# Dark SU(2) Stueckelberg portal

Valery E. Lyubovitskij  Institut für Theoretische Physik, Universität
Tübingen,
Kepler Center for Astro and Particle Physics,
Auf der Morgenstelle 14, D-72076 Tübingen, Germany  Departamento de Física y
Centro Científico Tecnológico de Valparaíso-CCTVal,
Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile
Millennium Institute for Subatomic Physics at the High-Energy Frontier
(SAPHIR) of ANID,
Fernández Concha 700, Santiago, Chile  Alexey S. Zhevlakov  Bogoliubov
Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia  Matrosov
Institute for System Dynamics and Control Theory SB RAS,
Lermontov str., 134, 664033, Irkutsk, Russia  Aliaksei Kachanovich  Rudjer
Bo$\breve{s}$kovi$\acute{c}$ Institute, Division of Theoretical Physics,
Bijeni$\acute{c}$ka 54, HR-10000 Zagreb, Croatia  Serguei Kuleshov  Millennium
Institute for Subatomic Physics at the High-Energy Frontier (SAPHIR) of ANID,
Fernández Concha 700, Santiago, Chile  Departamento de Ciencias Físicas,
Universidad Andres Bello, Sazié 2212, Piso 7, Santiago, Chile

###### Abstract

We study the non-Abelian $SU(2)_{D}$ extension of the $U(1)_{D}$ Stueckelberg
portal, which plays the role of mediator between the Standard Model (SM) and
dark sector (DS). This portal is specified by the Stueckelberg mechanism for
generation of dark gauge boson masses. The proposed $U(1)_{D}\otimes
SU(2)_{D}$ Stueckelberg portal has a connection with SM matter fields, in
analogy with the familon model. We derive bounds on the couplings of dark
portal bosons and SM particles, which govern diagonal and nondiagonal flavor
transitions of quarks and leptons.

††preprint: RBI-ThPhys-2022-37

##  I Introduction

The Standard Model (SM) of particle physics is a unified gauge theory of
strong and electroweak interactions, which allows one to perform a precise
description and explanation of most of data extracted at worldwide facilities
wherein, there are signals of new physics, which cannot be explained within
the SM and, therefore, require its extensions. These include, in particular,
the muon anomalous magnetic moment with last measurements at Fermilab Abi et
al. (2021), where the current deviation of the SM prediction and experimental
data lies at the $\sim 4.2\sigma$ level Aoyama et al. (2020). Besides this,
there are further unresolved puzzles, such as the strong $CP$ problem and rare
meson decays Pospelov and Tsai (2018); Zhevlakov and Lyubovitskij (2020);
Zhevlakov et al. (2019a, b); Gutsche et al. (2017), flavor nonuniversality Ema
et al. (2017); Buras et al. (2021); Crivellin and Hoferichter (2021), the
$b-s$ quark anomaly Aebischer et al. (2020), neutrino mass generation, and
etc. These puzzles initiated many efforts for SM extensions and new physics
searches.

The existence of dark matter (DM) is required by a wide spectrum of
gravitational, astrophysical, and cosmological phenomena. DM significantly
contributes to the mass of the Universe. However, its precise nature is not
yet known. The search for DM created an idea of portals between SM and DM
particles. In this respect the main popular portals are Higgs Arcadi et al.
(2020), axion Nomura and Thaler (2009), axion-like particles Bauer et al.
(2017, 2019, 2020); Zhevlakov et al. (2022); Kaneta et al. (2017), vector
Fortuna et al. (2021); Buras et al. (2021), and sterile neutrino Escudero et
al. (2017) ones. Recently, we proposed the $U(1)_{D}$ dark gauge invariant
vector portal between particles of the SM and Dark Sector (DS) Kachanovich et
al. (2022), where the of mass of dark photon $A^{\prime}$ is generated via the
Stueckelberg mechanism (see also Refs. Kribs et al. (2022); Ruegg and Ruiz-
Altaba (2004)). Additional scalar field $\sigma$ occurring in that approach is
an unphysical degree of freedom, which plays the role of ghost field. There is
a popular idea of interaction of gauge field and SM fermions based on the
ansatz of a familon (or flavons) in the literature Feng et al. (1998); Bauer
et al. (2020); Cornella et al. (2020). Such an interaction mechanism between
dark photon and SM fermions leads to a rich phenomenology including novel
information on couplings preserving and violating lepton symmetries, e.g.,
lepton flavor violation (LFV).

Phenomenological studies of the dark photon Holdom (1986) have been performed
using different scenarios and particle content (see, e.g., Refs. Buras et al.
(2021); Fayet (2017)). In particular, the dark photon could interact not only
with a QED photon-induced so-called kinetic mixing term Holdom (1986), but
also with leptonic pair including neutrinos (so-called $Z^{\prime}$ boson).
Additional vector gauge bosons have been extensively studied and searched
during three decades Altarelli et al. (1989). Especially, very promising
studies have been performed at CERN, Fermilab, and other experimental
facilities Khachatryan et al. (2017); Aaboud et al. (2017); Aad et al. (2019a,
2021); Banerjee et al. (2017, 2018a, 2018b); Cazzaniga et al. (2021); Andreev
et al. (2022a, b). One should stress that phenomenological models considered
before were not limited by simple kinetic mixing between SM and new gauge
bosons Pankov et al. (2020); Osland et al. (2022, 2021); Fuks and Ruiz (2017).
Indeed, other possible scenarios have been also considered.

The main motivation of present paper is an extension of the Abelian $U(1)_{D}$
Stueckelberg portal Kachanovich et al. (2022) by adding the non-Abelian
$SU(2)_{D}$ sector because it gives an opportunity to study the processes with
charged dark gauge bosons and currents. In particular, inclusion of additional
particles from the $SU(2)_{D}$ portal could give a chance to understand
existing deviations between SM and experiments. In particular, we introduce
the $SU(2)_{D}$ triplet of dark gauge bosons (DGBs). Such an extension is
natural and has the additional benefit of opening a window for couplings
between SM and DS particles via charged currents. Note that the Abelian
$U(1)_{D}$ Stueckelberg portal is based on the coupling of a dark photon with
neutral current formed by SM fermions, while inclusion of the $SU(2)_{D}$ DGBs
interacting with both charged and neutral currents can be performed in analogy
to the weak sector of SM. Thereby, all properties of the Stueckelberg portal
such as LFV are implemented. Besides, the LFV effect Heeck (2017), such
$SU(2)_{D}$ extension induces lepton number violation.

Discussion of the dark fermion sector, which is closely related to charge dark
gauge bosons $W^{\prime\pm}$, is beyond the scope of present paper. By the
way, we note that it is a very interesting topic (see, e.g., recent discussion
in Ref. Belyaev et al. (2022)). Instead, we focus on phenomenological aspects
of the $SU(2)_{D}$ vector Stueckelberg portal and derivation of bounds on its
couplings with SM fermions based on current deviations between the SM and
data. In our study, we intend to consider both LFV and non-LFV lepton decays
as a tool for such analysis. We also mention that the new $W^{\prime}$ bosons
have been intensively studied inn different theoretical approaches beyond the
SM. In particular, the $W^{\prime}s$ bosons have been proposed by several
theoretical approaches based on gauge extensions of the Standard Model
Altarelli et al. (1989); Eichten et al. (1984), extradimensional Randall and
Sundrum (1999a, b); Davoudiasl et al. (2001), technicolor Lane and Mrenna
(2003); Eichten and Lane (2008), and composite Higgs models Agashe et al.
(2005); Giudice et al. (2007) models. Experimental searches of the
$W^{\prime}$ bosons have been performed at LHC by the ATLAS and CMS
Collaborations Aaboud et al. (2019); Aad et al. (2019b); Sirunyan et al.
(2019); Tumasyan et al. (2022) based on the production of the $W^{\prime}$ in
proton-proton collisions, which led to the constraint of the $W^{\prime}$ mass
in the range 0.15-6 TeV. In our approach we assume that the dark $W^{\prime}$
boson masses do not exceed TeV region.

The paper is organized as follows. In Sec. II, we define the Lagrangian for
the non-Abelian extension of the dark sector. In Sec. III, we estimate bounds
on the couplings of the charged DGBs imposed by various muon decay processes,
with $\mu\to e\nu_{i}\bar{\nu}_{j}$ in Sec. III.1 and $\mu\to e\gamma$ in Sec.
III.2. The latter is relevant for the muon $g-2$ anomaly. Finally, in Sec. IV
we present our conclusions.

##  II Non-Abelian extension of Dark Sector

In this section, we discuss an extension of the Abelian dark Stueckelberg
portal to the non-Abelian case. In particular, we propose the existence of
three additional DGBs and three additional DSBs associated with $SU(2)_{D}$
group, which are required by gauge invariant principle. Our formalism is based
of effective SM+DS Lagrangian ${\cal L}_{\rm SM+DS}$, which after spontaneous
breaking of electroweak symmetry to electromagnetic group $U(1)_{Y}\otimes
SU(2)_{L}\to U(1)_{\rm em}$ is manifestly gauge-invariant under the product of
electromagnetic group and groups of DS and $U(1)_{\rm em}\otimes
U(1)_{D}\otimes SU(2)_{D}$. First we specify the fields of our Lagrangian. For
convenience, in this paper we use the notations introduced before for the
Abelian case Kachanovich et al. (2022) and introduce additional ones relevant
for the non-Abelian dark sector sector. The SM sector contains fundamental
fermions — left $q_{L}^{im}$ and right ($U_{R}^{i}$, $D_{R}^{i}$) quarks, left
and right leptons $(\ell_{L}^{im},\ell_{R}^{i})$ fields), gauge fields (weak
gauge bosons $W^{\pm}$, $Z^{0}$ and photon $A$) and scalar Higgs field $H$.
Left doublets and right singlets of quarks and leptons are specified as
$q_{L}^{1m}=(u_{L},d_{L})$, $q_{L}^{2m}=(c_{L},s_{L})$,
$q_{L}^{3m}=(t_{L},b_{L})$, $U_{R}^{1}=u_{R}$, $U_{R}^{2}=c_{R}$,
$U_{R}^{3}=t_{R}$, $D_{R}^{1}=u_{R}$, $D_{R}^{2}=c_{R}$, $D_{R}^{3}=b_{R}$,
$L^{1m}=(\nu_{eL},e_{L})$, $L^{2m}=(\nu_{\mu L},\mu_{L})$, $L^{3m}=(\nu_{\tau
L},\tau_{L})$, $R^{1}=e_{R}$, $R^{2}=\mu_{R}$, and $R^{3}=\tau_{R}$. The
indices $i,j=1,2,3$ number the fermion generations, while $m,n$ denote the
$SU(2)$ weak isospin and dark indices, respectively.

The DS sector contains singlet dark fermion $\chi$, gauge bosons (singlet
$A^{\prime}$ and triplet ${\bf W^{\prime}}$),and scalars (singlet $\sigma$ and
triplet ${\bf S}$). The matrices ${\bf S}$ and ${\bf W}^{\prime}$ are
specified as

$\displaystyle{\bf
S}=\left(\begin{array}[]{cc}S^{0}/\sqrt{2}&S^{+}\\\\[2.84526pt]
S^{-}&-S^{0}/\sqrt{2}\\\\[2.84526pt] \end{array}\right)\,,\quad{\bf
W}^{{}^{\prime}}=\left(\begin{array}[]{cc}W^{{}^{\prime}0}/\sqrt{2}&W^{{}^{\prime}+}\\\\[2.84526pt]
W^{{}^{\prime}-}&-W^{{}^{\prime}0}/\sqrt{2}\\\\[2.84526pt]
\end{array}\right)\,.$ (5)

Triplets of gauge bosons $W^{\prime}$ and scalars $S$ have electric charge. In
particular, $W^{{}^{\prime}\pm}$ and $S^{\pm}$ have charges $Q=\pm 1$,
respectively, while $W^{{}^{\prime}0}$, $S^{0}$, and dark fermions are
electrically neutral.

Now we define the covariant derivatives acting on fermions and scalars. In the
case of the SM sector the change will be in adding of the terms containing
dark gauge fields in the covariant derivatives acting on the left and right
fermions

$\displaystyle\Big{(}iD^{L}\Big{)}_{\mu}^{mn}\to\Big{(}iD^{L}\Big{)}_{\mu}^{mn}-\frac{g_{W^{\prime}}}{2}\,{\bf
W}^{{}^{\prime}mn}_{\mu}-g_{A^{\prime}}\,\delta^{mn}\,A^{\prime}_{\mu}\,,\qquad\Big{(}iD^{R}\Big{)}_{\mu}\to\Big{(}iD^{R}\Big{)}_{\mu}-g_{A^{\prime}}\,A^{\prime}_{\mu}\,,$
(6)

where $g_{W^{\prime}}$ and $g_{A^{\prime}}$ are the gauge couplings associated
with $SU(2)_{D}$ and $U(1)_{D}$ groups, respectively, In the case of the DS,
the covariant derivates acting on scalar fields $(D_{\mu}\sigma)$ and
$(D_{\mu}{\bf S})$ and fermions are defined respectively, as

$\displaystyle(D_{\mu}\sigma)=\partial_{\mu}{\bf\sigma}-M_{A^{\prime}}\,A^{\prime}_{\mu}\,,\qquad(D_{\mu}{\bf
S})^{mn}=\partial_{\mu}{\bf S}^{mn}-M_{W^{\prime}}\,{\bf
W}^{{}^{\prime}mn}_{\mu}$ (7)

and

$\displaystyle\Big{(}iD_{\chi}\Big{)}_{\mu}=i\,\partial_{\mu}-g_{A^{\prime}}\,A^{\prime}_{\mu}\,.$
(8)

Note that covariant derivatives acting on dark scalars $(D_{\mu}\sigma)$ and
$(D_{\mu}{\bf S})$ include dark gauge bosons (singlet $A^{\prime}$) and
triplet ($W^{{}^{\prime}\pm},W^{{}^{\prime}0}$) having finite masses
$M_{A^{\prime}}$ and $M_{W^{\prime}}$, respectively. The covariant derivatives
($D_{\mu}\sigma$) and ($D_{\mu}{\bf S}$) therefore contain DS gauge boson
masses, generated via the Stueckelberg mechanism Stueckelberg (1938); Ruegg
and Ruiz-Altaba (2004). The $U(1)_{D}$ gauge boson $A^{\prime}$ is called the
dark photon and it has the mass $M_{A^{\prime}}$. The scalar Stueckelberg
fields $\sigma$ and $\bf{S}$ play the role of supplementary Goldstone bosons
generating masses of dark photon and dark triplet gauge bosons ${\bf
W}^{\prime}$. Such an idea was considered before for the new $Z^{\prime}$
boson in Ref. Bell et al. (2017) (see also discussion in Ref. Gherghetta et
al. (2019)). DGBs acquire the masses in a manifestly gauge-invariant form.
Finite masses of scalars violate $U(1)_{D}\otimes SU(2)_{D}$ symmetry (for a
review see Ref. Ruegg and Ruiz-Altaba (2004)). Extension of the Stueckelberg
mechanism on non-Abelian field leads to known problems of renormalizability
and unitarity. Critical remarks and efforts to resolve these problems have
been discussed in detail in Ref. Ruegg and Ruiz-Altaba (2004). Note that, due
to the scale of new physics $\Lambda$ is much larger than the scale of SM
($\Lambda\gg\Lambda_{\rm SM})$; thus, we do not go to higher loops and
restrict ourselves to one-loop approximation. Therefore, the problems
mentioned above are not so critical for our purposes. Note that, for
convenience we write all terms in Lagrangian involving the ${\bf S}$ triplet
in a simplified form, which follows from Ref. Delbourgo and Thompson (1986),
where this field was derived using adjoint representation of the $SU(2)_{D}$
gauge group

$\displaystyle U(x)=\exp\biggl{[}-ig_{W^{\prime}}\frac{{\bf
S}(x)}{2M_{W^{\prime}}}\biggr{]}\,.$ (9)

In this approach,

$\displaystyle\partial_{\mu}{\bf
S}(x)=\frac{2M_{W^{\prime}}}{g_{W^{\prime}}}\,i\partial_{\mu}U(x)\,U^{-1}(x)\,.$
(10)

We propose that the Stueckelberg mechanism Stueckelberg (1938); Ruegg and
Ruiz-Altaba (2004) for generating masses of gauge fields is extended to the
group $U(1)_{D}\otimes SU(2)_{D}$.

The stress tensors for dark gauge bosons are defined as

$\displaystyle
A_{\mu\nu}^{\prime}=\partial_{\mu}A_{\nu}^{\prime}-\partial_{\nu}A_{\mu}^{\prime}\,,\qquad{\bf
W}_{\mu\nu}^{\prime}=\partial_{\mu}{\bf W}_{\nu}^{\prime}-\partial_{\nu}{\bf
W}_{\mu}^{\prime}+\frac{ig_{W^{\prime}}}{2}\,[{\bf W}_{\mu}^{\prime},{\bf
W}_{\nu}^{\prime}]\,.$ (11)

The $U(1)_{D}\otimes SU(2)_{D}$ gauge transformations of dark fermions,
scalars, and gauge bosons are specified as

$\displaystyle A^{\prime}_{\mu}\to
A^{\prime}_{\mu}+\frac{i}{g_{A^{\prime}}}\,\partial_{\mu}\Omega_{A^{\prime}}\,\Omega^{-1}_{A^{\prime}}\,,\qquad\hskip
56.9055ptA_{\mu\nu}^{\prime}\to A_{\mu\nu}^{\prime}\,,$ $\displaystyle{\bf
W}^{\prime}_{\mu}\to\Omega_{W^{\prime}}\,{\bf
W}^{\prime}_{\mu}\,\Omega^{-1}_{W^{\prime}}+\frac{2i}{g_{W^{\prime}}}\,\partial_{\mu}\Omega_{W^{\prime}}\,\Omega^{-1}_{W^{\prime}}\,,\qquad{\bf
W}_{\mu\nu}^{\prime}\to\Omega_{W^{\prime}}\,{\bf
W}_{\mu\nu}^{\prime}\,\Omega^{-1}_{W^{\prime}}\,,\ $
$\displaystyle\partial_{\mu}\sigma\to\partial_{\mu}\sigma+\frac{iM_{A^{\prime}}}{g_{A^{\prime}}}\,\partial_{\mu}\Omega_{A^{\prime}}\,\Omega^{-1}_{A^{\prime}}\,,\qquad\hskip
39.83368pt(D_{\mu}{\bf\sigma})\to(D_{\mu}{\bf\sigma})\,,$ $\displaystyle
U(x)\to\Omega_{W^{\prime}}(x)\,U(x)\,,\qquad\qquad\qquad\qquad\qquad\partial_{\mu}{\bf
S}\to\Omega_{W^{\prime}}\,\partial_{\mu}{\bf
S}\,\Omega_{W^{\prime}}^{-1}\,+\,\frac{2iM_{W^{\prime}}}{g_{W^{\prime}}}\,\partial_{\mu}\Omega_{W^{\prime}}\,\Omega^{-1}_{W^{\prime}}\,,$
(12) $\displaystyle(D_{\mu}{\bf S})\to\Omega_{W^{\prime}}\,(D_{\mu}{\bf
S})\,\Omega_{W^{\prime}}^{-1}\,,\qquad\qquad\qquad\qquad{\rm
Tr}\Big{[}(D_{\mu}{\bf S})(D_{\mu}{\bf S})\Big{]}\to{\rm
Tr}\Big{[}(D_{\mu}{\bf S})(D_{\mu}{\bf S})\Big{]}\,,$
$\displaystyle\chi\to\Omega_{A^{\prime}}\,\chi\,,\hskip
147.95424pti\not\\!\\!D_{\chi}\chi\to\Omega_{A^{\prime}}\,i\not\\!\\!D_{\chi}\,\chi\,,$

where

$\displaystyle\Omega_{W^{\prime}}(x)=\exp\Big{[}\frac{i}{2}\vec{\mathbf{\theta}}_{W^{\prime}}(x)\vec{\mathbf{\tau}}\Big{]}\,,\qquad\Omega_{A^{\prime}}(x)=\exp\Big{[}i{\theta}_{A^{\prime}}(x)\Big{]}$
(13)

are the matrices of the fundamental $SU(2)_{D}$ and $U(1)_{D}$
transformations.

Now we specify effective Lagrangian of our approach ${\cal L}_{\rm SM+DS}$
combining SM and DS:

$\displaystyle{\cal L}_{\rm SM+DS}={\cal L}_{\rm SM}+{\cal L}_{\rm
DS}+\Delta{\cal L}\,.$ (14)

This Lagrangian is by construction a low-energy Lagrangian, which is an
extrapolation of new physics Lagrangian including the DS sector to the SM
scale $\Lambda_{\rm SM}\simeq M_{W^{\pm}/Z^{0}}\simeq 100$ GeV. Here ${\cal
L}_{\rm SM}$ is the term describing dynamics of the SM sector including the
coupling of the SM fermions with DS gauge fields via extension of covariant
derivatives introduced above, ${\cal L}_{\rm DS}$ is the term describing
dynamics of the DS including the coupling of the dark fermions with SM gauge
fields via extension of covariant derivatives introduced above and two terms
describing the coupling of SM and DS - GB mixing term ${\cal L}_{\rm mix}$ and
a term describing additional coupling of DSBs with SM fields $\Delta{\cal L}$
allowing a violation of lepton flavor and number violation and violation of
the Glashow-Illiopulos-Maiani (GIM) mechanism Glashow et al. (1970) in the
quark sector.

Next we specify the terms ${\cal L}_{\rm DS}$ and ${\cal L}_{\rm int}$. The DS
Lagrangian ${\cal L}_{\rm DS}$ is given by:

$\displaystyle{\cal L}_{\rm DS}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\,{A}_{\mu\nu}^{\prime}{A}^{\prime\mu\nu}\,-\,\frac{1}{4}\,{\bf
W}_{\mu\nu}^{\prime}{\bf
W}^{\prime\mu\nu}\,+\,\frac{1}{2}\,(D_{\mu}\sigma)(D^{\mu}\sigma)+\frac{1}{2}\,{\rm
Tr}\Big{[}(D_{\mu}{\bf S})(D^{\mu}{\bf
S})\Big{]}+\bar{\chi}\,(i\not\\!\\!D_{\chi}-m_{\chi})\,\chi$ (15)
$\displaystyle-$
$\displaystyle\frac{1}{2\xi_{W^{\prime}}}\text{Tr}\left(\partial_{\mu}{\bf
W}^{\prime\mu}+\xi_{W^{\prime}}M_{W^{\prime}}{\bf
S}\right)^{2}-\frac{1}{2\xi_{A^{\prime}}}\left(\partial_{\mu}{A}^{\prime\mu}+\xi_{A^{\prime}}M_{A^{\prime}}{\bf\sigma}\right)^{2}\,.$

One should stress that quantization of the dark gauge field is required to add
the gauge-fixing term into the dark $U(1)_{D}$ and $SU(2)_{D}$ sectors Kors
and Nath (2005) (last two terms in ${\cal L}_{\rm DS}$), where
$\xi_{W^{\prime}(A^{\prime})}$ is an arbitrary gauge parameter corresponding
to the $W^{\prime}(A^{\prime})$ gauge boson. It provides “decoupling” of gauge
bosons and corresponding scalars particles with vanishing mixed terms:

$\displaystyle\mathcal{L}_{DS}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\,{A}_{\mu\nu}^{\prime}{A}^{\prime\mu\nu}\,-\,\frac{1}{4}\,{\bf
W}_{\mu\nu}^{\prime}{\bf
W}^{\prime\mu\nu}\,+\,\frac{1}{2}\,\text{Tr}\,(\partial_{\mu}{\bf
S}\partial^{\mu}{\bf S})\,+\,\frac{M_{W^{\prime}}^{2}}{2}\,\text{Tr}({\bf
W_{\mu}}{\bf
W^{\mu}})\,+\,\frac{M_{A^{\prime}}^{2}}{2}A^{\prime}_{\mu}A^{\prime\mu}$ (16)
$\displaystyle-$
$\displaystyle\frac{1}{2\xi_{W^{\prime}}}\text{Tr}\,(\partial_{\mu}{\bf
W}^{\prime}\partial^{\mu}{\bf
W}^{\prime})\,-\,\frac{\xi_{W^{\prime}}}{2}M_{W^{\prime}}^{2}\text{Tr}\left({\bf
S}^{2}\right)\,+\,\frac{1}{2}\,\partial_{\mu}\sigma\partial^{\mu}\sigma\,-\,\frac{1}{2\xi_{A^{\prime}}}(\partial_{\mu}A^{\prime\mu})^{2}\,-\,\frac{\xi_{A^{\prime}}}{2}M_{A^{\prime}}^{2}\sigma^{2}$
$\displaystyle+$
$\displaystyle\bar{\chi}_{L}\,(i\not\\!\\!D_{\chi}^{L}-m_{\chi})\,\chi_{L}\,+\,\bar{\chi}_{R}\,(i\not\\!\\!D_{\chi}^{R}-m_{\chi})\,\chi_{R}\,.$

We note that the masses of the $\sigma$ and triplet $\bf{S}$ are proportional
to the gauge parameter $\xi_{i}$, signaling that these fields are unphysical.
In the gauge, we are using, the dark boson propagator takes the form

$\displaystyle
D^{\mu\nu}(k;\xi_{i})=\frac{1}{k^{2}-M^{2}}\,\biggl{[}g^{\mu\nu}-\frac{k^{\mu}k^{\nu}\,(1-\xi_{i})}{k^{2}-\xi_{i}M^{2}}\biggr{]}\,.$
(17)

where $M$ is the mass of the dark gauge boson.

The interaction $\Delta{\cal L}$ Lagrangian is constructed by analogy with the
familon model proposed in Ref. Feng et al. (1998) for the hypothetical familon
field. The scalar fields are unphysical and are to be switched off by the
choice of gauge fixing:

$\displaystyle\Delta{\cal L}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda}\,(D_{\mu}{\bf\sigma})\,\sum_{ij}\biggl{[}\bar{q}_{L}^{i}c^{ij}_{\sigma}\gamma^{\mu}q_{L}^{j}\,+\,\bar{U}_{R}^{i}c^{ij}_{U\sigma}\gamma^{\mu}U_{R}^{j}\,+\,\bar{D}_{R}^{i}c^{ij}_{D\sigma}\gamma^{\mu}D_{R}^{j}\,+\,\bar{L}^{i}d^{ij}_{L\sigma}\gamma^{\mu}L^{j}\,+\,\bar{R}^{i}d^{ij}_{R\sigma}\gamma^{\mu}R^{j}\biggr{]}$
(18) $\displaystyle+$ $\displaystyle\frac{1}{\Lambda}\,(D_{\mu}{\bf
S})^{mn}\,\sum_{ij}\biggl{[}\bar{q}_{L}^{im}c^{ij}_{\bf
S}\gamma^{\mu}q_{L}^{jn}\,+\,\bar{L}^{im}d^{ij}_{\bf
S}\gamma^{\mu}L^{jn}\biggr{]}\,,$

where $\Lambda$ is the scale of new physics. Here, $c^{ij}$ and $d^{ij}$ are
the $3\times 3$ hermitian matrices containing the couplings of dark scalars
with the SM fermions, and include effects of lepton flavor and number
violation, and violation of the Glashow-Illiopulos-Maiani (GIM) mechanism
Glashow et al. (1970) in the quark sector. The parameter $\Lambda$ is the
characteristic scale of this effective operator, defining when it opens up in
terms of renormalizable interactions of an UV completion.

As was pointed out in Ref. Bauer et al. (2017) after spontaneous breaking of
electroweak symmetry in the SM, one should diagonalize the fermion mass
matrices by means of unitary transformations

$\displaystyle Y_{U}=(V_{L}^{q})^{\dagger}\,y_{U}\,W_{R}^{U}\,,\quad
Y_{D}=(V_{L}^{q})^{\dagger}\,y_{D}\,W_{R}^{D}\,,\quad
Y_{\ell}=(V_{L}^{\ell})^{\dagger}\,y_{\ell}\,W_{R}^{\ell}\,.$ (19)

Here $V_{L}^{q}$ and $V_{L}^{\ell}$ are the transformation matrices acting on
the left-handed quarks and leptons, respectively, and $W_{R}^{U}$,
$W_{R}^{D}$, and $W_{R}^{\ell}$ are the transformation matrices acting on
right singlets, and $y^{ij}$ are the $3\otimes 3$ Yukawa matrices of couplings
between two scalar doublets and SM fermions before spontaneous breaking of
electroweak symmetry. The matrices $V_{L}^{q}$, $V_{L}^{\ell}$ $W_{R}^{U}$,
and $W_{R}^{D}$ rotate the couplings of dark scalars with the SM fermions as

$\displaystyle c_{\sigma}$ $\displaystyle\to$ $\displaystyle
C_{\sigma}=(V_{L}^{q})^{\dagger}\,c_{\sigma}\,V_{L}^{q}\,,$ $\displaystyle
c_{U\sigma}$ $\displaystyle\to$ $\displaystyle
C_{U\sigma}=(W_{R}^{U})^{\dagger}\,c_{U\sigma}\,W_{R}^{U}\,,$ $\displaystyle
c_{D\sigma}$ $\displaystyle\to$ $\displaystyle
C_{D\sigma}=(W_{R}^{D})^{\dagger}\,c_{D\sigma}\,W_{R}^{D}\,,$ $\displaystyle
d_{L\sigma}$ $\displaystyle\to$ $\displaystyle
D_{L\sigma}=(V_{L}^{q})^{\dagger}\,c_{L\sigma}\,V_{L}^{q}\,,$ (20)
$\displaystyle d_{R\sigma}$ $\displaystyle\to$ $\displaystyle
D_{R\sigma}=(W_{R}^{D})^{\dagger}\,d_{R\sigma}\,W_{R}^{U}\,,$ $\displaystyle
c_{{\bf S}}$ $\displaystyle\to$ $\displaystyle C_{{\bf
S}}=(V_{L}^{q})^{\dagger}\,c_{{\bf S}}\,V_{L}^{q}\,,$ $\displaystyle d_{{\bf
S}}$ $\displaystyle\to$ $\displaystyle D_{{\bf
S}}=(V_{L}^{\ell})^{\dagger}\,d_{{\bf S}}\,V_{L}^{\ell}\,.$

Note that the resulting coupling of the SM fermions with DS fields is
contributed by three terms via minimal substitution of the covariant
derivatives acting on the SM fermions and via additional effective Lagrangian
(18). The first term does not mix the SM generations, does not violate certain
symmetries (like lepton flavor and lepton number), and preserves the GIM
mechanism:

$\displaystyle{\cal L}_{\rm int;1}$ $\displaystyle=$ $\displaystyle
g_{A^{\prime}}\,\sum_{i}\bar{\psi}_{i}\,\gamma^{\mu}\,A^{\prime}_{\mu}\,\psi_{i}+\frac{g_{W^{\prime}}}{2}\,\sum_{i}\,\bar{\psi}_{i}^{m}\,\gamma^{\mu}\,{\bf
W^{\prime}}_{\mu}^{mn}\,\left(1-\gamma_{5}\right)\psi_{i}^{n}\,,$ (21)

After the substitution of the covariant derivatives $(\partial\sigma)$ and
$(\partial\bf{S})$, we find that the gauge-invariant operator (18)
additionally generates dimension-4 interactions of the dark photon and
$\bf{W^{\prime}}$ bosons with the SM fermions $\psi$, in the form

$\displaystyle{\cal L}_{\rm int;2}$ $\displaystyle=$
$\displaystyle\sum_{ij}\bar{\psi}_{i}\,\gamma^{\mu}\,A^{\prime}_{\mu}\,\left(g^{V}_{ij}+g^{A}_{ij}\gamma_{5}\right)\psi_{j}+\sum_{ij}g^{V\\!A}_{ij}\bar{\psi}_{i}^{m}\,\gamma^{\mu}\,{\bf
W^{\prime}}_{\mu}^{mn}\,\left(1-\gamma_{5}\right)\psi_{j}^{n}\,,$ (22)

where vector $g^{V}$ and axial-vector $g^{A}$ dimensionless couplings are
defined as

$\displaystyle g_{ij}^{V}=\frac{m_{A^{\prime}}}{\Lambda}v_{ij}\,,$
$\displaystyle g_{ij}^{A}=\frac{m_{A^{\prime}}}{\Lambda}a_{ij}\,,$ (23)
$\displaystyle
v_{ij}=\frac{1}{2}\left(D^{ij}_{R\sigma}+D^{ij}_{L\sigma}\right)_{ij}\,,$
$\displaystyle
a_{ij}=\frac{1}{2}\left(D^{ij}_{R\sigma}-D^{ij}_{L\sigma}\right)_{ij}\,.$ (24)

Dimensionless couplings of $W^{\prime}$ with leptons have the simple form
$g^{V\\!A}_{ij}=D_{\bf S}^{ij}M_{W^{\prime}}/\Lambda$. Scalar nonphysical
fields can be switched off from consideration in the case of unitary gauge
($\xi\to\infty$), which corresponds to the limit of infinitely large masses of
scalars or vanishing of their contribution to physical processes. Therefore,
the full interaction Lagrangian of the SM fermions with DGBs reads

$\displaystyle{\cal L}_{\rm int}={\cal L}_{\rm int;1}+{\cal L}_{\rm
int;2}=\sum_{ij}\bar{\psi}_{i}\,\gamma^{\mu}\,A^{\prime}_{\mu}\,\left(G^{V}_{ij}+G^{A}_{ij}\gamma_{5}\right)\psi_{j}+\sum_{ij}G^{V\\!A}_{ij}\bar{\psi}_{i}^{m}\,\gamma^{\mu}\,{\bf
W^{\prime}}_{\mu}^{mn}\,\left(1-\gamma_{5}\right)\psi_{j}^{n}\,,$ (25)

where

$\displaystyle G^{V}_{ij}=g^{V}_{ij}+\delta_{ij}g_{A^{\prime}}\,,\quad
G^{A}_{ij}\equiv g^{A}_{ij}\,,\quad
G^{V\\!A}_{ij}=g^{V\\!A}_{ij}+\delta_{ij}\frac{g_{W^{\prime}}}{2}\,.$ (26)

##  III Bounds on $W^{\prime}$ couplings with Standard model particles

In this section, we discuss opportunities to estimate bounds on the couplings
of the charged dark gauge boson $W^{\prime}$ with the SM particles. On the one
hand, we base on data extracted from precise measurements, and on the other
hand we can involve rare decays in our analysis. First, we estimate diagonal
couplings $G^{V\\!A}_{ii}$, which give an additional contribution to the SM
processes. Hereinafter, we will concentrate on nondiagonal couplings
$G^{V\\!A}_{ij}$ ($i\neq j$), which can be responsible for a contribution to
LFV processes.

###  III.1 Dominant and LFV $\mu\to e\nu_{i}\bar{\nu}_{j}$ decays

To derive a limit for the $G^{V\\!A}_{ii}$ coupling we use one of the most
precise measurements in particle physics — decay rate $\mu\to
e\nu_{\mu}\bar{\nu}_{e}$ process. This decay gives a very accurate
determination of the Fermi constant $G^{\mu}_{F}$:

$\displaystyle G^{\mu}_{F}=1.1663787(6)\times 10^{-5}\text{GeV}^{-2}$ (27)

at the level of $0.5$ ppm Webber et al. (2011); Tishchenko et al. (2013);
Gorringe and Hertzog (2015).

It is clear that the diagonal coupling of dark gauge boson $W^{\prime}$ gives
an additional contribution to the $\mu\to e\nu_{\mu}\bar{\nu}_{e}$ decay rate.
In our benchmark scenario, we make a conjecture that there is an ambiguity for
the precise determination of the Fermi constant $G_{F}$. In particular, this
constant can be constrained using different data (see, e.g., Ref. Crivellin et
al. (2021)). In particular, $G_{F}$ can be determined from analysis of kaon or
$\tau$ decays using unitarity of the Cabibbo-Kobayashi-Maskawa matrix.
Alternatively, one can use the global electroweak (EW) fit. Obtained values
have a small deviation from the value of the $G_{F}$ extracted from weak muon
decay. Indeed, such uncertainty is a gap, which constrains possible
contribution of the $W^{\prime}$ to the $\mu\to e\nu_{\mu}\bar{\nu}_{e}$ muon
decay and provides bounds on its diagonal or nondiagonal couplings in
dependence on a type of dark vector boson.

For calculation of decay widths, we use the well-known formula

$\displaystyle d\Gamma=\frac{4\pi^{2}}{2m}|M|^{2}d\Phi_{n}\,,$ (28)

where $m$ is the mass of decaying particle, $|M|^{2}$ is the square of
amplitude of a process, which defines its dynamics, and $d\Phi_{n}$ is an
element of $n$-body phase space.

The muon decay rate $\mu\to e\nu_{\mu}\bar{\nu}_{e}$ in the SM is determined
by

$\displaystyle\Gamma(\mu^{-}\to\nu_{\mu}e^{-}\bar{\nu}_{e})=\left(\frac{\sqrt{2}g_{EW}^{2}}{8m_{W}^{2}}\right)^{2}\frac{m_{\mu}^{5}}{192\pi^{3}}(1+\Delta
q)=\frac{(G_{F}^{\mu})^{2}m_{\mu}^{5}}{192\pi^{3}}(1+\Delta q)\,,$ (29)

where the quantity $\Delta q$ includes the phase space, QED, and hadronic
radioactive corrections (see Ref. Crivellin et al. (2021)).

The square of the $\mu\to e\nu_{\mu}\bar{\nu}_{e}$ decay amplitude, taking
into account of the mixing of the SM $W$ and dark $W^{\prime}$ bosons, is
given by

$|M|^{2}=-\frac{(s+t)\left(s+t-m_{\mu}^{2}\right)\left(g_{EW}^{2}\left(t-m_{W^{\prime}}^{2}\right)+(G^{V\\!A})^{2}\left(t-m_{W}^{2}\right)\right)^{2}}{\left(m_{W^{\prime}}^{2}-t\right)^{2}\left(t-m_{W}^{2}\right)^{2}}\,,$
(30)

where the Mandelstam kinematical variables are defined as
$s=m_{\mu}^{2}-2(p_{\mu}p_{e})$, $t=m_{\mu}^{2}-2(p_{\mu}p_{\nu_{1}})$, and
$u=m_{\mu}^{2}-2(p_{\mu}p_{\nu_{2}})$ Zyla et al. (2020), and $g_{EW}$ is
electroweak coupling. We obtain an expression for the decay width as a
function of parameters of the new vector boson (mass of $m_{W^{\prime}}$ and
couplings with leptons $G_{ij}^{V\\!A}$).

Using the difference between data of the global EW fit presented in Ref.
Crivellin et al. (2021) $G^{EW}_{F}|_{full}=1.16716(39)\times
10^{-5}\,\,\text{GeV}^{-2}$ and $G^{\mu}_{F}$ from $\mu\to
e\nu_{\mu}\bar{\nu}_{e}$ decay, we can derive limits for the couplings of
$W^{\prime}$ with the SM particles using the muon width as input data
parameter. Bounds on the coupling ratio $G^{V\\!A}_{ij}/g_{EW}$ as function of
the $W^{\prime}$ mass are shown in Fig. 1. These bounds in the case of the
universality scenario of the coupling of the dark $W^{\prime}$ boson with
quarks and leptons can be compared with limits extracted by the CMS
Collaboration Tumasyan et al. (2022) (see Fig. 1). In this scenario at
$m_{W^{\prime}}\geq 0.5$ TeV, the CMS limits are more stronger and our bounds
established from study of the famous muon decay complement in this range of
the dark boson mass. In the case of the nonuniversal coupling of dark boson
$W^{\prime}$ with quark and leptons (general scenarios) our limits deduced
from the $\mu\to e\nu_{\mu}\bar{\nu}^{c}_{e}$ decay cannot be directly
compared with the CMS constraints. The difference between values of the
$G_{F}$ constant defined from different experimental data gives strong limits
on diagonal couplings of the dark charge boson $W^{{}^{\prime}\pm}$ with the
SM fermions.

We also want to mention about a landscape of study a possible dark charge
$W^{\prime}$ boson. The existed investigation is based on studies of different
process ATL (2023); Tumasyan et al. (2022) and different models Lindner et al.
(2018); Osland et al. (2021, 2022). As output of such studies, different
bounds on the masses, couplings, and other parameters of additional
$W^{\prime}$ and $Z^{\prime}$ bosons have been proposed in the literature.
Sometimes different bounds from different studies cannot be compared directly.
Here, we would like to stress that novel $W^{\prime}$ and $Z^{\prime}$ bosons
have been proposed in literature before in different extensions of SM at TeV
scales, including extradimensional models, technicolor, and composite Higgs
(for review see, e.g., Langacker (2009)). Also the existence of these bosons
has been searched at the LHC. Some constraints of the $W^{\prime}$ and
$Z^{\prime}$ bosons masses in the TeV region have been performed in dependence
of their coupling strengths with the SM fermions. On the other hand, searches
of light $W^{\prime}$ and $Z^{\prime}$ are also attracted a lot of interest
for resolving existing puzzles and anomalies. In particular, recently the
ATLAS Collaboration ATL (2023) set upper limits on the $Z^{\prime}$ production
cross section times the decay branching fraction of the $pp\to
Z^{\prime}\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ process, varying from 0.31 to 4.3 fb
at 95% C.L., in a $Z^{\prime}$ mass range 5-81 GeV, from which the coupling
strength of the $Z^{\prime}$ to muons above 0.003 to 0.2 (depending on the
$Z^{\prime}$ mass) are excluded in the same mass range.

Here we suppose scenarios that the $W^{\prime}$ is heavier than the $\tau$
lepton. It is necessary to forbid invisible decays of leptons into neutrinos
and dark bosons, which can further decay into light dark fermions. On the
other hand, $W^{\prime}$ should be heavier than the SM $W^{\pm}/Z$ bosons, as
otherwise one could see the $W^{\prime}$ pair creation in the final state of
physical reactions, e.g., in the $e^{+}e^{-}$ annihilation or hadron-hadron
collisions. Masses of dark fermions interacting with dark charge vector
current should be larger than masses of the SM leptons. We see that the
$U(1)_{D}$ portal can be connected with sub-GeV particles, whereas dark
mediators from the $SU(2)_{D}$ sector are weak interactive massive particles,
which have larger masses in comparison with one of the SM weak bosons.

In the universality scenario of interaction $W^{\prime}$ with all types of the
SM leptons, we have bound on $G^{V\\!A}_{ii}$ less than $10^{-1}\div 3\times
10^{-4}$ in the following range of the $W^{\prime}$ mass 2 GeV $\div$ 1 TeV.
Such limits correspond to the $G^{V\\!A}_{ij}$ nondiagonal coupling in the
case when the corresponding decay is induced by exchange of intermediate
neutral dark boson $W^{\prime 0}$. In the case of a neutral boson we have less
constraints as in case of the charge partners $W^{\prime\pm}$. Therefore, the
range of the neutral dark boson mass in Fig. 1 is extended to a region of
smaller values of mass. Moreover, we will further show that these bounds are
the strongest ones that can be obtained from pure LFV decay $\mu^{-}\to
e^{-}\bar{\nu}_{\mu}\nu_{e}$. Two LFV processes will be considered later in
our manuscript.

Figure 1: Bounds on the ratio $G^{V\\!A}_{ij}/g_{EW}$ from possible
contribution to the $\mu\to e\nu\nu$ decay rate (dirty blue line) as a
function of $m_{W^{\prime}}$ in the range 2 GeV $-$ 6 TeV. Black solid and
dashed lines are CMS observed and expected limit, respectively, with one
standard deviation and two standard deviations (green and yellow areas)
Tumasyan et al. (2022). These two limits can be compared only in the
universality case of diagonal coupling dark boson with quarks and leptons.
Figure 2: Bounds on the $\mu-e$ non-diagonal lepton flavor coupling from
existing LFV data $\mu^{-}\to\nu_{e}\bar{\nu}_{\mu}e^{-}$ Freedman et al.
(1993). The shaded area is a closed band for dark boson couplings.

Because of the structure of our interaction (25), we establish the same limits
for the nondiagonal couplings of both neutral and charged dark bosons with
leptons with specific flavor and mass. In case of nondiagonal couplings, we do
not have restrictions on the mass of neutral boson from charge conservation as
it occurs for charged dark bosons. It is different from a general scenario
with a dark $Z^{\prime}$ boson, where we can describe interaction with leptons
and neutrino independently: we consider neutral $W^{\prime 0}$ component in
couple with charged $W^{\prime\pm}$ bosons. In our approach dark photon
$A^{\prime}$ plays the role of neutral dark boson $Z^{\prime}$ Kachanovich et
al. (2022). Therefore, due to possible interference of $A^{\prime}$ and
$W^{\prime 0}$ one can establish new limits on couplings with leptons and
neutrinos, which could be different from those derived in Ref. Kachanovich et
al. (2022).

Next we can calculate decay width of the $W^{\prime}$ to $l\bar{\nu}_{l}$ pair
by analogy to the similar decay in SM:

$\displaystyle\Gamma(W^{\prime}\to
e^{-}\bar{\nu}_{e})\simeq\frac{(G^{V\\!A}_{e\bar{\nu}_{e}})^{2}m_{W^{\prime}}}{48\pi}$
(31)

and as example for $m_{W^{\prime}}=200$ GeV we have $\Gamma(W^{\prime}\to
e^{-}\bar{\nu}_{e})=6.5\times 10^{-4}$ MeV. It is by $\sim 6$ orders less than
we have for weak $W$ boson decay width.

Nondiagonal interaction couplings of the $W^{\prime}$ dark boson can be
constrained from LFV processes, e.g., from muon LFV decay $\mu^{-}\to
e^{-}\bar{\nu}_{\mu}\nu_{e}$. The best upper limit on the branching of this
decay is 1.2% Freedman et al. (1993). We note that this existing limit on the
$\mu^{-}\to e^{-}\bar{\nu}_{\mu}\nu_{e}$ decay rate is not very sensitive and
feeble competitive with many other experiments which limit most of new physics
scenarios. Contrariwise we stress that this LFV decay gives limitation
directly on one of the nondiagonal components, i.e., on coupling
$G^{V\\!A}_{\mu e}$. Bounds on this coupling are presented in Fig. 2. Because
the interaction couplings in our model for charged and neutral $W^{\prime}$
are universal, we see that famous weak muon decay $\mu\to
e\nu_{\mu}\bar{\nu}_{e}$ gives more stronger limits on the coupling constants
in comparison with limits derived from LFV decay, e.g., from the $\mu^{-}\to
e^{-}\bar{\nu}_{\mu}\nu_{e}$ decay mode.

###  III.2 LFV muon decay $\mu\to e\gamma$

Figure 3: Feynman diagrams with contribution of dark photon $A^{\prime}$ and
charge dark boson $W^{\prime}$ to the LFV process $l_{i}\to l_{k}\gamma$.

New dark vector bosons can potentially explain LFV effects. In this respect,
$\mu\to e\gamma$ decay is a good laboratory for the study of such effects.
Feynman diagrams contributing to this process taking into account $A^{\prime}$
and $W^{\prime\pm}$ bosons are presented in Fig. 3. Note that the diagram with
a pair of intermediate charged bosons $W^{\prime\pm}$ has an additional
suppression factor in comparison with the dark photon $A^{\prime}$ exchange.

Existing experimental limits for the branchings of the LFV lepton decays
$l_{i}\to l_{k}\gamma$ are Zyla et al. (2020)

$\displaystyle\mathrm{Br}(\mu\to e\gamma)$ $\displaystyle<$ $\displaystyle
4.2\times 10^{-13}\,,$ $\displaystyle\mathrm{Br}(\tau\to e\gamma)$
$\displaystyle<$ $\displaystyle 3.3\times 10^{-8}\,,$
$\displaystyle\mathrm{Br}(\tau\to\mu\gamma)$ $\displaystyle<$ $\displaystyle
4.2\times 10^{-8}\,.$ (32)

The general matrix element of this LFV process can be parametrized as

$\displaystyle
iM_{ik}=ie\,\epsilon^{\mu}(q)\,\bar{u}_{k}(p_{2},m_{k})\,\left[\frac{i}{2m_{i}}\sigma_{\mu\nu}q^{\nu}F_{M}+\frac{i}{2m_{i}}\sigma_{\mu\nu}q^{\nu}\gamma_{5}F_{D}\right]\,u_{i}(p_{1},m_{i})\,,$
(33)

where the square of matrix element is

$\displaystyle|M_{ik}|^{2}$ $\displaystyle=$ $\displaystyle
m_{i}^{2}\,\biggl{[}1-\frac{m_{k}^{2}}{m_{i}^{2}}\biggr{]}^{2}\,\biggl{(}|F_{M}|^{2}+|F_{D}|^{2}\biggr{)}$
(34)

and the decay width is given by

$\displaystyle\Gamma(l_{i}\to l_{k}\gamma)$ $\displaystyle=$
$\displaystyle\frac{1}{2m_{i}}\int\frac{d^{3}p_{2}\,d^{3}q}{4E_{2}E_{q}\,(2\pi)^{6}}\,(2\pi)^{4}\,\delta^{(4)}(p_{1}-p_{2}-q)\
|M_{ik}|^{2}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2}\,m_{i}\,\biggl{(}|F_{M}|^{2}+|F_{D}|^{2}\biggr{)}\,.$

Here, $F_{M}$ and $F_{D}$ are the magnetic and dipole form factors,
respectively, and $\alpha=e^{2}/(4\pi)=1/137.036$ is the fine-structure
constant. More explicitly, when the $A^{\prime}$ propagates in the loop
(right-hand diagram in Fig. 3), for a general $l_{i}\to l_{k}\gamma$ with an
$i$\- or $k$\- lepton in the loop, we obtain

$\displaystyle F_{M}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left[\left(G^{V}_{ik}G^{V}_{ii}+G^{A}_{ik}G^{A}_{ii}\right)h_{2}^{V}(x_{\mu})+\left(G^{V}_{ik}G^{V}_{kk}+G^{A}_{ik}G^{A}_{kk}\right)h_{3}^{V}(x_{\mu})\right]\,,$
$\displaystyle F_{D}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left[\left(G^{V}_{ik}G^{A}_{ii}+G^{A}_{ik}G^{V}_{ii}\right)h_{2}^{V}(x_{\mu})+\left(G^{V}_{ik}G^{A}_{kk}+G^{A}_{ik}G^{V}_{kk}\right)h_{3}^{V}(x_{\mu})\right]\,,$
(36)

whereas for $\mu\to e\gamma$, with the $\tau$ lepton propagating in the loop
and with double LFV coupling, we have:

$\displaystyle F_{M}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left(\frac{m_{\mu}}{m_{\tau}}\right)\left[G^{V}_{\mu\tau}G^{V}_{e\tau}h_{1}^{V}(x_{\tau})-G^{A}_{\mu\tau}G^{A}_{e\tau}h_{1}^{V}(x_{\tau})\right]\,,$
$\displaystyle F_{D}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left(\frac{m_{\mu}}{m_{\tau}}\right)\left[G^{V}_{\mu\tau}G^{A}_{e\tau}h_{1}^{V}(x_{\tau})-G^{A}_{\mu\tau}G^{V}_{e\tau}h_{1}^{V}(x_{\tau})\right]\,,$
(37)

where $x_{i}=m_{A^{\prime}}^{2}/m_{i}^{2}$. Expressions for the loop functions
$h_{i}^{V}(x_{i})$ in the approximation $m_{e}\ll m_{\mu}\ll m_{\tau}$ are
shown in the Appendix A.

Figure 4: Bounds to production $\mu-e$ nondiagonal lepton flavor coupling and
diagonal coupling from existing FV data $\mu^{-}\to e^{-}\gamma$ Zyla et al.
(2020). The shaded area is a closed band for dark boson couplings (blue for
dark photon and only $G^{V}$ contribution and green for $W^{\prime 0}$ neutral
dark boson due to full contribution from vector and axial-vector parts).

The matrix element corresponding to the loop LFV diagram induced by the
coupling of a neutral $W^{\prime}$ boson with a photon and contributing to the
$l_{i}\to l_{k}\gamma$ process is specified by two form factors

$\displaystyle F_{M}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left[\left(G^{VA}_{ik}G^{VA}_{ii}\right)h^{W^{0}}(x_{\mu})+2\left(G^{VA}_{ik}G^{VA}_{kk}\right)h_{3}^{V}(x_{\mu})\right]\,,$
$\displaystyle F_{D}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left[\left(G^{VA}_{ik}G^{VA}_{ii}\right)h^{W^{0}}(x_{\mu})+2\left(G^{VA}_{ik}G^{VA}_{kk}\right)h_{3}^{V}(x_{\mu})\right]\,,$
(38)

where $x=m_{W^{\prime 0}}^{2}/m_{l}^{2}$. Double LFV contributions are
neglected here.

Dependence on the masses of dark bosons $A^{\prime}$ and $W^{\prime}$ is
presented in Fig. 4. The peaks in Fig. 4 are connected with the behavior of
the loop integrals $h_{i}(x)$ near the point $x=1$ located in the vicinity of
the vector boson production threshold. To resolve this problem, one needs to
include in the analysis the finite width
$\Gamma_{A^{\prime}}\sim\tau_{A^{\prime}}^{-1}\sim G_{ij}^{2}$ of the dark
vector boson in the Breit-Wigner propagator. The latter is dominated by the
decay width of the $A^{\prime}$ to the leptonic pair.

We make conservative estimate for coupling production
$G^{V\\!A}_{ll}G^{V\\!A}_{e\mu}$ in proposition that diagonal couplings
$G^{V\\!A}_{ll}$ are equal to each other. Corresponding bounds are shown in
Fig. 4. It means that bounds on $G^{V\\!A}_{ll}G^{V\\!A}_{e\mu}$ from $\mu\to
e\gamma$ LFV decay for neutral vector bosons are strict for a light mass
boson. Limits on the $A^{\prime}$ couplings correspond to the fact that the
bounds for the neutral dark boson $W^{\prime 0}$ are divided by a factor of
two, because they have the same mechanism of interaction with SM matter
governed by both vector and axial couplings. Further discussion of bounds on
couplings $A^{\prime}$ dark photon has been done in Ref. Kachanovich et al.
(2022) (here, we note that in Ref. Kachanovich et al. (2022) in the plot of
the curve for limit on the LFV coupling derived from the decay $\mu\to
e\gamma$ the factor proportional to $10^{-6}$ was lost). With taking into
account with factor the limits on the LFV couplings of the dark gauge bosons
with SM fermions are consistent in this paper and Ref. Kachanovich et al.
(2022)). At the same time, it is important to stress a constantly increasing
research interest to study sub-GeV dark candidates which are the main goal in
running and planning experiments for searching dark matter at fixed target
regime such as NA64 SPS at CERN Andreev et al. (2022a, c); Cazzaniga et al.
(2021); Depero et al. (2020); Banerjee et al. (2020, 2019, 2018b, 2018a,
2017), M3 Kahn et al. (2018), and LDMX Mans (2017). LFV coupling between dark
boson and SM particles can explain existing experimental anomalies, wherein
different models provide different limits on mass and couplings of the
$Z^{\prime}$ boson Profumo and Queiroz (2014); Kelso et al. (2014); Alves et
al. (2015).

##  IV conclusion

We have proposed a phenomenological Lagrangian approach that combines the SM
and DM sectors based on the Stueckelberg mechanism for generation of masses of
dark $U(1)_{D}$ and $SU(2)_{D}$ gauge bosons due to the presence of dark
scalar Stueckelberg fields. These scalar fields play the role of Goldstone
bosons. A novel part consists in adding of non-Abelian part to the dark vector
$U(1)_{D}$ portal. We derived bounds on diagonal and nondiagonal interaction
couplings between $SU(2)_{D}$ dark gauge bosons and the SM leptons. In
particular, we established limits on the couplings using data from canonical
weak muon decay $\mu\to e\nu_{\mu}\bar{\nu}_{e}$ supposing that some
correction to its decay rate is possible and given the ambiguity in the
definition of value of $G_{F}$. Additionally, we have used the phenomenology
of lepton flavor-violating processes to derive limits on the $W^{\prime}$
couplings. In this paper, we concentrated on thee gauge boson sector of our
approach. It would be interesting to extend our formalism on the dark fermion
sector and consider applications on other rare lepton decays.

###### Acknowledgements.

This work was funded by BMBF (Germany) “Verbundprojekt 05P2021 (ErUM-FSP T01)
- Run 3 von ALICE am LHC: Perturbative Berechnungen von Wirkungsquerschnitten
für ALICE” (Förderkennzeichen: 05P21VTCAA), by ANID PIA/APOYO AFB180002 and
AFB220004 (Chile), by FONDECYT (Chile) under Grant No. 1191103 and by
ANID$-$Millennium Program$-$ICN2019_044 (Chile). The work of A.S.Z. is
supported by grant of AYSS JINR (22-302-02). The work was supported by the
Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS".

##  Appendix A Loop functions which is used at calculation $\mu\to e\gamma$
LFV decay

In this appendix, we present analytical expressions for the loop integrals
occurring in the amplitude of the LFV decays $l_{i}\to l_{k}\gamma$ for
different new particle channels and leptons propagating in the loop, used in
Eqs. (III.2)-(III.2). All results for the form factors have been numerically
and analytically cross-checked using the Mathematica packages Package-X Patel
(2015), FeynHelpers Shtabovenko (2017) and FeynCalc Shtabovenko et al. (2020,
2016); Mertig et al. (1991).

For the dark heavy neutral boson $W^{\prime}$, the loop integrals read

$\displaystyle h^{W^{\prime 0}}(x)$ $\displaystyle=$ $\displaystyle
8\left(\text{Li}_{2}\left(1-x\right)-\text{Li}_{2}\left(\frac{2}{2-x+\sqrt{(x-4)x}}\right)+\text{Li}_{2}\left(\frac{2}{x+\sqrt{(x-4)x}}\right)\right)+2x(2x^{2}-9x+9)\frac{\log(x)}{x-1}$
(39) $\displaystyle+$ $\displaystyle
2(-4x+\frac{1}{x}+8)+4\log^{2}\left(\frac{x+\sqrt{(x-4)x}-2)}{x+\sqrt{(x-4)x}}\right)-4\sqrt{(x-4)x}(2x-3)\log\left(\frac{x+\sqrt{(x-4)x}}{2\sqrt{x}}\right)\,.$

For the $A^{\prime}$ dark boson, the loop integrals are given by

$\displaystyle h^{V}_{1}(x)$ $\displaystyle=$
$\displaystyle-\frac{(4x^{3}-3x^{2}-6x^{2}\ln(x)-1)}{x(1-x)^{3}}\,,$ (40)
$\displaystyle h^{V}_{2}(x)$ $\displaystyle=$ $\displaystyle
2\left(2\text{Li}_{2}(1-x)-2\text{Li}_{2}\left(\frac{2}{-x+\sqrt{(x-4)x}+2}\right)+2\text{Li}_{2}\left(\frac{2}{x+\sqrt{(x-4)x}}\right)-2x+2\log(x)\right.$
$\displaystyle+$
$\displaystyle\left.\log^{2}\left(\frac{x+\sqrt{x-4)x}}{2x}\right)+\frac{(x+1)((x-4)x+2)\log\left(x\right)}{x-1}-2x\sqrt{(x-4)x}\log\left(\frac{\sqrt{x}+\sqrt{(x-4)}}{2}\right)+1\right)\,,$
$\displaystyle h^{V}_{3}(x)$ $\displaystyle=$
$\displaystyle-4x+4(x-1)^{2}\ln\left(\frac{x}{x-1}\right)+6\,.$ (42)

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