via a gauge boson, denoted as $\chi$. Their focus lies on an
ultralight gauge boson, which ensures finiteness in the massless $\chi$-boson
limit. Although the initial paper confines the models to two generations,
expanding them to encompass three generations is straightforward [31]. We are
especially intrigued by the tree-level model that induces cLFV transitions at
the tree level.

In the tree-level model, a $U(1)_{\chi}$ gauge symmetry was introduced
alongside complex scalar fields, $\phi_{ij}$, $i\,,j=1\,,2$. The model
postulated that leptons possess generation-dependent charges under
$U(1)_{\chi}$. Additionally, the tree-level model considered that doublet
scalars acquire a vacuum expectation value, $\langle\phi_{jk}\rangle=v_{jk}$,
where the hypercharge $Y_{jk}=1/2$, and $q_{\phi_{jk}}=q_{L_{j}}-q_{e_{k}}$.
Table 2 summarizes the particle content and the corresponding spins and
charges under the symmetries. Here, $L_{i}=(\nu_{L_{i}},e_{L_{i}})$ and
$e_{R_{i}}$, where $i=1\,,2$, represent the standard model $SU(2)_{L}$ lepton
doublets and singlets, respectively.

| $L_{1}$ | $L_{2}$ | $e_{R_{1}}$ | $e_{R_{2}}$ | $\phi_{11}$ | $\phi_{12}$ | $\phi_{21}$ | $\phi_{22}$
---|---|---|---|---|---|---|---|---
spin | 1/2 | 1/2 | 1/2 | 1/2 | 0 | 0 | 0 | 0
$SU(2)_{L}$ | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2
$U(1)_{Y}$ | $-1/2$ | $-1/2$ | $-1$ | $-1$ | $Y_{11}$ | $Y_{12}$ | $Y_{21}$ | $Y_{22}$
$U(1)_{\chi}$ | $q_{L_{1}}$ | $q_{L_{2}}$ | $q_{e_{1}}$ | $q_{e_{2}}$ | $q_{\phi_{11}}$ | $q_{\phi_{12}}$ | $q_{\phi_{21}}$ | $q_{\phi_{22}}$
Table 2: Spins and charges under $SU(2)_{L}\times U(1)_{Y}\times U(1)_{\chi}$
model that describes LFV transitions at tree level. All fields are assumed to
be singlets under $SU(3)_{C}$.

The kinetic and the Yukawa Lagrangian are formulated as follows:

$\displaystyle{\cal L}_{\rm kin}$
$\displaystyle=\sum_{j=1}^{2}i(\overline{L}_{j}\not{D}L_{j}+\overline{e}_{R_{j}}\not{D}e_{R_{j}})+\sum_{j,k=1}^{2}(D_{\mu}\phi_{jk})^{\dagger}(D^{\mu}\phi_{jk})\;,$
$\displaystyle-{\cal L}_{\rm Yuk}$
$\displaystyle=\sum_{j,k=1}^{2}y_{jk}\overline{L}_{j}\phi_{jk}e_{R_{k}}+{\rm
h.c.}\,,$ (1)

where $D_{\mu}$ denotes the covariant derivative, given by

$\displaystyle D_{\mu}$
$\displaystyle=\partial_{\mu}+igW_{\mu}^{a}T_{a}+ig^{\prime}YB_{\mu}+ig_{\chi}q\chi_{\mu}~{}~{}{\rm
for~{}the}~{}SU(2)_{L}~{}{\rm doublets}\;,$ $\displaystyle D_{\mu}$
$\displaystyle=\partial_{\mu}+ig^{\prime}YB_{\mu}+ig_{\chi}q\chi_{\mu}~{}~{}{\rm
for~{}the}~{}SU(2)_{L}~{}{\rm singlets}\;,$ (2)

being $g$, $g^{\prime}$ and $g_{\chi}$ the coupling constants of $SU(2)_{L}$,
$U(1)_{Y}$ and $U(1)_{\chi}$, respectively.

The LFV vertex arises from recasting the kinetic Lagrangian in terms of the
mass eigenstates

$\displaystyle-{\cal
L}\supset\overline{e_{R}}ig_{e\mu}^{RR}\gamma^{\rho}\chi_{\rho}\mu_{R}+\overline{e_{L}}ig_{e\mu}^{LL}\gamma^{\rho}\chi_{\rho}\mu_{L}+{\rm
h.c.}\;,$ (3)

with

$\displaystyle g_{e\mu}^{RR}$
$\displaystyle=g_{\chi}(q_{e_{1}}-q_{e_{2}})\sin\theta_{R}\cos\theta_{R}\;,$
$\displaystyle g_{e\mu}^{LL}$
$\displaystyle=g_{\chi}(q_{L_{1}}-q_{L_{2}})\sin\theta_{L}\cos\theta_{L}\;,$
(4)

where it can be deduced that the mixing angles $\theta_{L}\,,~{}\theta_{R}$
and the masses of the particle can be written as a function of $y_{jk}$,
$v_{jk}$ and $q_{\phi_{jk}}$, see Appendix A.

We can match this model with an Effective Field Theory (EFT), which offers the
advantage of greater generality and fewer constraints. Within the framework of
general EFT, the interaction responsible for lepton flavor violation can be
described by the effective Lagrangian

$\mathcal{L}\supset
F_{1}\bar{e}\gamma_{\alpha}\chi^{\alpha}\mu+G_{1}\bar{e}\gamma_{\alpha}\gamma_{5}\chi^{\alpha}\mu+{\rm
h.c.}\,,$ (5)

where compared to the Lagrangian in Eq. (3), we match the effective couplings,
$F_{1}$ and $G_{1}$, to correspond with the parameters of the model

$F_{1}=\frac{1}{2}\left(g_{e\mu}^{RR}+g_{e\mu}^{LL}\right)\,,\quad
G_{1}=\frac{1}{2}\left(g_{e\mu}^{RR}-g_{e\mu}^{LL}\right)\,.$ (6)

By simplicity, if we assumed that the Yukawa couplings satisfy
$y_{22}>>y_{11}>>y_{12},y_{21}$ and all $v_{jk}=v$, so $q_{\phi_{jk}}=Q$. The
relevant parameters of the tree-level model after the spontaneous breaking of
the gauge symmetries are:

$\displaystyle m_{\mu}^{2}\simeq y_{22}^{2}v^{2}\,,\quad m_{e}^{2}\simeq
y_{11}^{2}v^{2}\,\quad m_{\chi}^{2}\simeq 4g_{\chi}^{2}Q^{2}v^{2}\,,$
$\displaystyle\sin{2\theta_{L}}\simeq-2\frac{y_{12}}{y_{22}}\,,\quad\sin{2\theta_{R}}\simeq-2\frac{y_{21}}{y_{22}}\,.$
(7)

Based on the assumptions in Eqs. (2), it can be found that $F_{1}$ and $G_{1}$
can be expressed as:

$\displaystyle
F_{1}=\frac{1}{4Q}\left[\left(q_{e_{2}}-q_{e_{1}}\right)y_{21}+\left(q_{L_{1}}-q_{L_{2}}\right)y_{12}\right]\frac{m_{\chi}}{m_{\mu}}\equiv
c^{v}\frac{m_{\chi}}{m_{\mu}}\,,$ $\displaystyle
G_{1}=\frac{1}{4Q}\left[\left(q_{e_{2}}-q_{e_{1}}\right)y_{21}-\left(q_{L_{1}}-q_{L_{2}}\right)y_{12}\right]]\frac{m_{\chi}}{m_{\mu}}\equiv
c^{a}\frac{m_{\chi}}{m_{\mu}}\,.$ (8)

Expanding to three generations, the model that induces LFV transitions at the
tree level is included in the low-energy effective Lagrangian with monopole
operators, as follows:

$\mathcal{L}_{\rm
eff}=f_{ij}\bar{\ell}_{i}\gamma^{\alpha}\chi_{\alpha}\ell_{j}+g_{ij}\bar{\ell}_{i}\gamma^{\alpha}\gamma_{5}\chi_{\alpha}\ell_{j}+{\rm
h.c.},$ (9)

where $\ell_{i}\,,\ell_{j}=e\,,\mu\,,\tau$ and $f_{ij}\,,~{}g_{ij}$ are the
effective couplings in the three generations case. Taking into account the
matching with the parameters of the model in Eqs. (2), we can simplify and
define:

$f_{ij}=c^{v}_{ij}\frac{m_{\chi}}{m_{\ell_{i}^{j}}}~{}~{}{\rm
and}~{}~{}g_{ij}=c^{a}_{ij}\frac{m_{\chi}}{m_{\ell_{i}^{j}}}\,,$ (10)

where $m_{\ell_{i}^{j}}$, represents the mass of the highest-generation lepton
between $\ell_{i}$ and $\ell_{j}$, and $c^{v}_{ij}$ and $c^{a}_{ij}$ are
dimensionless independent coefficients.

In [30], a model inducing LFV interaction at the one-loop level is also
proposed. Although our study focuses on LFV interactions at the tree level, it
is noteworthy that the one-loop level model introduces both monopole and
dipole operators in an effective Lagrangian.

##  3 LFV Higgs decays

In the Standard Model, the flavor-conserving interaction
$\bar{\ell}_{i}\ell_{i}H$ emerges, while our effective Lagrangian, as outlined
in Eq. (9), describes transitions for all lepton interactions, encompassing
scenarios with flavor-conserving interaction $\ell_{i}=\ell_{j}$ and flavor-
violating interactions $\ell_{i}\neq\ell_{j}$. Using this effective
Lagrangian, we can induce LFV processes $H\to\ell_{i}\bar{\ell}_{j}$ at the
loop level mediated by $\chi$, as well as $H\to\ell_{i}\bar{\ell}_{j}\chi$ at
the tree level. The forthcoming two subsections will present the analytical
and phenomenological outcomes of these decay processes.

###  3.1 Higgs LFV decays with $\boldsymbol{\chi}$ off-shell

Figure 1: One-loop contribution to $H\to\ell_{i}\bar{\ell_{j}}$, with
$\ell_{i}\,,\ell_{j}\neq\ell_{k}$ and $\ell_{i}\,,\ell_{j}=\ell_{k}$.

Equation (9) describes the Lagrangian that induces one-loop level LFV decays
of $H\to\ell_{i}\bar{\ell}_{j}$ for all possible flavor combinations,
including $H\to\mu e$, $\mu\tau$, and $e\tau$, mediated by the $\chi$-boson,
as depicted in Figure 1. When the $\chi$-boson is ultralight, the triangle
diagram contribution becomes predominant in $H\to\ell_{i}\bar{\ell}_{j}$
processes, which justifies our focus on this particular contribution in the
analysis. The contribution from the triangle diagram to the branching ratio of
$H\to\ell_{i}\bar{\ell}_{j}$, with $\ell_{k}$ into the loop, is given by:

$\displaystyle{\rm BR}(H\to\ell_{i}\ell_{j})=$
$\displaystyle\frac{\Gamma(H\to\ell_{i}\bar{\ell}_{j})+\Gamma(H\to\bar{\ell}_{i}\ell_{j})}{\Gamma_{H}}$
(11)
$\displaystyle\simeq\frac{m_{\chi}^{2}}{m_{\ell_{i}^{k}}^{2}m_{\ell_{j}^{k}}^{2}}$
$\displaystyle\frac{M_{H}m_{\ell_{k}}^{2}}{4\pi~{}\Gamma_{H}~{}v^{2}}\bigl{[}\lvert
c_{jk}^{v}c_{ik}^{a}-c_{ik}^{v}c_{jk}^{a}\rvert^{2}+\lvert
c_{jk}^{v}c_{ik}^{v}-c_{ik}^{a}c_{jk}^{a}\rvert^{2}\bigr{]}~{}\bigl{\lvert}\mathcal{F}(m_{\ell_{i}}\,,m_{\ell_{j}}\,,m_{\ell_{k}})\bigr{\rvert}^{2}\,,$

where we have conveniently neglected the masses of the leptons in the
kinematic expression, $\Gamma_{H}$ represents the total Higgs decay width, and
the loop function $\mathcal{F}(m_{\ell_{i}},m_{\ell_{j}},m_{\ell_{k}})$ is
described by

$\displaystyle\mathcal{F}(m_{\ell_{i}}\,,m_{\ell_{j}}\,,m_{\ell_{k}})=$
$\displaystyle\frac{1}{m_{\chi}^{2}}{\rm
A0}\left[m_{\ell_{k}}^{2}\right]-\left(\frac{M_{H}^{2}}{2m_{\chi}^{2}}-3\right){\rm
B0}\left[M_{H}^{2}\,,m_{\ell_{k}}^{2}\,,m_{\ell_{k}}^{2}\right]-3\Bigl{(}{\rm
B0}\left[m_{\ell_{i}}^{2}\,,m_{\ell_{k}}^{2}\,,m_{\chi}^{2}\right]+$
$\displaystyle{\rm B0}$
$\displaystyle\left[m_{\ell_{j}}^{2}\,,m_{\ell_{k}}^{2}\,,m_{\chi}^{2}\right]-\left(M_{H}^{2}+m_{\chi}^{2}\right){\rm
C0}\left[M_{H}^{2}\,,m_{\ell_{i}}^{2}\,,m_{\ell_{j}}^{2}\,,m_{\ell_{k}}^{2}\,,m_{\ell_{k}}^{2}\,,m_{\chi}^{2}\right]\Bigr{)}+2\,.$
(12)

A0, B0, and C0 denote the Passarino-Veltman integrals, and $\Lambda$, encoded
within these integrals, represents the cutoff energy scale associated with our
EFT framework. The function
$\mathcal{F}(m_{\ell_{i}},m_{\ell_{j}},m_{\ell_{k}})$ depends on the masses
involved in the process, including the Higgs mass, the lepton masses, and the
$\chi$-boson mass, as well as $\Lambda$. It is important to note that the
$\chi$-boson mass in the denominator of
$\mathcal{F}(m_{\ell_{i}},m_{\ell_{j}},m_{\ell_{k}})$ is canceled by the
$m_{\chi}^{2}$ term in the expression for ${\rm BR}(H\to\ell_{i}\ell_{j})$ in
Eq. (11), thus ensuring finitude in the massless limit of $m_{\chi}$.

From Eq. (11), it is evident that the branching ratio ${\rm
BR}(H\to\ell_{i}\ell_{j})$ relies on six free parameters: $\Lambda$,
$m_{\chi}$, and the coefficients $c_{ik}^{v}$, $c_{jk}^{v}$, $c_{ik}^{a}$,
$c_{jk}^{a}$. Please be aware that when $c_{ik}^{v}$ and $c_{ik}^{a}$ are
equal, the model constraints dictate that $\Gamma(H\to\ell_{i}\bar{\ell}_{j})$
vanishes identically. This arises from the fact that if
$c_{ik}^{v}=c_{ik}^{a}$, the charges under $U(1)_{\chi}$ are generation-
independent, precluding any tree-level LFV interaction. Our focus lies in a
scenario where $\chi$ is a light boson, and the decay $\mu\to e\chi$ is
viable, implying $m_{\chi}\leq m_{\mu}$. Additionally, our attention is
directed toward low-energy processes, thus $\Lambda$ remains below the
electroweak scale.

In Eq. (11), we illustrate the contribution of a single lepton, $\ell_{k}$,
within the loop. However, it is crucial to note that the interaction
$\bar{\ell}_{i}\ell_{j}\chi$ can either conserve ($\ell_{i}=\ell_{j}$) or
violate ($\ell_{i}\neq\ell_{j}$) lepton flavor. Consequently, our analysis
incorporates three distinct contributions to the triangle diagram: electrons,
muons, and taus participating in the loop, along with the associated
interferences. However, the interference effects are suppressed and thus
neglected.

In Figures 2, we present the branching ratios ${\rm BR}(H\to\ell_{i}\ell_{j})$
as a function of $m_{\chi}$ for the three cLFV channels: $H\to\tau\mu$,
$H\to\tau e$, and $H\to\mu e$. For our numerical analysis, we fix
$\Lambda=M_{H}$, consider $m_{\chi}$ ranging from $0$ to $m_{\mu}$, and
utilize the experimental total Higgs decay width,
$\Gamma_{H}=3.7^{~{}+1.9}_{~{}-1.4}$ MeV [32]. To ensure clarity and ease of
interpretation, we adopt the simplifying assumptions $c_{jk}^{a}=0$, thereby
suppressing the axial contribution. Suitable values are assigned to
$c_{jk}^{v}$, as indicated in each figure. The red line corresponds to the
scenario where all three lepton contributions are active within the loop,
i.e., $\tau$, $\mu$, and $e$. Conversely, when only a single lepton
contribution is activated into the loop, we represent it with purple, green,
and blue lines for $e$, $\mu$, and $\tau$, respectively. To contextualize our
findings, the grey line denotes the current upper limit, as tabulated in Table
1. Dashed lines are employed to address cases of line overlap, ensuring
clarity in the visualization of results.

(a) ${\rm BR}(H\to\tau\mu)$

(b) ${\rm BR}(H\to\tau e)$

(c) ${\rm BR}(H\to\mu e)$

Figure 2: ${\rm BR}(H\to\ell_{i}\ell_{j})$ as a function to
$m_{\chi}~{}\in~{}[0,m_{\mu}]$.

Our observations demonstrate distinct dominant contributions for each decay
process, as depicted in Figures 2(a), 2(b), and 2(c). For the decay
$H\to\tau\mu$, the diagrams featuring $\tau$ and $\mu$ within the loop yield
the dominant contributions. In the case of $H\to\tau e$, all three diagrams,
with $\tau$, $\mu$, and $e$ in the loop, exhibit comparable significance.
Conversely, for $H\to\mu e$, the dominant diagrams involve $\mu$ and $e$. This
behavior is explained by Eq. (11), which states that
$\mathrm{BR}(H\to\ell_{i}\ell_{j})\propto\frac{m_{\ell_{k}}}{m_{\ell_{i}}m_{\ell_{j}}}$.
The masses in the denominator result from the relationship between the
effective couplings $f_{jk}$ and $g_{jk}$ with the coefficients $c_{jk}^{v}$
and $c_{jk}^{a}$ as described in Eq. (10).

If the lepton active is an electron, then
$\mathrm{BR}(H\to\tau\mu)\propto\frac{m_{e}^{2}}{m_{\tau}^{2}m_{\mu}^{2}}$.
Conversely, if $\mu$ or $\tau$ are active, then
$\mathrm{BR}(H\to\tau\mu)\propto\frac{1}{m_{\tau}^{2}}$. Given that
$\frac{1}{m_{\tau}^{2}}\gg\frac{m_{e}^{2}}{m_{\tau}^{2}m_{\mu}^{2}}$, it
follows that the dominant contribution to this decay channel arises when
$\tau$ and $\mu$ are active in the loop. Similarly, the dominant contribution
to the decay $H\to\mu e$ arises from scenarios where the muon and electron are
active in the loop. With these leptons in the loop, $\mathrm{BR}(H\to\mu
e)\propto\frac{1}{m_{\mu}^{2}}$. Conversely, if the tau lepton is active,
$\mathrm{BR}(H\to\mu e)\propto\frac{1}{m_{\tau}^{2}}$. Given that
$\frac{1}{m_{\mu}^{2}}>\frac{1}{m_{\tau}^{2}}$, it follows that the
contribution from the muon and electron is more significant. Lastly, for the
decay $H\to\tau e$, $\mathrm{BR}(H\to\tau e)\propto\frac{1}{m_{\tau}^{2}}$ for
all possible values of $m_{\ell_{k}}$.

Furthermore, it is noteworthy that in Figures 2, the branching ratio displays
minimal sensitivity to the $\chi$-boson mass and remains finite in the
massless limit. It is important to acknowledge that without defining
$f_{jk}\propto m_{\chi}c_{jk}^{v}$ and $g_{jk}\propto m_{\chi}c_{jk}^{a}$ (see
Eq. (10)), an unphysical divergence emerges as $m_{\chi}$ approaches zero.
This divergence would render the results unreliable for a light $\chi$-boson
mass [30].

By employing the upper limits provided for ${\rm BR}(H\to
e\mu\,,e\tau\,,\mu\tau)$ and the flavor-conserving processes ${\rm BR}(H\to
ee\,,\mu\mu\,,\tau\tau)$, which are detailed in Table 1, we can discern
constraints on the individual coefficients $|c_{ik}^{v(a)}|$. Under the
assumption that all $|c_{ik}^{v}|\neq 0~{}{\rm and}~{}|c_{ik}^{a}|=0$, and
with the parameters $m_{\chi}=m_{\mu}/2$ and $\Lambda=M_{H}$ fixed, we derive
restrictions for all $|c_{ik}^{v}|$.

When analyzing the branching fractions, we exclusively consider the dominant
contributions within each process, as illustrated in Figures 2(a), 2(b), and
2(c) for the cLFV decays. Regarding the flavor-conserving channels
$H\to\ell_{i}\bar{\ell}_{i}$, which although are not directly analyzed, can be
described by Eq. (11), the dominant contributions arise when
$m_{\ell_{k}}=m_{\ell_{i}}$. This deduction is evident, recalling that ${\rm
BR}(H\to\ell_{i}\ell_{i})\propto\frac{m_{\ell_{k}}^{2}}{m_{\ell_{i}^{k}}^{2}m_{\ell_{i}^{k}}^{2}}$.
The derived constraint regions for all $|c_{ik}^{v}|$ are as follows:

$\displaystyle 0<|c_{\mu\mu}^{v}|\lesssim 5.26\times 10^{-4}\,,\quad
0<|c_{\tau\tau}^{v}|\lesssim 8.41\times 10^{-3}\,,\quad 0<|c_{ee}^{v}|\lesssim
3.96\times 10^{-5}\,,$ $\displaystyle 0<|c_{\mu e}^{v}|\lesssim 5.35\times
10^{2}\sqrt{\frac{1}{4.41\times 10^{12}+1.59\times
10^{19}|c_{ee}^{v}|^{2}}}\,,\quad 0<|c_{\tau\mu}^{v}|\lesssim 1.33\times
10^{-3}\quad\text{and}$ $\displaystyle 0<|c_{\tau e}^{v}|\lesssim 1.76\times
10^{-3}\sqrt{\frac{4.58\times 10^{13}-4.4\times 10^{17}|c_{\mu
e}^{v}|^{2}}{5.5\times 10^{13}+7.78\times 10^{17}|c_{ee}^{v}|^{2}}}\,.$ (13)

It is important to emphasize that these constraint regions remain valid across
different values of $m_{\chi}$, as we have shown that ${\rm
BR}(H\to\ell_{i}\ell_{j})$ exhibits minimal dependence on $m_{\chi}$.
Furthermore, under the assumption that $c_{ik}^{a}\neq 0$ and $c_{ik}^{v}=0$,
with consistent values for $m_{\chi}$ and $\Lambda$, the resultant constraint
regions would remain consistent. This limit on the coefficients $c_{ii}^{v}$
can, in turn, be translated into constraints on the Yukawa couplings,
$U(1)_{\chi}$ charges and gauge coupling, and expectation values of the scalar
doublets $\phi_{ij}$ in the tree-level model, ensuring the correct couplings
with the Higgs are reproduced. Let us finish this section, noting that based
on the analysis performed in [30] for the tree-level model, it is anticipated
that the lepton three-body decays $\mu\to 3e$ and $\tau\to 3\ell$ (in six
possible combinations when $\ell=e$ or $\mu$) would provide the most stringent
constraints on the coefficients $c_{jk}^{v(a)}$. However, the primary focus of
this work is not on deriving the optimal constraints for the coefficients
$c_{jk}^{v(a)}$, but rather on examining the lepton-flavor violating decay
mediated by an ultralight boson within the context of Higgs decays.

###  3.2 Higgs decays with $\boldsymbol{\chi}$ on-shell

Utilizing the effective Lagrangian outlined in Eq. (9), we can induce the
decays $H\to\ell_{i}\bar{\ell}_{j}\chi$ at the tree level, as depicted in Fig.
3. Here, we introduce the Mandelstam variables
$t\equiv(q_{\ell_{j}}+q_{\chi})^{2}$ and $s\equiv(q_{\ell_{i}}+q_{\chi})^{2}$,
where $q_{\ell_{j}}$, $q_{\ell_{i}}$, and $q_{\chi}$ denote the four-momenta
of $\ell_{j}$, $\ell_{i}$, and $\chi$, respectively. The differential decay
rate is then expressed as:

$\frac{d^{2}\Gamma(H\to\ell_{i}\bar{\ell}_{j}\chi)}{ds~{}dt}=\frac{1}{32(2\pi)^{3}M_{H}^{3}}\overline{|\mathcal{M}_{H\to\ell_{i}\bar{\ell}_{j}\chi}(s,t)|^{2}}\,,$
(14)

where the squared amplitude as a function of $s$ and $t$ is given in Appendix
B. The kinematic limits are

$\displaystyle t_{\binom{{\rm max}}{{\rm
m_{\ell_{i}}n}}}=\frac{1}{4s}\biggl{[}\left(M_{H}^{2}+m_{\chi}^{2}-m_{\ell_{i}}^{2}-m_{\ell_{j}}^{2}\right)^{2}-\left(\lambda^{1/2}[s,m_{\ell_{i}}^{2},m_{\chi}^{2}]\mp\lambda^{1/2}[s,M_{H}^{2},m_{\ell_{j}}^{2}]\right)^{2}\biggr{]}\,,$
(15)

and $(m_{\ell_{i}}+m_{\chi})^{2}\leq s\leq(M_{H}-m_{\ell_{j}})^{2}$, where
$\lambda[s,M_{H}^{2},m_{\ell_{j}}^{2}]$ is the usual Källén function.

Figure 3: Contribution to $H\to\ell_{i}\bar{\ell}_{j}\chi$. Here
$\ell_{i}\,,\ell_{j}=e\,,\mu\,,\tau$ and $\ell_{i}=\ell_{j}$ or
$\ell_{i}\neq\ell_{j}$.

By leveraging the constraint regions on the coefficients $|c_{ik}^{v}|$
outlined in Eqs. (3.1), we can establish indirect upper bounds on ${\rm
BR}(H\to\ell_{i}\ell_{j}\chi)$, which is defined as:

${\rm
BR}(H\to\ell_{i}\ell_{j}\chi)=\frac{\Gamma(H\to\ell_{i}\bar{\ell}_{j}\chi)+\Gamma(H\to\bar{\ell}_{i}\ell_{j}\chi)}{\Gamma_{H}}\,.$
(16)

In Figure 4, we present the upper bound on ${\rm
BR}(H\to\ell_{i}\ell_{j}\chi)$ across the range $m_{\chi}\in[0,m_{\mu}]$.
These bounds are derived from constraints established by the upper limits of
$H\to\ell_{i}\bar{\ell}_{j}$ decays, as outlined in Eqs. (3.1), while assuming
$c_{ik}^{a}=0$. Notably, similar to the ${\rm BR}(H\to\ell_{i}\ell_{j})$
decays, the 3-body Higgs decay ${\rm BR}(H\to\ell_{i}\ell_{j}\chi)$ displays
minimal dependence on the $\chi$-boson mass. The upper limits on ${\rm
BR}(H\to\ell_{i}\ell_{j}\chi)$ are several orders of magnitude more stringent
than those reported in experimental searches for Higgs decays [32],
particularly for the decay $H\to\mu e\chi$. Detecting such decays would
require substantial experimental effort, including effectively managing the
decay background allowed by the Standard Model, such as
$H\to\ell_{i}\bar{\ell}_{i}\gamma$, especially when the photon has very low
energy.

Figure 4: Upper bound on ${\rm BR}(H\to\ell_{i}\ell_{j}\chi)$ as a function to
$m_{\chi}\in[0,m_{\mu}]$. We use the constraint regions in Eqs. (3.1) for the
$|c_{ij}^{v}|$.

####  3.2.1 Lepton Energy Spectrum

Analyzing the lepton energy spectrum for these processes,
$H\to\ell_{i}\bar{\ell}_{j}\chi$ is also interesting. In the Higgs rest frame,
$\vec{p}_{H}=\vec{0}$ then $E_{H}=M_{H}$, where $\vec{p}_{H}$ and $E_{H}$
represent the momentum and energy of the Higgs, respectively. Consequently,
the Mandelstam variable $t=M_{H}^{2}+m_{\ell_{i}}^{2}-2M_{H}E_{\ell_{i}}$. By
applying a variable change to Eq. (14), we can express the partial decay rate
as a function of $s$ and $E_{\ell_{i}}$ as follows:

$\frac{d^{2}\Gamma(H\to\ell_{i}\bar{\ell}_{j}\chi)}{dsdE_{\ell_{i}}}=\frac{1}{(2\pi)^{3}16M_{H}^{2}}\overline{|\mathcal{M}_{H\to\ell_{i}\bar{\ell}_{j}\chi}(s,E_{\ell_{i}})|^{2}}\,,$
(17)

where

$\displaystyle m_{\ell_{i}}\leq
E_{\ell_{i}}\leq\frac{M_{H}^{2}+m_{\ell_{i}}^{2}-(m_{\ell_{j}}+m_{\chi})^{2}}{2M_{H}}.$

In Figure 5, we present the lepton energy spectrum
$\frac{1}{\Gamma_{H}}\frac{d\Gamma(H\to\ell_{i}\bar{\ell}_{j}\chi)}{dE_{\ell_{i}}}$
as a function of $E_{\ell_{i}}$, where $\Gamma_{H}$ denotes the experimental
total Higgs decay width. This figure illustrates the decays
$H\to\tau\bar{\mu}\chi$ and $H\to\mu\bar{e}\chi$. The lepton energy spectrum
for $H\to\tau\bar{e}\chi$ resembles that shown in Fig. 5(a) for the analogous
decay $H\to\tau\bar{\mu}\chi$. We set $c_{ij}^{a}=0$ and apply the upper
constraints for $|c_{ij}^{v}|$ as outlined in Eqs. (3.1). We explore three
different choices of $\chi$-boson mass: $m_{\chi}=0$ (blue line),
$m_{\chi}=m_{\mu}/2$ (green line), and $m_{\chi}=m_{\mu}$ (purple line).
Notably, all scenarios exhibit overlapping spectra, indicating that the decays
$H\to\ell_{i}\bar{\ell}_{j}\chi$ display minimal dependence on the
$\chi$-boson mass. This outcome is consistent with the observations in Fig. 4
and the previous analysis of processes with $\chi$ off-shell:
$H\to\ell_{i}\bar{\ell}_{j}$, as illustrated in Figs. 2.

(a) $H\to\tau\bar{\mu}\chi$ as function of $E_{\tau}$.

(b) $H\to\mu\bar{e}\chi$ as function of $E_{\mu}$.

Figure 5: Energy spectrum
$\frac{1}{\Gamma_{H}}\frac{d\Gamma(H\to\ell_{i}\bar{\ell}_{j}\chi)}{dE_{\ell_{i}}}$
as a function of $E_{\ell_{i}}$. We assume $c_{ij}^{a}=0$ while $|c_{ij}^{v}|$
follows the constraints derived in Eqs. (3.1). We consider three options for
the $\chi$-boson mass: $m_{\chi}=0\,,$ $m_{\chi}=m_{\mu}/2$, and
$m_{\chi}=m_{\mu}$.

####  3.2.2 Angular Observables

We also examined the decays $H\to\ell_{i}\bar{\ell}_{j}\chi$ as functions of
lepton energy ($E_{\ell_{i}}$) and an angular variable
$\cos{\theta_{\ell_{i}\ell_{j}}}$. Here, $\theta_{\ell_{i}\ell_{j}}$
represents the angle between the momenta of the two leptons in the rest frame
of the $\ell_{i}-\chi$ system, where
$\vec{q}_{\ell_{i}}+\vec{q}_{\chi}=\vec{0}$. Consequently, we have
$|\vec{p}_{H}|=|\vec{q}_{\ell_{j}}|=\sqrt{E_{H}^{2}-M_{H}^{2}}$ and
$|\vec{q}_{\ell_{i}}|=|\vec{q}_{\chi}|=\sqrt{E_{\ell_{i}}^{2}-mi^{2}}$, with

$E_{H}=\frac{(E_{\ell_{i}}+E_{\chi})^{2}+M_{H}^{2}-m_{\ell_{j}}^{2}}{2(E_{\ell_{i}}+E_{\chi})}\,.$

Then $s=(E_{\ell_{i}}+E_{\chi})^{2}$ and
$t=m_{\ell_{j}}^{2}+m_{\chi}^{2}+2\left(E_{\ell_{j}}E_{\chi}+|\vec{q}_{\ell_{i}}||\vec{q}_{\ell_{j}}|\cos{\theta_{\ell_{i}\ell_{j}}}\right)$.
Thus, the partial decay rate as a function of $E_{\ell_{i}}$ and
$\cos{\theta_{\ell_{i}\ell_{j}}}$ can be expressed as:

$\displaystyle\frac{d^{2}\Gamma(H\to\ell_{i}\bar{\ell}_{j}\chi)}{dE_{\ell_{i}}d\cos{\theta_{\ell_{i}\ell_{j}}}}=$
$\displaystyle\frac{\lambda^{1/2}[\left(E_{\chi}+E_{\ell_{i}}\right)^{2}\,,M_{H}^{2}\,,m_{\ell_{j}}^{2}]}{(2\pi)^{3}16M_{H}^{2}E_{\chi}}(E_{\chi}+E_{\ell_{i}})|\vec{q}_{\ell_{i}}|~{}\overline{|\mathcal{M}_{H\to\ell_{i}\bar{\ell}_{j}\chi}(\cos{\theta_{\ell_{i}\ell_{j}}},E_{\ell_{i}})|^{2}}\,.$
(18)

where $\lambda$ is the Källén function.

We explored various angular observables for the processes with $\chi$ on-
shell: $H\to\tau\mu\chi\,,$ $H\to\tau e\chi\,,$ $H\to\mu e\chi$. For the
following analyses, we assumed $c_{ij}^{a}=0$ and determined the values of
$|c_{ij}^{v}|$ based on the constraints presented in Eqs. (3.1). Additionally,
we considered three scenarios for the $\chi$-boson mass: $m_{\chi}=0$,
$m_{\chi}=m_{\mu}/2$, and $m_{\chi}\sim m_{\mu}$.

(a) $m_{\chi}=0$

(b) $m_{\chi}=\frac{m_{\mu}}{2}$

(c) $m_{\chi}\sim m_{\mu}$

Figure 6: Dalitz Plot of the decay $H\to\tau\bar{\mu}\chi$ as a function of
$E_{\tau}$ and $\cos\theta_{\tau\mu}$.

In Figure 6, we illustrate the normalized partial decay rate
$\frac{d^{2}\Gamma(H\to\ell_{i}\bar{\ell}_{j}\chi)}{dE_{\ell_{i}}d\cos{\theta_{\ell_{i}\ell_{j}}}}$
as a Dalitz plot. The distributions of Dalitz plots for the other two
processes, $H\to\tau\bar{e}\chi$ and $H\to\mu\bar{e}\chi$, resemble that shown
in Fig. 6. Please be aware that the Dalitz plot distribution is not sensitive
to the $\chi$-boson mass, consistent with the other observables explored.

To investigate whether differential distributions in
$H\to\ell_{i}\bar{\ell}_{j}\chi$ decays can provide insights into the mass of
the $\chi$-boson, we consider the Lepton Charge Asymmetry as a function of
$\cos\theta_{\ell_{i}\ell_{j}}$. This asymmetry is defined as follows:

$\mathcal{A}^{L-C}(H\to\ell_{i}\ell_{j}\chi)=\frac{\frac{d\Gamma(H\to\ell_{i}\bar{\ell}_{j}\chi)}{d\cos\theta_{\ell_{i}\ell_{j}}}-\frac{d\Gamma(H\to\bar{\ell}_{i}\ell_{j}\chi)}{d\cos\theta_{\ell_{i}\ell_{j}}}}{\frac{d\Gamma(H\to\ell_{i}\bar{\ell}_{j}\chi)}{d\cos\theta_{\ell_{i}\ell_{j}}}+\frac{d\Gamma(H\to\bar{\ell}_{i}\ell_{j}\chi)}{d\cos\theta_{\ell_{i}\ell_{j}}}}\,.$
(19)

(a) $H\to\tau\mu\chi$ as function of $\cos\theta_{\tau\mu}$.

(b) $H\to\tau e\chi$ as function of $\cos\theta_{\tau e}$.

(c) $H\to\mu e\chi$ as function of $\cos\theta_{\mu e}$.

Figure 7: Lepton Charged Asymmetry
$\mathcal{A}^{L-C}(H\to\ell_{i}\ell_{j}\chi)$ as a function of
$\cos\theta_{\ell_{i}\ell_{j}}$. We assume $c_{ij}^{a}=0$ while $|c_{ij}^{v}|$
follows the constraints derived in Eqs. (3.1). We consider three options for
the $\chi$-boson mass: $m_{\chi}=0$ (olive line), $m_{\chi}=m_{\mu}/2$ (cyan
line), and $m_{\chi}=m_{\mu}$ (magenta line).

This observable is plotted in Fig. 7 for the lepton flavor-violating channels:
$H\to\tau\mu\chi$, $H\to\tau e\chi$, $H\to\mu e\chi$. In Fig. 7, we took the
heaviest lepton as the particle ($\ell_{i}$) in the positive partial decay
rate for the asymmetry calculations. To maintain consistency with the
definition in Eq. (19), we list the heaviest lepton first in the notation.

For decays involving the lepton $\tau$, the asymmetry $\mathcal{A}^{L-C}$ is
generally negative, except when $m_{\chi}\sim 0$ in the decay $H\to\tau
e\chi$. The asymmetry $\mathcal{A}^{L-C}$ is close to zero for $H\to\mu e\chi$
when $m_{\chi}$ is large but becomes negative at $m_{\chi}=0$ and positive for
intermediate values of the $\chi$-boson mass. As shown in Fig. 7(a), for
$H\to\tau\mu\chi$, the asymmetry can distinguish between intermediate and
large values of $m_{\chi}$ with about $9\%$ precision, and between small and
large values with about $25\%$ precision for small values of
$\cos\theta_{\tau\mu}$. In the decay $H\to\tau e\chi$, this observable is
quite sensitive to the $\chi$-boson mass for small values of $m_{\chi}$ but
can differentiate between intermediate and large values with about $5\%$
precision. The decay $H\to\mu e\chi$ shows the greatest sensitivity to
$m_{\chi}$, particularly for large values of $\cos\theta_{\mu e}$. Remarkably,
the Lepton Charge Asymmetry exhibits significant sensitivity to the
$\chi$-boson mass, especially when $m_{\chi}\sim 0$ and in decays involving an
electron, such as $H\to\tau e\chi$ and $H\to\mu e\chi$.

We also analyzed the Forward-Backward Asymmetry as a function to
$E_{\ell_{i}}$ defined as follow

$\mathcal{A}^{F-B}(H\to\ell_{i}\bar{\ell}_{j}\chi)=\frac{\int_{-1}^{0}\frac{d\Gamma(H\to\ell_{i}\bar{\ell}_{j}\chi)}{dE_{\ell_{i}}d\cos\theta_{\ell_{i}\ell_{j}}}d\cos\theta_{\ell_{i}\ell_{j}}-\int_{0}^{1}\frac{d\Gamma(H\to\ell_{i}\bar{\ell}_{j}\chi)}{dE_{\ell_{i}}d\cos\theta_{\ell_{i}\ell_{j}}}d\cos\theta_{\ell_{i}\ell_{j}}}{\int_{-1}^{0}\frac{d\Gamma(H\to\ell_{i}\bar{\ell}_{j}\chi)}{dE_{\ell_{i}}d\cos\theta_{\ell_{i}\ell_{j}}}d\cos\theta_{\ell_{i}\ell_{j}}+\int_{0}^{1}\frac{d\Gamma(H\to\ell_{i}\bar{\ell}_{j}\chi)}{dE_{\ell_{i}}d\cos\theta_{\ell_{i}\ell_{j}}}d\cos\theta_{\ell_{i}\ell_{j}}}$
(20)

We illustrate this observable in Fig. 8 as a function of $E_{\ell_{i}}$. It is
apparent that this observable is sensitive to the $\chi$-boson mass,
particularly in the low energy range. This sensitivity is more pronounced in
channels involving an electron and when $\chi$ is ultralight. For the channel
$H\to\bar{\mu}\tau\chi$, the sensitivity is slight; at best, there is a $\sim
10\%$ difference between the lightest and heaviest values of the $\chi$-boson.
In the decays $H\to e\bar{\tau}\chi$ and $H\to e\bar{\mu}\chi$, these
observables exhibit significant sensitivity to the $\chi$-boson mass at small
$m_{\chi}$ values. Additionally, in the best-case scenario, they can
distinguish between intermediate and large $m_{\chi}$ values with a precision
of about $15\%$ and $13\%$, respectively.

Remarkably, the asymmetries are the only analyzed observables that exhibit
sensitivity to the $\chi$-boson mass. While this analysis assumes a
background-free scenario, it is important to note that achieving the required
precision for such measurements, particularly for
$H\to\ell_{i}\bar{\ell}_{j}\gamma$ decays, would be challenging. Attaining
this precision would necessitate meticulous and dedicated analysis to ensure
accurate results.

(a) $H\to\mu\bar{\tau}\chi$ as function of $E_{\mu}$.

(b) $H\to e\bar{\tau}\chi$ as function of $E_{e}$.

(c) $H\to e\bar{\mu}\chi$ as function of $E_{e}$.

Figure 8: Forward-Backward Asymmetry
$\mathcal{A}^{F-B}(H\to\ell_{i}\bar{\ell}_{j}\chi)$ as a function of $E_{i}$.
We assume $c_{ij}^{a}=0$ while $|c_{ij}^{v}|$ follows the constraints derived
in Eqs. (3.1). We consider three options for the $\chi$-boson
mass:$m_{\chi}=0$ (olive line), $m_{\chi}=m_{\mu}/2$ (cyan line), and
$m_{\chi}=m_{\mu}$ (magenta line).

##  4 Conclusions

Lepton Flavor Violation is highly suppressed in the SM extended with right-
handed neutrinos, but it may manifest at measurable rates in several well-
motivated BSM models. The discovery of the Higgs boson at the LHC has
introduced a new avenue for exploring LFV in the decays of this scalar boson.

In this study, we explored the influence of an ultralight gauge boson, $\chi$,
in mediating cLFV transitions. Our analysis incorporates a model that
generates the lepton flavor-violating interaction $\bar{\ell}_{i}\ell_{j}\chi$
at the tree level into an effective field theory, ensuring the $\chi$-boson
mass remains well-defined as it approaches zero. We examined the LFV Higgs
decay phenomenology under both on-shell and off-shell $\chi$ conditions. The
results indicate that these processes exhibit minimal dependence on the
$\chi$-boson mass, due to the established conditions for the observables to
remain finite in the massless limit.

However, the Lepton Charge and Forward-Backward Asymmetries, display
sensitivity to $m_{\chi}$. These asymmetries indirectly reveal information
about the range of the $\chi$-boson mass as a function of the lepton energy or
the angle $\theta_{\ell_{i}\ell_{j}}$. This sensitivity provides a potential
method for constraining the mass of the $\chi$-boson in experimental
observations.

Utilizing the upper bounds and measurements of Higgs decays into leptons,
$H\to\ell_{i}\bar{\ell}_{j}$ from CMS and ATLAS Collaborations, we derived
indirect upper limits on the decays $H\to\ell_{i}\bar{\ell}_{j}\chi$. These
constraints provide critical insights into the potential signatures of LFV in
Higgs decays mediated by an ultralight gauge boson. Collider experiments with
enhanced sensitivity to LFV Higgs decay could potentially uncover the presence
of an ultralight gauge boson.

## Acknowledgments

The research presented herein has been supported by the UNAM Postdoctoral
Program (POSDOC) and the PAPIIT project IN102122. We are grateful to Pablo
Roig for his invaluable comments and feedback.

##  Appendix A Appendix: mixing angles and masses in the tree level model

In this appendix, we provide expressions for the $\chi$-boson mass, the masses
of the leptons ($m_{\mu}$ and $m_{e}$), and the mixing angles ($\sin
2\theta_{L}$ and $\sin 2\theta_{R}$) as functions of $y_{jk}$, $v_{jk}$, and
$q_{\phi_{jk}}$ within the context of the tree-level model. After the
spontaneous symmetry breaking of the symmetry $U(1)_{\chi}$, the non-zero
expectation values for $\phi_{jk}$ generate a mass for the $\chi$ boson:

$\displaystyle
m^{2}_{\chi}=g_{\chi}^{2}(q_{\phi_{11}}^{2}v_{11}^{2}+q_{\phi_{12}}^{2}v_{12}^{2}+q_{\phi_{21}}^{2}v_{21}^{2}+q_{\phi_{22}}^{2}v_{22}^{2})\;.$
(21)

The expectation value of the doublet scalars generates a mass term for the
charged leptons, $-{\cal L}_{\rm
mass}\supset\overline{e_{L_{j}}}M_{jk}e_{R_{k}}+{\rm h.c.}$, with

$\displaystyle M=\begin{pmatrix}y_{11}v_{11}&y_{12}v_{12}\\\
y_{21}v_{21}&y_{22}v_{22}\\\ \end{pmatrix}\;.$ (22)

We now rotate the fields to express the Lagrangian on the mass eigenstate
basis:

$\displaystyle\begin{pmatrix}e_{L}\\\
\mu_{L}\end{pmatrix}=\begin{pmatrix}\cos\theta_{L}&\sin\theta_{L}\\\
-\sin\theta_{L}&\cos\theta_{L}\end{pmatrix}\begin{pmatrix}e_{L_{1}}\\\
e_{L_{2}}\end{pmatrix}\,,~{}~{}~{}~{}~{}~{}~{}\begin{pmatrix}e_{R}\\\
\mu_{R}\end{pmatrix}=\begin{pmatrix}\cos\theta_{R}&\sin\theta_{R}\\\
-\sin\theta_{R}&\cos\theta_{R}\end{pmatrix}\begin{pmatrix}e_{R_{1}}\\\
e_{R_{2}}\end{pmatrix}$ (23)

so that $-{\cal
L}_{mass}\supset\overline{e_{L}}m_{e}e_{R}+\overline{\mu_{L}}m_{\mu}\mu_{R}\,+$
h.c., with

$\displaystyle m_{\mu}^{2}$ $\displaystyle\simeq
y_{11}^{2}v_{11}^{2}+y_{12}^{2}v_{12}^{2}+y_{21}^{2}v_{21}^{2}+y_{22}^{2}v_{22}^{2}\;,$
$\displaystyle m_{e}^{2}$
$\displaystyle\simeq\frac{(y_{11}v_{11}y_{22}v_{22}-y_{12}v_{12}y_{21}v_{21})^{2}}{y_{11}^{2}v_{11}^{2}+y_{12}^{2}v_{12}^{2}+y_{21}^{2}v_{21}^{2}+y_{22}^{2}v_{22}^{2}}\;,$
$\displaystyle\sin 2\theta_{L}$
$\displaystyle\simeq-2\frac{y_{11}v_{11}y_{21}v_{21}+y_{12}v_{12}y_{22}v_{22}}{y_{11}^{2}v_{11}^{2}+y_{12}^{2}v_{12}^{2}+y_{21}^{2}v_{21}^{2}+y_{22}^{2}v_{22}^{2}}\;,$
$\displaystyle\sin 2\theta_{R}$
$\displaystyle\simeq-2\frac{y_{11}v_{11}y_{12}v_{12}+y_{21}v_{21}y_{22}v_{22}}{y_{11}^{2}v_{11}^{2}+y_{12}^{2}v_{12}^{2}+y_{21}^{2}v_{21}^{2}+y_{22}^{2}v_{22}^{2}}\;,$
(24)

where we have used that empirically $m_{\mu}\gg m_{e}$.

##  Appendix B Appendix: Squared Amplitude

In this appendix, we provide the squared amplitude for the decay with $\chi$
on-shell, $H\to\ell_{i}\bar{\ell}_{j}\chi$, as a function of the Mandelstam
variables $s$ and $t$:

$\displaystyle\overline{|\mathcal{M}_{H\to\ell_{i}\bar{\ell}_{j}\chi}(s,t)|^{2}}$
$\displaystyle\simeq\frac{2}{m_{\ell_{i}^{j}}^{2}v^{2}}\Biggl{[}\frac{m_{\ell_{j}}^{2}}{\Gamma_{\ell_{j}}^{2}m_{\ell_{j}}^{2}+(m_{\ell_{j}}^{2}-s)^{2}}\biggl{[}\left(\left|c^{a}_{ij}\right|^{2}+\left|c^{v}_{ij}\right|^{2}\right)\Bigl{(}M_{H}^{2}(m_{\ell_{i}}^{4}+m_{\ell_{i}}^{2}(m_{\chi}^{2}-2s)$
$\displaystyle+m_{\ell_{j}}^{2}m_{\chi}^{2}-2m_{\chi}^{4}+s^{2})-m_{\ell_{i}}^{4}(3m_{\ell_{j}}^{2}+s)+m_{\ell_{i}}^{2}(m_{\ell_{j}}^{4}-m_{\ell_{j}}^{2}(2m_{\chi}^{2}-5s+t)$
$\displaystyle+s(-2m_{\chi}^{2}+2s+t))+t(m_{\ell_{j}}^{2}-s)(s-2m_{\chi}^{2})+(m_{\chi}^{2}-s)((m_{\ell_{j}}^{2}+s)^{2}+8m_{\ell_{j}}^{2}m_{\chi}^{2})\Bigr{)}$
$\displaystyle+6m_{\ell_{i}}m_{\ell_{j}}m_{\chi}^{2}\left(\left|c^{a}_{ij}\right|^{2}-\left|c^{v}_{ij}\right|^{2}\right)(M_{h}^{2}-2(m_{\ell_{j}}^{2}+s))\biggr{]}+{\scriptstyle\left\\{\begin{matrix}s\leftrightarrow
t\\\ m_{\ell_{i}}\leftrightarrow m_{\ell_{j}}\\\
\Gamma_{\ell_{j}}\rightarrow\Gamma_{\ell_{i}}\end{matrix}\right\\}}\Biggr{]}$
(25)

In this expression, the interference terms are subdominant and have been
neglected. Here, $\Gamma_{\ell_{k}}$ denotes the total decay width of the
lepton $\ell_{k}$. It is important to note that the squared amplitude does not
exhibit divergences for $m_{\chi}$, ensuring the finiteness of the decay rate
in the massless limit.

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