# Charged Lepton Flavor Violating Radiative Decays $l_{i}\rightarrow
l_{j}\gamma$ in G2HDM

Van Que Tran<EMAIL_ADDRESS>Tsung Dao Lee Institute $\&$ School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
Faculty of Fundamental Sciences, PHENIKAA University, Yen Nghia, Ha Dong,
Hanoi 12116, Vietnam Tzu-Chiang Yuan<EMAIL_ADDRESS>Institute of
Physics, Academia Sinica, Nangang, Taipei 11529, Taiwan

###### Abstract

We compute the electromagnetic form factors of the $l_{i}l_{j}\gamma$ vertex
at one-loop level in the minimal G2HDM which has a sub-GeV vector dark matter
candidate. The results are applied to the radiative decay rates for the
charged lepton flavor violating processes $l_{i}\rightarrow l_{j}\gamma$, and
the anomalous magnetic dipole moment and the electric dipole moment of the
charged lepton. To numerically compute the branching ratio for $\mu\rightarrow
e\gamma$ and compare with the latest experimental limit from MEG, we adapt our
previous parameter space scan that is consistent with the relic density and
constraints from direct searches of dark matter, $W$ and $Z$ mass
measurements, as well as the LHC Higgs signal strengths and invisible width.
While the extra contributions are at least an order of magnitude smaller than
required to explain the $\sim 4.2\sigma$ discrepancy in the muon anomaly, the
existing MEG limit imposes stringent constraint on the parameter space. The
remaining viable parameter space can be further probed by the MEG II
sensitivity for $\mu\rightarrow e\gamma$ as well as from the direct searches
of sub-GeV dark matter in foreseeable future. Higher loop contributions may be
significant to resolve the discrepancy in the muon anomaly and generate a non-
vanishing electric dipole moments for the standard model quarks and leptons in
G2HDM.

##  I Introduction

Charged lepton flavor violating processes like $\mu\rightarrow e\gamma$,
$\mu\rightarrow 3e$ or $\mu-e$ conversion in nuclei, etc. without any neutrino
in the final states are absent at tree level in the standard model (SM) of
particle physics. However they are not strictly forbidden by symmetry and can
be induced by one-loop diagram with the $W$ boson exchange. Thus their
branching ratios are vanishingly small as they are proportional to the
neutrino masses Petcov:1976ff ; Cheng:1976uq ; Wilczek:1977wb . The most
stringent experimental constraint is for $\mu\rightarrow e\gamma$, with the
following limit on its branching ratio published in 2016 by the MEG
collaboration MEG:2016leq ,

$\mathcal{B}(\mu^{+}\rightarrow e^{+}\gamma)<4.2\times 10^{-13}\;(90\%{\rm
C.L.})$ (1)

and the projected future sensitivity is expected to improve about an order of
magnitude $\sim 6\times 10^{-14}$ by MEG II Meucci:2022qbh . For reviews on
the charged lepton flavor violation, see for example Refs. Kuno:1999jp ;
deGouvea:2013zba ; Lindner:2016bgg ; Bernstein:2013hba .

The process $\mu\rightarrow e\gamma$ (or in general $l_{i}\rightarrow
l_{j}\gamma$ with the Latin indices $i,j,k=1,2,3$ labeling the generation (or
flavor) here and henceforth) has been widely studied beyond the standard model
(bSM) Kuno:1999jp ; deGouvea:2013zba ; Lindner:2016bgg ; Hung:2015hra ;
Hung:2017voe  with predictions on the branching ratios that are more reachable
experimentally than the SM one in foreseeable future. Here we will study this
process in the context of minimal gauged two-Higgs-doublet model (G2HDM)
Ramos:2021omo ; Ramos:2021txu ; Tran:2022yrh  which has a hidden SM-like dark
sector of $SU(2)_{H}\times U(1)_{X}$ with a sub-GeV
$\mathcal{W}^{\prime(p,m)}$ dark matter candidate. The stability of the dark
matter in the model is due to a hidden $h$-parity which emerges naturally
without introducing it on ad hoc basis. Under the $h$-parity, all the SM
particles and extra neutral gauge bosons are even while other new particles in
G2HDM are odd.

The new contributions to the one-loop process $l_{i}\rightarrow l_{j}\gamma$
in G2HDM involve the new gauge or Yukawa couplings between a $h$-parity odd
particle like the dark gauge boson $\mathcal{W}^{\prime(p,m)}$, complex scalar
$\mathcal{D}$ or charged Higgs $\mathcal{H}^{\pm}$ couple to another
$h$-parity odd heavy hidden leptons $l^{H}_{k}$ or $\nu^{H}_{k}$ and the
external SM charged leptons $l_{i}$ and $l_{j}$. These new couplings are in
general off-diagonal in the generation space and hence can give rise to
$l_{i}\rightarrow l_{j}$ transition with $i>j$ (in particular muon
$\rightarrow$ electron) at one-loop. While the contribution from the dark
charged Higgs $\mathcal{H}^{\pm}$ is suppressed by the neutrino masses like
the SM $W^{\pm}$, the other new contributions are not and therefore can give
rise to a branching ratio that is more accessible experimentally. Turning the
argument around, one can use the present and future experimental limits on the
charged lepton violating processes to constrain our model parameters in G2HDM.

As a byproduct of our computation of the form factors for $l_{i}\rightarrow
l_{j}\gamma$, we can also extract the anomalous magnetic dipole moment
$a_{l_{i}}$ and the electric dipole moment $d_{l_{i}}$ easily by setting $i=j$
in our results. The muon anomalous magnetic dipole moment,

$a_{\mu}\equiv\left(\frac{g_{\mu}-2}{2}\right),$ (2)

where $g_{\mu}$ is the $g-$factor of the muon, is the most precise measured
quantity in SM, with a value measured at the Brookhaven National Laboratory
(BNL) E821 experiment (1997–2001) Muong-2:2002wip ; Muong-2:2004fok ;
Muong-2:2006rrc ,

$a_{\mu}({\rm BNL})=(11\,659\,208.9\pm 5.4_{\rm stat}\pm 3.3_{\rm sys})\times
10^{-10}\;.$ (3)

Recently, the Fermilab (FNAL) Muon $g-2$ Collaboration, based on the analysis
of data set from Run 1 and Run 2, announced the first result on the
measurement Muong-2:2021ojo

$a_{\mu}({\rm FNAL})=(11\,659\,204.0\pm 5.4)\times 10^{-10}\;.$ (4)

The average value of $a_{\mu}$ from the two experiments is given by
Muong-2:2021ojo

$a_{\mu}({\rm BNL+FNAL})=(11\,659\,206.1\pm 4.1)\times 10^{-10}\;.$ (5)

For recent reviews of the muon $g-2$, see for example Refs. Lindner:2016bgg ;
Aoyama:2020ynm ; Keshavarzi:2021eqa . The recommended value for the SM
prediction of the muon $g-2$ is Aoyama:2020ynm

$a^{\rm SM}_{\mu}=(11\,659\,181.0\pm 4.3)\times 10^{-10}\;.$ (6)

Hence the discrepancy between the experimental and theoretical values amounts
to Muong-2:2021ojo

$\Delta a_{\mu}\equiv a_{\mu}({\rm BNL+FNAL})-a_{\mu}^{\rm SM}=(25.1\pm
5.9)\times 10^{-10}\;,$ (7)

which implies a significance at the 4.2$\sigma$ level, slightly under the
standard criterion of 5$\sigma$ to claim a discovery. Nevertheless, this
discrepancy is as large as the SM electroweak contribution to the muon $g-2$
Aoyama:2020ynm ,

$a_{\mu}^{\rm EW}=(15.4\pm 0.1)\times 10^{-10}\;,$ (8)

which provides strong hints of bSM physics around the electroweak scale be
responsible for it. Future goal of the ongoing efforts at FNAL Muong-2:2021ojo
is to further reduce the existing uncertainty in the muon anomaly measurement
by a factor of 1/4.

For the electric dipole moment of the SM charged leptons, we show that they
vanish identically at one-loop in minimal G2HDM due to the lack of CP
violating phases in the products of related complex couplings as well as
vanishing combinations of loop integrals. Similar conclusions can be obtained
for the SM quarks. Higher loop contributions are needed to anticipate to
achieve a nonzero electric dipole moments for the SM fermions in minimal
G2HDM. We will not address this issue in this work.

Current experimental status of $a_{\mu}$, $\mathcal{B}(\mu^{+}\rightarrow
e^{+}\gamma)$ and $|d_{e,\mu}/e|$ are summarized in Table 1.

Observable | Experimental Result/Limit | Future Goal
---|---|---
$a_{\mu}({\rm BNL})$ |  $(11\,659\,208.9\pm 5.4_{\rm stat}\pm 3.3_{\rm sys})\times 10^{-10}$ Muong-2:2002wip ; Muong-2:2004fok ; Muong-2:2006rrc  | –
$a_{\mu}({\rm FNAL})$ |  $(11\,659\,204.0\pm 5.4)\times 10^{-10}$ Muong-2:2021ojo  | Uncertainty $\sim$1/4 of BNL
$a_{\mu}({\rm BNL+FNAL})$ |  $(11\,659\,206.1\pm 4.1)\times 10^{-10}$ Muong-2:2021ojo  | Uncertainty $\sim$1/4 of BNL
$\mathcal{B}(\mu^{+}\rightarrow e^{+}\gamma$) (MEG) |  $<4.2\times 10^{-13}(90\%{\rm C.L.})$ MEG:2016leq  |  $\sim 6\times 10^{-14}$ (MEG II Meucci:2022qbh )
$|\frac{d_{\mu}}{e}|$ [cm] |  $<1.8\times 10^{-19}(95\%{\rm C.L.})$ Muong-2:2008ebm  |  $\sim 6\times 10^{-23}$ (PSI Adelmann:2021udj )
$|\frac{d_{e}}{e}|$ [cm] (ACME) |  $<1.1\times 10^{-29}(90\%{\rm C.L.})$ ACME:2018yjb  | (Advanced ACME Advanced-ACME )

Table 1: Experimental results for $a_{\mu}$ and upper limits for
$\mathcal{B}(\mu\rightarrow e\gamma)$, $|d_{\mu}/e|$ and $|d_{e}/e|$.

We lay out the paper as follows. In the next section, we will review the
minimal particle content in G2HDM and write down the relevant interactions
required for the one-loop computation for the form factors of the radiative
decays $l_{i}\rightarrow l_{j}\gamma$. In Section 3, we compute the magnetic
and electric dipole form factors for the radiative decays. In the case of
$i=j$ we also obtain the anomalous magnetic dipole moment and electric dipole
moment for the lepton $l_{i}$. We will show that the electric dipole moment of
the lepton vanishes identically at one-loop in G2HDM. Numerical analysis for
$\mu\rightarrow e\gamma$ and $\Delta a_{\mu}$ is presented in Section 4. We
also present the impact of the viable parameter space on the spin-independent
cross section for the sub-GeV dark matter direct search experiments. We
conclude in Section 5.

Analytical formulas for the form factors and the associated loop integrals are
given in Appendix A. In Appendix B, we show that the form factors of the SM
$W$ boson loop obtained in the unitary and ’t Hooft-Feynman gauges are
equivalent. Discrepancies between our results and existing ones in the
literature are clarified in the Appendices. Some relevant Feynman rules in
G2HDM are shown in Appendix C. In Appendix D, we demonstrate the well-known
fact that only the magnetic and electric dipole moment form factors are
relevant for the computations of the on-shell amplitude of $l_{i}\rightarrow
l_{j}\gamma$.

##  II Minimal G2HDM

In this section, we will briefly review the minimal G2HDM studied recently in
Ramos:2021omo ; Ramos:2021txu ; Tran:2022yrh . The original model based on
augmenting the SM electroweak gauge group $SU(2)_{L}\times U(1)_{Y}$ by a
hidden gauge sector $SU(2)_{H}\times U(1)_{X}$ was introduced in Ref.
Huang:2015wts . The main idea of G2HDM is to group the two Higgs doublets
$H_{1}$ and $H_{2}$ in inert 2HDM (I2HDM) together to form a bifundamental
irreducible representation of $SU(2)_{L}\times SU(2)_{H}$. Various refinements
Arhrib:2018sbz ; Huang:2019obt ; Chen:2019pnt  and collider phenomenology
Chen:2018wjl ; Huang:2017bto ; Huang:2015rkj  were pursued subsequently with
the same particle content as the original model where the DM candidate is a
complex scalar $\mathcal{D}$. In this work, as in Ramos:2021omo ;
Ramos:2021txu ; Tran:2022yrh , we will drop the triplet field $\Delta_{H}$ of
the extra $SU(2)_{H}$ in the original model and propose the complex gauge
boson field $\mathcal{W}^{\prime(p,m)}$ as DM candidate rather than the
complex scalar $\mathcal{D}$. For convenience, the scalar and fermion contents
and their quantum numbers as well as $h$-parity in the model are tabulated in
Table 2 and 3 respectively. Our convention for the electric charge $Q$ (in
unit of $e$) is $Q=T^{3}_{L}+Y$ where $T^{3}_{L}$ is the third component of
the $SU(2)_{L}$ generators and $Y$ is the hypercharge. $S$ is the scalar field
introduced to implement the Stueckelberg mechanism to provide a mass for the
$U(1)_{X}$ gauge boson Kors:2004dx ; Kors:2005uz ; Feldman:2006wb ;
Feldman:2007wj .

Scalar | $SU(2)_{L}$ | $SU(2)_{H}$ | $U(1)_{Y}$ | $U(1)_{X}$ |  $h$-parity
---|---|---|---|---|---
$H=\left(H_{1}\;\;H_{2}\right)^{\rm T}$ | 2 | 2 | $\frac{1}{2}$ | $\frac{1}{2}$ | $(+,-)$
$\Phi_{H}=\left(\Phi_{1}\;\;\Phi_{2}\right)^{\rm T}$ | 1 | 2 | 0 | $\frac{1}{2}$ | $(-,+)$
$S$ | 1 | 1 | 0 | 0 | $+$
Table 2: Higgs scalars in the minimal G2HDM and their quantum number assignments.  Fermion | $SU(3)_{C}$ | $SU(2)_{L}$ | $SU(2)_{H}$ | $U(1)_{Y}$ | $U(1)_{X}$ |  $h$-parity
---|---|---|---|---|---|---
$Q_{L}=\left(u_{L}\;\;d_{L}\right)^{\rm T}$ | 3 | 2 | 1 | $\frac{1}{6}$ | 0 | $(+,+)$
$U_{R}=\left(u_{R}\;\;u^{H}_{R}\right)^{\rm T}$ | 3 | 1 | 2 | $\frac{2}{3}$ | $\frac{1}{2}$ | $(+,-)$
$D_{R}=\left(d^{H}_{R}\;\;d_{R}\right)^{\rm T}$ | 3 | 1 | 2 | $-\frac{1}{3}$ | $-\frac{1}{2}$ | $(-,+)$
$u_{L}^{H}$ | 3 | 1 | 1 | $\frac{2}{3}$ | 0 | $-$
$d_{L}^{H}$ | 3 | 1 | 1 | $-\frac{1}{3}$ | 0 | $-$
$L_{L}=\left(\nu_{L}\;\;e_{L}\right)^{\rm T}$ | 1 | 2 | 1 | $-\frac{1}{2}$ | 0 | $(+,+)$
$N_{R}=\left(\nu_{R}\;\;\nu^{H}_{R}\right)^{\rm T}$ | 1 | 1 | 2 | 0 | $\frac{1}{2}$ | $(+,-)$
$E_{R}=\left(e^{H}_{R}\;\;e_{R}\right)^{\rm T}$ | 1 | 1 | 2 | $-1$ | $-\frac{1}{2}$ | $(-,+)$
$\nu_{L}^{H}$ | 1 | 1 | 1 | 0 | 0 | $-$
$e_{L}^{H}$ | 1 | 1 | 1 | $-1$ | 0 | $-$
Table 3: Fermions in the minimal G2HDM and their quantum number assignments.

###  II.1 Higgs Potential and Spontaneous Symmetry Breaking

The most general Higgs potential which is invariant under both
$SU(2)_{L}\times U(1)_{Y}\times SU(2)_{H}\times U(1)_{X}$ can be written down
as follows

$\displaystyle V={}$
$\displaystyle-\mu^{2}_{\Phi}\Phi_{H}^{\dagger}\Phi_{H}+\lambda_{\Phi}\left(\Phi_{H}^{\dagger}\Phi_{H}\right)^{2}-\mu^{2}_{H}\left(H^{\alpha
i}H_{\alpha i}\right)+\lambda_{H}\left(H^{\alpha i}H_{\alpha i}\right)^{2}$
$\displaystyle+\frac{1}{2}\lambda^{\prime}_{H}\epsilon_{\alpha\beta}\epsilon^{\gamma\delta}\left(H^{\alpha
i}H_{\gamma i}\right)\left(H^{\beta j}H_{\delta
j}\right)+\lambda_{H\Phi}\left(H^{\dagger}H\right)\left(\Phi_{H}^{\dagger}\Phi_{H}\right)$
(9)
$\displaystyle+\lambda^{\prime}_{H\Phi}\left(H^{\dagger}\Phi_{H}\right)\left(\Phi_{H}^{\dagger}H\right),$

where ($\alpha$, $\beta$, $\gamma$, $\delta$) and ($i$, $j$) refer to the
$SU(2)_{H}$ and $SU(2)_{L}$ indices respectively, all of which run from one to
two, and $H^{\alpha i}=H^{*}_{\alpha i}$.

To study spontaneous symmetry breaking (SSB) in the model, we parameterize the
Higgs fields according to standard practice

$\displaystyle H_{1}=\begin{pmatrix}G^{+}\\\ \frac{v+h_{\rm
SM}}{\sqrt{2}}+i\frac{G^{0}}{\sqrt{2}}\end{pmatrix},\,H_{2}=\begin{pmatrix}\mathcal{H}^{+}\\\
\mathcal{H}_{2}^{0}\end{pmatrix},\,\Phi_{H}=\begin{pmatrix}G_{H}^{p}\\\
\frac{v_{\Phi}+\phi_{H}}{\sqrt{2}}+i\frac{G_{H}^{0}}{\sqrt{2}}\end{pmatrix}\;\;\;$
(10)

where $v$ and $v_{\Phi}$ are the only non-vanishing vacuum expectation values
(VEVs) in $H_{1}$ and $\Phi_{H}$ fields respectively. $H_{2}$ does not develop
VEV as in the case of I2HDM.

Theoretical constraints like bounded from below and perturbative unitarity of
the above scalar potential can be found in our previous works Ramos:2021omo ;
Ramos:2021txu .

###  II.2 Interaction Lagrangian

Besides the unitary Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing
matrix

$V_{\rm PMNS}\equiv\left(U_{\nu}^{L}\right)^{\dagger}U^{L}_{l}\;,$ (11)

defined in the left-handed lepton sector, we also need to introduce the
following unitary mixing matrices in the right-handed lepton sector in G2HDM,

$\displaystyle V^{H}_{l}$
$\displaystyle\equiv\left(U^{R}_{l}\right)^{\dagger}U^{R}_{l^{H}}\,,$
$\displaystyle V_{\nu}^{H}$
$\displaystyle\equiv\left(U^{R}_{\nu}\right)^{\dagger}U^{R}_{\nu^{H}}\;.$ (12)

There are altogether 6 one-loop contributions to the $l_{i}-l_{j}-\gamma$
vertex in the minimal G2HDM. The Feynman diagrams are shown in Figs. (1) and
(2). Here the self-energy diagrams are not explicitly shown. However they
contribute to the $\gamma^{\mu}$ and $\gamma^{\mu}\gamma_{5}$ form factors in
the amplitude which are important for the cancellation of ultraviolet
divergences and the maintenance of gauge invariance. Fig. (1) is the SM-like
contributions with all $h$-parity even particles circulating inside the loop,
while Fig. (2) is the new contributions from G2HDM with all $h$-parity odd
particles circulating inside the loop. The QED vertex for a photon couples
with $W^{\pm}$, $l_{i}$, $l_{i}^{H}$ and $\mathcal{H}^{\pm}$ are standard,
they can be found in many textbooks and will be omitted in what follows.

Figure 1:  The one-loop SM-like contribution to $l_{i}-l_{j}-\gamma$ vertex
from the SM $W$ boson diagram (left panel), and contributions to
$l_{i}-l_{i}-\gamma$ vertex from $\left\\{Z_{n}\right\\}$ diagram (center
panel) and $\left\\{h_{n}\right\\}$ diagram (right panel) in G2HDM.

The first diagram in Fig. (1) is the contribution from the SM charged
$W^{\pm}$ boson. The relevant interaction Lagrangian is

$\mathcal{L}^{W}\supset\frac{g}{2\sqrt{2}}\sum_{i,k}\left(V_{\rm
PMNS}\right)_{ki}\bar{\nu}_{k}\gamma^{\mu}\left(1-\gamma_{5}\right)l_{i}W^{+}_{\mu}+{\rm
H.c.}\;.$ (13)

The second diagram in Fig. (1) is the contribution from the neutral gauge
bosons $\\{Z_{n}\\}$. The relevant interaction Lagrangian is

$\mathcal{L}^{\\{Z_{n}\\}}\supset\sum_{n}\sum_{i}\bar{l}_{i}\gamma_{\mu}\left(C_{Vn}+C_{An}\gamma_{5}\right)l_{i}Z^{\mu}_{n}\;,$
(14)

where $C_{Vn}$ and $C_{An}$ are the vector and axial-vector coupling
constants. Based on lepton universality, these couplings are independent of
the charged lepton flavor $i$. Their expressions are given by
$C_{Vn}=(C_{Ln}+C_{Rn})/2$ and $C_{An}=(-C_{Ln}+C_{Rn})/2$ with

$\displaystyle C_{Ln}$ $\displaystyle=$ $\displaystyle\frac{g}{\cos\theta_{\rm
W}}\left(-\frac{1}{2}+\sin^{2}\theta_{\rm W}\right)O^{N}_{1n}\;,$ (15)
$\displaystyle C_{Rn}$ $\displaystyle=$ $\displaystyle\frac{g}{\cos\theta_{\rm
W}}\sin^{2}\theta_{\rm
W}O^{N}_{1n}-\frac{1}{2}g_{H}O^{N}_{2n}-\frac{1}{2}g_{X}O^{N}_{3n}\;,$ (16)

where $\theta_{\rm W}$ is the weak mixing angle, $g_{X}$ and $g_{H}$ are gauge
couplings of the $U(1)_{X}$ and $SU(2)_{H}$, respectively. $O^{N}$ is a
$3\times 3$ orthogonal matrix that diagonalizes the following mass matrix in
the basis of $(Z^{\rm SM},\mathcal{W}^{\prime}_{3},X)$

$\mathcal{M}_{Z}^{2}=\begin{pmatrix}m^{2}_{Z}&-\frac{1}{2}g_{H}vm_{Z}&-{\frac{1}{2}}g_{X}vm_{Z}\\\
-\frac{1}{2}g_{H}vm_{Z}&m^{2}_{\mathcal{W}^{\prime}}&{\frac{1}{4}}g_{H}g_{X}v_{-}^{2}\\\
-{\frac{1}{2}}g_{X}vm_{Z}&{\frac{1}{4}}g_{H}g_{X}v_{-}^{2}&{\frac{1}{4}}g_{X}^{2}v_{+}^{2}+M_{X}^{2}\end{pmatrix}\;,$
(17)

where

$\displaystyle m_{Z}$ $\displaystyle=$
$\displaystyle\frac{1}{2}v\sqrt{g^{2}+g^{\prime\,2}}\;,$ (18) $\displaystyle
m_{\mathcal{W}^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}g_{H}\sqrt{v^{2}+v_{\Phi}^{2}}\;,$ (19)
$\displaystyle v_{\pm}^{2}$ $\displaystyle=$ $\displaystyle\left(v^{2}\pm
v^{2}_{\Phi}\right)\;,$ (20)

and $M_{X}$ is the Stueckelberg mass for the $U(1)_{X}$. We denote the
physical mass eigenstates as $Z_{n}\,(n=1,2,3)$ with the mass ordering
$M_{Z_{1}}\geq M_{Z_{2}}\geq M_{Z_{3}}$. In the parameter space choice in our
numerical work, $Z_{1}$ will be identified as the $Z$ boson of 91.1876 GeV
Zyla:2020zbs  observed at LEP, $Z_{2}$ is the dark $Z^{\prime}$ and $Z_{3}$ is
the dark photon $\gamma^{\prime}$ (or $A^{\prime}$ in some literature). They
all have even $h$-parity.

The third diagram in Fig. (1) is the contribution from the neutral Higgs
bosons $\\{h_{n}\\}$. The relevant interaction Lagrangian is

$\mathcal{L}^{\\{h_{n}\\}}\supset-\sum_{n}\sum_{i}\left(O^{H}\right)_{1n}\frac{m_{i}}{v}\bar{l}_{i}l_{i}h_{n}\;,$
(21)

where $O^{H}$ is the mixing matrix between $h_{\rm SM}$ and $\phi_{H}$,

$\left(\begin{matrix}h_{\rm SM}\\\
\phi_{H}\end{matrix}\right)=O^{H}\cdot\left(\begin{matrix}h_{1}\\\
h_{2}\end{matrix}\right)=\left(\begin{matrix}\cos\theta_{1}&\sin\theta_{1}\\\
-\sin\theta_{1}&\cos\theta_{1}\end{matrix}\right)\cdot\left(\begin{matrix}h_{1}\\\
h_{2}\end{matrix}\right)\;.$ (22)

The mixing angle $\theta_{1}$ is given by

$\tan
2\theta_{1}=\frac{\lambda_{H\Phi}vv_{\Phi}}{\lambda_{\Phi}v^{2}_{\Phi}-\lambda_{H}v^{2}}\;.$
(23)

The masses of $h_{1}$ and $h_{2}$ are given by

$\displaystyle m_{h_{1},h_{2}}^{2}$ $\displaystyle=$
$\displaystyle\lambda_{H}v^{2}+\lambda_{\Phi}v_{\Phi}^{2}\mp\sqrt{\lambda_{H}^{2}v^{4}+\lambda_{\Phi}^{2}v_{\Phi}^{4}+\left(\lambda^{2}_{H\Phi}-2\lambda_{H}\lambda_{\Phi}\right)v^{2}v_{\Phi}^{2}}\;.$
(24)

Depending on its mass, either $h_{1}$ or $h_{2}$ is identified as the observed
Higgs boson $h$ at the Large Hadron Collider (LHC). Currently the most precise
measurement of the Higgs boson mass is $m_{h}=125.38\pm 0.14$ GeV CMS:2020xrn
. In this work, we will identify the lighter state $h_{1}$ as $h$.

Since the gauge and Yukawa couplings in (14) and (21) respectively are all
real and flavor diagonal, there are no contributions to $l_{i}\rightarrow
l_{j}\gamma\,(i\neq j)$ and electric dipole moment of $l_{i}$ from the
interactions $\mathcal{L}^{\\{Z_{n}\\}}$ and $\mathcal{L}^{\\{h_{n}\\}}$. The
only non-vanishing contribution to $l_{i}\rightarrow l_{j}\gamma\,(i\neq j)$
in SM at one-loop is the charged $W^{\pm}$ from $\mathcal{L}^{W}$ in (13).
However it is well known that its amplitude is suppressed by the squared of
neutrino masses due to GIM-like mechanism in the lepton sector. Furthermore,
due to the unitarity of $V_{\rm PMNS}$, $d_{l_{i}}(W^{\pm})$ also vanishes at
one-loop. See Appendix A.

Figure 2: Three new contributions of $\mathcal{D}$, $\mathcal{H}^{+}$ and
$\mathcal{W}^{\prime}$ to $l_{i}-l_{j}-\gamma$ vertex in G2HDM.

Next we turn to the new contributions in G2HDM.

The first diagram in Fig. (2) is the contribution from the dark Higgs
$\mathcal{D}$ which is a linear combination of two odd $h$-parity components
$\mathcal{H}_{2}^{0}$ and $G_{H}^{m}$,

$\mathcal{D}=\cos\theta_{2}\mathcal{H}_{2}^{0}+\sin\theta_{2}G_{H}^{m}\;,$
(25)

where $\theta_{2}$ is a mixing angle giving by

$\tan 2\theta_{2}=\frac{2vv_{\Phi}}{v_{\Phi}^{2}-v^{2}}\;.$ (26)

The mass of $\mathcal{D}$ is

$m_{\mathcal{D}}^{2}=\frac{1}{2}\lambda^{\prime}_{H\Phi}v_{+}^{2}\;,$ (27)

where $v^{2}_{+}$ is defined in (20).

The relevant interaction Lagrangian is given by

${\mathcal{L}}^{\mathcal{D}}\supset\sum_{i,j}\overline{l^{H}_{i}}\left(y^{\mathcal{D}}_{S\,ij}+y^{\mathcal{D}}_{P\,ij}\gamma_{5}\right)l_{j}\mathcal{D}^{*}+{\rm
H.c.}\;,$ (28)

where the scalar and pseudoscalar Yukawa couplings $y^{\mathcal{D}}_{S\,ij}$
and $y^{\mathcal{D}}_{P\,ij}$ are given by

$y^{\mathcal{D}}_{S/P\,ij}=\pm\frac{\sqrt{2}}{2v}\cos\theta_{2}\left(V_{l}^{H\,\dagger}M_{l}\right)_{ij}+\frac{\sqrt{2}}{2v_{\Phi}}\sin\theta_{2}\left(M_{l^{H}}V_{l}^{H\,\dagger}\right)_{ij}\;,$
(29)

with $M_{l}={\rm diag}\left(m_{e},m_{\mu},m_{\tau}\right)$ and $M_{l^{H}}={\rm
diag}\left(m_{l^{H}_{1}},m_{l^{H}_{2}},m_{l^{H}_{3}}\right)$. Note that the
ordering of the mass matrices are important in the Yukawa couplings (29). From
(29), one obtains

$y^{\mathcal{D}\,*}_{P\,ki}y^{\mathcal{D}}_{S\,ki}=\frac{1}{2}\left|\left(V_{l}^{H}\right)_{ik}\right|^{2}\left(\frac{m_{l^{H}_{k}}^{2}}{v_{\Phi}^{2}}\sin^{2}\theta_{2}-\frac{m_{l_{i}}^{2}}{v^{2}}\cos^{2}\theta_{2}\right)\,.$
(30)

Thus ${\rm
Im}\left(y^{\mathcal{D}\,*}_{P\,ki}y^{\mathcal{D}}_{S\,ki}\right)=0$. We don’t
expect the complex Yukawa couplings in $\mathcal{L}^{\mathcal{D}}$ to give
rise a non-vanishing electric dipole moment $d_{l_{i}}$ at one-loop, as shown
in Appendix A.

The second diagram in Fig. (2) is the contribution from the dark charged Higgs
$\mathcal{H}^{\pm}$ which has odd $h$-parity and a mass given by

$m^{2}_{\mathcal{H}^{\pm}}=\frac{1}{2}\left(\lambda^{\prime}_{H\Phi}v^{2}_{\Phi}-\lambda^{\prime}_{H}v^{2}\right)\;.$
(31)

The relevant interaction Lagrangian is given by

${\mathcal{L}}^{\mathcal{H}}\supset\sum_{i,j}\overline{\nu^{H}_{i}}\left(y^{\mathcal{H}}_{S\,ij}+y^{\mathcal{H}}_{P\,ij}\gamma_{5}\right)l_{j}\mathcal{H}^{+}+{\rm
H.c.}\;,$ (32)

where the scalar and pseudoscalar Yukawa couplings $y^{\mathcal{H}}_{S\,ij}$
and $y^{\mathcal{H}}_{P\,ij}$ are given by

$y^{\mathcal{H}}_{S/P\,ij}=\pm\frac{\sqrt{2}}{2v}\left(V_{\nu}^{H\,\dagger}M_{\nu}V_{\rm
PMNS}\right)_{ij}\;,$ (33)

with $M_{\nu}={\rm diag}\left(m_{\nu_{1}},m_{\nu_{2}},m_{\nu_{3}}\right)$.
Since the Yukawa couplings $y^{\mathcal{H}}_{S\,ij}$ and
$y^{\mathcal{H}}_{P\,ij}$ are related, we expect
$d_{l_{i}}(\mathcal{H}^{\pm})=0$ at one-loop. See Appendix A for detail.

The third diagram in Fig. (2) is the contribution from the vector dark matter
$\mathcal{W}^{\prime(p,m)}$ $(\equiv(\mathcal{W}^{\prime}_{1}\mp
i\mathcal{W}^{\prime}_{2})/\sqrt{2})$ which is assumed to be the lightest
$h$-parity odd particle in G2HDM. Its mass is given by (19). The relevant
interaction Lagrangian is given by

${\mathcal{L}}^{\mathcal{W}^{\prime}}\supset\sum_{i,j}\overline{l^{H}_{i}}\gamma^{\mu}\left(g^{\mathcal{W}^{\prime}}_{V\,ij}+g^{\mathcal{W}^{\prime}}_{A\,ij}\,\gamma_{5}\right)l_{j}\mathcal{W}^{\prime\,p}_{\mu}+{\rm
H.c.}\;,$ (34)

where the vector and axial gauge couplings $g^{\mathcal{W}^{\prime}}_{V\,ij}$
and $g^{\mathcal{W}^{\prime}}_{A\,ij}$ are given by

$g^{\mathcal{W}^{\prime}}_{V\,ij}=g^{\mathcal{W}^{\prime}}_{A\,ij}=\frac{g_{H}}{2\sqrt{2}}\left(V^{H}_{l}\right)^{\dagger}_{ij}\;.$
(35)

Since the vector and axial vector couplings $g^{\mathcal{W}^{\prime}}_{V\,ij}$
and $g^{\mathcal{W}^{\prime}}_{A\,ij}$ are the same, we expect
$d_{l_{i}}(\mathcal{W}^{\prime})=0$ at one-loop. (See Appendix A.) This is
analogous to the SM charged $W^{\pm}$ case where the vector and axial vector
couplings are opposite sign to each other, there as is well-known we have
$d_{l_{i}}(W^{\pm})=0$ at one-loop too.

In summary, we expect all the new flavor non-diagonal complex couplings from
$\mathcal{L}^{\mathcal{D}}$, $\mathcal{L}^{\mathcal{H}}$ and
$\mathcal{L}^{\mathcal{W}^{\prime}}$ in G2HDM can give rise to contributions
to $l_{i}\rightarrow l_{j}\gamma\,(i\neq j)$. Certainly they will all give
non-vanishing contributions to $a_{l_{i}}$ but not $d_{l_{i}}$ at one-loop.
The relevant Feynman rules are given in Appendix C.

Figure 3: Feynman Diagram for the $l_{i}-l_{j}-\gamma$ vertex.

##  III Magnetic and Electric Dipole Form Factors

The Lorentz invariant amplitude for a charged lepton $l_{i}$ of flavor $i$
scatters with an electromagnetic background field
$\widetilde{\mathcal{A}}^{\rm ext}_{\mu}(q)$ to become another charged lepton
$l_{j}$ of flavor $j$ as depicted in Fig. (3) is given by 111In general, the
amplitude has six Lorentz decomposition form factors, however, due to the
gauge invariance, only $\sigma^{\mu\nu}q_{\nu}$ and
$\sigma^{\mu\nu}q_{\nu}\gamma_{5}$ terms are retained for on-shell photon
where $q^{2}=0$ (see Appendix D for a detailed discussion).

$i\mathcal{M}_{ji}=\overline{u_{j}}\left(p^{\prime}\right)\left(-ie\Gamma_{ji}^{\mu}\right)u_{i}\left(p\right)\widetilde{\mathcal{A}}^{\rm
ext}_{\mu}\left(q\right)\;,$ (36)

with $-e\,(e>0)$ and $m_{i}$ are the electric charge and mass of $l_{i}$
respectively, $q=(p^{\prime}-p)$ is the momentum transfer, and the vertex
function $\Gamma^{\mu}_{ji}$ can be decomposed as

$\Gamma_{ji}^{\mu}=i\sigma^{\mu\nu}q_{\nu}\frac{m_{i}}{2}\left(A^{M}_{ji}+iA^{E}_{ji}\gamma_{5}\right)\;,$
(37)

with $A^{M}_{ji}$ and $A^{E}_{ji}$ related to the transition magnetic and
electric dipole form factors respectively 222For ease of comparisons of their
analytical expressions presented in the Appendices, we use the same notations
$A^{M}_{ji}$ and $A^{E}_{ji}$ as in Lindner:2016bgg . And they are understood
to be evaluated at $q^{2}=0$..

The effective Lagrangian that can reproduce the matrix element (36) with the
associated vertex (37) is

$\mathcal{L}_{\rm
eff}=-\frac{1}{4}e\,m_{i}\overline{l_{j}}\sigma^{\mu\nu}\left(A^{M}_{ji}+i\gamma_{5}A^{E}_{ji}\right)l_{i}F^{\rm
ext}_{\mu\nu}\;,$ (38)

where $F^{\rm ext}_{\mu\nu}$ is the electromagnetic background field strength.

The above form factors $A^{M}_{ji}$ and $A^{E}_{ji}$ enable us to compute the
decay rate for the process $l_{i}\rightarrow l_{j}\gamma\;(i\neq j)$ with the
following spin-averaged matrix element squared

$\overline{\sum}|\mathcal{M}_{ji}|^{2}=\frac{e^{2}}{2}m_{i}^{6}\left(1-\frac{m_{j}^{2}}{m_{i}^{2}}\right)^{2}\left(|A^{M}_{ji}|^{2}+|A^{E}_{ji}|^{2}\right)\;.$
(39)

We thus obtain the decay rate and branching ratio for $l_{i}\rightarrow
l_{j}\gamma$

$\displaystyle\Gamma\left(l_{i}\rightarrow l_{j}\gamma\right)$
$\displaystyle=\frac{1}{32\pi}m^{5}_{i}\left(1-\frac{m^{2}_{j}}{m^{2}_{i}}\right)^{3}e^{2}\left(|A^{M}_{ji}|^{2}+|A^{E}_{ji}|^{2}\right)\;,$
(40) $\displaystyle\mathcal{B}\left(l_{i}\rightarrow l_{j}\gamma\right)$
$\displaystyle=\frac{\Gamma\left(l_{i}\rightarrow
l_{j}\gamma\right)}{\Gamma\left(l_{i}\rightarrow
l_{j}\nu_{i}\overline{\nu_{j}}\right)}\cdot\frac{\Gamma\left(l_{i}\rightarrow
l_{j}\nu_{i}\overline{\nu_{j}}\right)}{\Gamma_{l_{i}}}\;,$ (41)

where 333See for example the Appendix in the textbook Collider Physics,
Updated Edition, CRC Press 1996, by Barger and Phillips.

$\Gamma\left(l_{i}\rightarrow
l_{j}\nu_{i}\overline{\nu_{j}}\right)=\frac{G_{F}^{2}m_{i}^{5}}{192\pi^{3}}f\left(\frac{m_{j}}{m_{i}}\right)\;,\\\
$ (42)

with $G_{F}$ is the Fermi constant and

$f\left(x\right)=\left(1-8\,x^{2}+8\,x^{6}-x^{8}-24\,x^{4}\log x\right)\;.$
(43)

Therefore

$\mathcal{B}\left(l_{i}\rightarrow
l_{j}\gamma\right)=\frac{3\left(4\pi\right)^{3}\alpha_{\rm
EM}}{2G_{F}^{2}}\cdot\frac{\left(1-\frac{m^{2}_{j}}{m^{2}_{i}}\right)^{3}}{f\left(\frac{m_{j}}{m_{i}}\right)}\cdot\left(|A^{M}_{ji}|^{2}+|A^{E}_{ji}|^{2}\right)\cdot\mathcal{B}\left(l_{i}\rightarrow
l_{j}\nu_{i}\overline{\nu_{j}}\right)\;,$ (44)

where $\alpha_{\rm EM}=e^{2}/(4\pi)$. For $\mu\rightarrow
e\overline{\nu_{e}}\nu_{\mu}$,

$\mathcal{B}\left(\mu\rightarrow e\overline{\nu_{e}}\nu_{\mu}\right)\approx
100\%\;.$ (45)

For $l_{i}\rightarrow l_{j}\gamma$ ($i\neq j$) in G2HDM there are 4 distinct
non-vanishing contributions to each $A^{M}$ and $A^{E}$,

$A^{M/E}_{ji}=A^{M/E}_{ji}\left(W\right)+A^{M/E}_{ji}\left(\mathcal{D}\right)+A^{M/E}_{ji}\left(\mathcal{H}\right)+A^{M/E}_{ji}\left(\mathcal{W}^{\prime}\right)\;.$
(46)

As is well known the SM contribution $A^{M/E}_{ji}\left(W\right)$ for $i\neq
j$ from the $W$ boson loop is vanishingly small and many orders below the
current experimental sensitivities.

The anomalous magnetic dipole moment $a_{l_{i}}$ of the charged lepton $l_{i}$
can be identified as the coefficient of
$\left(e/2\,m_{i}\right)i\sigma^{\mu\nu}q_{\nu}$ in the vertex
$e\Gamma^{\mu}_{ii}$ of (37), i.e.

$a_{l_{i}}=m_{i}^{2}A^{M}_{ii}\;,\;\;\;\;\;\;({\rm no\;sum\;on\;}i)\;.$ (47)

The electric dipole moment $d_{i_{i}}$ of the charged lepton $l_{i}$ is given
by

$\frac{d_{l^{i}}}{e}=\frac{m_{i}}{2}A^{E}_{ii}\;,\;\;\;\;\;\;({\rm
no\;sum\;on\;}i)\;.$ (48)

For the anomalous magnetic dipole moment of $l_{i}$ in G2HDM, besides the
well-known QED contribution $a_{l_{i}}(\gamma)=\alpha_{\rm EM}/2\pi$, there
are in general 6 distinct electroweak contributions to the $a_{l_{i}}$,

$\displaystyle a_{l_{i}}$ $\displaystyle=\frac{\alpha_{\rm
EM}}{2\pi}+m_{i}^{2}A^{M}_{ii}\;,\;\;({\rm no\;sum\;on}\;i)\;,$ $\displaystyle
A^{M}_{ii}$
$\displaystyle=A^{M}_{ii}\left(W\right)+A^{M}_{ii}\left(\left\\{Z_{n}\right\\}\right)+A^{M}_{ii}\left(\left\\{h_{n}\right\\}\right)+A^{M}_{ii}\left(\mathcal{D}\right)+A^{M}_{ii}\left(\mathcal{H}\right)+A^{M}_{ii}\left(\mathcal{W}^{\prime}\right)\;.$
(49)

Analytical one-loop expressions for $A^{M/E}_{ji}$ are given in Appendix A.
There one will see all the $A^{E}_{ii}$s vanish at one-loop in G2HDM, hence
the electric dipole moment $d_{l_{i}}$ of $l_{i}$ vanish too according to
(48). These form factors $A^{M/E}_{ji}$ were also computed for general
couplings in Lindner:2016bgg . Aside from an overall factor of 2 in the form
factors, we will discuss some minor discrepancies in the loop integrals
between our results and Lindner:2016bgg  in Appendix A.

##  IV Numerical results

In this section, we show numerical results for the cLFV process
$\mu\rightarrow e\gamma$ and muon anomalous magnetic dipole moment with the
parameter space in the model chosen to satisfy the current constraints for a
sub-GeV non-abelian vector DM $\mathcal{W}^{\prime}$. In particular, the scan
data are adapted from Ref. Tran:2022yrh  in which the theoretical constraints
on the scalar potential Ramos:2021omo ; Ramos:2021txu , signal strength
measurements from the LHC Sirunyan:2018koj ; Aad:2019mbh ; ATLAS:2021vrm ,
dark photon physics ATLAS:2019erb ; Fabbrichesi:2020wbt , electroweak
precision measurements Zyla:2020zbs  including the recent $W$ boson mass
measurement at the CDF II CDF:2022hxs  and constraints from DM searches
including the DM relic density measured from Planck collaboration
Aghanim:2018eyx , DM direct detections Angloher:2017sxg ; Agnes:2018ves ;
Aprile:2019xxb  and Higgs invisible decays constraint from the LHC
ATLAS:2022yvh . For the data points that satisfy the above mentioned
constraints, the total DM annihilation cross section is of order
$10^{-32}\,{\rm cm^{3}}\cdot s^{-1}$ or below that is much lower than the
current DM indirect detection constraints.

We set the mixing matrices in the right-handed lepton sector in the model to
be

$V_{l}^{H}=V_{\nu}^{H}=V_{\rm PMNS},$ (50)

where $V_{\rm PMNS}$ is parameterized as

$V_{\rm
PMNS}\equiv\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{\rm
CP}}\\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{\rm
CP}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{\rm CP}}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{\rm
CP}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{\rm
CP}}&c_{23}c_{13}\end{pmatrix},$ (51)

where $s_{ij}$ and $c_{ij}$ stand for $\sin\theta_{ij}$ and $\cos\theta_{ij}$
respectively, and $\delta_{\rm CP}$ is a Dirac CP violating phase. The current
best-fit values using a normal ordering are given by Esteban:2020cvm :
$\theta_{12}={33.44^{\circ}}^{+0.77^{\circ}}_{-0.74^{\circ}}$,
$\theta_{23}={49.2^{\circ}}^{+1.0^{\circ}}_{-1.3^{\circ}}$,
$\theta_{13}={8.57^{\circ}}^{+0.13^{\circ}}_{-0.12^{\circ}}$, and $\delta_{\rm
CP}={194^{\circ}}^{+52^{\circ}}_{-25^{\circ}}$. We also set the heavy hidden
lepton masses to be

$M_{\nu^{H}}=M_{l^{H}}={\rm diag}\left(m_{l^{H}},m_{l^{H}}+\Delta
m_{l^{H}},m_{l^{H}}+\Delta m_{l^{H}}\right)$ (52)

where the second and third generations are assumed degenerate, and $\Delta
m_{l^{H}}$ is a mass splitting between the first and second (third)
generations.

Figure 4:  Branching ratio of $\mu\rightarrow e\gamma$ as a function of the
heavy hidden lepton mass $m_{l^{H}}$. Other parameters in the model are set to
be $m_{h_{2}}=292.50$ GeV, $m_{\mathcal{D}}=766.07$ GeV,
$m_{\mathcal{H}^{\pm}}=848.13$ GeV, $m_{\mathcal{W}^{\prime}}=1.0$ GeV,
$\theta_{1}=0.030$ rad, $\theta_{2}=0.056$ rad, $M_{X}=1.96$ GeV and
$g_{X}=2.5\times 10^{-4}$. From the left to right and top to bottom panels,
the mass splitting between the heavy hidden lepton generations is set to be
$\Delta m_{l^{H}}=1$ GeV, $50$ GeV and $500$ GeV, respectively. The dashed
red, black and green lines represent the contributions from $\mathcal{D}$
boson, $\mathcal{W}^{\prime}$ boson multiplied by $10^{6}$ and
$\mathcal{H}^{\pm}$ boson multiplied by $10^{66}$ respectively. The orange
region is the excluded region at $90\%$ C.L. from MEG collaboration
MEG:2016leq  and the dotted blue line indicates the future sensitivity from
MEG II Meucci:2022qbh .

Fig. 4 shows the branching ratio of $\mu\rightarrow e\gamma$ as a function of
the heavy hidden lepton mass $m_{l^{H}}$. Here we fixed other parameters in
the model to be $m_{h_{2}}=292.50$ GeV, $m_{\mathcal{D}}=766.07$ GeV,
$m_{\mathcal{H}^{\pm}}=848.13$ GeV, $m_{\mathcal{W}^{\prime}}=1.0$ GeV,
$\theta_{1}=0.030$ rad, $\theta_{2}=0.056$ rad, $M_{X}=1.96$ GeV and
$g_{X}=2.5\times 10^{-4}$. We note that this benchmark point satisfies all
current constraints mentioned above. The mass splitting $\Delta m_{l^{H}}$ is
fixed to be $1$ GeV, $50$ GeV and $500$ GeV as respectively shown from the
left to right and top to bottom panels in Fig. 4. From Fig. 4 one can see that
the contribution from the $\mathcal{D}$ boson diagram to the branching ratio
of $\mu\rightarrow e\gamma$ is dominant. The SM $W$ boson contribution is
suppressed by the sums over of $(\Delta m_{i1}/m_{W})^{4}$ with $i=2,3$ and
$\Delta m_{i1}^{2}$ is the mass difference between the neutrino generations.
Using global fit values for $\Delta m_{i1}^{2}$ from Esteban:2020cvm , one can
obtain $\mathcal{B}(\mu\rightarrow e\gamma)_{W}\simeq 4.4\times 10^{-55}$. The
contribution from $\mathcal{H}^{\pm}$ is similarly suppressed by the mass of
neutrinos, whereas the contribution from $\mathcal{W}^{\prime}$ is negligible
due to the smallness of the gauge coupling $g_{H}$ that is $g_{H}\simeq
4.58\times 10^{-4}$ for this benchmark point. For a fixed value of the mass
splitting $\Delta m_{l^{H}}$, the total branching ratio of $\mu\rightarrow
e\gamma$ decreases when $m_{l^{H}}$ increases. When $\Delta m_{l^{H}}$
increases, the branching ratio of $\mu\rightarrow e\gamma$ from $\mathcal{D}$
and $\mathcal{W}^{\prime}$ bosons increase, while the contribution from
charged Higgs is almost unchanged. For large values of $\Delta m_{l^{H}}$, the
current limit from the MEG experiment can put a lower bound on $m_{l^{H}}$. A
larger mass splitting $\Delta m_{l^{H}}$ requires a larger $m_{l^{H}}$. In
particular, as shown on the top-right and bottom panel in Fig. 4, the heavy
hidden lepton mass $m_{l^{H}}$ is required $\gtrsim$ 1 and 3 TeV for fixing
$\Delta m_{l^{H}}$ at 50 and 500 GeV, respectively. For small values of
$\Delta m_{l^{H}}$, the branching ratio of $\mu\rightarrow e\gamma$ is
suppressed and thus escaping the MEG constraint (see the top-left panel in
Fig. 4). We note that for the degenerate mass case, i.e. $\Delta m_{l^{H}}=0$,
the contributions from new particles to the branching ratio of $\mu\rightarrow
e\gamma$ vanishes. This is because, in this case, the form factors from
$\mathcal{D},\mathcal{W}^{\prime}$ and $\mathcal{H}^{\pm}$ are proportional to
the following factors 444 In general, we have $A_{ji}^{M/E}(\mathcal{D\,{\rm
or}\,W^{\prime}\,{\rm or}\,H^{\pm}})|_{{\rm degenerate}\,f^{H}}=0$ for $i\neq
j$, which is just the manifestation of the well-known GIM-mechanism in SM.

$\displaystyle A_{e\mu}^{M/E}(\mathcal{D}\,{\rm
or}\,\mathcal{W}^{\prime})\bigg{|}_{{\rm degenerate}\,l^{H}}$
$\displaystyle\sim$
$\displaystyle\sum_{k=1}^{3}\left({V_{l^{H}}}\right)_{2k}^{*}\left({V_{l^{H}}}\right)_{1k}=0\,,$
(53) $\displaystyle A_{e\mu}^{M/E}(\mathcal{H}^{\pm})\bigg{|}_{{\rm
degenerate}\,\nu^{H}}$ $\displaystyle\sim$
$\displaystyle\sum_{k=1}^{3}\left({V_{\nu^{H}}}\right)_{2k}^{*}\left({V_{\nu^{H}}}\right)_{1k}=0\,.$
(54)

Figure 5:  Viable DM parameter points spanned in
(${m_{\mathcal{W}^{\prime}},g_{H}}$) plane (left) and
(${m_{\mathcal{D}},|\sin\theta_{2}|}$) plane (right). The color of circle
points indicates the contribution to ${\cal B}(\mu\rightarrow e\gamma)$ from
the $\mathcal{W}^{\prime}$ diagram (left panel) and $\mathcal{D}$ diagram
(right panel). Here we fixed $m_{l^{H}}=1$ TeV and $\Delta m_{l^{H}}=50$ GeV.
The solid red and dashed blue lines on the right panel are the MEG limit and
future sensitivity from MEG II Meucci:2022qbh , respectively. The crossed
purple points on the left panel are allowed by the current limit from MEG
MEG:2016leq  which are obtained from the bound on $|\sin\theta_{2}|$ (solid
red on the right panel) due to the relation given in Eq. (55).

Fig. 5 shows $2\sigma$ favored parameter space for $\mathcal{W}^{\prime}$ as a
sub-GeV M candidate in the model. The data points are projected on
(${m_{\mathcal{W}^{\prime}},g_{H}}$) plane (left panel) and (${m_{\cal
D},|\sin\theta_{2}|}$) plane (right panel). The colors of circle points in the
left and right panels of Fig. 5 indicate the values of
$\mathcal{B}(\mu\rightarrow e\gamma)$ calculated from $\mathcal{W}^{\prime}$
diagram and $\mathcal{D}$ diagram, respectively. Here we fixed $m_{l^{H}}=1$
TeV and $\Delta m_{l^{H}}=50$ GeV. The contribution from the
$\mathcal{W}^{\prime}$ diagram to $\mathcal{B}(\mu\rightarrow e\gamma)$ is
linearly proportional to the gauge coupling $g_{H}^{2}$. However due to the
constraints from the dark matter direct detection and dark photon physics that
required $g_{H}\lesssim 10^{-3}$ Ramos:2021omo ; Ramos:2021txu ; Tran:2022yrh
, the $\mathcal{B}(\mu\rightarrow e\gamma)$ from the $\mathcal{W}^{\prime}$
diagram is suppressed. In particular, $\mathcal{B}(\mu\rightarrow
e\gamma)_{\mathcal{W}^{\prime}}\lesssim 10^{-21}$ as shown in the left panel
of Fig. 5. On the other hand, the contribution to $\mathcal{B}(\mu\rightarrow
e\gamma)$ from the $\mathcal{D}$ diagram is significant. The branching ratio
is enhanced in the region of large mixing angle $\theta_{2}$ and heavy mass
region of $\mathcal{D}$ boson. The current experimental data requires
$0.018\lesssim|\sin\theta_{2}|\lesssim 0.21$ for the $2\sigma$ favored region
Tran:2022yrh , which results $10^{-17}\lesssim\mathcal{B}(\mu\rightarrow
e\gamma)\lesssim 10^{-10}$. We note that the ${\cal B}(\mu\rightarrow
e\gamma)$ from the $\cal D$ boson diagram peaks at a certain value of $m_{\cal
D}$ depending on the mass of heavy hidden leptons. For $m_{l^{H}}=1$ TeV and
$\Delta m_{l^{H}}=50$ GeV, the peak is at $m_{\cal D}\sim 1.8$ TeV. The
current limit from the MEG experiment MEG:2016leq  can exclude a large portion
of parameter space ($\sim 50\%$ of data points) in this enhanced region as
shown by the red line on the right panel of Fig. 5. The future sensitivity
from MEG II Meucci:2022qbh , as shown by the dashed blue line on the right
panel of Fig. 5, can probe lower values of the mixing angle $\theta_{2}$ and a
smaller region of $\mathcal{D}$ boson mass in the model. We note that the
upper bound on $|\sin\theta_{2}|$ from the MEG experiment can be translated
into a bound on the DM mass $m_{\mathcal{W}^{\prime}}$ and the gauge coupling
$g_{H}$ due to the following relation

$g_{H}=\frac{2\,m_{W^{\prime}}}{v}\times\begin{cases}\begin{matrix}|\sin\theta_{2}|\,,&&{\rm
for}\;\theta_{2}>0\;,\\\ |\cos\theta_{2}|\,,&&{\rm for}\;\theta_{2}\leq
0\;,\end{matrix}\end{cases}$ (55)

which can be derived from Eqs. (19) and (26). The allowed points after taking
into account the MEG constraint projected on
($m_{\mathcal{W}^{\prime}},g_{H}$) plane is shown as the crossed purple points
in the left panel of Fig. 5.

Figure 6:  Viable DM parameter points spanned in the plane of the total
branching ratio of $\mu\rightarrow e\gamma$ and muon anomalous magnetic dipole
moment $\Delta a_{\mu}$ . Here we fixed $m_{l^{H}}=1$ TeV and $\Delta
m_{l^{H}}=50$ GeV. The solid red and dashed blue lines are the current limit
from MEG MEG:2016leq  and future sensitivity from MEG II Meucci:2022qbh ,
respectively. The shaded light blue band represents the $2\sigma$ region of
$\Delta a_{\mu}$ measured at BNL Muong-2:2002wip ; Muong-2:2004fok ;
Muong-2:2006rrc  and FNAL Muong-2:2021ojo .

In Fig. 6, we show the $2\sigma$ favored parameter space on the plane of the
total branching ratio of $\mu\rightarrow e\gamma$ and muon anomalous magnetic
dipole moment $\Delta a_{\mu}$. As mentioned above, the main contribution to
the $\mathcal{B}(\mu\rightarrow e\gamma)$ is from the $\mathcal{D}$ boson
diagram and a large portion of parameter space can be excluded by the current
MEG experiment. On the other hand, the main contributions to $\Delta a_{\mu}$
in the model are from $\mathcal{D}$ and $Z_{2,3}$ diagrams. The contribution
from the $\mathcal{W}^{\prime}$ diagram gives a negative value for $\Delta
a_{\mu}$, whereas the neutral Higgs and charged Higgs contributions are both
suppressed for the current viable parameter space in the model. One can see in
Fig. 6, the total contribution to $\Delta a_{\mu}$ is not reaching the
$2\sigma$ region (shaded light blue) for the muon anomalous magnetic dipole
moment measured at BNL Muong-2:2002wip ; Muong-2:2004fok ; Muong-2:2006rrc
and FNAL Muong-2:2021ojo . We expect the $\Delta a_{\mu}$ can be enhanced in
higher loop diagrams such as the two-loop Barr-Zee mechanism Barr:1990um .
Calculation of these two-loop Barr-Zee contributions is thus highly desirable
but nevertheless beyond the scope of this study. We hope to return to this
issue in the future.

Figure 7:  Favored data projected on the plane of the DM mass and spin
independent DM-proton scattering cross section. Here we fixed $m_{l^{H}}=1$
TeV and $\Delta m_{l^{H}}=50$ GeV. The crossed purple points indicate the data
satisfied the MEG constraint MEG:2016leq , while the circle green points
indicate the data that can be probed by future experiment from MEG II
Meucci:2022qbh . The gray regions are the exclusion from CRESST-III
Angloher:2017sxg , DarkSide-50 Agnes:2018ves  and XENON1T Aprile:2019xxb
experiments. The dashed blue, red and light blue lines represent the future
sensitivities from DM direct detection experiments at NEWS-G
Battaglieri:2017aum , SuperCDMS Agnese:2016cpb  and CDEX Ma:2017nhc ,
respectively. Orange region is the neutrino floor background.

Fig. 7 shows the allowed data points by the current MEG constraints (crossed
purple) and the points that can be probed by future experiment from MEG II
(circle green) on the DM direct detection plane. The predicted DM mass range
in $2\sigma$ favored region is $\sim(0.02-3)$ GeV. The future sensitivity from
MEG II can probe almost the entire viable range of DM mass. Interestingly,
some data points with the DM mass at around $1$ GeV can be probed
complementarily by various future DM direct detection experiments including
NEWS-G Battaglieri:2017aum , SuperCDMS Agnese:2016cpb  and CDEX Ma:2017nhc .

##  V Conclusion

We computed the one-loop radiative decay rates for the charged lepton flavor
violation processes $l_{i}\rightarrow l_{j}\gamma$, the anomalous magnetic
dipole moment $\Delta a_{l_{i}}$ and the electric dipole moment $d_{l_{i}}$ of
the charged lepton $l_{i}$ at one-loop level in a minimal G2HDM. Besides the
contribution from the SM $W$ boson, the charged lepton flavor violation
processes occurred at one-loop in G2HDM due to the new interactions of SM
charged leptons with two $h$-parity odd particles – heavy hidden leptons
($l^{H}$ or $\nu^{H}$) and hidden dark scalars or gauge bosons ($\mathcal{D}$,
$\mathcal{H}^{\pm}$,$\mathcal{W}^{\prime}$). The contributions from these new
interactions vanish when the heavy hidden lepton masses among generations
degenerate.

We analyzed the $\mu\rightarrow e\gamma$ process and $\Delta a_{\mu}$ using a
parameter space that favors a sub-GeV non-abelian vector dark matter
$\mathcal{W}^{\prime(p,m)}$ in the model. The scan data is adapted from Ref.
Tran:2022yrh  and they satisfy various constraints including the theoretical
constraints on the scalar potential, the Higgs signal strength measurements
from the LHC, the dark photon physics, the electroweak precision measurements
including the recent $W$ boson mass measurement at the CDF II, DM relic
density measured from Planck collaboration, and from DM searches including the
DM direct detections and the Higgs invisible width from the LHC.

We found that the branching ratio of $\mu\rightarrow e\gamma$ is significantly
dependent on the heavy hidden lepton mass and the mass splitting between its
generations. In particular, a heavier hidden lepton mass results in a smaller
branching ratio of $\mu\rightarrow e\gamma$ while a larger mass splitting
gives a larger branching ratio as shown in Fig. 4.

Among the new contributions to $\mu\rightarrow e\gamma$ in the model, the
contribution from $\mathcal{D}$ boson diagram is dominant. The branching ratio
can be enhanced in the heavy $\mathcal{D}$ boson mass region and the region of
large mixing angle $\theta_{2}$ between two odd $h$-parity bosons,
$\mathcal{H}^{0}_{2}$ and $G_{H}^{m}$, which compose $\mathcal{D}$. The
current constraint on the cLFV process from MEG can exclude a significant
portion of the parameter space in the favored region obtained in previous
studies. Although the contribution from the DM candidate
$\mathcal{W}^{\prime}$ to the branching ratio of $\mu\rightarrow e\gamma$ is
suppressed due to the smallness of new gauge coupling $g_{H}$, the DM
parameters can be affected indirectly by the cLFV processes due to the
relation between parameters in the model, especially the relation (55) between
the mixing angle $\theta_{2}$, new gauge coupling $g_{H}$ and DM mass
$m_{\mathcal{W}^{\prime}}$. We found that the future measurement at MEG II can
probe almost the entire viable range of the DM mass which is $\sim(0.02-3)$
GeV and interestingly, the region at the DM mass around $1$ GeV can be also
probed by future DM direct detection experiments such as NEWS-G, SuperCDMS and
CDEX.

In the viable DM parameter space, the total one-loop level contribution to
$\Delta a_{\mu}$ in the model is not big enough to explain the $4.2\sigma$
level discrepancy between the theoretical value and the experimental results
measured at the BNL and FNAL. We expect an enhanced contribution to $\Delta
a_{\mu}$ at higher loop corrections, such as the Barr-Zee two-loop mechanism
Barr:1990um  for the neutrino magnetic dipole moments, can be anticipated to
address the muon anomaly in the model.

In Appendix A, we also showed that the electric dipole moment of charged
lepton vanishes at one-loop in G2HDM. This is due to (1) the lack of CP
violating phases (or in general imaginary parts) in products of generic but
related complex vector and axial vector gauge couplings or scalar and
pseudoscalar Yukawa couplings, and (2) vanishing combinations of Feynman loop
integrals. Same conclusion can be drawn for the SM quarks in the model. Thus
it is also interesting to investigate if the two-loop mechanisms like the
Weinberg three-gluon operator Weinberg:1989dx ; Braaten:1990gq ;
Weinberg:1990me  for the gluon chromo-electric dipole moment Braaten:1990zt
and the Barr-Zee diagrams for the charged leptons Barr:1990vd  can generate a
non-vanishing result for the electric dipole moments for the neutron and SM
leptons respectively in G2HDM. For these two-loop calculations, we will
reserve them for future tasks.

## Acknowledgments

We would like to thank Michael J. Ramsey-Musolf for encouragements and useful
discussions . This work is supported in part by the Ministry of Science and
Technology (MOST) of Taiwan under Grant No 111-2112-M-001-035 (TCY) and by
National Natural Science Foundation of China under Grant No. 19Z103010239
(VQT). VQT would like to thank the High Energy Theory Group at the Institute
of Physics, Academia Sinica, Taiwan for its hospitality.

## Appendix A. One-Loop Analytical Formulae of $A^{M}_{ji}$ and $A^{E}_{ji}$

In this Appendix, we present the one-loop expressions for the transition
magnetic and electric dipole form factors $A^{M}_{ji}$ and $A^{E}_{ji}$ from
the six different contributions in G2HDM. For the gauge particle loops of
$W^{\pm}$, $\\{Z_{n}\\}$ and $\mathcal{W}^{\prime(p,m)}$, we use unitary gauge
in the computation. For the computation of $W^{\pm}$ loop in the ’t Hooft-
Feynman gauge, see Appendix B. 555The issue of gauge fixings in the model has
been studied as given in an Appendix in Ref. Ramos:2021txu .  For convenience,
we define $z=1-x-y$ in what follows.

###  V.1 $W$ contribution – Left diagram in Fig. (1)

The transition magnetic and electric dipole form factors are given by

$\displaystyle A^{M}_{ji}\left(W\right)$
$\displaystyle=+\frac{1}{8\pi^{2}}\left(\frac{g}{2\sqrt{2}}\right)^{2}\sum_{k}\left(V_{\rm
PMNS}\right)^{*}_{kj}\left(V_{\rm PMNS}\right)_{ki}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\times\left[\mathcal{I}\left(m_{i},m_{j},m_{\nu_{k}},m_{W}\right)+\mathcal{I}\left(m_{i},m_{j},-m_{\nu_{k}},m_{W}\right)\right]\;,$
(A.1) $\displaystyle A^{E}_{ji}\left(W\right)$
$\displaystyle=-\frac{i}{8\pi^{2}}\left(\frac{g}{2\sqrt{2}}\right)^{2}\sum_{k}\left(V_{\rm
PMNS}\right)^{*}_{kj}\left(V_{\rm PMNS}\right)_{ki}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\times\left[\mathcal{I}\left(m_{i},-m_{j},m_{\nu_{k}},m_{W}\right)+\mathcal{I}\left(m_{i},-m_{j},-m_{\nu_{k}},m_{W}\right)\right]\;,$
(A.2)

respectively where the loop integral
$\mathcal{I}\left(m_{i},m_{j},m_{k},m_{X}\right)$ is

$\displaystyle\mathcal{I}$ $\displaystyle\left(m_{i},m_{j},m_{k},m_{X}\right)$
$\displaystyle=\int_{0}^{1}{\rm d}x\int_{0}^{1-x}{\rm
d}y\Biggl{\\{}\frac{1}{-xzm_{i}^{2}-xym_{j}^{2}+xm_{k}^{2}+(1-x)m_{X}^{2}}\Biggr{.}$
$\displaystyle\;\;\;\;\times\Biggl{[}\left(\bigl{(}y+2z\left(1-x\right)\bigr{)}+\bigl{(}z+2y\left(1-x\right)\bigr{)}\frac{m_{j}}{m_{i}}-\,3\left(1-x\right)\frac{m_{k}}{m_{i}}\right)\Biggr{.}$
$\displaystyle\;\;\;\;\;\;\;\;\;+\Biggl{.}\frac{m_{i}^{2}}{m^{2}_{X}}x^{2}\left(z^{2}+y^{2}\frac{m^{3}_{j}}{m_{i}^{3}}+yz\frac{m_{j}}{m_{i}}\left(1+\frac{m_{j}}{m_{i}}\right)-\frac{m_{j}m_{k}}{m_{i}^{2}}\right)\Biggr{]}$
$\displaystyle\;\;\;\;+\frac{1}{m^{2}_{X}}\left(x(1-z)+y+\bigl{(}x\left(1-y\right)+z\bigr{)}\frac{m_{j}}{m_{i}}-\frac{m_{k}}{m_{i}}\right)$
$\displaystyle\;\;\;\;+\frac{1}{m^{2}_{X}}\left(2-x\left(3-4z\right)-3y-z+\bigl{(}2-x\left(3-4y\right)-y-3z\bigr{)}\frac{m_{j}}{m_{i}}\right)$
$\displaystyle\;\;\;\;\;\;\;\;\;\Biggl{.}\times\log\left(\frac{m^{2}_{X}}{-xzm_{i}^{2}-xym_{j}^{2}+xm_{k}^{2}+(1-x)m_{X}^{2}}\right)\Biggr{\\}}\;.$
(A.3)

We note that this integral $\mathcal{I}$ is for the diagram with two internal
charged vector bosons $X$ coupled to the external photon computed using the
unitary gauge. The third line of Eq. (V.1) comes from the product of the
transverse pieces of the two vector boson propagators, while all the remaining
terms are due to the product of the transverse and longitudinal pieces of
these two propagators. The product of longitudinal pieces do not give rise to
the contributions for the transition magnetic and electric dipole form
factors. Our integral $\mathcal{I}$ is denoted as $I^{++}_{k,3}$ in Eq. (A.5)
of Lindner:2016bgg . Except for the fourth line of Eq. (V.1), our formula
agrees 666Note that there are also a couple of trivial typos in the
coefficients of the log term of $I^{++}_{k,3}$ in Eq. (A.5) of Lindner:2016bgg
.. The difference between our result of Eq. (V.1) and Eq. (A.5) of
Lindner:2016bgg  is

$\displaystyle{\rm Diff}$
$\displaystyle=\frac{\left(m_{i}-m_{j}\right)^{2}}{m_{X}^{2}}\int_{0}^{1}{\rm
d}x\int_{0}^{1-x}{\rm
d}y\frac{xyz\left(x+y+\left(1-y\right)\frac{m_{j}}{m_{i}}-\frac{m_{k}}{m_{i}}\right)}{-xzm_{i}^{2}-xym_{j}^{2}+xm_{k}^{2}+(1-x)m_{X}^{2}}\;.$
(A.4)

Since the difference disappears in the case of $m_{i}=m_{j}$, one can’t use
the known result of charged lepton anomaly Leveille:1977rc  to discriminate
the two answers. However, see Appendix B.

For $i=j$, there is no CP violating phase arise from the product
$|\left(V_{\rm PMNS}\right)_{ki}|^{2}$ in (V.1), which implies the electric
dipole moment $d_{l_{i}}$ should be vanishing from the one-loop $W$ diagram in
SM. Indeed the sum of the two integrals in (V.1) vanishes when $m_{i}=m_{j}$!

###  V.2 $\left\\{Z_{n}\right\\}$ contribution – Middle diagram in Fig. (1)

The transition magnetic dipole form factor is given by

$\displaystyle A^{M}_{ji}\left(\left\\{Z_{n}\right\\}\right)$
$\displaystyle=+\frac{\delta_{ij}}{8\pi^{2}}\sum_{n}\left[\left(C_{V\,n}\right)^{2}\mathcal{J}\left(m_{i},m_{i},m_{i},m_{Z_{n}}\right)\right.$
$\displaystyle\quad\quad\quad\quad\quad\left.+\left(C_{A\,n}\right)^{2}\mathcal{J}\left(m_{i},m_{i},-m_{i},m_{Z_{n}}\right)\right]\;,$
(A.5)

where

$\displaystyle\mathcal{J}$ $\displaystyle\left(m_{i},m_{j},m_{k},m_{X}\right)$
$\displaystyle=-\int_{0}^{1}{\rm d}x\int_{0}^{1-x}{\rm
d}y\Biggl{\\{}\frac{1}{-xzm_{i}^{2}-xym_{j}^{2}+(1-x)m_{k}^{2}+xm_{X}^{2}}\Biggr{.}$
$\displaystyle\;\;\;\;\times\Biggl{\lceil}2x\left((1-z)+(1-y)\frac{m_{j}}{m_{i}}-2\frac{m_{k}}{m_{i}}\right)\Biggr{.}$
$\displaystyle\;\;\;\;\;\;\;\;\;+\frac{m_{i}^{2}}{m^{2}_{X}}\Biggl{(}(1-x)\left(\frac{m_{j}}{m_{i}}-\frac{m_{k}}{m_{i}}\right)\left(z+y\frac{m_{j}}{m_{i}}\right)\left(1-\frac{m_{k}}{m_{i}}\right)\Biggr{.}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-z\left(\frac{m_{j}}{m_{i}}-\frac{m_{k}}{m_{i}}\right)\left((1-x(1-z))+xy\frac{m_{j}^{2}}{m_{i}^{2}}\right)$
$\displaystyle\Biggl{.}\Biggl{.}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\,y\left(1-\frac{m_{k}}{m_{i}}\right)\left(xz+(1-x(1-y))\frac{m_{j}^{2}}{m_{i}^{2}}\right)\Biggr{)}\Biggr{\rfloor}$
$\displaystyle\;\;\;\;+\frac{1}{m^{2}_{X}}\left(y+z\frac{m_{j}}{m_{i}}-\left(1-x\right)\frac{m_{k}}{m_{i}}\right)$
$\displaystyle\;\;\;\;+\frac{1}{m^{2}_{X}}\left((1-3y)+(1-3z)\frac{m_{j}}{m_{i}}+\left(1-3\,x\right)\frac{m_{k}}{m_{i}}\right)$
$\displaystyle\;\;\;\;\;\;\;\;\;\Biggl{.}\times\log\left(\frac{m^{2}_{X}}{-xzm_{i}^{2}-xym_{j}^{2}+(1-x)m_{k}^{2}+xm_{X}^{2}}\right)\Biggr{\\}}\;.$
(A.6)

We note that this integral of $\mathcal{J}$ is for the diagram with one
internal neutral gauge boson $X$ exchange computed using the unitary gauge.
The third line of Eq. (V.2) comes from the transverse piece of the vector
boson propagator, while the remaining terms come entirely from the
longitudinal piece of the propagator. Our loop integral $(-\mathcal{J})$
corresponds to $I^{++}_{k,4}$ in Eq. (A.6) of Lindner:2016bgg . 777 We note
that in the fourth line of our Eq. (V.2), instead of the factor of $(1-x)$,
Eq. (A.6) of Lindner:2016bgg  got $(x-1)$. Using our expression of
$\mathcal{J}$ in Eq. (V.2) for the equal mass case of $m_{i}=m_{j}=m_{\mu}$
and setting $m_{k}=m_{F}$, one can easily reproduce the well-known expression
of muon anomaly for a neutral gauge boson $X$ with a general gauge coupling of
a muon and another fermion $F$ first obtained in Leveille:1977rc .

For the transition electric dipole from factor, one finds

$\displaystyle A^{E}_{ji}\left(\left\\{Z_{n}\right\\}\right)$
$\displaystyle=0\;,$ (A.7)

which implies $d_{l_{i}}(\\{Z_{n}\\})=0$.

###  V.3 $\left\\{h_{n}\right\\}$ contribution – Right diagram in Fig. (1)

The transition magnetic dipole form factor is

$\displaystyle A^{M}_{ji}\left(\left\\{h_{n}\right\\}\right)$
$\displaystyle=\frac{\delta_{ij}}{\pi^{2}}\frac{m_{i}^{2}}{v^{2}}\sum_{n}\left(O^{H}\right)_{1n}^{2}\mathcal{K}\left(m_{i},m_{i},m_{i},m_{h_{n}}\right)\;,$
(A.8)

with

$\displaystyle\mathcal{K}\left(m_{i},m_{j},m_{k},m_{X}\right)$
$\displaystyle=\int_{0}^{1}{\rm d}x\int_{0}^{1-x}{\rm d}y$
$\displaystyle\;\;\;\;\times\left[\frac{x\left(y+z\frac{m_{j}}{m_{i}}\right)+(1-x)\frac{m_{k}}{m_{i}}}{-xym_{i}^{2}-xzm_{j}^{2}+(1-x)m_{k}^{2}+xm_{X}^{2}}\right]\;.$
(A.9)

This loop integral $\mathcal{K}$ is the same as $I^{++}_{k,1}$ in Eq. (A.1) of
Lindner:2016bgg .

As in the $\\{Z_{n}\\}$ case, one finds that the transition electric dipole
form factor vanishes

$A^{E}_{ji}\left(\left\\{h_{n}\right\\}\right)=0\;,$ (A.10)

which implies $d_{l_{i}}(\\{h_{n}\\})=0$ as well.

###  V.4 $\mathcal{D}$ contribution – Left diagram in Fig. (2)

The transition magnetic and electric dipole form factors are given by

$\displaystyle A^{M}_{ji}\left(\mathcal{D}\right)$
$\displaystyle=\frac{1}{8\pi^{2}}\left[\sum_{k}y^{\mathcal{D}\,*}_{S\,kj}y^{\mathcal{D}}_{S\,ki}\mathcal{K}\left(m_{i},m_{j},m_{l^{H}_{k}},m_{\mathcal{D}}\right)\right.$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left.\sum_{k}y^{\mathcal{D}\,*}_{P\,kj}y^{\mathcal{D}}_{P\,ki}\mathcal{K}\left(m_{i},m_{j},-m_{l^{H}_{k}},m_{\mathcal{D}}\right)\right]\;,$
(A.11) $\displaystyle A^{E}_{ji}\left(\mathcal{D}\right)$
$\displaystyle=\frac{i}{8\pi^{2}}\left[\sum_{k}y^{\mathcal{D}\,*}_{P\,kj}y^{\mathcal{D}}_{S\,ki}\mathcal{K}\left(m_{i},-m_{j},m_{l^{H}_{k}},m_{\mathcal{D}}\right)\right.$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left.\sum_{k}y^{\mathcal{D}\,*}_{S\,kj}y^{\mathcal{D}}_{P\,ki}\mathcal{K}\left(m_{i},-m_{j},-m_{l^{H}_{k}},m_{\mathcal{D}}\right)\right]\;,$
(A.12)

where the summation is over all the heavy hidden charged leptons $l^{H}_{k}$
running inside the loop. $\mathcal{K}$ is defined already in (V.3).

For $i=j$, (V.4) reduces to

$\displaystyle A^{M}_{ii}\left(\mathcal{D}\right)$
$\displaystyle=\frac{1}{8\pi^{2}}\left[\sum_{k}|y^{\mathcal{D}}_{S\,ki}|^{2}\mathcal{K}\left(m_{i},m_{i},m_{l^{H}_{k}},m_{\mathcal{D}}\right)\right.$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\
+\left.\sum_{k}|y^{\mathcal{D}}_{P\,ki}|^{2}\mathcal{K}\left(m_{i},m_{i},-m_{l^{H}_{k}},m_{\mathcal{D}}\right)\right]\;,$
(A.13)

with

$\displaystyle|y^{\mathcal{D}}_{(S,P)\,ki}|^{2}$
$\displaystyle=\frac{1}{2}|\left(V^{H}_{l}\right)_{ik}|^{2}\left(\frac{m_{l_{i}}^{2}}{v^{2}}\cos^{2}\theta_{2}+\frac{m_{l^{H}_{k}}^{2}}{v_{\Phi}^{2}}\sin^{2}\theta_{2}\pm\frac{m_{l_{i}}m_{l^{H}_{k}}}{vv_{\Phi}}\sin
2\theta_{2}\right)\;.$ (A.14)

We note that the possible new CP violating phase in $V^{H}_{l}$ is cancelled
out in $|y^{\mathcal{D}}_{(S,P)\,ki}|^{2}$.

On the other hand, for $i=j$, (V.4) reduces to

$\displaystyle A^{E}_{ii}\left(\mathcal{D}\right)$
$\displaystyle=\frac{i}{8\pi^{2}}\sum_{k}{\rm
Re}\left(y^{\mathcal{D}\,*}_{P\,ki}y^{\mathcal{D}}_{S\,ki}\right)\left[\mathcal{K}\left(m_{i},-m_{i},m_{l^{H}_{k}},m_{\mathcal{D}}\right)+\mathcal{K}\left(m_{i},-m_{i},-m_{l^{H}_{k}},m_{\mathcal{D}}\right)\right]$
$\displaystyle-\frac{1}{8\pi^{2}}\sum_{k}{\rm
Im}\left(y^{\mathcal{D}\,*}_{P\,ki}y^{\mathcal{D}}_{S\,ki}\right)\left[\mathcal{K}\left(m_{i},-m_{i},m_{l^{H}_{k}},m_{\mathcal{D}}\right)-\mathcal{K}\left(m_{i},-m_{i},-m_{l^{H}_{k}},m_{\mathcal{D}}\right)\right]$
$\displaystyle=0\;,$ (A.15)

due to the fact that
$\mathcal{K}\left(m_{i},-m_{i},m_{l^{H}_{k}},m_{\mathcal{D}}\right)+\mathcal{K}\left(m_{i},-m_{i},-m_{l^{H}_{k}},m_{\mathcal{D}}\right)=0$
and from (30) we have ${\rm
Im}\left(y^{\mathcal{D}\,*}_{P\,ki}y^{\mathcal{D}}_{S\,ki}\right)=0$.

###  V.5 $\mathcal{H}^{\pm}$ contribution – Middle diagram in Fig. (2)

The transition magnetic and electric dipole form factors are

$\displaystyle A^{M}_{ji}\left(\mathcal{H}\right)$
$\displaystyle=+\frac{1}{8\pi^{2}}\sum_{k}y^{\mathcal{H}\,*}_{S\,kj}y^{\mathcal{H}}_{S\,ki}\left[\mathcal{L}\left(m_{i},m_{j},m_{\nu^{H}_{k}},m_{\mathcal{H}}\right)\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\left.+\mathcal{L}\left(m_{i},m_{j},-m_{\nu^{H}_{k}},m_{\mathcal{H}}\right)\right]\;,$
(A.16) $\displaystyle A^{E}_{ji}\left(\mathcal{H}\right)$
$\displaystyle=-\frac{i}{8\pi^{2}}\sum_{k}y^{\mathcal{H}\,*}_{S\,kj}y^{\mathcal{H}}_{S\,ki}\left[\mathcal{L}\left(m_{i},-m_{j},m_{\nu^{H}_{k}},m_{\mathcal{H}}\right)\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\left.+\mathcal{L}\left(m_{i},-m_{j},-m_{\nu^{H}_{k}},m_{\mathcal{H}}\right)\right]\;,$
(A.17)

where we have used $y^{\mathcal{H}}_{P\,kj}=-y^{\mathcal{H}}_{S\,kj}$ from
(33) and the summation is over all the heavy hidden neutrinos $\nu^{H}_{k}$
running inside the loop. The loop integral $\mathcal{L}$ is given by

$\displaystyle\mathcal{L}\left(m_{i},m_{j},m_{k},m_{X}\right)$
$\displaystyle=-\int_{0}^{1}{\rm d}x\int_{0}^{1-x}{\rm d}y$
$\displaystyle\;\;\;\;\times\left[\frac{x\left(y+z\frac{m_{j}}{m_{i}}+\frac{m_{k}}{m_{i}}\right)}{-xym_{i}^{2}-xzm_{j}^{2}+xm_{k}^{2}+(1-x)m_{X}^{2}}\right]\;.$
(A.18)

Our loop integral $(-\mathcal{L})$ is the same as $I^{++}_{k,2}$ in Eq. (A.2)
of Lindner:2016bgg .

For $i=j$, each term in both $A^{M}_{ii}(\mathcal{H})$ and
$A^{E}_{ii}(\mathcal{H})$ is proportional to

$|y^{\mathcal{H}}_{S\,ki}|^{2}=\frac{|\left(V^{H\,\dagger}_{\nu}M_{\nu}V_{\rm
PMNS}\right)_{ki}|^{2}}{2v^{2}}\;,$ (A.19)

which is real but may contain CP-violating phases from $V_{\rm PMNS}$ and
$V^{H}_{\nu}$. The effects from these CP-violating phases in $\Delta
a_{l_{i}}(\mathcal{H}^{\pm})$ are small due to the suppression from the
neutrino masses. The important role of CP violating phases in the muon anomaly
in MSSM coming from the charginos and neutrinos sectors has been emphasized
previously in Ibrahim:1999hh ; Ibrahim:1999aj ; Ibrahim:2001ym . The electric
dipole moment $d_{l_{i}}(\mathcal{H}^{\pm})$ should be vanishing since the sum
of the two integrals in (V.5) vanishes when $m_{i}=m_{j}$!

###  V.6 $\mathcal{W}^{\prime(p,m)}$ contribution – Right diagram in Fig. (2)

The transition magnetic and electric dipole form factors are

$\displaystyle A^{M}_{ji}\left(\mathcal{W}^{\prime}\right)$
$\displaystyle=+\frac{1}{8\pi^{2}}\sum_{k}g^{\mathcal{W}^{\prime}\,*}_{V\,kj}g^{\mathcal{W}^{\prime}}_{V\,ki}\left[\mathcal{J}\left(m_{i},m_{j},m_{l^{H}_{k}},m_{\mathcal{W}^{\prime}}\right)\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\left.+\mathcal{J}\left(m_{i},m_{j},-m_{l^{H}_{k}},m_{\mathcal{W}^{\prime}}\right)\right]\;,$
(A.20) $\displaystyle A^{E}_{ji}\left(\mathcal{W}^{\prime}\right)$
$\displaystyle=+\frac{i}{8\pi^{2}}\sum_{k}g^{\mathcal{W}^{\prime}\,*}_{V\,kj}g^{\mathcal{W}^{\prime}}_{V\,ki}\left[\mathcal{J}\left(m_{i},-m_{j},m_{l^{H}_{k}},m_{\mathcal{W}^{\prime}}\right)\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\left.+\mathcal{J}\left(m_{i},-m_{j},-m_{l^{H}_{k}},m_{\mathcal{W}^{\prime}}\right)\right]\;,$
(A.21)

where we have used
$g^{\mathcal{W}^{\prime}}_{A\,kj}=g^{\mathcal{W}^{\prime}}_{V\,kj}$ from (35)
and the summation is over all the heavy hidden charged leptons $l^{H}_{k}$
running inside the loop. The loop integral $\mathcal{J}$ is given in (V.2). In
the case of $i=j$, each term in both
$A^{M}_{ii}\left(\mathcal{W}^{\prime}\right)$ and
$A^{E}_{ii}\left(\mathcal{W}^{\prime}\right)$ is proportional to

$|g^{\mathcal{W}^{\prime}}_{V\,ki}|^{2}=\frac{g_{H}^{2}}{8}|\left(V^{H}_{l}\right)_{ik}|^{2}\;,$
(A.22)

which is real and contains no CP-violating phase. The electric dipole moment
$d_{l_{i}}(\mathcal{W}^{\prime})$ should be vanishing as one can check that
the sum of the two integrals in (V.6) vanishes when $m_{i}=m_{j}$!

We note that all our results for the charged lepton anomalous magnetic dipole
moments (where $i$ and $j$ are the same charged lepton) are consistent with
Eqs. (3), (4), (10) and (11) in Leveille:1977rc  if we choose $q_{\rm
F}=q_{\rm X}=q_{\rm H}=-1$ in these formulas.

## Appendix B. Expression for $\mathcal{I}(m_{i},m_{j},m_{k},m_{X})$ in ’t
Hooft-Feynman gauge.

In the SM, the $W$ loop contribution can be evaluated in the ’t Hooft-Feynman
gauge. The longitudinal contributions from the two $W$ propagators will be
‘simulated’ by three extra diagrams involving the couplings $\gamma
G^{+}G^{-}$ and $\gamma G^{\pm}W^{\mp}$ where $G^{\pm}$ are the charged
Goldstone bosons of $W^{\pm}$. The expression of
$\mathcal{I}(m_{i},m_{j},m_{k},m_{X})$ in the ’t Hooft-Feynman gauge is

$\displaystyle\mathcal{I}_{\,\rm{}^{\prime}t\,Hooft-Feynman}$
$\displaystyle\left(m_{i},m_{j},m_{k},m_{X}\right)$
$\displaystyle=\int_{0}^{1}{\rm d}x\int_{0}^{1-x}{\rm
d}y\Biggl{(}\frac{1}{-xzm_{i}^{2}-xym_{j}^{2}+xm_{k}^{2}+(1-x)m_{X}^{2}}\Biggr{)}$
$\displaystyle\;\;\;\;\times\Biggl{\\{}\Biggl{[}\bigl{(}y+2z\left(1-x\right)\bigr{)}+\bigl{(}z+2y\left(1-x\right)\bigr{)}\frac{m_{j}}{m_{i}}-\,3\left(1-x\right)\frac{m_{k}}{m_{i}}\Biggr{]}\Biggr{.}$
$\displaystyle\;\;\;\;\;\;\;\;\;-\frac{m_{i}^{2}}{m_{X}^{2}}\Biggl{[}x\Biggl{(}1-\frac{m_{k}}{m_{i}}\Biggr{)}\Biggl{(}\frac{m_{j}}{m_{i}}-\frac{m_{k}}{m_{i}}\Biggr{)}\Biggl{(}z+y\frac{m_{j}}{m_{i}}+\frac{m_{k}}{m_{i}}\Biggr{)}\Biggr{]}$
$\displaystyle\;\;\;\;\;\;\;\;\;+\Biggl{.}y\Biggl{(}1-\frac{m_{k}}{m_{i}}\Biggr{)}+z\Biggl{(}\frac{m_{j}}{m_{i}}-\frac{m_{k}}{m_{i}}\Biggr{)}\Biggl{\\}}\;,$
(B.1)

where we have defined $z=1-x-y$ as before.

One can integrate over the $y$ variable in (Appendix B. Expression for
$\mathcal{I}(m_{i},m_{j},m_{k},m_{X})$ in ’t Hooft-Feynman gauge.) and obtain
a 1-dimension integral representation

$\displaystyle\mathcal{I}_{\,\rm{}^{\prime}t\,Hooft-
Feynman}\left(m_{i},m_{j},m_{k},m_{X}\right)$ $\displaystyle=\int_{0}^{1}{\rm
d}x\Biggl{\\{}A(1-x)\Biggr{.}$
$\displaystyle\;\;\;\;\;\;\;\;\;\Biggl{.}+\frac{B}{x}\biggl{(}C_{0}+C_{1}x+C_{2}x^{2}\biggr{)}{\rm
log}\Biggl{[}\frac{m_{X}^{2}(1-x)-x(m_{i}^{2}(1-x)-m_{k}^{2})}{m_{X}^{2}(1-x)-x(m_{j}^{2}(1-x)-m_{k}^{2})}\Biggr{]}\Biggr{\\}}\;,$
(B.2)

with

$\displaystyle A$
$\displaystyle=\frac{\left(m_{i}-m_{k}\right)\left(m_{j}-m_{k}\right)+2m_{X}^{2}}{m_{i}\left(m_{i}+m_{j}\right)m_{X}^{2}}\;,$
(B.3) $\displaystyle B$
$\displaystyle=\frac{1}{m_{i}(m_{i}-m_{j})(m_{i}+m_{j})^{2}m_{X}^{2}}\;,$
(B.4) $\displaystyle C_{0}$
$\displaystyle=\left(-2m_{i}^{2}-3m_{i}m_{j}-2m_{j}^{2}+3(m_{i}+m_{j})m_{k}+m_{k}^{2}+2m_{X}^{2}\right)m_{X}^{2}\;,$
(B.5) $\displaystyle C_{1}$
$\displaystyle=(m_{i}^{2}-m_{k}^{2})(m_{j}^{2}-m_{k}^{2})+(2m_{i}+m_{j}-m_{k})(m_{i}+2m_{j}-m_{k})m_{X}^{2}-2m_{X}^{4}\;,$
(B.6) $\displaystyle C_{2}$
$\displaystyle=-m_{i}m_{j}\left((m_{i}-m_{k})(m_{j}-m_{k})+2m_{X}^{2}\right)\;.$
(B.7)

Taking the limit of $m_{j}\rightarrow m_{i}$, (Appendix B. Expression for
$\mathcal{I}(m_{i},m_{j},m_{k},m_{X})$ in ’t Hooft-Feynman gauge.) reduces to

$\displaystyle\mathcal{I}_{\,\rm{}^{\prime}t\,Hooft-
Feynman}\left(m_{i},m_{i},m_{k},m_{X}\right)$ $\displaystyle=\int_{0}^{1}{\rm
d}x(1-x)\Biggl{[}\frac{2\left(1-x\right)\bigl{(}\left(2-x\right)-2\frac{m_{k}}{m_{i}}\bigr{)}-\frac{m_{i}^{2}}{m_{X}^{2}}x\left(1-\frac{m_{k}}{m_{i}}\right)^{2}\left(\left(1-x\right)+\frac{m_{k}}{m_{i}}\right)}{m_{X}^{2}(1-x)-x(m_{i}^{2}(1-x)-m_{k}^{2})}\Biggr{]}\;.$
(B.8)

Multiplying the above result by $m_{i}^{2}C_{V}^{2}/8\pi^{2}$ and let
$m_{i}\rightarrow m_{\mu}$ reproduces the first vector coupling piece in Eq.
(4) (with $x\rightarrow 1-x$, $q_{X}=-1$ and $m_{F}=m_{k}$) of Ref.
Leveille:1977rc , who first computed the anomaly $a_{\mu}$ for a charged
$X$-loop with general gauge couplings of a muon and another neutral fermion
$F$ in the unitary gauge. The contribution from the axial coupling can be
obtained by setting $C_{V}\rightarrow C_{A}$ and flipping the sign of the mass
$m_{k}$ in the above loop integral Leveille:1977rc . One can also reproduce
the result of Ref. Leveille:1977rc  by starting directly from our result (V.1)
in the unitary gauge, as mentioned earlier.

An 1-dimension integral representation for
$\mathcal{I}(m_{i},m_{j},m_{k},m_{X})$ in the ’t Hooft-Feynman gauge had been
obtained previously in Ref. Ma:1980gm . Our result disagrees with this earlier
result. One can show analytically that both of our expressions of
$\mathcal{I}(m_{i},m_{j},m_{k},m_{X})$ in (V.1) and (Appendix B. Expression
for $\mathcal{I}(m_{i},m_{j},m_{k},m_{X})$ in ’t Hooft-Feynman gauge.) (or
equivalently (Appendix B. Expression for
$\mathcal{I}(m_{i},m_{j},m_{k},m_{X})$ in ’t Hooft-Feynman gauge.)) from the
unitary and ’t Hooft-Feynman gauges respectively agree with each other. This
can be done by integrating the integrand in (V.1) over $y$ first and then
subtract it with (Appendix B. Expression for
$\mathcal{I}(m_{i},m_{j},m_{k},m_{X})$ in ’t Hooft-Feynman gauge.). The
difference can then be shown to be zero by applying the following identity

$\int_{0}^{1}{\rm d}x\,g^{\prime}(x)\log f(x)=g(x)\log
f(x)\bigg{|}_{0}^{1}-\int_{0}^{1}{\rm d}x\frac{f^{\prime}(x)g(x)}{f(x)}$ (B.9)

to the log terms. The intermediate steps are tedious and not illuminative, we
will omit them here.

## Appendix C. Feynman Rules

Some relevant Feynman rules for this work are listed as follows.

$\displaystyle-i(O^{H})_{1n}\frac{m_{i}}{v}$ (C.1)

$\displaystyle i\gamma_{\mu}\left(C_{Vn}+C_{An}\gamma_{5}\right)$ (C.2)

$\displaystyle
i\left(y^{\mathcal{D}}_{S\,ji}+y^{\mathcal{D}}_{P\,ji}\gamma_{5}\right)$ (C.3)

$\displaystyle
i\left(y^{\mathcal{H}}_{S\,ji}+y^{\mathcal{H}}_{P\,ji}\gamma_{5}\right)$ (C.4)

$\displaystyle
i\gamma_{\mu}\left(g^{\mathcal{W}^{\prime}}_{V\,ji}+g^{\mathcal{W}^{\prime}}_{A\,ji}\,\gamma_{5}\right)$
(C.5)

## Appendix D. Amplitude of the on-shell $l_{i}\rightarrow l_{j}\gamma$
process

In general, the Lorentz invariant amplitude for $l_{i}\rightarrow
l_{j}\gamma(i\neq j)$ as depicted in Fig. (3) is given by Cheng:1984vwu

$i{\cal M}_{ji}=\left<l_{j}|J^{\mu}_{\rm
em}|l_{i}\right>\widetilde{\mathcal{A}}^{\rm ext}_{\mu}(q),$ (D.1)

where $\widetilde{\mathcal{A}}^{\rm ext}_{\mu}(q)$ is the electromagnetic
background field and

$\left<l_{j}|J^{\mu}_{\rm
em}|l_{i}\right>=\overline{u_{j}}(p^{\prime})\left[i\sigma^{\mu\nu}q_{\nu}(A+B\gamma_{5})+\gamma^{\mu}(C+D\gamma_{5})+q^{\mu}(E+F\gamma_{5})\right]u_{i}(p)$
(D.2)

with $q=p^{\prime}-p$ and $A,\,B,\,C,\,D,\,E$ and $F$ are the form factors.
Using the electromagnetic gauge invariance, one has

$\partial_{\mu}J^{\mu}_{\rm em}=0$ (D.3)

which yields the condition

$-m_{i}(C-D\gamma_{5})+m_{j}(C+D\gamma_{5})+q^{2}(E+F\gamma_{5})=0$ (D.4)

or $C=D=0$ for the case of $m_{i}\neq m_{j}$ and on-shell photon ($q^{2}=0$).
Furthermore, since $q^{\mu}\widetilde{\mathcal{A}}^{\rm ext}_{\mu}=0$, the
amplitude for the on-shell $l_{i}\rightarrow l_{j}\gamma$ process is then
given as

$i{\cal
M}_{ji}=\overline{u_{j}}(p^{\prime})\left[i\sigma^{\mu\nu}q_{\nu}(A+B\gamma_{5})\right]u_{i}(p)\widetilde{\mathcal{A}}^{\rm
ext}_{\mu}(q).$ (D.5)

To compare with the conventions established in Eq. (36), we can identify

$\displaystyle A$ $\displaystyle=$ $\displaystyle-
ie\frac{m_{i}}{2}A_{ji}^{M}\,,$ (D.6) $\displaystyle B$ $\displaystyle=$
$\displaystyle e\frac{m_{i}}{2}A_{ji}^{E}\,.$ (D.7)

It is important to note that the amplitude in Eq. (D.5) corresponds to a
dimension-five operator and as such, it can only be induced from loop
diagrams. Furthermore, as there can be no counterterm to absorb infinities, it
is imperative that the amplitude in Eq. (D.5) must be finite Cheng:1984vwu .
Since the self-energy diagrams contribute only to the $C$ and $D$ form
factors, they are not relevant to the on-shell amplitude of $l_{i}\rightarrow
l_{j}\gamma$. Of course these self-energy diagrams are necessarily included
along with the 1PI diagrams to maintain the gauge invariance of QED!

## References

  * (1) S. T. Petcov, Sov. J. Nucl. Phys. 25, 340 (1977) [erratum: Sov. J. Nucl. Phys. 25, 698 (1977); erratum: Yad. Fiz. 25, 1336 (1977)] JINR-E2-10176.
  * (2) T. P. Cheng and L. F. Li, Phys. Rev. Lett. 38, 381 (1977)
  * (3) F. Wilczek and A. Zee, Phys. Rev. Lett. 38, 531 (1977)
  * (4) A. M. Baldini et al. [MEG], Eur. Phys. J. C 76, no.8, 434 (2016) [arXiv:1605.05081 [hep-ex]].
  * (5) M. Meucci [MEG II], PoS NuFact2021, 120 (2022) [arXiv:2201.08200 [hep-ex]].
  * (6) Y. Kuno and Y. Okada, Rev. Mod. Phys. 73, 151-202 (2001) [arXiv:hep-ph/9909265 [hep-ph]].
  * (7) A. de Gouvea and P. Vogel, Prog. Part. Nucl. Phys. 71, 75-92 (2013) [arXiv:1303.4097 [hep-ph]].
  * (8) M. Lindner, M. Platscher and F. S. Queiroz, Phys. Rept. 731, 1-82 (2018) [arXiv:1610.06587 [hep-ph]].
  * (9) R. H. Bernstein and P. S. Cooper, Phys. Rept. 532, 27-64 (2013) [arXiv:1307.5787 [hep-ex]].
  * (10) P. Q. Hung, T. Le, V. Tran and T. C. Yuan, JHEP 12, 169 (2015) [arXiv:1508.07016 [hep-ph]].
  * (11) P. Q. Hung, T. Le, V. Tran and T. C. Yuan, Nucl. Phys. B 932, 471-504 (2018) [arXiv:1701.01761 [hep-ph]].
  * (12) R. Ramos, Van Que Tran and T. C. Yuan, Phys. Rev. D 103, no.7, 075021 (2021) [arXiv:2101.07115 [hep-ph]].
  * (13) R. Ramos, V. Tran and T. C. Yuan, JHEP 11, 112 (2021) [arXiv:2109.03185 [hep-ph]].
  * (14) V. Tran, T. T. Q. Nguyen and T. C. Yuan, [arXiv:2208.10971 [hep-ph]].
  * (15) G. W. Bennett et al. [Muon g-2], Phys. Rev. Lett. 89, 101804 (2002) [erratum: Phys. Rev. Lett. 89, 129903 (2002)] [arXiv:hep-ex/0208001 [hep-ex]].
  * (16) G. W. Bennett et al. [Muon g-2], Phys. Rev. Lett. 92, 161802 (2004) [arXiv:hep-ex/0401008 [hep-ex]].
  * (17) G. W. Bennett et al. [Muon g-2], Phys. Rev. D 73, 072003 (2006) [arXiv:hep-ex/0602035 [hep-ex]].
  * (18) B. Abi et al. [Muon g-2], Phys. Rev. Lett. 126, no.14, 141801 (2021) [arXiv:2104.03281 [hep-ex]].
  * (19) 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, 1-166 (2020) [arXiv:2006.04822 [hep-ph]].
  * (20) A. Keshavarzi, K. S. Khaw and T. Yoshioka, Nucl. Phys. B 975, 115675 (2022) [arXiv:2106.06723 [hep-ex]].
  * (21) G. W. Bennett et al. [Muon (g-2)], Phys. Rev. D 80, 052008 (2009) [arXiv:0811.1207 [hep-ex]].
  * (22) A. Adelmann, M. Backhaus, C. C. Barajas, N. Berger, T. Bowcock, C. Calzolaio, G. Cavoto, R. Chislett, A. Crivellin and M. Daum, et al. [arXiv:2102.08838 [hep-ex]].
  * (23) V. Andreev et al. [ACME], Nature 562, no.7727, 355-360 (2018)
  * (24) Advanced ACME: http://doylegroup.harvard.edu/edm/index.html
  * (25) W. C. Huang, Y. L. S. Tsai and T. C. Yuan, JHEP 1604, 019 (2016) [arXiv:1512.00229 [hep-ph]].
  * (26) A. Arhrib, W. C. Huang, R. Ramos, Y. L. S. Tsai and T. C. Yuan, Phys. Rev. D 98, no. 9, 095006 (2018) [arXiv:1806.05632 [hep-ph]].
  * (27) C. T. Huang, R. Ramos, V. Q. Tran, Y. L. S. Tsai and T. C. Yuan, JHEP 1909, 048 (2019) [arXiv:1905.02396 [hep-ph]].
  * (28) C. R. Chen, Y. X. Lin, C. S. Nugroho, R. Ramos, Y. L. S. Tsai and T. C. Yuan, Phys. Rev. D 101, no. 3, 035037 (2020) [arXiv:1910.13138 [hep-ph]].
  * (29) C. R. Chen, Y. X. Lin, V. Q. Tran and T. C. Yuan, Phys. Rev. D 99, no. 7, 075027 (2019) [arXiv:1810.04837 [hep-ph]].
  * (30) W. C. Huang, H. Ishida, C. T. Lu, Y. L. S. Tsai and T. C. Yuan, Eur. Phys. J. C 78, no. 8, 613 (2018) [arXiv:1708.02355 [hep-ph]].
  * (31) W. C. Huang, Y. L. S. Tsai and T. C. Yuan, Nucl. Phys. B 909, 122-134 (2016) [arXiv:1512.07268 [hep-ph]].
  * (32) B. Kors and P. Nath, Phys. Lett. B 586, 366-372 (2004) [arXiv:hep-ph/0402047 [hep-ph]].
  * (33) B. Kors and P. Nath, JHEP 07, 069 (2005) [arXiv:hep-ph/0503208 [hep-ph]].
  * (34) D. Feldman, Z. Liu and P. Nath, JHEP 11, 007 (2006) [arXiv:hep-ph/0606294 [hep-ph]].
  * (35) D. Feldman, Z. Liu and P. Nath, Phys. Rev. D 75, 115001 (2007) [arXiv:hep-ph/0702123 [hep-ph]].
  * (36) P. A. Zyla et al. [Particle Data Group], PTEP 2020, no.8, 083C01 (2020)
  * (37) A. M. Sirunyan et al. [CMS], Phys. Lett. B 805, 135425 (2020) [arXiv:2002.06398 [hep-ex]].
  * (38) A. M. Sirunyan et al. [CMS], Eur. Phys. J. C 79, no.5, 421 (2019) [arXiv:1809.10733 [hep-ex]].
  * (39) G. Aad et al. [ATLAS], Phys. Rev. D 101, no.1, 012002 (2020) [arXiv:1909.02845 [hep-ex]].
  * (40) [ATLAS], ATLAS-CONF-2021-053.
  * (41) G. Aad et al. [ATLAS], Phys. Lett. B 796, 68-87 (2019) [arXiv:1903.06248 [hep-ex]].
  * (42) M. Fabbrichesi, E. Gabrielli and G. Lanfranchi, doi:10.1007/978-3-030-62519-1 [arXiv:2005.01515 [hep-ph]].
  * (43) T. Aaltonen et al. [CDF], Science 376, no.6589, 170-176 (2022)
  * (44) N. Aghanim et al. [Planck], Astron. Astrophys. 641, A6 (2020) [erratum: Astron. Astrophys. 652, C4 (2021)] [arXiv:1807.06209 [astro-ph.CO]].
  * (45) G. Angloher et al. [CRESST], Eur. Phys. J. C 77, no.9, 637 (2017) [arXiv:1707.06749 [astro-ph.CO]].
  * (46) P. Agnes et al. [DarkSide], Phys. Rev. Lett. 121, no.8, 081307 (2018) [arXiv:1802.06994 [astro-ph.HE]].
  * (47) E. Aprile et al. [XENON], Phys. Rev. Lett. 123, no.25, 251801 (2019) [arXiv:1907.11485 [hep-ex]].
  * (48) G. Aad et al. [ATLAS], [arXiv:2202.07953 [hep-ex]].
  * (49) I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz and A. Zhou, JHEP 09, 178 (2020) doi:10.1007/JHEP09(2020)178 [arXiv:2007.14792 [hep-ph]].
  * (50) S. M. Barr, E. M. Freire and A. Zee, Phys. Rev. Lett. 65, 2626-2629 (1990)
  * (51) M. Battaglieri, A. Belloni, A. Chou, P. Cushman, B. Echenard, R. Essig, J. Estrada, J. L. Feng, B. Flaugher and P. J. Fox, et al. [arXiv:1707.04591 [hep-ph]].
  * (52) R. Agnese et al. [SuperCDMS], Phys. Rev. D 95, no.8, 082002 (2017) [arXiv:1610.00006 [physics.ins-det]].
  * (53) H. Ma et al. [CDEX], J. Phys. Conf. Ser. 1342, no.1, 012067 (2020) [arXiv:1712.06046 [hep-ex]].
  * (54) S. Weinberg, Phys. Rev. Lett. 63, 2333 (1989)
  * (55) E. Braaten, C. S. Li and T. C. Yuan, Phys. Rev. Lett. 64, 1709 (1990)
  * (56) S. Weinberg, Phys. Rev. D 42, 860-866 (1990)
  * (57) E. Braaten, C. S. Li and T. C. Yuan, Phys. Rev. D 42, 276-278 (1990)
  * (58) S. M. Barr and A. Zee, Phys. Rev. Lett. 65, 21-24 (1990) [erratum: Phys. Rev. Lett. 65, 2920 (1990)]
  * (59) J. P. Leveille, Nucl. Phys. B 137, 63-76 (1978)
  * (60) T. Ibrahim and P. Nath, Phys. Rev. D 61, 095008 (2000) [arXiv:hep-ph/9907555 [hep-ph]]
  * (61) T. Ibrahim and P. Nath, Phys. Rev. D 62, 015004 (2000) [arXiv:hep-ph/9908443 [hep-ph]]
  * (62) T. Ibrahim, U. Chattopadhyay and P. Nath, Phys. Rev. D 64, 016010 (2001) [arXiv:hep-ph/0102324 [hep-ph]]
  * (63) E. Ma and A. Pramudita, Phys. Rev. D 24, 1410 (1981)
  * (64) T. P. Cheng and L. F. Li, “Gauge theory of elementary particle physics,” Oxford University Press, 1984.