# Lepton Phenomenology of Stueckelberg Portal to Dark Sector

Aliaksei Kachanovich  Institute for Theoretical Particle Physics (TTP),
Karlsruhe Institute of Technology (KIT), Wolfgang-Gaede-Straße 1, 76131
Karlsruhe, Germany  Sergey Kovalenko  Departamento de Ciencias Físicas,
Universidad Andres Bello, Sazié 2212, Piso 7, Santiago, Chile  Millennium
Institute for Subatomic Physics at the High-Energy Frontier (SAPHIR) of ANID,
Fernández Concha 700, Santiago, Chile  Centro Científico Tecnológico de
Valparaíso-CCTVal,
Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile
Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia
Serguei Kuleshov  Departamento de Ciencias Físicas, Universidad Andres Bello,
Sazié 2212, Piso 7, Santiago, Chile  Millennium Institute for Subatomic
Physics at the High-Energy Frontier (SAPHIR) of ANID,
Fernández Concha 700, Santiago, Chile  Valery E. Lyubovitskij  Institut für
Theoretische Physik, Universität Tübingen,
Kepler Center for Astro and Particle Physics,
Auf der Morgenstelle 14, D-72076 Tübingen, Germany  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

###### Abstract

We propose an extension of the Standard Model (SM) with a $U_{A^{\prime}}(1)$
gauge invariant Dark Sector connected to the SM via a new portal arising in
the framework of dark photon $A^{\prime}$ mass generation via Stueckelberg
mechanism. This mechanism implies the existence of a scalar field $\sigma$,
which is shift-transformed under this group and resembles an axion-like
particle (ALP) widely addressed in the literature in different contexts. The
effective dim=5 operators constructed of the covariant derivative of the
$\sigma$ field generate flavor non-diagonal renormalizable couplings of both
$\sigma$ and $A^{\prime}$ to the SM fermions $\psi$. Contrary to the
conventional kinetic mixing portal, in our scenario flavor diagonal
$A^{\prime}$-$\psi$ couplings are not proportional to the fermion charges.
These features drastically change the phenomenology of dark photon
$A^{\prime}$ relaxing or avoiding some previously established experimental
constraints. We focus on the phenomenology of the described scenario of the
Stueckelberg portal in the lepton sector and analyze the contribution of the
dark sector fields $A^{\prime}$ and $\sigma$ to the anomalous magnetic moment
of muon $(g-2)_{\mu}$, Lepton Flavor Violating decays $l_{i}\to l_{k}\gamma$
and $\mu-e$ conversion in nuclei. We obtain limits on the model parameters
from the existing experimental data on the corresponding observables.

††preprint: TTP21-049

##  I Introduction

The idea of the Dark Sector (DS) of the Universe, existing almost
independently of the Standard Model (SM) sector, has attracted growing
interest in recent years. Originally, DS was thought to be populated by only
one dark species, necessary to make up for the lack of matter in the universe
with dark matter (DM). Extensions of DS was, in particular, motivated by the
popular scenario of Light sub-GeV DM. It was realized that in this case a dark
boson, known as the Dark Photon, would need to be introduced to prevent the
Universe from over-closing. An extended DS can have not only cosmological, but
also interesting phenomenological consequences. This DS physics beyond the SM
can manifest itself in the phenomena observable experimentally (for a status
report see, e.g., Ref. Alexander:2016aln ).

Presently, there are a number of experiments to search for DS physics and
others are planned for the near future. Among them we mention CERN based
experiments NA64 Banerjee:2016tad ; Banerjee:2017hhz , NA62 CortinaGil:2019nuo
, SHiP Alekhin:2015byh ; Anelli:2015pba ; SHiP:2020noy , LHCb Aaij:2017rft ,
ATLAS Aad:2020arf , CMS Sirunyan:2020zow  and BaBar experiment at SLAC
Lees:2017lec , HPS at JLab Adrian:2018scb , Belle at KEK Park:2020lyz . So far
no signal of DS or other kind of new physics beyond the SM (BSM) is observed.

An encouraging indication of new physics has recently come from measurements
of the anomalous magnetic moment (AMM) of the muon $(g-2)_{\mu}$. The Fermilab
Muon g-2 Collaboration published Muong-2:2021ojo  the observation of 4.2
$\sigma$ deviation of the $(g-2)_{\mu}$ from its SM value and stimulated an
explosion of the BSM literature. As is known, measurements of $(g-2)_{\mu}$
are a very sensitive probe of BSM physics. The Fermilab Muon $g-2$ result with
such unprecedented precision can severely limit or refute many BSM models.

On the other hand, there is no doubt that the SM is an incomplete theory,
requiring some physics beyond its scope to explain a number of problems that
cannot be addressed in the SM. Among them, the DM problem is one of the most
obvious. As we already mentioned, DM hints at the existence of a DS of the
Universe, which not only provides DM particle candidates, but is also
populated by other particles involved in interactions governed by some dark
symmetries. The DS with possible non-trivial physics could have a
phenomenological impact on the SM sector through portals such as the well-
known kinetic mixing of dark and normal photons. Other hypothetical DS
particles can have access to the SM sector through different portals and
contribute to various observables, in particular, to $(g-2)_{\mu}$, allowing
one to probe the DS.

We should stress that there is much evidence of deviation from SM. Besides
$(g-2)$ of muon Aoyama:2020ynm  they are strong CP problem and rare meson
decays Pospelov:2017kep -Zhevlakov:2020bvr , flavor non-universality
Ema:2016ops ; Buras:2021btx , $b-s$ quark anomaly, and others. It motivates
theoretical study/constructing of effective Lagrangians beyond Standard Model
(BSM) trying to involve new particles/portals, like axion, dark photon,
vectors, pseudoscalars, scalars, and axials, etc. Alexander:2016aln ;
Alekhin:2015byh ,Kahlhoefer:2017dnp -Buen-Abad:2021fwq .

Here, we propose an extension of the SM by inclusion of DS with a softly-
broken $U_{A^{\prime}}(1)$ symmetry. The corresponding gauge boson
$A^{\prime}$, also known as Dark Photon, acquires a non-zero gauge invariant
mass via the Stueckelberg mechanism Stueckelberg:1938zz ; Ruegg:2003ps , which
implies the existence of a scalar Stueckelberg field $\sigma$.

This field opens a new portal from the SM to the Dark sector via the effective
dimension-5 operator with the covariant derivative of the $\sigma$-field. We
call it the Stueckelberg portal. In our setup this portal coexists with the
conventional kinetic mixing portal and leads to new phenomenological effects
in the SM sector, in particular, flavor violation both in lepton and quark
sectors. In the present work we focus on the Lepton Flavor Violation (LFV) and
the corresponding experimental observables.

We also introduce one dark fermion, $\chi$, charged under $U(1)_{D}$, which is
a viable Light DM particle candidate. We postulate that DSB plays important
role in this model: (1) generate mass of dark gauge boson (DGB) (Dark photon)
via Stueckelberg mechanism Stueckelberg:1938zz ; Ruegg:2003ps ; (2) generate a
mixing of DS with SM fermion including couplings preserving and violating
symmetries of SM (e.g., lepton flavor violation (LFV)). Interaction of DGB and
DSB with fermions is based on idea of familon (or flavons) Feng:1997tn ;
Bauer:2019gfk ; Cornella:2019uxs .

The paper is organized as follows. In Sec. II we describe our theoretical
setup. In Sec. III, IV and V we consider application of the Stueckelberg
portal to phenomenological aspects of the $g-2$ lepton anomaly, lepton
conversion and rare lepton-flavor violating decays $l_{i}\to l_{k}\gamma$
which were used to derive limits for couplings occurring in the Stueckelberg
portal. In Sec. VI we discuss boundary to couplings from DSB and DGB and a
possible contribution to $g-2$ lepton from obtained restrictions for different
channels. The Sec. VII is the conclusion. Technical details of our
calculations are placed in Appendices.

##  II Theoretical setup

We consider the conventional Dark Sector scenario with the $U_{A^{\prime}}(1)$
extension of the SM gauge symmetry. We suppose that Dark Sector (DS), blind to
the SM interactions, is populated with Dirac fermions $\chi_{i}$ charged under
$U_{A^{\prime}}(1)$. The lightest of which is stable and plays the role of
Dark Matter. By definition, all the SM fields are neutral with respect to this
group. The gauge boson, $A^{\prime}$, of the Dark sector $U_{A^{\prime}}(1)$
group is conventionally called Dark Photon. In the conventional Dark Photon
scenario $A^{\prime}$ acquires its mass $M_{A^{\prime}}$ from spontaneous
breaking of $U_{A^{\prime}}(1)$ group. In contrast, in our approach its mass
is a gauge invariant quantity generated by the Stueckelberg mechanism. The
latter requires the introduction of a scalar Stueckelberg field $\sigma$.
Gauge invariant Lagrangian of the model reads

$\displaystyle{\cal L}_{\rm SM+DS}={\cal L}_{\rm SM}+{\cal L}_{\rm DS}+{\cal
L}_{\rm mix}+{\cal L}_{\rm int}\,,$ (1)

where ${\cal L}_{\rm SM}$ is the SM Lagrangian. The terms denoted by ${\cal
L}_{\rm DS}$ describe Dark Sector according to

$\displaystyle{\cal L}_{\rm
DS}=-\frac{1}{4}\,{A}_{\mu\nu}^{\prime}{A}^{\prime\mu\nu}\,+\,\frac{1}{2}D_{\mu}\sigma
D^{\mu}\sigma\,+\,\bar{\chi}\,(i\not\\!\\!D_{\chi}-m_{\chi})\,\chi\,,$ (2)

where, as usual,
$A_{\mu\nu}^{\prime}=\partial_{\mu}A_{\nu}^{\prime}-\partial_{\nu}A_{\mu}^{\prime}$
and

$\displaystyle\Big{(}iD_{\chi}\Big{)}_{\mu}=i\partial_{\mu}-g_{A^{\prime}}A^{\prime}_{\mu}\,.$
(3)

is the conventional covariant derivative. The Stueckelberg covariant
derivative is defined as

$\displaystyle
D_{\mu}{\bf\sigma}=\partial_{\mu}{\bf\sigma}-M_{A^{\prime}}A^{\prime}_{\mu}$
(4)

The $U_{A^{\prime}}(1)$-symmetry is realized on the Dark Sector fields
according to the transformations

$\displaystyle A^{\prime}_{\mu}$ $\displaystyle\to$ $\displaystyle
A^{\prime}_{\mu}+\frac{i}{g_{A^{\prime}}}\,\partial_{\mu}\Omega_{A^{\prime}}\,\Omega^{-1}_{A^{\prime}}\,,\qquad
A_{\mu\nu}^{\prime}\,\to\,A_{\mu\nu}^{\prime}\,,$ (5) $\displaystyle\sigma$
$\displaystyle\to$
$\displaystyle\sigma-\frac{M_{A^{\prime}}}{g_{A^{\prime}}}\theta_{A^{\prime}}\,,\qquad\partial_{\mu}\sigma\,\to\,\partial_{\mu}\sigma+\frac{iM_{A^{\prime}}}{g_{A^{\prime}}}\,\partial_{\mu}\Omega_{A^{\prime}}\,\Omega^{-1}_{A^{\prime}}\,,\qquad
D_{\mu}\sigma\to D_{\mu}\sigma\,,$ (6) $\displaystyle\chi_{D}$
$\displaystyle\to$ $\displaystyle\Omega_{A^{\prime}}\,\chi_{D}\,,\qquad
i\not\\!\\!D_{\chi_{D}}\,\chi_{D}\,\to\,\Omega_{A^{\prime}}\,i\not\\!\\!D_{\chi_{D}}\,\chi_{D}\,,$
(7)

where

$\displaystyle\Omega_{A^{\prime}}(x)=\exp\Big{[}i{\theta}_{A^{\prime}}(x)\Big{]}\,.$
(8)

As seen from from the last relation in (7), the $\sigma$ is an axion-like
field shift-transformed under the $U_{A^{\prime}}(1)$. In our framework the
$U_{A^{\prime}}(1)$-symmetry is softly broken by the mass $M_{\sigma}$ of the
$\sigma$-field

$\displaystyle{\cal L}_{\rm DS}^{\rm
mass}=-\frac{1}{2}\,M_{\sigma}^{2}\sigma^{2}\,.$ (9)

We introduce this term in order to avoid massless scalars.

We also introduce the conventional portal to Dark sector via kinetic mixing of
the Dark and the SM photons, $A-A^{\prime}$, according to

$\displaystyle{\cal L}_{\rm mix}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon_{A}}{2}\,F_{\mu\nu}A^{\prime\mu\nu}\,,$ (10)

where $\epsilon_{A}$ is the mixing parameter. This term is necessary in our
model in in order to close it under renormalization after we introduce the
Stueckelberg portal interactions denoted in Eq. (1) as ${\cal L}_{\rm int}$.
These interactions appear at the level effective operators of dim=4 (in case
of couplings of Dark photon with SM fermions) and dim=5 (in case of coupling
of DSB with SM fermions) and have the form

$\displaystyle{\cal L}_{\rm
int}=\frac{1}{\Lambda}\,D_{\mu}\sigma\,\sum_{ij}\,\biggl{[}\bar{Q}^{i}\chi^{ij}_{\sigma}\gamma^{\mu}Q^{j}\,+\,\bar{u}_{R}^{i}\chi^{ij}_{u\sigma}\gamma^{\mu}u_{R}^{j}\,+\,\bar{d}_{R}^{i}\chi^{ij}_{d\sigma}\gamma^{\mu}d_{R}^{j}\,+\,\bar{L}^{im}\kappa^{ij}_{L\sigma}\gamma^{\mu}L^{jm}\,+\,\bar{\ell}_{R}^{i}\kappa^{ij}_{R\sigma}\gamma^{\mu}\ell_{R}^{j}\biggr{]}+{\cal
L}_{\sigma FF}\,,$ (11)

which is invariant under the $SU_{c}(3)\times SU_{2L}(2)\times U_{Y}(1)\times
U_{A^{\prime}}(1)$ gauge group. The fields belong to the following
representations of this group $Q(3,2;1/3;0)$, $u_{R}(3,1;4/3;0)$,
$d_{R}(3,1;-2/3;0)$, $L(1,2;-1;0)$, $\ell_{R}(1,1;-2;0)$. The parameter
$\Lambda$ is the characteristic scale of this effective operator, defining
when it opens up in terms of renormalizabale interactions of an UV completion.

The term ${\cal L}_{\sigma FF}$ is defined as

$\displaystyle{\cal L}_{\sigma FF}$ $\displaystyle=$
$\displaystyle\sigma\,\biggl{[}g_{\sigma\gamma\gamma}\,F_{\mu\nu}\tilde{F}^{\mu\nu}\,+\,\bar{g}_{\sigma\gamma\gamma}\,F_{\mu\nu}F^{\mu\nu}\biggr{]}\,.$
(12)

where $\tilde{F}^{\mu\nu}$ is the dual tensor. We included this term for the
similar reason as the kinetic mixing term (10): they both are generated at
loop-level from the interactions of $\sigma$ with the SM fermions and are
needed for closure of the model under renormalization.

In Eq. (11) the parameters $\chi^{ij}$ and $\kappa^{ij}$ form $3\otimes 3$
hermitian matrices leading to the neutral current flavor violation both in
quark and lepton sectors. In the present work we focus only on the lepton
sector and make ad hoc assumption
$\chi_{\sigma}=\chi_{u\sigma}=\chi_{d\sigma}=0$. As it was pointed out in Ref.
Bauer:2017ris  after spontaneous breaking of electroweak symmetry in SM one
should diagonalize the fermions mass matrices by means of unitary
transformations involving the matrices $V_{L}^{q}$ and $V_{L}^{\ell}$ acting
on the left-handed quarks and leptons, respectively, and $W_{R}^{U}$,
$W_{R}^{D}$, $W_{R}^{\ell}$ are the transformation matrices acting on right
singlets. It will have impact on the couplings of dark scalars with SM
fermions. In particular, they will rotate as

$\displaystyle\chi_{\sigma}$ $\displaystyle\to$ $\displaystyle
X_{\sigma}=(V_{L}^{q})^{\dagger}\,\chi_{\sigma}\,V_{L}^{q}\,,$
$\displaystyle\chi_{U\sigma}$ $\displaystyle\to$ $\displaystyle
X_{U\sigma}=(W_{R}^{U})^{\dagger}\,\chi_{U\sigma}\,W_{R}^{U}\,,$
$\displaystyle\chi_{D\sigma}$ $\displaystyle\to$ $\displaystyle
X_{D\sigma}=(W_{R}^{D})^{\dagger}\,\chi_{D\sigma}\,W_{R}^{D}\,,$
$\displaystyle\kappa_{L\sigma}$ $\displaystyle\to$ $\displaystyle
K_{L\sigma}=(V_{L}^{q})^{\dagger}\,\kappa_{L\sigma}\,V_{L}^{q}\,,$
$\displaystyle\kappa_{R\sigma}$ $\displaystyle\to$ $\displaystyle
K_{R\sigma}=(W_{R}^{D})^{\dagger}\,\kappa_{R\sigma}\,W_{R}^{U}\,.$ (13)

It is convenient to remove the kinetic mixing term (10) by the field
redefinition. As usual, we shift the SM photon field

$\displaystyle A_{\mu}\to A_{\mu}-\epsilon_{A}\,A_{\mu}^{\prime}\,$ (14)

and, then, rescale the Dark photon field

$\displaystyle A^{\prime}_{\mu}\to
A^{\prime}_{\mu}\,(1-\epsilon_{A}^{2})^{-1/2}\,.$ (15)

These redefinitions generate flavor diagonal couplings of the SM fermions,
$\psi$, to the dark photon originating from the kinetic mixing term:

$\displaystyle{\cal L}_{\rm mix}^{\rm
A^{\prime}-\psi}=e\,\epsilon_{A}\,A^{\prime}_{\mu}\,\bar{\psi}\,\gamma^{\mu}\,T_{Q}\,\psi\,,$
(16)

where $T_{Q}$ is the charge matrix. Now let us examine the first term in (11).
It generates interactions of $\sigma$ and $A^{\prime}$ fields with the SM
fermions. The latter takes form

$\displaystyle{\cal L}^{\rm A^{\prime}-\psi}$ $\displaystyle=$ $\displaystyle
A^{\prime}_{\mu}\sum_{ij}\bar{\psi}_{i}\gamma^{\mu}\left(g^{V}_{ij}+g^{A}_{ij}\gamma_{5}\right)\psi_{j}\,,$
(17)

where we defined vector $g^{V}$ and axial-vector $g^{A}$ dimensionless
couplings, which are proportional to the factor $M_{A^{\prime}}/\Lambda$,
which multiplies linear combinations of the dimensionless couplings
$\chi,\kappa$ present in Eq. (11). Such mass behavior is crucial for analysis
of the $A^{\prime}$ contribution and setting limits on the couplings taking
into account the dependence on the intermediate state mass.

It is clear that the terms (16), originating from the kinetic mixing, are
completely absorbed by the redefinition of the flavor diagonal matrix elements
of the coupling $g^{V}_{ii}$. Thus, in our approach the Lagrangian (17)
describes interactions of the Dark Photon $A^{\prime}$ to the SM fermions with
arbitrary dimensionless flavor non-diagonal couplings $g^{V,A}_{ij}$. Let us
highlight two principal differences between the conventional kinetic portal
and the Stueckelberg portal scenarios of DS. First, in the latter case
contrary to the former one the $A^{\prime}$ couplings to the SM fermions are
not proportional to the SM fermion electric charges. Second, these couplings
are flavor non-diagonal leading to reach LFV phenomenology. Note that the
first point can significantly affect the conclusions of the existing searches
of the Dark Photon for the case of the Stueckelberg scenario. In particular,
the conventional dark photon from the kinetic portal scenario has been
strongly constrained from the data of NA64 experiment at SPS CERN
Banerjee:2016tad ; Banerjee:2017hhz .

Finally, let us consider the interaction terms of the $\sigma$-field. As
follows from (11), we have

$\displaystyle{\cal L}_{\sigma}={\cal L}_{\sigma
FF}+\frac{1}{\Lambda}\partial_{\mu}\sigma\sum_{ij}\bar{\psi}_{i}\gamma^{\mu}\left(v_{ij}+a_{ij}\gamma_{5}\right)\psi_{j}\,.$
(18)

Where $v_{ij}$ and $a_{ij}$ are generic hermitian matrices of couplings. Using
equations of motion for fermion fields one can rewrite derivative coupling of
scalar field with fermions in the equivalent non-derivative form:

$\displaystyle{\cal
L}_{\sigma\bar{\psi}\psi}=i\sigma\sum_{ij}\bar{\psi}_{i}\left(g^{S}_{ij}-g^{PS}_{ij}\gamma_{5}\right)\psi_{j}\,,$
(19)

were we introduced dimensionless scalar and pseudoscalar couplings

$\displaystyle g^{S}_{ij}=\frac{v_{ij}(m_{i}-m_{j})}{\Lambda};\qquad\qquad
g^{PS}_{ij}=\frac{a_{ij}(m_{i}+m_{j})}{\Lambda}\,.$ (20)

Note that matrices $v_{ij}$ and $a_{ij}$ are Hermitian: $v_{ij}=v^{\ast}_{ji}$
and $a_{ij}=a^{\ast}_{ji}$. From Eq. (19) one can see, that the flavor
diagonal scalar couplings of $\sigma$ to the SM fermions vanish
$g^{s}_{ii}$=0. Note that the similar Lagrangian was obtained in Ref.
Feng:1997tn  in the framework of a flavor model with $\sigma$ in the role of
familon. Currently, this part of the effective Lagrangian is considered as the
part of low-energy action for axion-like particles (ALPs) Endo:2020mev ;
Cornella:2019uxs ; Bauer:2017ris ; Bauer:2021mvw .

In the subsequent sections we will study contributions of the dark sector
fields $A^{\prime}$ and $\sigma$ to muon anomalous magnetic moment
$(g-2)_{\mu}$ and LFV decays $l_{i}\to l_{j}\gamma$ and $\mu-e$ conversion in
nuclei.

##  III Anomalous magnetic moment

---
Figure 1: Feynman diagrams which gives a contribution to AMM of leptons due to
the exchange by intermediate DGB or DSB.

To derive the boundaries on coupling constants we use existing difference
between theoretical prediction based on SM and available actual data on
$(g-2)$ of leptons PDG20 ; Aoyama:2020ynm ; Muong-2:2021ojo ; Dorokhov:2014iva
:

$\displaystyle\Delta a_{e}=a_{e}^{exp}-a_{e}^{SM}=8.7\times 10^{-13},$
$\displaystyle\Delta a_{\mu}=a_{\mu}^{exp}-a_{\mu}^{SM}=251\times 10^{-11},$
$\displaystyle\Delta a_{\tau}=a_{\tau}^{exp}-a_{\tau}^{SM}=279\times
10^{-6}\,.$ (21)

Discrepancy between theory and experiments for $(g-2)$ in case of $\tau$
lepton is huge because the lifetime $\tau$ lepton is short and experimental
measurements of $(g-2)$ is very difficult.

Study of $(g-2)$ anomaly helps to derive the upper limits for couplings of DGB
and DSB with SM fermions. Here DM particles propagate in the loops, see Fig.
1.

###  III.1 Vector boson contribution

Contribution massive vector boson to $(g-2)$ with taking into account LFV
effect in the covariant gauge has a form:

$\displaystyle\delta
a_{f}=U_{fl}\frac{g_{lf}^{V}g_{fl}^{V}}{4\pi^{2}}\,\left(\frac{m_{f}}{m_{l}}\right)\int\limits_{0}^{1}\,dy\,y(1-y)\frac{\dfrac{m_{f}}{m_{l}}(y-1)-1}{\dfrac{m_{A}^{2}}{m_{l}^{2}}y+(1-y)\dfrac{m_{f}^{2}}{m_{l}^{2}}}\,,$
(22)

where $m_{l}$ is mass of a fermion which propagates in the loop, $m_{A}$ is
the mass of the dark photon $A^{\prime}$. In case when $m_{l}=m_{f}$ and in
the limits $m_{A}^{2}\to 0$ and $g^{V}_{ij}\to e$ we reproduce a famous
Schwinger term contribution to fermion anomalous magnetic moment.

The upper limits on dimensionless coupling
$g_{lf}^{V}=m_{A}\dfrac{v_{lf}}{\Lambda}$ of vector dark photon with fermion
are given in Fig. 2. Contribution induced by LFV coupling and effect and light
particle propagating in the loop is negative and cannot be considered as a
limit.

Figure 2: Upper boundaries on the couplings $g^{V}$ as function of dark photon
mass $m_{A}^{\prime}$ derived using data on leptonic $(g-2)$ coupling. The
shaded area is prohibited by data on $(g-2)$.

###  III.2 Axial-vector contribution

Figure 3: Bounds from $(g-2)$ of leptons on the couplings of dark boson in the
vector+axial-vector channel in dependence on different ratio of the channel
constants $d_{V}$ and $d_{A}$. The shaded area is prohibited from $(g-2)$
lepton data.

The contribution to AMM from axial-vector channel is

$\displaystyle\delta
a_{f}^{AV}=-\frac{(g^{AV}_{fl}g^{AV}_{lf})}{4\pi^{2}}\left(\frac{m_{f}}{m_{l}}\right)\int\limits_{0}^{1}\,dx\,\,x(1-x)\frac{\dfrac{m_{f}}{m_{l}}(1+x)+2}{\dfrac{m_{A}^{2}}{m_{l}^{2}}x+(1-x)(1-\dfrac{m_{f}^{2}}{m_{l}^{2}}x)}\,,$
(23)

where $g_{lf}^{AV}=m_{A}\dfrac{a_{lf}}{\Lambda}$ is axial-vector dimensionless
coupling which, in general, is not equal to $g_{lf}^{V}$. If $d_{L_{ij}}$ and
$d_{R_{ij}}$ coupling constants are positive and real when axial-vector
coupling should be less than vector coupling on magnitude.

The contribution from axial-vector channel is negative. In this case we obtain
that sum of vector and axial-vector channels is analogue of the contribution
of electroweak $Z$-boson to $(g-2)$, which is negative. In case, when $g^{AV}$
couplings are suppressed, vector contribution is dominant.

In Fig. 3 we show boundaries for couplings $g_{lf}^{V}$ as function of
$m_{A}^{\prime}$ and in dependence on ratio $g^{AV}/g^{V}$. It is very
important to note that the combination of vector and negative axial-vector
contributions to $(g-2)_{\mu}$ makes limits less stringent in comparison with
the case of using pure vector term. This is a chance for existence of light
dark vector particles in more wide band of possible masses and couplings. It
is worth noting that for the case of boundary without LFV difference between
$g_{lf}^{V}$ and $g_{lf}^{AV}$ couplings can be less but allowable area will
be wide. In the case when we have heavy particle propagating in the loop,
negative contribution from axial-vector part are strongly suppressed.

###  III.3 Scalar contribution

As we stressed before, scalar contribution is defined by new boson $\sigma$
occurs only when LFV effects are taken into account and has the following form

$\displaystyle\delta
a_{f}^{S}=\frac{g^{S}_{fl}g^{S}_{lf}}{8\pi^{2}}\left(\frac{m_{f}}{m_{l}}\right)\int\limits_{0}^{1}\,dx\,(1-x)^{2}\,\frac{1-\dfrac{m_{f}}{m_{l}}x}{\dfrac{m_{S}^{2}}{m_{l}^{2}}x+(1-x)(1+\dfrac{m_{f}^{2}}{m_{l}^{2}}x)}\,,$
(24)

where $m_{S}$ and $g^{S}_{fl}$ are the mass and coupling of a new scalar boson
introduced before. In the case when particle propagating in the loop is
lighter then initial particle, the contribution is negative and we cannot
constraint couplings and should use boundary from $(g-2)$ of more heavy
leptons. Scalar dimensionless coupling boundaries from $(g-2)$ of lepton are
presented in Fig. 4.

Figure 4: Bounds from $g-2$ of leptons for couplings $g^{S}_{ij}$ of scalar
boson with fermions in dependence on $m_{\sigma}$. The shaded area is
prohibited from $(g-2)$ lepton data.

###  III.4 Pseudoscalar contribution

Figure 5: Bounds from $g-2$ of leptons on the couplings $g^{PS}_{ij}$ of
scalar boson with fermions in dependence on $m_{\sigma}$. The shaded area is
prohibited from $(g-2)$ lepton data.

Pseudoscalar dimensionless coupling boundaries from $(g-2)$ of lepton are
presented in Fig.5. Upper boundary from $\tau$ lepton $(g-2)$ is huge. It is
connected with the difficulty of measurements of this value for $\tau$ lepton.
Mass of $\tau$ lepton is more heavy them the masses other leptons and main
band of mass range of new scalar meson because the boundary for lepton
conserving coupling and coupling which include LFV effects have the same
restriction.

The form of pseudoscalar contribution to $(g-2)$ of lepton from new $\sigma$
DSB is:

$\displaystyle\delta
a_{f}^{PS}=\frac{(g^{PS}_{fl})^{2}}{8\pi^{2}}\,\left(\frac{m_{f}}{m_{l}}\right)\,\int\limits_{0}^{1}\,dx\,(1-x)^{2}\,\frac{1+\dfrac{m_{f}}{m_{l}}x}{\dfrac{m_{S}^{2}}{m_{l}^{2}}x+(1-x)(1+\dfrac{m_{f}^{2}}{m_{l}^{2}}x)},$
(25)

which have a difference with scalar only in sign at term
$\dfrac{m_{f}}{m_{l}}x$ in numerator. The light lepton state in loop non play
a role in dependence of AMM of one and we have a similar behavior for
different couplings from same AMM quantity.

##  IV Nuclear Lepton Conversion

Here we estimate our LFV couplings between dark photon and leptons using data
(uppers limits) of the SINDRUM II searches for $\mu-e$ conversion on 198Au
Bertl:2006up . Together with these limits we consider the limits corresponding
to the future experiment PRISM/PRIME Witte:2012zza  with titanium 48Ti target
aiming at the sensitivity of $10^{-18}$. We have for these two cases:

$\displaystyle R_{\mu e}^{{\rm Au}}\leq 4.3\times 10^{-12}\ \
\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Bertl:2006up}{\@@citephrase{(}}{\@@citephrase{)}}}}:\ \ \
\alpha_{A^{\prime}}^{V(A)}\left(\frac{1\,\mbox{GeV}}{\Lambda_{\rm
LFV}}\right)^{2}\leq 8.5\times 10^{-13}\,,$ (26) $\displaystyle R_{\mu
e}^{{\rm Ti}}\ \lesssim 10^{-18}\ \ \ \ \ \ \ \
\mbox{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Witte:2012zza}{\@@citephrase{(}}{\@@citephrase{)}}}}:\ \ \
\alpha_{A^{\prime}}^{V(A)}\left(\frac{1\,\mbox{GeV}}{\Lambda_{\rm
LFV}}\right)^{2}\leq 1.6\times 10^{-15}\,,$ (27)

where $\alpha_{A^{\prime}}^{V(A)}$ is defined from effective Lagrangian which
describes nucleon-lepton LFV coupling

$\displaystyle{\cal L}_{\rm eff}^{\ell N}=\ \bar{N}\gamma^{\mu}N\
\bar{e}\,\Big{[}\alpha_{A^{\prime}}^{V}\gamma_{\mu}+\alpha_{A^{\prime}}^{A}\gamma_{\mu}\gamma_{5}\Big{]}\,\mu+{\rm
H.c.}$ (28)

These couplings $\alpha_{A^{\prime}}^{V(A)}$ are related to couplings
$g^{V}_{ij}$ of dark photon as [see details in Ref. Gonzalez:2013rea ]:

$\displaystyle\alpha_{A^{\prime}}^{V(A)}\simeq
g^{V(A)}_{NN}g^{V(A)}_{e\mu}\frac{m_{A}^{2}}{m_{A}^{2}+m_{\mu}^{2}},$ (29)

where $g^{V(A)}_{NN}$ is the dark photon $A^{\prime}$-nucleon coupling, which
we can put equal $g^{V(A)}_{NN}=m_{A}/\Lambda$ in flavor conservation scenario
as for lepton. The vector couplings with leptons were defined before. Using
this definition and approximation proposed in Gonzalez:2013rea , we can get
boundary for LFV coupling as

$\displaystyle\bigg{|}v_{e\mu}\bigg{|}\simeq\left\\{\begin{array}[]{cl}8.5\times
10^{-13}\Big{[}\dfrac{m_{A}}{\Lambda}\Big{]}^{-2}\left(\dfrac{m_{A}^{2}}{m_{A}^{2}+m_{\mu}^{2}}\right)^{-1}&\
\ \ {\rm SINDRUM}\\\\[8.53581pt] 1.6\times
10^{-15}\,\Big{[}\dfrac{m_{A}}{\Lambda}\Big{]}^{-2}\left(\dfrac{m_{A}^{2}}{m_{A}^{2}+m_{\mu}^{2}}\right)^{-1}&\
\ \ {\rm PRISM/PRIME}\end{array}\right.$ (32)

Similar limits we can establish for the $t$-channel scalar boson exchange and
obtain boundary for $a_{ij}$ LFV couplings from $\mu-e$ conversion restriction
using that $g^{PS}_{NN}=2m_{N}/\Lambda$, where $m_{N}$ is the nucleon mass:

$\displaystyle\bigg{|}a_{e\mu}\bigg{|}\simeq\left\\{\begin{array}[]{cl}8.5\times
10^{-13}\Big{[}\dfrac{m_{\mu}m_{N}}{\Lambda^{2}}\Big{]}^{-1}\left(\dfrac{m_{\sigma}^{2}}{m_{\sigma}^{2}+m_{\mu}^{2}}\right)^{-1}&\
\ \ {\rm SINDRUM}\\\\[8.53581pt] 1.6\times
10^{-15}\,\Big{[}\dfrac{m_{\mu}m_{N}}{\Lambda^{2}}\Big{]}^{-1}\left(\dfrac{m_{\sigma}^{2}}{m_{\sigma}^{2}+m_{\mu}^{2}}\right)^{-1}&\
\ \ {\rm PRISM/PRIME}\end{array}\right.$ (35)

Mass dependence of DGB coupling $g^{V}_{ij}=\dfrac{v_{ij}m_{A}}{\Lambda}$
gives mass square suppression for upper limits $|v_{e\mu}|$ from lepton
conversion by DGB exchange. The case DSB from Stueckelberg portal has other
suppression factor which is not connected with mass of mediator state.

##  V LFV decays $l_{i}\to l_{k}\gamma$

Rare LFV decays $l_{i}\to l_{k}\gamma$ due to LFV effect generated by dark
vector or scalar bosons vanish. Scalar boson have two different types of
Feynman diagrams which generate this rare decay. First is analog Barr-Zee
diagram by accounting scalar two photon transition [see diagrams (a-b) in Fig.
6] and second type is triangle diagram with two additional diagrams which
needed for fulfillment of gauge invariance [see diagrams (c-e) in Fig. 6].
Dark photon $A^{\prime}$ also gives contribution to this rare LFV decay [see
diagrams in Fig. 6(c-e)].

Figure 6: Feynman diagrams of gauge invariant matrix elements of interaction
lepton with external electromagnetic field accounting LFV effect generated by
new scalar and vector fields.

In general matrix element describing this LFV process can be parameterized as

$\displaystyle
iM_{ik}=ie\epsilon^{\mu}(q)\bar{u}_{i}(p_{2},m_{e})\left[\frac{i}{2m_{i}}\sigma_{\mu\nu}q^{\nu}F^{LFV}_{M}+\frac{i}{2m_{i}}\sigma_{\mu\nu}q^{\nu}\gamma_{5}F^{LFV}_{D}\right]u_{k}(p_{1},m_{\mu})\,.$
(36)

Square of the amplitude is

$\displaystyle|M|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{i}^{2}}((F^{LFV}_{M})^{2}+(F^{LFV}_{D})^{2})(m_{i}^{4}-2m_{k}^{2}m_{i}^{2}+m_{k}^{4})\,.$
(37)

Using approximation $m_{e}\ll m_{\mu}\ll m_{\tau}$, decay width for this rare
decay is

$\displaystyle\Gamma(l_{i}\to el_{k})$ $\displaystyle=$
$\displaystyle\frac{1}{2m_{i}}\int\frac{d^{3}qd^{3}p_{2}}{4E_{2}E_{q}(2\pi)^{6}}|M|^{2}(2\pi)^{4}\delta^{(4)}(p_{1}-p_{2}-q)$
$\displaystyle=$
$\displaystyle\frac{m_{i}}{8\pi}e^{2}\left(|F^{LFV}_{M}|^{2}+|F^{LFV}_{D}|^{2}\right)\,,$

where $F_{M}$ and $F_{D}$ are dimensionless form-factors in analogy of
magnetic and dipole which goes from different channel of interaction and with
different intermediate leptons in loops.

Contributions induced by diagrams in Figs. 6(a) and 6(b) for CP even and CP-
odd vertices are connected with CP-even and CP-odd couplings with gauge bosons
of SM [see Eq. 12]. These contributions have the form

$\displaystyle F_{M}$ $\displaystyle=$
$\displaystyle-\frac{m_{i}}{8\pi^{2}}g^{PS}_{ik}g_{\sigma\gamma\gamma}h^{1}_{\sigma\gamma\gamma}(x_{\mu})=-\frac{m_{i}^{2}}{8\pi^{2}\Lambda}a_{ik}g_{\sigma\gamma\gamma}h^{1}_{\sigma\gamma\gamma}(x_{\mu})\,,$
$\displaystyle F_{D}$ $\displaystyle=$
$\displaystyle-\frac{m_{i}}{8\pi^{2}}g^{S}_{ik}g_{\sigma\gamma\gamma}h^{1}_{\sigma\gamma\gamma}(x_{\mu})=-\frac{m_{i}^{2}}{8\pi^{2}\Lambda}v_{ik}g_{\sigma\gamma\gamma}h^{1}_{\sigma\gamma\gamma}(x_{\mu})$
(39)

for CP-even coupling and

$\displaystyle F_{M}$ $\displaystyle=$
$\displaystyle-\frac{m_{i}}{8\pi^{2}}g^{S}_{ik}\bar{g}_{\sigma\gamma\gamma}h^{2}_{\sigma\gamma\gamma}(x_{\mu})=-\frac{m_{i}^{2}}{8\pi^{2}\Lambda}v_{ik}\bar{g}_{\sigma\gamma\gamma}h^{2}_{\sigma\gamma\gamma}(x_{\mu})\,,$
$\displaystyle F_{D}$ $\displaystyle=$
$\displaystyle-\frac{m_{i}}{8\pi^{2}}g^{PS}_{ik}\bar{g}_{\sigma\gamma\gamma}h^{2}_{\sigma\gamma\gamma}(x_{\mu})=-\frac{m_{i}^{2}}{8\pi^{2}\Lambda}a_{ik}\bar{g}_{\sigma\gamma\gamma}h^{2}_{\sigma\gamma\gamma}(x_{\mu})\,,$
(40)

for CP-odd coupling, where $h^{i}_{\sigma\gamma\gamma}(x_{\mu})$ is sum of
different contributions from diagrams in Figs. 6(a) and 6(b) for CP even and
CP-odd case of vertex interaction scalar with SM photon in Barr-Zee diagram
contribution. We remind that $g_{\sigma\gamma\gamma}$ and
$\bar{g}_{\sigma\gamma\gamma}$ couplings are gone from fermion loop. When this
contribution to decay width is $\alpha^{2}$ suppressed for CP-even type
coupling contribution and $\alpha^{2}$ and LFV suppressed for CP-odd coupling.

The contributions to $l_{i}\to l_{k}\gamma$ rare decay involving only one LFV
coupling where [case when index $j=i$ or $j=k$ in diagrams in Figs. 6(c)-(e)]
are leading and have the form:

$\displaystyle F_{M}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}g^{PS}_{ik}\left[g^{PS}_{ii}h_{2}^{S}(x_{\mu})+g^{PS}_{kk}h_{3}^{S}(x_{\mu})\right]=-\frac{2m_{i}}{16\pi^{2}\Lambda^{2}}a_{ik}\left[m_{i}a_{ii}h_{2}^{S}(x_{\mu})+m_{k}a_{kk}h_{3}^{S}(x_{\mu})\right]\,,$
$\displaystyle F_{D}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}g^{S}_{ik}\left[g^{PS}_{ii}h_{2}^{S}(x_{\mu})+g^{PS}_{kk}h_{3}^{S}(x_{\mu})\right]=-\frac{2m_{i}}{16\pi^{2}\Lambda^{2}}v_{ik}\left[m_{i}a_{kk}h_{2}^{S}(x_{\mu})-m_{k}a_{ii}h_{3}^{S}(x_{\mu})\right]\,,$
(41)

and for the process $\mu\to e\gamma$ with $\tau$ lepton in loop with double
LFV coupling

$\displaystyle F_{M}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}\left(\frac{m_{\mu}}{m_{\tau}}\right)\left[g^{PS}_{\mu\tau}g^{PS}_{\tau
e}+g^{S}_{\mu\tau}g^{S}_{\tau
e}\right]h_{1}^{S}(x_{\tau})=-\frac{m_{\mu}m_{\tau}}{16\pi^{2}\Lambda^{2}}\left[a_{\mu\tau}a_{\tau
e}-v_{\mu\tau}v_{\tau e}\right]h_{1}^{S}(x_{\tau})\,,$ $\displaystyle F_{D}$
$\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}\left(\frac{m_{\mu}}{m_{\tau}}\right)\left[g^{S}_{\mu\tau}g^{PS}_{\tau
e}+g^{PS}_{\mu\tau}g^{S}_{\tau
e}\right]h_{1}^{S}(x_{\tau})=-\frac{m_{\mu}m_{\tau}}{16\pi^{2}\Lambda^{2}}\left[a_{\mu\tau}v_{\tau
e}-v_{\mu\tau}a_{\tau e}\right]h_{1}^{S}(x_{\tau})\,.$ (42)

For dark photon vector boson channel we also have a contribution to LFV lepton
decay. Dimensionless parameters $g_{ij}^{V}$ and $g_{ij}^{AV}$ are defined
before. They don’t have dependence from lepton mass (it makes them different
from scalar couplings) and are proportional to the factor $m_{A}/\Lambda$:

$\displaystyle F_{M}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}g^{V}_{ik}\left[g^{V}_{ii}h_{2}^{V}(x_{\mu})+g^{V}_{kk}h_{3}^{V}(x_{\mu})\right]=-\frac{m_{A}^{2}}{16\pi^{2}\Lambda^{2}}v_{ik}\left[v_{ii}h_{2}^{V}(x_{\mu})+v_{kk}h_{3}^{V}(x_{\mu})\right]\,,$
$\displaystyle F_{M}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}g^{AV}_{ik}\left[g^{AV}_{ii}h_{2}^{V}(x_{\mu})+g^{AV}_{kk}h_{3}^{V}(x_{\mu})\right]=-\frac{m_{A}^{2}}{16\pi^{2}\Lambda^{2}}a_{ik}\left[a_{ii}h_{2}^{V}(x_{\mu})+a_{kk}h_{3}^{V}(x_{\mu})\right]$
(43)

and

$\displaystyle F_{D}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}g^{V}_{ik}\left[g^{AV}_{ii}h_{2}^{V}(x_{\mu})+g^{AV}_{kk}h_{3}^{V}(x_{\mu})\right]=\frac{m_{A}^{2}}{16\pi^{2}\Lambda^{2}}v_{ik}\left[a_{\mu\mu}h_{2}^{V}(x_{\mu})+a_{kk}h_{3}^{V}(x_{\mu})\right]\,,$
$\displaystyle F_{D}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}g^{AV}_{ik}\left[g^{V}_{ii}h_{2}^{V}(x_{\mu})+g^{V}_{kk}h_{3}^{V}(x_{\mu})\right]=\frac{m_{A}^{2}}{16\pi^{2}\Lambda^{2}}a_{ik}\left[v_{ii}h_{2}^{V}(x_{\mu})+v_{kk}h_{3}^{V}(x_{\mu})\right]$
(44)

for $\mu\to e\gamma$ process with $\tau$ lepton in loop with double LFV
coupling

$\displaystyle F_{M}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}\left(\frac{m_{\mu}}{m_{\tau}}\right)\left[g^{V}_{\mu\tau}g^{V}_{\tau
e}h_{1}^{V}(x_{\tau})+g^{AV}_{\mu\tau}g^{AV}_{\tau
e}h_{1}^{V}(x_{\tau})\right]=-\frac{m_{A}^{2}}{16\pi^{2}}\left(\frac{m_{\mu}}{m_{\tau}}\right)\left[v_{\mu\tau}v_{\tau
e}h_{1}^{V}(x_{\tau})+a_{\mu\tau}a_{\tau e}h_{1}^{V}(x_{\tau})\right]\,,$
$\displaystyle F_{D}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left(\frac{m_{\mu}}{m_{\tau}}\right)\left[g^{V}_{\mu\tau}g^{AV}_{\tau
e}h_{1}^{V}(x_{\tau})+g^{AV}_{\mu\tau}g^{V}_{\tau
e}h_{1}^{V}(x_{\tau})\right]=\frac{m_{A}^{2}}{16\pi^{2}}\left(\frac{m_{\mu}}{m_{\tau}}\right)\left[v_{\mu\tau}a_{\tau
e}h_{1}^{V}(x_{\tau})+a_{\mu\tau}v_{\tau e}h_{1}^{V}(x_{\tau})\right]\,,$ (45)

where $x_{i}=m_{i}^{2}/m_{S}^{2}$ for scalar channel and
$x_{i}=m_{i}^{2}/m_{A}^{2}$ for dark vector channel, $h_{i}^{S}(x_{i})$ and
$h_{i}^{V}(x_{i})$ are loop integration functions arising from form factors at
certain limit presented in Appendix B, full expressions for the form factors
are also presented in this Appendix B. We note that form factors $h_{i}$
generated by diagrams in Figs. 6(c)-(e) do not have divergences unlike
contribution due to the Barr-Zee like diagrams displayed in Figs. 6(a) and
6(b) having CP-even coupling with photon. Herewith we note that contributions
from Figs. 6(d) and 6(e) is needed for cancellation divergences a contribution
from diagram in Fig. 6(c) and for gauge invariance of interaction photon with
lepton trough loop diagrams with intermediate DSB or DGB states.

Similarly we can write the contribution to $F_{M}$ and $F_{D}$ form factors of
the for $\tau\to\mu\gamma$ and $\tau\to e\gamma$ LFV rare decays in the case
when initial/final leptons are differed from a lepton propagating in the loop:

$\tau\to\mu\gamma$ process

$\displaystyle F_{M}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}(g^{PS}_{\tau e}g^{PS}_{e\mu}+g^{S}_{\tau
e}g^{S}_{e\mu})h_{3}^{S}(x_{\tau})=-\frac{m_{\tau}m_{\mu}}{16\pi^{2}\Lambda^{2}}(a_{\tau
e}a_{\mu e}+v_{\tau e}v_{\mu e})h_{3}^{S}(x_{\tau})\,,$ $\displaystyle F_{D}$
$\displaystyle=$ $\displaystyle-\frac{1}{16\pi^{2}}(g^{PS}_{\tau
e}g^{S}_{e\mu}+g^{S}_{\tau
e}g^{PS}_{e\mu})h_{3}^{S}(x_{\tau})=\frac{m_{\tau}m_{\mu}}{16\pi^{2}\Lambda^{2}}(a_{\tau
e}v_{\mu e}+v_{\tau e}a_{\mu e})h_{3}^{S}(x_{\tau})$ (46)

from scalar and

$\displaystyle F_{M}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}\left[g^{V}_{\tau
e}g^{V}_{e\mu}+g^{AV}_{\tau
e}g^{AV}_{e\mu}\right]h_{3}^{V}(x_{\tau})=-\frac{m_{A}^{2}}{16\pi^{2}\Lambda^{2}}\left[v_{\tau
e}v_{e\mu}+a_{\tau e}a_{e\mu}\right]h_{3}^{V}(x_{\tau}),\,,$ $\displaystyle
F_{D}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\left[g^{V}_{\tau
e}g^{AV}_{e\mu}+g^{AV}_{\tau
e}g^{V}_{e\mu}\right]h_{3}^{V}(x_{\tau})=\frac{m_{A}^{2}}{16\pi^{2}\Lambda^{2}}\left[v_{\tau
e}a_{e\mu}+a_{\tau e}v_{e\mu}\right]h_{3}^{V}(x_{\tau})$ (47)

from vector $A^{\prime}$ boson exchange.

$\tau\to e\gamma$ process

$\displaystyle F_{M}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}(g^{PS}_{\tau\mu}g^{PS}_{e\mu}+g^{S}_{\tau\mu}g^{S}_{e\mu})h_{3}^{S}(x_{\tau})=-\frac{m_{\tau}m_{\mu}}{16\pi^{2}\Lambda^{2}}(a_{\tau\mu}a_{e\mu}-v_{\tau\mu}v_{e\mu})h_{3}^{S}(x_{\tau}),\,,$
$\displaystyle F_{D}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}(g^{PS}_{\tau\mu}g^{S}_{e\mu}+g^{S}_{\tau\mu}g^{PS}_{e\mu})h_{3}^{S}(x_{\tau})=-\frac{m_{\tau}m_{\mu}}{16\pi^{2}\Lambda^{2}}(a_{\tau\mu}v_{e\mu}-v_{\tau\mu}a_{e\mu})h_{3}^{S}(x_{\tau})$
(48)

from scalar and

$\displaystyle F_{M}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}\left[g^{V}_{\tau\mu}g^{V}_{\mu
e}+g^{AV}_{\tau\mu}g^{AV}_{\mu
e}\right]h_{3}^{V}(x_{\tau})=-\frac{m_{A}^{2}}{16\pi^{2}\Lambda^{2}}\left[v_{\tau\mu}v_{\mu
e}+a_{\tau\mu}a_{\mu e}\right]h_{3}^{V}(x_{\tau}),\,,$ $\displaystyle F_{D}$
$\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left[g^{V}_{\tau\mu}g^{AV}_{\mu
e}+g^{AV}_{\tau\mu}g^{V}_{\mu
e}\right]h_{3}^{V}(x_{\tau})=\frac{m_{A}^{2}}{16\pi^{2}\Lambda^{2}}\left[v_{\tau\mu}a_{\mu
e}+a_{\tau\mu}v_{\mu e}\right]h_{3}^{V}(x_{\tau}),$ (49)

from vector $A^{\prime}$ boson exchange.

Here we find a full agreement for scalar contribution with results reported in
Ref. Cornella:2019uxs  with precision to a factor 2 that is hidden in the
definition of form factors. It is same because we use the same idea of familon
as in Refs. Cornella:2019uxs ; Bauer:2020jbp  for ALPs physics. Gordon
identities which are used here for LFV process are listed in Appendix A.

##  VI Analysis of current limits

Current limits for the branchings of the LFV lepton decays $l_{i}\to
l_{k}\gamma$ are PDG20

$\displaystyle\mathrm{Br}(\mu\to e\gamma)$ $\displaystyle<$ $\displaystyle
4.2\times 10^{-13}\,,$ $\displaystyle\mathrm{Br}(\tau\to e\gamma)$
$\displaystyle<$ $\displaystyle 3.3\times 10^{-8}\,,$
$\displaystyle\mathrm{Br}(\tau\to\mu\gamma)$ $\displaystyle<$ $\displaystyle
4.4\times 10^{-8}\,.$ (50)

Future experimental prospects aim to measure these quantities with more high
precision in order to low the upper boundary by one order of magnitude.

We analyze LFV couplings by focusing onto scenario of lepton-flavor
universality for lepton conserving couplings $a_{ii}$ or $v_{ii}$. For
estimation of bounds we put a scale of New Physics $\Lambda$ equal to 1 TeV.
Scalar and vector contributions to LFV rare decay widths are considered
independently. The results for the lepton-flavor universality scenario using
$\mu\to e\gamma$ and $\tau\to e\gamma$ processes are presented in Fig. 7 in
case of equal $a_{ii}$ couplings. Bounds from $\tau\to\mu\gamma$ are the same
as for $\tau\to e\gamma$ because in approximation $m_{e}\ll m_{\mu}\ll
m_{\tau}$ contributions from the loops are same. We consider independent
contributions from diagrams in Fig.6(c)-(e). Contributions from the Barr-Zee
diagram are suppressed as pointed before. The contributions of heavy lepton
state in loop was omitted here in the analysis of certain couplings because
these contributions have double LFV suppression.

The peaks in the Fig. 7 is connected to behavior of loop integrals $h_{i}(x)$
near the point $x=1$ where vector boson production in vicinity of the
threshold. For solution of this problem one needs to include the decay width
of dark vector boson to lepton pair with
$\tau_{A}^{-1}\sim(v_{ij}/\Lambda)^{2}$ in the Breit-Wigner propagator.

Boundaries to LFV couplings on Fig. 7 include restriction from lepton $(g-2)$
and rare LFV $l_{i}\to l_{k}\gamma$ lepton decays. In case of $e-\mu$ LFV
transition, left pictures in Figs. 7, we add bound from $e-\mu$ conversion.
Suppression at heavy masses for $e-\mu$ conversion is due of heavy bosons
exchange in the $t$-channel. Inclusion of boundaries from $e-\mu$ conversion
gives suppression from scalar $a_{ij}$ coupling in all ranges of mass. The
dark vector photon keeps a window for huge LFV couplings $v_{ij}$ at light
masses of one.

Figure 7: Bounds from $g-2$ of leptons, from $\mu\to e\gamma$ and $\tau\to
e\gamma$ LFV decay width and lepton conversion to interaction couplings to
$a_{ij}$ scalar boson coupling and $v_{ij}$ dark vector $A^{\prime}$ boson
with fermions in dependence from $m_{\sigma}$ or $m_{A}$. Bound is constructed
for the scale $\Lambda=1$ TeV and at assumption that $a_{ii}=1$ or $v_{ii}=1$.
The shaded area is prohibited from data.

Boundaries for $a_{ij}$ from leptonic $(g-2)$ are strong for more massive
lepton propagating in the loop and less suppressed lepton conserving
couplings. From LFV decays $l_{i}\to l_{k}\gamma$ the behavior of bound is
similar but suppression is more strong. It also occurs for scalar channel and
dark vector boson channels. Wherein that vector and scalar coupling have the
same hermitian matrices $v_{ij}$ and $a_{ij}$. The boundary from the vector
channel gives limits for scalar LFV dimensionless couplings. Such interference
of couplings is due to the Stueckelberg mechanism.

Figure 8: Estimate of contribution to lepton AMM in dependence on masses of
scalar boson or dark photon. Boundary is made for the scale $\Lambda=1$ TeV
and in assumption that $a_{ii}=1$ or $v_{ii}=1$ with taking into account
restriction for LFV couplings.

Using boundary from rare decays $l_{i}\to l_{k}\gamma$ in universal lepton
conserving couplings scenario for $a_{ii}=1$ or $v_{ii}=1$ we can deduce
possible lepton contribution to $(g-2)$ in dependence on masses of new scalar
or dark vector bosons. We estimate contribution to AMM due to sum of the loops
with lightest leptons $\mu$ and $e$ and with taking into account lepton flavor
conservation (LFC) couping in conserving scenario and LFV coupling are
constrained from LFV lepton decay $l_{i}\to l_{k}\gamma$

$\displaystyle\Delta a_{l_{i}}=(\Delta a_{l_{i}})_{\mathrm{LFC}}+(\Delta
a_{l_{i}})_{\mathrm{LFV}}\,.$ (51)

For vector contribution we also include restriction from $\mu-e$ conversion.
In case of $(g-2)$ of electron, restriction from conversion it makes no sense.
For $(g-2)$ of muon, vector channel has a suppression at mass of dark photon
larger than $10^{-3}$ GeV. Such additional restriction for $(g-2)$ of muon
gives the same behavior of vector channel for this anomaly as for the case of
$(g-2)$ of electron. The dependence is presented in Fig. 8.

As mentioned in Ref. Bauer:2019gfk , a various of lepton conserving couplings
we can tuck up mass (or mass and LFC couplings) when difference between theory
and experimental data of $(g-2)$ of leptons due to contributions from ALPs or
new scalar in our case. In the case where we take into account dark vector
photon, the problem of lepton $(g-2)$ anomaly can be solved by accounting for
contributions from vector channel and scalar can have more suppressed coupling
in dependence of mass.

##  VII Conclusions

We constructed a phenomenological Lagrangian approach which combines SM and DM
sectors based on the Stueckelberg mechanism for generation mass of dark
$U_{D}(1)$ gauge boson (or dark photon). The DM sector contains dark photon,
dark scalar, and generic dark fermion fields. DM scalar could be identified as
the Stueckelberg ALPs. Couplings of DSB and DGB with fermions of SM have
interference between $v_{ij}$ and $a_{ij}$ and coupling of DGB with fermions
is proportional to ratio mass DGB to scale of New Physics $\Lambda$.
Stueckelberg Portal opens new possibilities for study of phenomenology of BSM
Physics and can be important for running and planning experiments at world-
wide facilities (e.g., for the NA64 Experiment at SPS CERN Banerjee:2016tad ;
Banerjee:2017hhz ).

We derived boundaries on the effective couplings of our Lagrangian using data
on lepton AMMs, LFV lepton decays $l_{i}\to\gamma l_{k}$, and $\mu-e$
conversion. It is known that the latter are very useful because they give more
stringent limits on the couplings of effective Lagrangian. We also founded
that the $(g-2)$ anomaly can be preferably solved by light dark photon then by
scalar boson.

In future we plan to study a possible role of the Stueckelberg portal in
different rare LFV processes including semileptonic decays. In forthcoming
paper we also plan to extend this portal by inclusion of the $SU_{D}(2)$ gauge
sector and matter fields.

###### Acknowledgements.

This work was funded by ANID PIA/APOYO AFB180002 (Chile), by FONDECYT (Chile)
under Grants No. 1191103 and No. 1190845 and by ANID$-$Millennium
Program$-$ICN2019_044 (Chile). A.K. acknowledges support by DFG through CRC
TRR 257, _Particle Physics Phenomenology after the Higgs Discovery_ (grant no.
396021762).

##  Appendix A Gordon Identities

The Gordon identities the matrix elements describing the coupling of external
gauge field with fermions having different masses read:

$\displaystyle i\sigma_{\mu\nu}q^{\nu}$ $\displaystyle=$ $\displaystyle-
P_{\mu}+(m_{i}+m_{j})\gamma_{\mu};\qquad\qquad\quad
i\sigma_{\mu\nu}P^{\nu}=-q_{\mu}+(m_{j}-m_{i})\gamma_{\mu};$ (52)
$\displaystyle i\sigma_{\mu\nu}q^{\nu}\gamma_{5}$ $\displaystyle=$
$\displaystyle-P_{\mu}\gamma_{5}+(m_{j}-m_{i})\gamma_{\mu}\gamma_{5};\qquad
i\sigma_{\mu\nu}P^{\nu}\gamma_{5}=-q_{\mu}\gamma_{5}+(m_{i}+m_{j})\gamma_{\mu}\gamma_{5}.$
(53)

##  Appendix B Loop function occuring in the form factors

In this Appendix, we present the analytical expressions of the loop integrals
occuring in the amplitude of the LFV decays $l_{i}\to\gamma l_{k}$ for
different new particle channel and lepton propagating in the loop. Full
results for the form factors in form of Feynman integrals by Feynman are
presented too. All results for the form factors have been numerically and
analytically cross-checked using the Mathematica Package-X Patel:2015tea  and
packages FeynHelpers Shtabovenko:2016whf  and FeynCalc FeynCalc .

Loop function from CP-even and CP-odd coupling of scalar field with photons
Fig.6(a) and (b):

$\displaystyle h^{1}_{\sigma\gamma\gamma}$ $\displaystyle=$ $\displaystyle
2\ln\left(\frac{\Lambda^{2}}{m_{S}^{2}}\right)-\frac{\ln
x}{x-1}-(x-1)\ln\left(\frac{x}{x-1}\right)-2\,,$ $\displaystyle
h^{2}_{\sigma\gamma\gamma}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1-\ln(x)+\frac{x\ln(x)}{x-1}-(x-1)\ln\left(\frac{x}{x-1}\right)\right)\,,$
(54)

where $x=m_{S}^{2}/m_{\mu}^{2}$.

Full loop function occuring in the form factors read:

Scalar or pseudoscalar case:

$\displaystyle
F^{PS/S}=2m_{i}\int_{0}^{1}dx_{1}\int^{1-x_{1}}_{0}\\!\\!\\!\\!\\!dx_{2}\frac{(m_{f}x_{2}+m_{i}x_{1})(x_{1}+x_{2}-1)\pm
m_{l}(x_{1}+x_{2})}{(x_{1}+x_{2})(m_{f}^{2}x_{2}+m_{i}^{2}x_{1}+m_{l}^{2})-m_{f}^{2}x_{2}-m_{i}^{2}x_{1}-m_{S}^{2}(x_{1}+x_{2}-1)}.$
(55)

Vector intermediate state case:

$\displaystyle
F^{V/AV}=2m_{i}\int_{0}^{1}dx_{1}\int^{1-x_{1}}_{0}\\!\\!\\!\\!\\!dx_{2}\frac{2(x_{1}+x_{2}-1)(m_{f}(x_{2}-1)+m_{i}(x_{1}-1)\pm
2m_{l})}{(x_{1}+x_{2})(m_{i}^{2}x_{2}+m_{f}^{2}x_{1}+m_{l}^{2})-m_{e}^{2}x_{2}-m_{i}^{2}x_{1}-m_{S}^{2}(x_{1}+x_{2}-1)}.$
(56)

Here, $m_{i}$ is a mass off initial lepton state, $m_{f}$ is a mass of final
lepton state, and $m_{l}$ is mass of loop lepton.

In the approximation when $m_{e}\ll m_{\mu}\ll m_{\tau}$ the loop integrals
reduce to simple functions $h^{S}$.

Case: scalar intermediate state and $\tau$ lepton in the loop

$\displaystyle h^{S}_{1}(x)=\frac{3x^{2}-4x+2x^{2}\ln(x)+1}{(1-x)^{3}}\,,$
(57)

where $x=m_{S}^{2}/m_{\tau}^{2}$.

Case: scalar intermediate state and muon in the loop

$\displaystyle
h^{S}_{2}(x)=\frac{2x^{2}-(x-3)x^{2}\ln(x)-3x+1}{1-x}-2x\sqrt{x^{2}-4x}\ln\left(\frac{\sqrt{x}+\sqrt{x-4}}{2}\right)\,,$
(58)

where $x=m_{S}^{2}/m_{\mu}^{2}$.

Case: scalar intermediate state and electron in the loop

$\displaystyle
h^{S}_{3}(x)=\left(1-2x-2(x-1)x\ln\left(\frac{x-1}{x}\right)\right)\,,,$ (59)

where $x=m_{S}^{2}/m_{\mu}^{2}$.

Case: vector intermediate state and $\tau$ lepton in the loop

$\displaystyle h^{V}_{1}(x)=4\frac{x^{2}-1-2x\ln(x)}{(1-x)^{3}}\,,$ (60)

where $x=m_{S}^{2}/m_{\tau}^{2}$.

Case: vector intermediate state and muon in the loop

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

where $x=m_{S}^{2}/m_{\mu}^{2}$,

Case: vector intermediate state and electron in the loop

$\displaystyle h^{V}_{3}=2x-2(x-1)\,x\ln\left(\frac{x}{x-1}\right)-1,$ (62)

where $x=m_{S}^{2}/m_{\mu}^{2}$.

## References

  * (1) J. Alexander et al., arXiv:1608.08632 [hep-ph].
  * (2) D. Banerjee et al. (NA64 Collaboration), Phys. Rev. Lett. 118, 011802 (2017); 123, 121801 (2019); 125, 081801 (2020).
  * (3) D. Banerjee et al. (NA64 Collaboration), Phys. Rev. D 97, 072002 (2018); 101, 071101 (2020).
  * (4) E. Cortina Gil et al. (NA62 Collaboration), JHEP 1905, 182 (2019).
  * (5) S. Alekhin et al., Rept. Prog. Phys. 79, 124201 (2016).
  * (6) M. Anelli et al. (SHiP Collaboration), arXiv:1504.04956 [physics.ins-det].
  * (7) C. Ahdida et al. (SHiP Collaboration), JHEP 2104, 199 (2021).
  * (8) R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 120, 061801 (2018).
  * (9) G. Aad et al. (ATLAS Collaboration), JHEP 2102, 226 (2021).
  * (10) A. M. Sirunyan et al. [CMS Collaboration], JHEP 2103, 011 (2021).
  * (11) J. P. Lees et al. (BaBar Collaboration), Phys. Rev. Lett. 119, 131804 (2017).
  * (12) P. H. Adrian et al. (HPS Collaboration), Phys. Rev. D 98, 091101 (2018).
  * (13) S.-H. Park et al. (Belle Collaboration), JHEP 2104, 191 (2021).
  * (14) B. Abi et al. (Muon g-2), Phys. Rev. Lett. 126, 141801 (2021)
  * (15) T. Aoyama, N. Asmussen, M. Benayoun, J. Bijnens, T. Blum, M. Bruno, I. Caprini, C. M. Carloni Calame, M. Cè, and G. Colangelo, et al. Phys. Rept. 887, 1 (2020).
  * (16) M. Pospelov and Y. D. Tsai, Phys. Lett. B 785, 288 (2018).
  * (17) T. Gutsche, A. N. Hiller Blin, S. Kovalenko, S. Kuleshov, V. E. Lyubovitskij, M. J. Vicente Vacas, and A. Zhevlakov, Phys. Rev. D 95, 036022 (2017); Few Body Syst. 59, 66 (2018).
  * (18) A. S. Zhevlakov, M. Gorchtein, A. N. Hiller Blin, T. Gutsche, and V. E. Lyubovitskij, Phys. Rev. D 99, 031703(R) (2019).
  * (19) A. S. Zhevlakov, T. Gutsche, and V. E. Lyubovitskij, Phys. Rev. D 99, 115004 (2019);
  * (20) A. S. Zhevlakov and V. E. Lyubovitskij, Phys. Rev. D 101, 115041 (2020).
  * (21) Y. Ema, K. Hamaguchi, T. Moroi, and K. Nakayama, JHEP 01, 096 (2017)
  * (22) A. J. Buras, A. Crivellin, F. Kirk, C. A. Manzari, and M. Montull, JHEP 06, 068 (2021)
  * (23) F. Kahlhoefer, Int. J. Mod. Phys. A 32, 1730006 (2017).
  * (24) S. Gninenko, S. Kovalenko, S. Kuleshov, V. E. Lyubovitskij, and A. S. Zhevlakov, Phys. Rev. D 98, 015007 (2018).
  * (25) S. N. Gninenko, N. V. Krasnikov, and V. A. Matveev, Phys. Part. Nucl. 51, 829 (2020).
  * (26) G. Lanfranchi, M. Pospelov, and P. Schuster, Ann. Rev. Nucl. Part. Sci. Annu. Rev. Nucl. Part. Sci. 71, 279 (2021).
  * (27) P. Agrawal et al., Eur. Phys. J. C 81, 1015 (2021).
  * (28) H. Georgi, D. B. Kaplan, and L. Randall, Phys. Lett. 169B, 73 (1986).
  * (29) B. Holdom, Phys. Lett. 166B, 196 (1986).
  * (30) P. Fayet, Eur. Phys. J. C 77, 53 (2017).
  * (31) M. Bauer, M. Neubert and A. Thamm, JHEP 1712, 044 (2017).
  * (32) M. Bauer, M. Heiles, M. Neubert, and A. Thamm, Eur. Phys. J. C 79, 74 (2019).
  * (33) M. Bauer, M. Neubert, S. Renner, M. Schnubel, and A. Thamm, Phys. Rev. Lett. 124, 211803 (2020).
  * (34) M. Bauer, M. Neubert, S. Renner, M. Schnubel, and A. Thamm, JHEP 2104, 063 (2021).
  * (35) T. Gherghetta, J. Kersten, K. Olive, and M. Pospelov, Phys. Rev. D 100, 095001 (2019).
  * (36) J. Martin Camalich, M. Pospelov, P. N. H. Vuong, R. Ziegler, and J. Zupan, Phys. Rev. D 102, 015023 (2020).
  * (37) R. Delbourgo and G. Thompson, Phys. Rev. Lett. 57, 2610 (1986).
  * (38) C. D. Carone and H. Murayama, Phys. Rev. D 52, 484 (1995).
  * (39) C. Cheung, J. T. Ruderman, L. T. Wang, and I. Yavin, Phys. Rev. D 80, 035008 (2009).
  * (40) M. Duerr, F. Kahlhoefer, K. Schmidt-Hoberg, T. Schwetz, and S. Vogl, JHEP 1609, 042 (2016).
  * (41) M. Bauer, M. Neubert, S. Renner, M. Schnubel and A. Thamm, [arXiv:2110.10698 [hep-ph]].
  * (42) C. Cornella, P. Paradisi, and O. Sumensari, JHEP 2001, 158 (2020).
  * (43) D. V. Kirpichnikov, V. E. Lyubovitskij, and A. S. Zhevlakov, Phys. Rev. D 102, 095024 (2020).
  * (44) D. V. Kirpichnikov, V. E. Lyubovitskij, and A. S. Zhevlakov, Particles 3, 719 (2020).
  * (45) M. Endo, S. Iguro, and T. Kitahara, JHEP 2006, 040 (2020).
  * (46) S. Iguro, Y. Omura, and M. Takeuchi, JHEP 2009, 144 (2020).
  * (47) L. Calibbi, M. L. López-Ibáñez, A. Melis, and O. Vives, JHEP 2006, 087 (2020).
  * (48) B. Barman, S. Bhattacharya, and B. Grzadkowski, JHEP 12 (2020), 162.
  * (49) P. Escribano and A. Vicente, JHEP 2103, 240 (2021).
  * (50) L. Calibbi, D. Redigolo, R. Ziegler, and J. Zupan, JHEP 09, 173 (2021).
  * (51) K. Ma, arXiv:2104.11162 [hep-ph].
  * (52) A. E. Cárcamo Hernández, S. Kovalenko, M. Maniatis, and I. Schmidt, JHEP 10, 036 (2021).
  * (53) M. A. Buen-Abad, J. Fan, M. Reece, and C. Sun, JHEP 09, 101 (2021).
  * (54) E. C. G. Stueckelberg, Helv. Phys. Acta 11, 225 (1938).
  * (55) H. Ruegg and M. Ruiz-Altaba, Int. J. Mod. Phys. A 19, 3265 (2004).
  * (56) J. L. Feng, T. Moroi, H. Murayama, and E. Schnapka, Phys. Rev. D 57, 5875 (1998).
  * (57) A. E. Dorokhov, A. E. Radzhabov, and A. S. Zhevlakov, JETP Lett. 100, 133 (2014).
  * (58) P. A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020).
  * (59) W. H. Bertl et al. (SINDRUM II Collaboration), Eur. Phys. J. C47, 337 (2006).
  * (60) H. Witte et al., Conf. Proc. C1205201, 79 (2012).
  * (61) M. Gonzalez, T. Gutsche, J. C. Helo, S. Kovalenko, V. E. Lyubovitskij, and I. Schmidt, Phys. Rev. D 87, 096020 (2013).
  * (62) H. H. Patel, Comput. Phys. Commun. 197, 276 (2015).
  * (63) V. Shtabovenko, Comput. Phys. Commun. 218, 48 (2017).
  * (64) V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun. 256, 107478 (2020); V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun. 207, 432 (2016); R. Mertig, M. Böhm, and A. Denner, Comput. Phys. Commun. 64, 345 (1991).