#  Electron EDM and LFV decays in the light of
Muon $(g-2)_{\mu}$ with U(2) flavor symmetry

Morimitsu Tanimoto 1 111email<EMAIL_ADDRESS>and Kei
Yamamoto 2 222e-mail<EMAIL_ADDRESS>
1Department of Physics, Niigata University, Niigata 950-2181, Japan
2Department of Global Environment Studies, Hiroshima Institute of Technology,
Hiroshima 731-5193, Japan

(

Abstract We study the interplay of New Physics (NP) among the lepton magnetic
moment, the lepton flavor violation (LFV) and the electron electric dipole
moment (EDM) in the light of recent data of the muon $(g-2)_{\mu}$. The NP is
discussed in the leptonic dipole operator with the $U(2)$ flavor symmetry of
the charged leptons, where possible CP violating phases of the three family
space are taken into account. It is remarked that the third-family contributes
significantly to the LFV decay, $\mu\to e\gamma$, and the electron EDM. The
experimental upper-bound of the $\mu\to e\gamma$ decay gives a severe
constraint for parameters of the flavor model. The predicted electron EDM is
rather large due to CP violating phases in the three family space. In
addition, we also study $(g-2)_{e,\tau}$ of the electron and tauon, and EDMs
of the muon and tauon as well as the $\tau\to e\gamma$ and $\tau\to\mu\gamma$
decays. The $\tau_{R}\to\mu_{L}\gamma$ decay is predicted close to the
experimental upper-bound. The $\tau_{R}\to e_{L}\gamma$ decay is not
suppressed. )

##  1 Introduction

The electric and magnetic dipole moments of the electron and the muon are low-
energy probes of New Physics (NP) beyond the Standard Model (SM). Recently,
the muon $(g-2)_{\mu}$ experiment at Fermilab reported a new measurement of
the muon magnetic anomaly using data collected in 2019 (Run-2) and 2020
(Run-3) [1]. Improvements of the analysis and run condition lead to more than
a factor of two reduction in the systematic uncertainties, which is compared
with the E989 experiment at Fermilab [2] and the previous BNL result [3]. This
result indicates the discrepancy of $5.1\,\sigma$ with the SM prediction by
"Muon $g-2$ Theory Initiative" [4] (see also [5, 6, 7, 8, 9, 10, 11, 12, 13,
14]).

While there is a debatable point on the precise value of the SM prediction.
The problem is the contribution of the hadronic vacuum polarization (HVP). The
current situation is complicated. The CMD-3 collaboration [15] released
results on the cross section that disagree at the $(2.5-5)\sigma$ level with
all previous measurements including those from the previous CMD-2
collaborations. The origin of this discrepancy is currently unknown. The BMW
collaboration published the first complete lattice-QCD result with subpercent
precision [16]. Their result is closer to the experimental average
($1.7\sigma$ tension for no NP in the muon $(g-2)_{\mu}$). Further studies are
underway to clarify these theoretical differences. The white paper (SM
prediction) is expected to be updated including the HVP problem in 2024.

If the muon $(g-2)_{\mu}$ anomaly comes from NP, it possibly appears in other
observables of the charged lepton sector. The interesting one is the electric
dipole moments (EDM) of the electron. In this year, the JILA group has
reported a new upper-bound on the electron EDM, which is $|d_{e}|<4.1\times
10^{-30}\rm e\,cm\,(90\%\,{confidence})$, by using the
$\rm{}^{18}Hf^{19}f^{+}$ ions trapped by the rotating electric field [17]. It
overcame the latest ACME collaboration result obtained in 2018 [18]. Precise
measurements of the electron EDM will be rapidly being updated. The future
sensitivity at ACME $\rm I\hskip-1.13791ptI\hskip-1.13791ptI$ is expected to
be $|d_{e}|<0.3\times 10^{-30}\,\text{e\,cm}$ [19, 20]. In contrast, the
present upper-bound of the muon [21] and tauon EDM’s [22, 23, 24] are not so
tight.

The lepton flavor violation (LFV) is also possible NP phenomena of the charged
leptons. The tightest constraint for LFV is the branching ratio of the $\mu\to
e\gamma$ decay. The experimental bound is $\mathcal{B}({\mu^{+}\to
e^{+}\gamma})<4.2\times 10^{-13}$ in the MEG experiment [25]. On the other
hands, the present upper-bounds of $\mathcal{B}\,({\tau\to\mu\gamma})$ and
$\mathcal{B}\,({\tau\to e\gamma})$ are $4.4~{}\times~{}10^{-8}$ and
$3.3~{}\times~{}10^{-8}$ [26, 27], respectively.

Comprehensive studies of the electric and magnetic dipole moments of leptons
are given in the SM Effective Field Theory (SMEFT) [28, 29, 30], i.e., under
the hypothesis of new degrees of freedom above the electroweak scale [31, 32,
33, 34]. The phenomenological discussion of NP has presented taking the
anomaly of the muon $(g-2)_{\mu}$ and the LFV bound in the SMEFT with imposing
$U(2)_{L_{L}}\otimes U(2)_{E_{R}}$ flavor symmetry [35], which is the
corresponding subgroup acting only on the first two (light) families [36, 37,
38]. Since this flavor symmetry reduces the number of independent parameters
of the flavor sector [39, 40], we can expect the interplay among possible NP
evidence in the observables of different flavors, which is independent of
details of NP. Thus, the SMEFT with flavor symmetry can probe NP without
discussing its dynamics in the flavor space.

We have already studied the interplay of NP among the muon $(g-2)_{\mu}$, the
electron EDM and the $\mu\to e\gamma$ decay in the modular symmetry of flavors
[41, 42], which is realized in the context of string effective field theory
[43] (see also [44]). The obtained result depends on the modular weights of
the charged leptons and the relevant discrete symmetry, considerably.
Therefore, more general discussions are required to confirm the role of flavor
symmetry for the NP search without depending on details of the flavor model.

There are many possible discrete symmetries for the leptons flavor mixing [45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. Some of them are subgroups of
continuous groups [48]. On the other hand, the NP analysis of SMEFT has been
performed attractively by imposing the flavor symmetry of continuous group,
especially $U(2)$ [39, 40]. We also adopt the $U(2)$ flavor symmetry for
discussing the dipole operator of leptons to investigate NP [35]. In this
paper, we present numerical discussions of the interplay of NP in the light of
recent data of the muon $(g-2)_{\mu}$ and the electron EDM by taking
$U(2)_{L_{L}}\otimes U(2)_{E_{R}}$ flavor symmetry of the charged leptons.
Since we discuss the electron EDM, we analyze NP in the three family space of
$U(2)_{L_{L}}\otimes U(2)_{E_{R}}$ flavor symmetry to take into account of
possible CP violating phases. It is remarked that the third-family contributes
significantly to the $\mu\to e\gamma$ decay and the electron EDM.

In addition to the muon $(g-2)_{\mu}$ and the electron EDM, we also study
$(g-2)_{e,\tau}$ of the electron and tauon, and EDM of the muon and tauon as
well as LFV processes $\tau\to e\gamma$ and $\tau\to\mu\gamma$.

The paper is organized as follows. In section 2, we discuss the experimental
constraints for Wilson coefficients of the leptonic dipole operator. In
section 3, we present our framework of the $U(2)$ flavor model. In section 4,
numerical discussions are presented. The summary and discussion are devoted to
section 5. In Appendix A, the experimental constraints of the relevant Wilson
coefficients of the leptonic dipole operator are presented. In Appendix B, the
$3\times 3$ Yukawa matrices, $YY^{\dagger}$ and $Y^{\dagger}Y$, are given
explicitly.

##  2 Constraints of Wilson coefficients of the dipole operator

###  2.1 Input experimental data

The combined result from the E989 experiment at Fermilab [1, 2] and the E821
experiment at BNL [3] on $a_{\mu}=(g-2)_{\mu}/2$, together with the SM
prediction $a_{\mu}^{\mathrm{SM}}$ in [4], implies

$\displaystyle\Delta
a_{\mu}=a_{\mu}^{\mathrm{Exp}}-a_{\mu}^{\mathrm{SM}}=\left(249\pm
49\right)\times 10^{-11}~{}.$ (1)

We suppose that the contribution of NP appears in $\Delta a_{\mu}$.

However, the precise value of the SM prediction of HVP is still unclear.
Further studies will clarify the theoretical differences. If $\Delta a_{\mu}$
is significantly lower, less than or of order $1\,\sigma$, one may obtain
somewhat loose bound in phenomenological studies of NP [56]. In our study, we
take a following reference value (the discrepancy of $5.1\,\sigma$ with the SM
prediction) as the input in our numerical analysis:

$\displaystyle\Delta a_{\mu}$ $\displaystyle=249\times 10^{-11}\,.$ (2)

We will comment on our result in the case that the discrepancy with the SM
prediction is reduced in section 5.

We also input the upper-bound of the electron EDM by the JILA group [17]:

$\displaystyle|{d_{e}}|<4.1\times 10^{-30}\,\rm e\,cm=6.3\times
10^{-14}\,TeV^{-1}\,.$ (3)

On the other hand, the upper-bound of muon EDM is [21]:

$\displaystyle|{d_{\mu}}|<1.8\times 10^{-19}\,\rm e\,cm=2.76\times
10^{-3}\,TeV^{-1}\,.$ (4)

The tauon EDM can be evaluated through the measurement of CP-violating
correlations in tauon-pair production such as $e^{+}e^{-}\to\tau^{+}\tau^{-}$
[22] (see also [23]). The present upper-bound on the tauon EDM $d_{\tau}$ is
given as: [24]:

$\displaystyle-1.85\times 10^{-17}{\rm e\,cm}<{\rm Re}\,d_{\tau}<0.61\times
10^{-17}{\rm e\,cm},\ \ -1.03\times 10^{-17}{\rm e\,cm}<{\rm
Im}\,d_{\tau}<0.23\times 10^{-17}{\rm e\,cm}.$ (5)

Taking the bound of ${\rm Re}\,d_{\tau}$, we have

$\displaystyle|d_{\tau}|<1.85\times 10^{-17}{\rm e\,cm}=2.84\,{\rm
TeV}^{-1}\,.$ (6)

The experimental upper-bound for the branching ratio of $\mu\to e\gamma$ is
[25]:

$\displaystyle\mathcal{B}({\mu^{+}\to e^{+}\gamma})<4.2\times 10^{-13}\,.$ (7)

We also take account of the upper-bound for LFV comes from the branching
ratios of $\tau\to\mu\gamma$ and $\tau\to e\gamma$ [26, 27] :

$\displaystyle\mathcal{B}({\tau\to\mu\gamma})<4.2\times
10^{-8}\,,\quad\qquad\mathcal{B}({\tau\to e\gamma})<3.3\times 10^{-8}\,.$ (8)

These input data are converted into the magnitudes of the Wilson coefficients
of the leptonic dipole operator in the next subsection.

###  2.2 Wilson coefficients of the leptonic dipole operator

We make the assumption that NP is heavy and can be given by the SMEFT
Lagrangian. Let us focus on the dipole operators of leptons and their Wilson
coefficients at the weak scale as:

$\displaystyle\mathcal{O}_{\underset{LR}{e\gamma}}=\frac{v}{\sqrt{2}}\overline{E}_{L}\sigma^{\mu\nu}E_{R}F_{\mu\nu}\,,\qquad\qquad\mathcal{C}_{\underset{LR}{e\gamma}}^{\prime}=\begin{pmatrix}\mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}&\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}&\mathcal{C}_{\underset{e\tau}{e\gamma}}^{\prime}\\\
\mathcal{C}_{\underset{\mu
e}{e\gamma}}^{\prime}&\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}&\mathcal{C}_{\underset{\mu\tau}{e\gamma}}^{\prime}\\\
\mathcal{C}_{\underset{\tau
e}{e\gamma}}^{\prime}&\mathcal{C}_{\underset{\tau\mu}{e\gamma}}^{\prime}&\mathcal{C}_{\underset{\tau\tau}{e\gamma}}^{\prime}\\\
\end{pmatrix}\,,$
$\displaystyle\mathcal{O}_{\underset{RL}{e\gamma}}=\frac{v}{\sqrt{2}}\overline{E}_{R}\sigma^{\mu\nu}E_{L}F_{\mu\nu}\,,\qquad\qquad\mathcal{C}_{\underset{RL}{e\gamma}}^{\prime}=\mathcal{C}_{\underset{LR}{e\gamma}}^{\prime\,\dagger}\,,$
(9)

where $E_{L}$ and $E_{R}$ denote three flavors of the left-handed and right-
handed leptons, respectively, and $v$ denotes the vacuum expectation value
(VEV) of the Higgs field $H$. The prime of the Wilson coefficient indicates
the flavor basis corresponding to the mass-eigenstate basis of the charged
leptons. The relevant effective Lagrangian is written as:

$\displaystyle\mathcal{L}_{\rm
dipole}=\frac{1}{\Lambda^{2}}\,\left(\mathcal{C}_{\underset{LR}{e\gamma}}^{\prime}\mathcal{O}_{\underset{LR}{e\gamma}}+\mathcal{C}_{\underset{RL}{e\gamma}}^{\prime}\mathcal{O}_{\underset{RL}{e\gamma}}\right)\,,$
(10)

where $\Lambda$ is a certain mass scale of NP in the effective theory. Here
the Wilson coefficient is understood to be evaluated at the weak scale (we
neglect the small effect of running below the weak scale) 333The one-loop
effect is small as seen in [57]..

Inputting the value in Eq. (2), the Wilson coefficient is obtained as[35] (see
Appendix A):

$\displaystyle\frac{1}{\Lambda^{2}}\text{Re}\
[\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}]=1.0\times
10^{-5}\,\mathrm{TeV}^{-2}\,.$ (11)

The LFV process $\mu\to e\gamma$ gives us more severe constraint for the
Wilson coefficient by the experimental data in Eq. (7). The upper-bound is
obtained [35] (see Appendix A):

$\displaystyle\frac{1}{\Lambda^{2}}|\mathcal{C}_{\underset{e\mu(\mu
e)}{e\gamma}}^{\prime}|<2.1\times 10^{-10}\,\mathrm{TeV}^{-2}\,.$ (12)

Taking into account Eqs. (11) and (12), one has the ratio [35] :

$\displaystyle\left|\frac{\mathcal{C}_{\underset{e\mu(\mu
e)}{e\gamma}}^{\prime}}{\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}}\right|<2.1\times
10^{-5}\,.$ (13)

Thus, the magnitude of $\mathcal{C}_{\underset{e\mu(\mu
e)}{e\gamma}}^{\prime}$ is much suppressed compared with
$\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}$. This gives the severe
constraint for parameters of the flavor model.

The $\tau\to\mu\gamma$ process also gives us another constraint for the Wilson
coefficient by the experimental data in Eq. (8). The upper-bounds are obtained
as seen in Appendix A:

$\displaystyle\frac{1}{\Lambda^{2}}|\mathcal{C}_{\underset{\mu\tau(\tau\mu)}{e\gamma}}^{\prime}|<2.65\times
10^{-6}\,\mathrm{TeV}^{-2}\,,\qquad\frac{1}{\Lambda^{2}}|\mathcal{C}_{\underset{e\tau(\tau
e)}{e\gamma}}^{\prime}|<2.35\times 10^{-6}\,\mathrm{TeV}^{-2}.$ (14)

The electron EDM, $d_{e}$ is defined in the operator:

$\displaystyle\mathcal{O}_{\mathrm{edm}}=-\frac{i}{2}\,d_{e}(\mu)\,\overline{e}\sigma^{\mu\nu}\gamma_{5}eF_{\mu\nu}\,,$
(15)

where $d_{e}=d_{e}(\mu=m_{e})$. Therefore, the electron EDM is extracted from
the effective Lagrangian

$\displaystyle\mathcal{L}_{\rm
EDM}=\frac{1}{\Lambda^{2}}\mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}\mathcal{O}_{\underset{LR}{e\gamma}}=\frac{1}{\Lambda^{2}}\mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}\frac{v}{\sqrt{2}}\overline{e}_{L}\sigma^{\mu\nu}e_{R}F_{\mu\nu}\,,$
(16)

which leads to

$\displaystyle d_{e}=-\sqrt{2}\,\frac{v}{\Lambda^{2}}\,{\rm
Im}\,[\mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}]\,,$ (17)

at tree level, where the small effect of running below the electroweak scale
is neglected.

Inputting the experimental upper-bound of the electron EDM in Eq. (3) [17], we
obtain the constraint of the Wilson coefficient:

$\frac{1}{\Lambda^{2}}{\mathrm{Im}}\,[\mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}]<1.8\times
10^{-13}\,\mathrm{TeV^{-2}}\,.$ (18)

On the other hand, the experimental upper-bound of the muon EDM in Eq. (4)
gives:

$\frac{1}{\Lambda^{2}}{\mathrm{Im}}\,[\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}]<7.9\times
10^{-3}\,\mathrm{TeV^{-2}}\,.$ (19)

Observables | Exp.$-$ SM / upper bound | Wilson Coef. in $1/\Lambda^{2}\ [\mathrm{TeV}^{-2}]$
---|---|---
$\Delta a_{\mu}$ |  $249\times 10^{-11}$[1, 2, 3] | $\text{Re}\ [\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}]=1.0\times 10^{-5}$
$\mathcal{B}({\mu^{+}\to e^{+}\gamma})$ |  $<4.2\times 10^{-13}$ [25] | $|\mathcal{C}_{\underset{e\mu(\mu e)}{e\gamma}}^{\prime}|<2.1\times 10^{-10}$
$\mathcal{B}({\tau\to\mu\gamma})$ |  $<4.2\times 10^{-8}$ [26, 27] | $|\mathcal{C}_{\underset{\mu\tau(\tau\mu)}{e\gamma}}^{\prime}|<2.65\times 10^{-6}$
$\mathcal{B}({\tau\to e\gamma})$ |  $<3.3\times 10^{-8}$ [26, 27] | $|\mathcal{C}_{\underset{e\tau(\tau e)}{e\gamma}}^{\prime}|<2.35\times 10^{-6}$
$|d_{e}|$ |  $<4.1\times 10^{-30}\,\rm e\,cm$ [17] | $\text{Im}\ [\mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}]<1.8\times 10^{-13}$
$|d_{\mu}|$ |  $<1.80\times 10^{-19}\,\rm e\,cm$[21] | $\text{Im}\ [\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}]<7.9\times 10^{-3}$
$|d_{\tau}|$ |  $<1.85\times 10^{-17}\,\rm e\,cm$ [22] | $\text{Im}\ [\mathcal{C}_{\underset{\tau\tau}{e\gamma}}^{\prime}]<8.2\times 10^{-1}$
Table 1: Relevant observables and the corresponding values of Wilson
coefficients, which are presented in $1/\Lambda^{2}\ (\mathrm{TeV}^{-2})$
unit.

The upper-bound of the tauon EDM in Eq. (6) also gives:

$\frac{1}{\Lambda^{2}}{\mathrm{Im}}\,[\mathcal{C}_{\underset{\tau\tau}{e\gamma}}^{\prime}]<8.2\times
10^{-1}\,\mathrm{TeV^{-2}}\,.$ (20)

By using these bounds of Wilson coefficients, we analyze the $U(2)$ flavor
model in the next section. These data are listed in Table 1.

##  3 U(2) flavor model

###  3.1 Flavor structure of Yukawa and dipole operator in charged leptons

In the lepton sector, we impose $U(2)_{L_{L}}\otimes U(2)_{E_{R}}$ flavor
symmetry, which is the corresponding subgroup acting only on the first two
light families. The minimal set of the breaking terms, which are so called
spurions, are

$\displaystyle
V_{\ell}\sim(\bf{2,\,1})\,,\qquad\Delta_{e}\sim(\bf{2,\,\bar{2}})\,.$ (21)

The two suprions can be parameterized by

$\displaystyle V_{\ell}=\begin{pmatrix}0\\\
{\epsilon_{\ell}}\,\end{pmatrix},\qquad\Delta_{e}=O_{e}^{T}\
\begin{pmatrix}\delta^{\prime}_{e}&0\\\ 0&\delta_{e}\\\ \end{pmatrix}\,,\qquad
O_{e}=\begin{pmatrix}c_{e}&s_{e}\\\ -s_{e}&c_{e}\\\ \end{pmatrix}\,,$ (22)

where $\delta^{\prime}_{e}$, $\delta_{e}$ and $\epsilon_{\ell}$ are taken to
be real without loss of generality, while $c_{e}$ and $s_{e}$ denote
$\cos\theta_{e}$ and $\sin\theta_{e}$, respectively. By using the spurions in
Eq. (22), the $3\times 3$ Yukawa coupling of leptons $Y$ is written in the
right-left (LR) convention at order ${\cal O}(V_{\ell}^{2}\Delta_{e})$ and
${\cal O}(V_{\ell}^{3})$ of spurion couplings:

$\displaystyle
Y=\begin{pmatrix}C^{y}_{\Delta}(\Delta_{e})_{\alpha\beta}+C^{y}_{VV\Delta}(V_{\ell})_{\alpha}(V_{\ell}^{\dagger})_{\gamma}(\Delta_{e})_{\gamma\beta}&C^{y}_{V}(V_{\ell})_{\alpha}+C^{y}_{VVV}(V_{\ell})(V_{\ell}^{\dagger})(V_{\ell})_{\alpha}\\\
C^{y}_{V\Delta}(V_{\ell}^{\dagger})_{\alpha}(\Delta_{e})_{\alpha\beta}&C^{y}+C^{y}_{VV}(V_{\ell}^{\dagger}V_{\ell})\end{pmatrix}_{LR}\,,$
(23)

where $\alpha,\beta$ are the first or second component in Eq. (22). Taking
suprion parameters in Eq. (22), the Yukawa matrix is written explicitly as:

$\displaystyle
Y=\begin{pmatrix}C^{y}_{\Delta}c_{e}\delta_{e}^{\prime}&-C^{y}_{\Delta}s_{e}\delta_{e}&0\\\
s_{e}\delta_{e}^{\prime}(C^{y}_{\Delta}+C^{n}_{VV\Delta}\epsilon_{\ell}^{2})&c_{e}\delta_{e}(C^{y}_{\Delta}+C^{y}_{VV\Delta}\epsilon_{\ell}^{2})&(C^{y}_{V}\epsilon_{\ell}+C^{y}_{VVV}\epsilon_{\ell}^{3})\\\
C^{y}_{V\Delta}(s_{e}\epsilon_{\ell}\delta_{e}^{\prime})&C^{y}_{V\Delta}(c_{e}\epsilon_{\ell}\delta_{e})&C^{y}+C^{y}_{VV}\epsilon_{\ell}^{2}\end{pmatrix}_{LR}\,,$
(24)

where $C^{y}$, $C^{y}_{\Delta}$, $C^{y}_{V}$, $C^{y}_{V\Delta}$,
$C^{y}_{VV\Delta}$ and $C^{y}_{VVV}$ are complex parameters of order $1$.

The flavor structure of the leptonic dipole operator in Eq. (9) is given by
$3\times 3$ matrix $X^{e\gamma}$. It is also given in terms of spurion
couplings like in Eq. (23) as follows:

$\displaystyle X^{e\gamma}$
$\displaystyle=\begin{pmatrix}C^{e\gamma}_{\Delta}(\Delta_{e})_{\alpha\beta}+C^{e\gamma}_{VVV}(V_{\ell})_{\alpha}(V_{\ell}^{\dagger})_{\gamma}(\Delta_{e})_{\gamma\beta}&C^{e\gamma}_{V}(V_{\ell})_{\alpha}+C^{e\gamma}_{VVV}(V_{\ell})(V_{\ell}^{\dagger})(V_{\ell})_{\alpha}\\\
C^{e\gamma}_{V\Delta}(V_{\ell}^{\dagger})_{\alpha}(\Delta_{e})_{\alpha\beta}&C^{e\gamma}+C^{e\gamma}_{VV}(V_{\ell}^{\dagger}V_{\ell})\end{pmatrix}_{LR}\,$
$\displaystyle=\begin{pmatrix}C^{e\gamma}_{\Delta}c_{e}\delta_{e}^{\prime}&-C^{e\gamma}_{\Delta}s_{e}\delta_{e}&0\\\
s_{e}\delta_{e}^{\prime}(C^{e\gamma}_{\Delta}+C^{e\gamma}_{VV\Delta}\epsilon_{\ell}^{2})&c_{e}\delta_{e}(C^{e\gamma}_{\Delta}+C^{e\gamma}_{VV\Delta}\epsilon_{\ell}^{2})&(C^{e\gamma}_{V}\epsilon_{\ell}+C^{e\gamma}_{VVV}\epsilon_{\ell}^{3})\\\
C^{e\gamma}_{V\Delta}(s_{e}\epsilon_{\ell}\delta_{e}^{\prime})&C^{e\gamma}_{V\Delta}(c_{e}\epsilon_{\ell}\delta_{e})&C^{e\gamma}+C^{e\gamma}_{VV}\epsilon_{\ell}^{2}\end{pmatrix}_{LR}\,,$
(25)

where $C^{e\gamma}$, $C^{e\gamma}_{\Delta}$, $C^{e\gamma}_{V}$,
$C^{e\gamma}_{V\Delta}$, $C^{e\gamma}_{VV\Delta}$ and $C^{e\gamma}_{VVV}$ are
also complex parameters of order $1$. Since the Wilson coefficients in Eq. (9)
are written in the mass-eigenstate basis of the charged leptons, the matrix
$X^{e\gamma}$ in Eq.(25) should be transformed into the diagonal basis of $Y$
in Eq.(24).

The eigenvalues of Yukawa matrix $Y$ in Eq.(24) ($y_{e}\ll y_{\mu}\ll
y_{\tau}$) are obtained by solving the eigenvalue equation. For the
determinant and trace of $YY^{\dagger}$, one finds in the leading order:

$\displaystyle{\rm
Det}\,[YY^{\dagger}]=y^{2}_{e}\,y^{2}_{\mu}\,y^{2}_{\tau}\simeq|C^{y}_{\Delta}|^{4}\,|C^{y}|^{2}\,c_{e}^{4}\,\delta_{\ell}^{2}\,{\delta^{\prime}_{\ell}}^{2}\,,$
$\displaystyle{\rm
Tr}\,[YY^{\dagger}]=y^{2}_{e}+y^{2}_{\mu}+y^{2}_{\tau}\simeq|C^{y}|^{2}\,,$
(26)

where $\delta_{e}$ and $\delta^{\prime}_{e}$ are much smaller than $1$ to
reproduce the charged lepton mass hierarchy. An expression for
$y_{\mu}^{2}y_{\tau}^{2}$ is given by the determinant of the 2-3 submatrix:

$\displaystyle
y_{\mu}^{2}\,y_{\tau}^{2}\simeq|C^{y}_{\Delta}|^{2}\,|C^{y}|^{2}\,c_{e}^{2}\,\delta_{\ell}^{2}\,.$
(27)

Then, one gets

$\displaystyle y_{\tau}^{2}\simeq|C^{y}|^{2}\,,\qquad
y_{\mu}^{2}\simeq|C^{y}_{\Delta}|^{2}c_{e}^{2}\,\delta_{\ell}^{2}\,,\qquad
y_{e}^{2}\simeq|C^{y}_{\Delta}|^{2}c_{e}^{2}\,{\delta^{\prime}_{\ell}}^{2}\,,\qquad$
(28)

which lead to

$\displaystyle\frac{y_{e}^{2}}{y_{\mu}^{2}}\simeq\frac{{\delta^{\prime}_{\ell}}^{2}}{\delta_{\ell}^{2}}\,,\qquad\frac{y_{\mu}^{2}}{y_{\tau}^{2}}\simeq\frac{\delta_{\ell}^{2}}{|C^{y}_{\Delta}|^{2}}\,.$
(29)

These ratio indicates $\delta_{\ell}\gg\delta^{\prime}_{\ell}$ to reproduce
the Yukawa hierarchy of the charged leptons. Especially,
${y_{e}^{2}}/{y_{\mu}^{2}}$ is independent of other coefficients of order one.
The parameters $\epsilon_{\ell}$ and $s_{e}$ are not constrained by the
charged lepton mass spectrum.

The neutrino mass matrix was discussed in $U(2)$ and $U(3)$ flavor model [38].
Indeed, the large lepton mixing angles come from the neutrino mass matrix.
However, we do not address details of neutrino mass matrix because its
contribution to our result is negligibly small due to small neutrino masses.

We can presume the magnitudes of $s_{e}$ and $\epsilon_{\ell}$ from the CKM
mixing angles of the quark sector. In our numerical analysis, we scan $s_{e}$
and $\epsilon_{\ell}$ as variant parameters without fixing in advance as:

$\displaystyle s_{e}=0.01-0.1\,,\qquad\epsilon_{\ell}=0.01-0.1\,.$ (30)

These ranges are set to be consistent with the viewpoint of quark sector in
Refs.[38, 40].

###  3.2 Mass-eigenstate basis of the charged leptons

The Yukawa matrix $Y$ in Eq. (24) is diagonalized by the unitary
transformation $U_{L}^{\dagger}YU_{R}$, where the unitary matrices are given
in terms of $2\times 2$ orthogonal matrices and phase matrices:

$\displaystyle U_{L}=P_{L23}U_{L23}P_{L13}U_{L13}P_{L12}U_{L12}\,,\qquad
U_{R}=P_{R23}U_{R23}P_{R13}U_{R13}P_{R12}U_{R12}\,,$ (31)

where the rotation matrices are

$\displaystyle U_{12}=\begin{pmatrix}c_{12}&s_{12}&0\\\ -s_{12}&c_{12}&0\\\
0&0&1\\\ \end{pmatrix}\,,\qquad U_{13}=\begin{pmatrix}c_{13}&0&s_{13}\\\
0&1&0\\\ -s_{13}&0&c_{13}\\\ \end{pmatrix}\,,\qquad
U_{23}=\begin{pmatrix}1&0&0\\\ 0&c_{23}&s_{23}\\\ 0&-s_{23}&c_{23}\\\
\end{pmatrix}\,,$ (32)

and phase matrices are

$\displaystyle P_{12}=\begin{pmatrix}1&0&0\\\ 0&e^{i\phi_{12}}&0\\\ 0&0&1\\\
\end{pmatrix}\,,\qquad P_{13}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&e^{i\phi_{13}}\\\ \end{pmatrix}\,,\qquad P_{23}=\begin{pmatrix}1&0&0\\\
0&1&0\\\ 0&0&e^{i\phi_{23}}\\\ \end{pmatrix}\,,$ (33)

where notations $L$ and $R$ are omitted. The $s_{ij}$ and $\phi_{ij}$ are
obtained approximately from $YY^{\dagger}$ or $Y^{\dagger}Y$ in Appendix B for
the left-handed or the right-handed sector.

In the following expression, we take $C^{y}_{\Delta}$ to be real positive and
$C^{y}$, $C^{y}_{V}$, $C^{y}_{VV\Delta}$ and $C^{y}_{VVV}$ to be complex
without loss of generality. The mixing angles $s_{Lij}$ and $s_{Rij}$ are
obtained approximately (see Appendix B). The mixing angles $s_{L12}$ and
$s_{R12}$ are :

$\displaystyle
s_{L12}\simeq-\frac{s_{e}}{c_{e}}\left[1-\left|\frac{C^{y}_{VV\Delta}}{C^{y}_{\Delta}}\right|\epsilon_{\ell}^{2}\cos(\arg
C^{y}_{VV\Delta})\right]\,,$ $\displaystyle s_{R12}\simeq
2\frac{s_{e}}{c_{e}}\frac{\delta^{\prime}_{e}}{\delta_{e}}\left[\cos(\arg
C^{y}_{VV\Delta})+\left|\frac{C^{y}_{VV\Delta}}{2C^{y}_{\Delta}}\right|\epsilon_{\ell}^{2}\right]\left|\frac{C^{y}_{VV\Delta}}{C^{y}_{\Delta}}\right|\epsilon_{\ell}^{2}\,,$
(34)

respectively. The phase $\phi_{L12}$ and $\phi_{R12}$ are:

$\displaystyle\phi_{L12}\simeq-\left|\frac{C^{y}_{VV\Delta}}{C^{y}_{\Delta}}\right|\epsilon_{\ell}^{2}\sin(\arg
C^{y}_{VV\Delta})\,,\qquad\phi_{R12}\simeq 0\,,$ (35)

which are much smaller than $1$. The mixing angles $s_{L23}$ and $s_{R23}$ are
rather simple in the leading order as:

$\displaystyle
s_{L23}\simeq\left|\frac{C^{y}_{V}}{C^{y}}\right|\epsilon_{\ell}\,,\qquad
s_{R23}\simeq\left|\frac{C^{y\,*}_{\Delta}C^{y}_{V}+C^{y\,*}_{V\Delta}C^{y}}{C^{y\,2}}\right|c_{e}\delta_{e}\epsilon_{\ell}\,,$
(36)

respectively. The phase $\phi_{L12}$ and $\phi_{R12}$ are:

$\displaystyle\phi_{L23}\simeq\arg\frac{C^{y}_{V}}{C^{y}}\,,\qquad\phi_{R23}\simeq\arg\left[\frac{C^{y\,*}_{\Delta}C^{y}_{V}+C^{y\,*}_{V\Delta}C^{y}}{|C^{y}|^{2}}\right]\,,$
(37)

which are of order $1$.

The mixing angles $s_{L13}$ and $s_{R13}$ are also simple in the leading order
as:

$\displaystyle
s_{L13}\simeq\left|\frac{C^{y}_{\Delta}C^{y\,*}_{V\Delta}}{C^{y\,2}}\right|c_{e}s_{e}(\delta_{e}^{2}-{\delta^{\prime}_{e}}^{2})\epsilon_{\ell}\,,\qquad
s_{R13}\simeq\left|\frac{C^{y\,*}_{\Delta}C^{y}_{V}+C^{y\,*}_{V\Delta}C^{y}}{C^{y\,2}}\right|s_{e}\delta^{\prime}_{e}\epsilon_{\ell}\,,$
(38)

respectively. The phase $\phi_{L13}$ and $\phi_{R13}$ are:

$\displaystyle\phi_{L13}\simeq\arg\frac{C^{y}_{\Delta}C^{y\,*}_{V\Delta}}{|C^{y}|^{2}}\,,\qquad\qquad\phi_{R13}\simeq
0\,.$ (39)

It is noticed that the $(1,3)$ and $(2,3)$ components of $(Y^{\dagger}Y)_{RR}$
have the same phase as seen in Eq.(65). Therefore, the phase matrix $P_{R23}$
removes phases of both $(1,3)$ and $(2,3)$ components, that is,
$\phi_{R13}\simeq 0$ is derived.

In mass-eigenstate basis, the matrix $X^{e\gamma}$ in Eq. (25) is transformed
by the unitary matrix in Eq. (31) as $U_{L}^{\dagger}X^{e\gamma}U_{R}$, whose
elements correspond to the Wilson coefficients of Eq. (9). Those are given as:

$\displaystyle\mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}=[U_{L}^{{\dagger}}X^{e\gamma}U_{R}]_{11}\simeq|C^{e\gamma}_{\Delta}|\delta_{e}^{\prime}\left[\frac{1}{c_{e}}+\frac{s_{e}^{2}}{c_{e}}\epsilon_{\ell}^{2}\left(\frac{C^{e\gamma}_{VV\Delta}}{C^{e\gamma}_{\Delta}}-\frac{C^{y}_{VV\Delta}}{C^{y}_{\Delta}}\right)\right]+\delta_{e}^{\prime}\frac{s_{e}^{2}}{c_{e}}\epsilon_{\ell}^{2}\,C_{3rd}\,,$
$\displaystyle\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}=[U_{L}^{{\dagger}}X^{e\gamma}U_{R}]_{22}$
$\displaystyle\quad\
\simeq|C^{e\gamma}_{\Delta}|\delta_{e}\left[\frac{1}{c_{e}}+c_{e}\epsilon_{\ell}^{2}\left(\left|\frac{C^{e\gamma}_{VV\Delta}}{C^{e\gamma}_{\Delta}}\right|\cos{(\arg{C^{e\gamma}_{VV\Delta}})}-\left|\frac{C^{y}_{VV\Delta}}{C^{y}_{\Delta}}\right|\frac{s_{e}^{2}}{c_{e}^{2}}\cos{(\arg{C^{y}_{VV\Delta}})}\right)\right]$
$\displaystyle\quad\
+i|C^{e\gamma}_{\Delta}|c_{e}\delta_{e}\epsilon_{\ell}^{2}\left(\left|\frac{C^{e\gamma}_{VV\Delta}}{C^{e\gamma}_{\Delta}}\right|\sin{(\arg{C^{e\gamma}_{VV\Delta}})}-\left|\frac{C^{y}_{VV\Delta}}{C^{y}_{\Delta}}\right|\sin{(\arg{C^{y}_{VV\Delta}})}\right)+c_{e}\delta_{e}\epsilon_{\ell}^{2}\,C_{3rd}\,,$
$\displaystyle\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}=[U_{L}^{{\dagger}}X^{e\gamma}U_{R}]_{12}\simeq|C^{e\gamma}_{\Delta}|s_{e}\delta_{e}\epsilon_{\ell}^{2}\left(\frac{C^{e\gamma}_{VV\Delta}}{C^{e\gamma}_{\Delta}}-\frac{C^{y}_{VV\Delta}}{C^{y}_{\Delta}}\right)+s_{e}\delta_{e}\epsilon_{\ell}^{2}\,C_{3rd}\,,$
$\displaystyle\mathcal{C}_{\underset{\mu
e}{e\gamma}}^{\prime}=[U_{L}^{{\dagger}}X^{e\gamma}U_{R}]_{21}\simeq|C^{e\gamma}_{\Delta}|s_{e}\delta^{\prime}_{e}\epsilon_{\ell}^{2}\left(\frac{C^{e\gamma}_{VV\Delta}}{C^{e\gamma}_{\Delta}}-\frac{C^{y}_{VV\Delta}}{C^{y}_{\Delta}}\right)+s_{e}\delta^{\prime}_{e}\epsilon_{\ell}^{2}\,C_{3rd}\,,$
(40)

where $C_{3rd}$ is given in terms of third-family parameters as:

$\displaystyle
C_{3rd}=\frac{C^{y*}_{\Delta}C^{y}_{V}+C^{y*}_{V\Delta}C^{y}}{|C^{y}|^{2}}\left(\frac{C^{e\gamma}}{C^{y}}C^{y}_{V}-C^{e\gamma}_{V}\right)+\frac{C^{y}_{V}}{C^{y}}\left(\frac{C^{e\gamma}_{V}}{C^{y}_{V}}C^{y}_{V\Delta}-C^{e\gamma}_{V\Delta}\right)\,.$
(41)

This factor is of order 1 and does not vanish unless
$C^{e\gamma}_{V}=C^{y}_{V}$, $C^{e\gamma}_{V\Delta}=C^{y}_{V\Delta}$ and
$C^{e\gamma}=C^{y}$. It is remarked that the off-diagonal component of
$\mathcal{C}_{\underset{e\mu(\mu e)}{e\gamma}}^{\prime}$ are not suppresses
due to $C_{3rd}$ even if the alignment
$C^{e\gamma}_{V\Delta}/C^{e\gamma}_{\Delta}=C^{y}_{V\Delta}/C^{y}_{\Delta}$ is
imposed [35]. The third family contribution on the $\mu\to e\gamma$ decay is
comparable to the one of the first- and second-family. It is also remarked
that the non-vanishing electron EDM is realized from the third family even if
there is no CP phase in first- and second-family as seen in the imaginary part
of $\mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}$ in Eq. (40).

The Wilson coefficients with respect to the third-family are given as:

$\displaystyle\mathcal{C}_{\underset{e\tau}{e\gamma}}^{\prime}=[U_{L}^{{\dagger}}X^{e\gamma}U_{R}]_{13}\simeq\left({C^{e\gamma}_{V}-\frac{C^{e\gamma}}{C^{y}}C^{y}_{V}}\right)\frac{s_{e}}{c_{e}}\epsilon_{\ell}\,,$
$\displaystyle\mathcal{C}_{\underset{\tau
e}{e\gamma}}^{\prime}=[U_{L}^{{\dagger}}X^{e\gamma}U_{R}]_{31}\simeq\left({C^{e\gamma}_{V\Delta}-\frac{C^{e\gamma}}{C^{y}}C^{y}_{V\Delta}}\right)s_{e}\delta^{\prime}\epsilon_{\ell}\,,$
$\displaystyle\mathcal{C}_{\underset{\mu\tau}{e\gamma}}^{\prime}=[U_{L}^{{\dagger}}X^{e\gamma}U_{R}]_{23}\simeq\left({C^{e\gamma}_{V}-\frac{C^{e\gamma}}{C^{y}}C^{y}_{V}}\right)\epsilon_{\ell}\,,$
$\displaystyle\mathcal{C}_{\underset{\tau\mu}{e\gamma}}^{\prime}=[U_{L}^{{\dagger}}X^{e\gamma}U_{R}]_{32}\simeq\left({C^{e\gamma}_{V\Delta}-\frac{C^{e\gamma}}{C^{y}}C^{y}_{V\Delta}}\right)c_{e}\delta\epsilon_{\ell}\,,$
$\displaystyle\mathcal{C}_{\underset{\tau\tau}{e\gamma}}^{\prime}=[U_{L}^{{\dagger}}X^{e\gamma}U_{R}]_{33}\simeq
C^{e\gamma}+\left(C^{e\gamma}_{VV}+C^{e\gamma}_{V}\frac{C^{y}_{V}}{C^{y}}\right)\epsilon_{\ell}^{2}\,.$
(42)

As seen in Eq. (40), the electron EDM (${\rm
Im}\,\mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}$) is suppressed by the
factor of $s_{e}^{2}\epsilon_{\ell}^{2}\delta_{e}^{\prime}/\delta_{e}$ while
$\mu\to e\gamma$ ($|\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}|$) is
suppressed by $s_{e}\epsilon_{\ell}^{2}$ compared with the muon $(g-2)_{\mu}$.
It is noted that the $\tau_{R}\to e_{L}\gamma$ transition is not suppressed
due to the $e_{L}$-$\mu_{L}$ mixing of $s_{e}=0.01-0.1$ as seen in Eq. (42)
although $(X^{e\gamma})_{13}$ vanishes in the flavor basis as seen in Eq. (25)

In the next section, we show the numerical results without approximations.
Indeed, it is found that the approximate Wilson coefficients in Eqs. (40) and
(42) are agree with exact ones with $10\,\%$ numerically.

##  4 Numerical analyses

Let us discuss Wilson coefficients numerically in mass-eigenvalue basis. The
parameters $\delta_{e}$ and $\delta_{e}^{\prime}$ are determined according to
the observed charged lepton mass ratios, $m_{e}/m_{\mu}$ and
$m_{\mu}/m_{\tau}$ as in Eq. (29). On the other hand, we presume the
magnitudes of $s_{e}$ and $\epsilon_{\ell}$ from the CKM mixing angles of the
quark sector. The magnitude of $s_{e}$ is expected to be $0.01-0.1$ while
$\epsilon_{\ell}$ is ${\cal O}(|V_{cb}|)$ [38, 40, 35]. It is most natural
choice for similar treatment of quark and lepton sector. We scan both in the
region of Eq. (30), $[0.01-0.1]$ with equal weights, that is the random scan
in the linear space since we have no other information to fix $s_{e}$ and
$\epsilon_{\ell}$ in the lepton sector.

Other parameters are $C^{y}$, $C^{y}_{\Delta}$, $C^{y}_{V}$, $C^{y}_{V\Delta}$
$C^{y}_{VV\Delta}$, $C^{y}_{VVV}$, $C^{e\gamma}$, $C^{e\gamma}_{\Delta}$,
$C^{e\gamma}_{V}$, $C^{e\gamma}_{V\Delta}$, $C^{e\gamma}_{VV\Delta}$ and
$C^{e\gamma}_{VVV}$ which are complex parameters of order $1$. Since the
relative phases contribute on the observables except for EDM, we take
$C^{y}_{\Delta}$ and $C^{e\gamma}_{\Delta}$ to be real in order to predict the
electron EDM properly removing unphysical CP phases. Those parameters are put
in the normal distribution with an average $1$ and standard deviation $0.25$,
which are given in the work of the flavor structure of quarks and leptons with
reinforcement learning [62]. (We have checked that our numerical results are
not so changed even if the standard deviation $0.5$ is taken.) The phases are
scanned to be random in $0\sim 2\pi$.

Then, we construct the matrices Eqs. (24) and (25) numerically. After
diagonalizing Yukawa matrices in Eq.(24), we can obtain
$\mathcal{C}_{\underset{\alpha\beta}{e\gamma}}^{\prime}$’s in the mass-
eigenvalues basis. In the following analysis, we have only taken the regions
where $\text{Re}\
[\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}]=1.0\times 10^{-5}$,
corresponding to the $(g-2)_{\mu}$ anomaly, can be reproduced. It is noted
that the $U(2)$ flavor symmetry is supposed in the electroweak scale.
Therefore, the contribution on the leptonic dipole operator of the
renormalization group equations (RGEs) is neglected (see also the study in the
subsection 4.3 of Ref.[41].).

###  4.1 Prediction of $(g-2)_{e}$ and $(g-2)_{\tau}$

The NP effect in $(g-2)_{\ell}$ is occurred by the diagonal components of the
Wilson coefficient of the leptonic dipole operator at mass-eigenstate basis.
We have the ratios of the diagonal coefficients from Eqs. (40) and (42) as:

$\displaystyle\frac{\mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}}{\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}}\simeq\frac{\delta_{e}^{\prime}}{\delta_{e}}\simeq\frac{m_{e}}{m_{\mu}}\simeq
4.8\times
10^{-3}\,,\qquad\quad\frac{\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}}{\mathcal{C}_{\underset{\tau\tau}{e\gamma}}^{\prime}}=\frac{C^{e\gamma}_{\Delta}}{C^{e\gamma}}\frac{1}{c_{e}}\delta_{e}\simeq\frac{m_{\mu}}{m_{\tau}}\simeq
5.9\times 10^{-2}\,.$ (43)

By inputting the observed charged lepton masses with $3\,\sigma$ error-bar, we
have obtained

$\displaystyle\delta_{e}=(5.0-6.0)\times
10^{-2}\,,\qquad\quad\delta^{\prime}_{e}=(2.3-3.0)\times 10^{-4}\,,$ (44)

in the numerical results.

Suppose that the leptonic dipole operator is responsible for the observed
anomaly of $(g-2)_{\mu}$. By inputting the reference value of Eq. (2), we can
estimate the magnitude of the electron $(g-2)_{e}$ by using the relation in
Eq. (43) as:

$\displaystyle\Delta a_{e}$
$\displaystyle=\frac{4m_{e}}{e}\frac{v}{\sqrt{2}}\,\frac{1}{\Lambda^{2}}\text{Re}\,[\mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}]\simeq
6.2\times 10^{-14}\,,$ (45)

where $v\approx 246$ GeV. It is easily seen that $\Delta a_{e}$ and $\Delta
a_{\mu}$ are proportional to the lepton masses squared. This result is
agreement with the naive scaling $\Delta a_{\ell}\propto m^{2}_{\ell}$[58].

In the electron anomalous magnetic moment, the experiments [59] give

$\displaystyle a_{e}^{\mathrm{Exp}}=1\,159\,652\,180.73(28)\times 10^{-12}\,,$
(46)

while the SM prediction crucially depends on the input value for the fine-
structure constant $\alpha$. Two latest determination [60, 61] based on Cesium
and Rubidium atomic recoils differ by more than $5\sigma$. Those observations
lead to the difference from the SM prediction

$\displaystyle\Delta a_{e}^{Cs}$
$\displaystyle=a_{e}^{\mathrm{Exp}}-a_{e}^{\mathrm{SM,CS}}=\left(-8.8\pm
3.6\right)\times 10^{-13}~{}\,,$ $\displaystyle\Delta a_{e}^{Rb}$
$\displaystyle=a_{e}^{\mathrm{Exp}}-a_{e}^{\mathrm{SM,Rb}}=\left(4.8\pm
3.0\right)\times 10^{-13}~{}\,.$ (47)

Our predicted value is small of one order compared with the observed one at
present. We wait for the precise determination of the fine structure constant
to test our flavor model.

The tauon $(g-2)_{\tau}$ is also predicted by using the relation in Eq.(43)
as:

$\displaystyle\Delta a_{\tau}$
$\displaystyle=\frac{4m_{\tau}}{e}\frac{v}{\sqrt{2}}\,\frac{1}{\Lambda^{2}}\text{Re}\,[\mathcal{C}_{\underset{\tau\tau}{e\gamma}}^{\prime}]\simeq
7.5\times 10^{-7}\,,$ (48)

which is also proportional to the lepton masses squared.

###  4.2 $(g-2)_{\mu}$ versus $\mu\to e\gamma$, $\tau\to\mu\gamma$ and
$\tau\to e\gamma$

The NP in the LFV process is severely constrained by the experimental bound of
the $\mu\to e\gamma$ decay in the MEG experiment [25]. It is also constrained
by the $\tau\to\mu\gamma$ decay in the Belle experiment [27]. We can discuss
the correlation between the anomaly of the muon $(g-2)_{\mu}$ and the LFV
process $\mu\to e\gamma$ by using the Wilson coefficients in Eqs. (40). The
ratios are given as:

$\displaystyle\left|\frac{\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}}{\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}}\right|\simeq
c_{e}s_{e}\epsilon_{\ell}^{2}\left|\frac{C^{e\gamma}_{VV\Delta}}{C^{e\gamma}_{\Delta}}-\frac{C^{y}_{VV\Delta}}{C^{y}_{\Delta}}+\frac{C_{3rd}}{C^{e\gamma}_{\Delta}}\right|\,,$
(49)

and

$\displaystyle\left|\frac{\mathcal{C}_{\underset{\mu
e}{e\gamma}}^{\prime}}{\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}}\right|\simeq
c_{e}s_{e}\frac{\delta^{\prime}_{e}}{\delta_{e}}\epsilon_{\ell}^{2}\left|\frac{C^{e\gamma}_{VV\Delta}}{C^{e\gamma}_{\Delta}}-\frac{C^{y}_{VV\Delta}}{C^{y}_{\Delta}}+\frac{C_{3rd}}{C^{e\gamma}_{\Delta}}\right|\,.$
(50)

We also obtain ratios for $\tau\to\mu\gamma$ and $\tau\to e\gamma$ processes
as:

$\displaystyle\left|\frac{\mathcal{C}_{\underset{\mu\tau}{e\gamma}}^{\prime}}{\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}}\right|\simeq
c_{e}\epsilon_{\ell}\frac{1}{\delta_{e}}\frac{1}{|C^{e\gamma}_{\Delta}|}\left|C^{e\gamma}_{V}-\frac{C^{e\gamma}}{C^{y}}C^{y}_{V}\right|\,,\qquad\left|\frac{\mathcal{C}_{\underset{\tau\mu}{e\gamma}}^{\prime}}{\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}}\right|\simeq
c_{e}^{2}\epsilon_{\ell}\frac{1}{|C^{e\gamma}_{\Delta}|}\left|C^{e\gamma}_{V}-\frac{C^{e\gamma}}{C^{y}}C^{y}_{V}\right|\,,$
(51)

and

$\displaystyle\left|\frac{\mathcal{C}_{\underset{e\tau}{e\gamma}}^{\prime}}{\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}}\right|\simeq
s_{e}\epsilon_{\ell}\frac{1}{\delta_{e}}\frac{1}{|C^{e\gamma}_{\Delta}|}\left|C^{e\gamma}_{V}-\frac{C^{e\gamma}}{C^{y}}C^{y}_{V}\right|\,,\qquad\left|\frac{\mathcal{C}_{\underset{\tau
e}{e\gamma}}^{\prime}}{\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}}\right|\simeq
c_{e}s_{e}\frac{\delta^{\prime}_{e}}{\delta_{e}}\epsilon_{\ell}\frac{1}{|C^{e\gamma}_{\Delta}|}\left|C^{e\gamma}_{V}-\frac{C^{e\gamma}}{C^{y}}C^{y}_{V}\right|\,.$
(52)

For the $\tau_{R}\to\mu_{L}\gamma$ process, the ratio in Eq. (51) is
proportional to $\epsilon_{\ell}/\delta_{e}$, which is expected to be of order
1. Since $\delta_{e}$ is fixed in Eq. (44), we have a strong constraint for
$\epsilon_{\ell}$ from the experimental upper-bound of the Wilson coefficient
in Eq. (14). On the other hand, for the $\mu_{R}\to e_{L}\gamma$ process, the
ratio in Eq. (49) is proportional to $s_{e}\epsilon_{\ell}^{2}$. Therefore,
$s_{e}$ is also constrained considerably as well as $\epsilon_{\ell}$.

In order to see the predicted range of $\mu\to e\gamma$, we plot the magnitude
of $\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}$ versus $\epsilon_{\ell}$,
which corresponds to $\mu_{R}\to e_{L}\gamma$ process, in Fig. 2. In this
figure, $s_{e}=0.03$ is put as a benchmark. It is found that $\epsilon_{\ell}$
should be smaller than around $0.05$ in order to obtain
$\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}$ below the experimental
upper-bound. The expected value of the branching ratio $\mathcal{B}({\mu\to
e\gamma})$ will be discussed in subsection 4.4.

We also show the magnitude of $\mathcal{C}_{\underset{\mu
e}{e\gamma}}^{\prime}$ versus $\epsilon_{\ell}$ with $s_{e}=0.03$, which
corresponds $\mu_{L}\to e_{R}\gamma$ process in Fig. 2. The magnitude of
$\mathcal{C}_{\underset{\mu e}{e\gamma}}^{\prime}$ is suppressed compared with
$\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}$. It is remarked that the NP
signal of the $\mu\to e\gamma$ process mainly comes from the operator
$\bar{e}_{L}\sigma_{\mu\nu}\mu_{R}$ in $U(2)$ flavor model. Indeed, the ratio
is given as:

$\displaystyle\left|\frac{\mathcal{C}_{\underset{\mu
e}{e\gamma}}^{\prime}}{\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}}\right|\simeq\frac{\delta_{e}^{\prime}}{\delta_{e}}\simeq\frac{m_{e}}{m_{\mu}}\,.$
(53)

The angular distribution with respect to the muon polarization can distinguish
between $\mu_{R}\to e_{L}\gamma$ and $\mu_{L}\to e_{R}\gamma$ [63].

Figure 1: $|\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}|$ in
$1/\Lambda^{2}\ [\mathrm{TeV}^{-2}]$ unit versus $\epsilon_{\ell}$, where
$s_{e}=0.03$ is put and the grey region is excluded by the experiment of
$\mu\to e\gamma$.

Figure 2:  $|\mathcal{C}_{\underset{\mu e}{e\gamma}}^{\prime}|$ in
$1/\Lambda^{2}\ [\mathrm{TeV}^{-2}]$ unit versus $\epsilon_{\ell}$, where
$s_{e}=0.03$ is put and the grey region is excluded by the experiment of
$\mu\to e\gamma$.

Figure 3: $|\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}|$ in
$1/\Lambda^{2}\ [\mathrm{TeV}^{-2}]$ unit versus $\epsilon_{\ell}$ for case
(a) (excluding third-family contribution), where $s_{e}=0.03$ is put and the
grey region is excluded by the experiment of $\mu\to e\gamma$.

Figure 4: $|\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}|$ in
$1/\Lambda^{2}\ [\mathrm{TeV}^{-2}]$ unit versus $\epsilon_{\ell}$ for case
(b) (excluding first- and second family contribution ), where $s_{e}=0.03$ is
put and the grey region is excluded by the experiment of $\mu\to e\gamma$.

In order to see the contribution of the third-family to
$\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}$, we show the magnitude of
$\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}$ for both cases of alignment
of coefficients (a) $C^{e\gamma}_{V}=C^{y}_{V}$,
$C^{e\gamma}_{V\Delta}=C^{y}_{V\Delta}$, $C^{e\gamma}=C^{y}$ and (b)
$C^{e\gamma}_{V\Delta}/C^{e\gamma}_{\Delta}=C^{y}_{V\Delta}/C^{y}_{\Delta}$.
The case (a) corresponds to the case excluding the third-family contribution
and (b) to the case excluding the first- and second-family contribution. The
numerical results are presented in Fig. 4 for case (a) and Fig. 4 for case
(b). Thus, the third-family contribution is comparable or rather large
compared with the one from the first- and second-family.

The $\tau\to\mu\gamma$ decay is an interesting process in $U(2)$ flavor
symmetry because it is not suppressed [35]. The upper-bound of the branching
ratio of $\tau\to\mu\gamma$ constrains the magnitude of $\epsilon_{\ell}$ as
discussed below Eq. (52). We show the magnitude of
$\mathcal{C}_{\underset{\mu\tau}{e\gamma}}^{\prime}$ versus $\epsilon_{\ell}$
in Fig. 6, where the predicted value is almost independent of $s_{e}$ as seen
in Eq. (51). It is found that small $\epsilon_{\ell}$ (much less than $0.1$)
is favored by the experimental upper-bound of the $\tau\to\mu\gamma$ decay. On
the other hand, $\mathcal{C}_{\underset{\tau\mu}{e\gamma}}^{\prime}$ is
suppressed in more than one order by $\delta_{e}$. We omit a figure for this
process.

Figure 5: $|\mathcal{C}_{\underset{\mu\tau}{e\gamma}}^{\prime}|$ in
$1/\Lambda^{2}\ [\mathrm{TeV}^{-2}]$ unit versus $\epsilon_{\ell}$, where the
grey region is excluded by the experiment of $\tau\to\mu\gamma$.

Figure 6: Allowed region in the $\epsilon_{\ell}$-$s_{e}$ plane by the upper-
bound of $\mu\to e\gamma$ and $\tau\to\mu\gamma$ branching ratios.

Figure 7:  $|\mathcal{C}_{\underset{\mu\tau}{e\gamma}}^{\prime}|$ in
$1/\Lambda^{2}\ [\mathrm{TeV}^{-2}]$ unit versus
$|\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}|$. The grey region is
excluded by the experiments.

Figure 8: $|\mathcal{C}_{\underset{e\tau}{e\gamma}}^{\prime}|$ in
$1/\Lambda^{2}\ [\mathrm{TeV}^{-2}]$ unit versus
$|\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}|$. The grey region is
excluded by the experiments.

As seen in Figs.2 and 6, the substantial region of predicted Wilson
coefficients exceeds the experimental upper-bounds. Imposing the upper-bounds
of the branching ratios of $\mu\to e\gamma$ and $\tau\to\mu\gamma$ decays, we
obtain the allowed region in $\epsilon_{\ell}$-$s_{e}$ plane in Fig. 6. It is
found that $\epsilon_{\ell}$ is almost smaller than $0.05$ while $s_{e}$ is
allowed in rather wide range of $[0.01-0.1]$.

In order to see the correlation between
$|\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}|$ and
$|\mathcal{C}_{\underset{\mu\tau}{e\gamma}}^{\prime}|$, we plot their
predictions in Fig. 8. It is found that the allowed region is restricted. In
the next subsection, we discuss the expectation value of $\mu\to e\gamma$ and
$\tau\to\mu\gamma$.

We also show the predicted region of
$|\mathcal{C}_{\underset{e\tau}{e\gamma}}^{\prime}|$ versus
$|\mathcal{C}_{\underset{\mu\tau}{e\gamma}}^{\prime}|$ in Fig. 8. The
magnitude of $\mathcal{C}_{\underset{e\tau}{e\gamma}}^{\prime}$ is almost
proportional to $|\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}|$. The
prediction of $\mathcal{C}_{\underset{e\tau}{e\gamma}}^{\prime}$ is still low
in one order compared with the experimental upper-bound. It is noted that
$\mathcal{C}_{\underset{\tau e}{e\gamma}}^{\prime}$ is suppressed in
$\delta^{\prime}_{e}$ as seen in Eq. (42). We omit a figure for this process.

###  4.3  Predictions of EDM

In the allowed region of $\epsilon_{\ell}$ and $s_{e}$ in Fig. 6, we discuss
the EDM of the charged leptons. The electron EDM comes from the imaginary part
of $\mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}$. The magnitude is estimated
approximately from Eq. (40). This ratio is bounded by the observed constraints
in Eqs. (11) and(18) as:

$\displaystyle\left|\frac{{\rm Im}\
\mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}}{{\rm Re}\
\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}}\right|\simeq
s_{e}^{2}\frac{\delta^{\prime}}{\delta}\epsilon_{\ell}^{2}\,{\rm
Im}\,\left(\frac{C^{e\gamma}_{VV\Delta}}{C^{e\gamma}_{\Delta}}-\frac{C^{y}_{VV\Delta}}{C^{y}_{\Delta}}+\frac{C_{3rd}}{|C^{e\gamma}_{\Delta}|}\right)<1.8\times
10^{-8}\,.$ (54)

By putting $\epsilon_{\ell}=0.03$ and $s_{e}=0.03$ with
$\delta^{\prime}_{e}/\delta_{e}\simeq m_{e}/m_{\mu}$ in Eq. (53), the factor
in front of middle side equation,
$s_{e}^{2}\epsilon_{\ell}^{2}\delta^{\prime}/\delta$ is $4\times 10^{-9}$.
Thus, the ratio is possibly predicted to be ${\cal O}(10^{-8})$, which
corresponds to $|d_{e}|\sim 10^{-30}\,{\rm e\,cm}$. We expect it to be
detectable in the near future.

We show the plot of ${\rm Im}\ \mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}$
versus $\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}$ in Fig. 10. Since we
have a ratio approximately:

$\displaystyle\frac{{\rm Im}\
\mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}}{|\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}|}\simeq\frac{s_{e}}{c_{e}}\frac{\delta^{\prime}}{\delta}\frac{{\rm
Im}\,\left(\frac{C^{e\gamma}_{VV\Delta}}{C^{e\gamma}_{\Delta}}-\frac{C^{y}_{VV\Delta}}{C^{y}_{\Delta}}+\frac{C_{3rd}}{|C^{e\gamma}_{\Delta}|}\right)}{\left|\frac{C^{e\gamma}_{VV\Delta}}{C^{e\gamma}_{\Delta}}-\frac{C^{y}_{VV\Delta}}{C^{y}_{\Delta}}+\frac{C_{3rd}}{|C^{e\gamma}_{\Delta}|}\right|}\,,$
(55)

we estimate it to be ${\cal O}(10^{-4})$, which is confirmed in Fig. 10. The
predicted electron EDM is below the experimental upper-bound after taking
account of the upper-bound of $\mathcal{B}({\mu\to e\gamma})$. In the next
subsection, we discuss the expectation value of the electron EDM.

We also present the predicted region of EDMs of muon and tauon in Fig. 10.
Those are still far from the present experimental upper-bounds,
$|{d_{\mu}}|<1.8\times 10^{-19}\,\rm e\,cm$ [21] and $|d_{\tau}|<1.85\times
10^{-17}{\rm e\,cm}$ [24] as seen in Eqs. (4) and (6).

Figure 9: ${\rm Im}\ \mathcal{C}_{\underset{ee}{e\gamma}}^{\prime}$ versus
$|\mathcal{C}_{\underset{e\mu}{e\gamma}}^{\prime}|$ (electron EDM versus
$\mu\to e\gamma$) in $1/\Lambda^{2}\ [\mathrm{TeV}^{-2}]$ unit. The grey
region is excluded by the experiments.

Figure 10: Predicted EDMs of muon and tauon in $[\rm e\,cm]$ unit. The grey
region is excluded by the experiments.

###  4.4 Expectation of $\mu\to e\gamma$, $\tau\to\mu\gamma$, $\tau\to
e\gamma$ and electron EDM

Since we have performed the random scan in the linear space of $s_{e}$ and
$\epsilon_{\ell}$ in $[0.01-0.1]$ as given in Eq. (30), we can show the
frequency distribution of the predicted LFV decays and the electron EDM.
Indeed, we obtain the frequency distribution of the predicted
$\mathcal{B}({\mu\to e\gamma})$ with imposing contraints of the upper-bound of
$\mathcal{B}(\tau\to\mu\gamma)$ and the electron EDM in Fig. 12. The frequency
is normalized by total sum being $1$. The grey region is already excluded by
the experimental upper-bound of $\mathcal{B}({\mu\to e\gamma})$. The maximal
peak is almost inside the grey region. Since the future sensitivity at the MEG
II experiment is $\mathcal{B}({\mu\to e\gamma})=6\times 10^{-14}$ [66], we
expect the observation of $\mu\to e\gamma$ decay in the near future.

In Fig. 12, we plot the frequency distribution of the predicted
$\mathcal{B}({\tau\to\mu\gamma})$ with imposing constraints of the upper-bound
of $\mathcal{B}(\mu\to e\gamma)$ and the electron EDM. The peak is also inside
the grey region. We also expect the observation of $\tau\to\mu\gamma$ decay in
the near future.

In Fig. 14, we plot the frequency distribution of the predicted
$\mathcal{B}({\tau\to e\gamma})$ with (without) imposing the upper-bounds of
$\mu\to e\gamma$, $\tau\to\mu\gamma$ and the electron EDM with the color of
orange (blue). Due to those constraints, the predicted $\mathcal{B}({\tau\to
e\gamma})$ is far from the present experimental upper-bound, smaller than
three order.

Finally, we plot the frequency distribution of the predicted electron EDM with
(without) imposing the upper-bounds of $\mu\to e\gamma$ and $\tau\to\mu\gamma$
with the color of orange (blue) in Fig. 14. Due to those constraints, the peak
appears arpond $|d_{e}|\simeq 3\times 10^{-31}$e cm. Since the future
sensitivity at ACME $\rm I\hskip-1.13791ptI\hskip-1.13791ptI$ is expected to
be $|d_{e}|<3\times 10^{-31}\,\text{e\,cm}$ [19, 20], the electron EDM will be
possibly observed in the near future.

Figure 11: The frequency distribution of the predicted $\mathcal{B}({\mu\to
e\gamma})$ with imposing the upper-bounds of $\tau\to\mu\gamma$ decay and the
electron EDM. The grey region is excluded by the experimental data of
$\mathcal{B}({\mu\to e\gamma})$.

Figure 12: The frequency distribution of the predicted
$\mathcal{B}({\tau\to\mu\gamma})$ with imposing the upper-bound of the $\mu\to
e\gamma$ decay and the electron EDM. The grey region is excluded by the
experimental data of $\mathcal{B}({\tau\to\mu\gamma})$.

Figure 13: The orange (blue) frequency distribution of the predicted
$\mathcal{B}({\tau\to e\gamma})$ with (without) imposing the upper-bounds of
the $\mu\to e\gamma$ decay, $\tau\to\mu\gamma$ decay and the electron EDM. The
grey region is excluded by the experimental data of $\mathcal{B}({\tau\to
e\gamma})$.

Figure 14: The orange (blue) frequency distribution of the predicted electron
EDM $[\rm e\,cm]$ with (without) imposing the $\mu\to e\gamma$ decay and
$\tau\to\mu\gamma$ decay. The grey region is excluded by the experimental data
of the electron EDM.

##  5 Summary and discussions

We have studied the interplay of NP among the lepton magnetic moments, LFV and
the electron EDM in the light of recent data of the muon $(g-2)_{\mu}$. The NP
is discussed in the leptonic dipole operator with the $U(2)_{L_{L}}\otimes
U(2)_{E_{R}}$ flavor symmetry of the charged leptons, where possible CP
violating phases of the three family space are taken into account. It is
remarked that the the third-family contributes significantly to the $\mu\to
e\gamma$ decay and the electron EDM. Indeed, the $\mu\to e\gamma$ process is
not suppressed even if the contribution of the first- and second-family
vanishes. The electron EDM is also predicted to be rather large due to the CP
violating phase in the third-family even if the CP phase of the first- and
second-family is zero.

In addition, we have also discussed the electron and tauon $(g-2)_{e,\tau}$
and the EDM of the muon and tauon as well as LFV processes $\tau\to\mu\gamma$
and $\tau\to e\gamma$. In particular, the $\tau_{R}\to\mu_{L}\gamma$ process
is predicted close to the experimental upper-bound due to the
$\mu_{L}$-$\tau_{L}$ mixing of ${\cal O}(\epsilon_{\ell})$. This decay will be
possibly observed in the near future in Belle II experiment [64, 65]. The
$\tau_{R}\to e_{L}\gamma$ decay is not suppressed due to the $e_{L}$-$\mu_{L}$
mixing of ${\cal O}(s_{e})$ together with the $\mu_{L}$-$\tau_{L}$ mixing. On
the other hand, the $\mu_{L}\to e_{R}\gamma$, $\tau_{L}\to\mu_{R}\gamma$ and
$\tau_{L}\to e_{R}\gamma$ decays are suppressed. The angular distribution with
respect to the muon polarization can distinguish between $\mu_{R}\to
e_{L}\gamma$ and $\mu_{L}\to e_{R}\gamma$.

The expected values of LFV decays and the electron EDM are presented in the
frequency distribution of them. The electron EDM will be possibly observed as
well as the $\mu\to e\gamma$ and $\tau\to\mu\gamma$ decays in the near future.

In our numerical analyses, we have put $\Delta a_{\mu}=249\times 10^{-11}$ for
the NP effect. If $\Delta a_{\mu}$ is significantly lower, less than or of
order $1\,\sigma$, we obtain somewhat different prediction for the interplay
among the muon $(g-2)_{\mu}$, the electron EDM and the $\mu\to e\gamma$ decay.
For example, if we take the input $\Delta a_{\mu}\simeq 50\times 10^{-11}$,
which is of order $1\,\sigma$. Our predicted LFV branching ratios are reduced
by $1/25$ while predictions of $(g-2)_{e,\tau}$ and EDM’s are reduced by
$1/5$. We need the precise value of the SM prediction of HVP to make our
numerical result of NP contributions to LFV and EDM reliable.

Acknowledgement
This work was supported by JSPS KAKENHI Grant Number JP21K13923 (KY).

## Appendix

##  Appendix A Experimental constraints on the dipole operators

From the experimental data of the muon $(g-2)_{\mu}$ and ${\mu\to e\gamma}$,
Ref. [35] gave the constraints on the dipole operators. We summarize briefly
them on the dipole operators in Eq. (9). Below the scale of electroweak
symmetry breaking, the leptonic dipole operators are given as:

$\displaystyle\mathcal{O}_{\underset{rs}{e\gamma}}=\frac{v}{\sqrt{2}}\overline{e}_{L_{r}}\sigma^{\mu\nu}e_{R_{s}}F_{\mu\nu}\,,$
(56)

where $\\{r,s\\}$ are flavor indices $e,\mu,\tau$ and $F_{\mu\nu}$ is the
electromagnetic field strength tensor. The effective Lagrangian is

$\displaystyle\mathcal{L}_{\rm
dipole}=\frac{1}{\Lambda^{2}}\,\left(\mathcal{C}_{\underset{rs}{e\gamma}}^{\prime}\mathcal{O}_{\underset{rs}{e\gamma}}+\mathcal{C}_{\underset{es}{e\gamma}}^{\prime}\mathcal{O}_{\underset{rs}{e\gamma}}\right)\,,$
(57)

where $\Lambda$ is a certain mass scale of NP in the effective theory. The
corresponding Wilson coefficient
$\mathcal{C}_{\underset{rs}{e\gamma}}^{\prime}$ is denoted in the mass-
eigenstate basis of leptons.

The tree-level expression for $\Delta a_{\mu}$ in terms of the Wilson
coefficient of the dipole operator is

$\displaystyle\Delta a_{\mu}$
$\displaystyle=\frac{4m_{\mu}}{e}\frac{v}{\sqrt{2}}\,\frac{1}{\Lambda^{2}}\text{Re}\,[\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}]\,,$
(58)

where $v\approx 246$ GeV. Let us input the value

$\displaystyle\Delta a_{\mu}=249\times 10^{-11}~{}\,,$ (59)

then, we obtain the Willson coefficient as:

$\displaystyle\frac{1}{\Lambda^{2}}\text{Re}\
[\mathcal{C}_{\underset{\mu\mu}{e\gamma}}^{\prime}]=1.0\times
10^{-5}\,\mathrm{TeV}^{-2}\,,$ (60)

where $e\simeq 0.3028$ is put in the natural unit.

The tree-level expression of a radiative LFV rate in terms of the Wilson
coefficients is

$\displaystyle\mathcal{B}(\ell_{r}\to\ell_{s}\gamma)=\frac{m_{\ell_{r}}^{3}v^{2}}{8\pi\Gamma_{\ell_{r}}}\frac{1}{\Lambda^{4}}\left(|\mathcal{C}_{\underset{rs}{e\gamma}}^{\prime}|^{2}+|\mathcal{C}_{\underset{sr}{e\gamma}}^{\prime}|^{2}\right)\,.$
(61)

Taking the experimental bound $\mathcal{B}\\!\left(\mu^{+}\to
e^{+}\gamma\right)<4.2\times 10^{-13}$ (90% C.L.) obtained by the MEG
experiment [25] in Eq. (7), we obtain the upper-bound of the Wilson
coefficient as:

$\displaystyle\frac{1}{\Lambda^{2}}|\mathcal{C}_{\underset{e\mu(\mu
e)}{e\gamma}}^{\prime}|<2.1\times 10^{-10}\,\mathrm{TeV}^{-2}\,.$ (62)

On the other hand, by taking the following experimental upper-bound of the
branching ratios, $\mathcal{B}\\!\left(\tau\to\mu+\gamma\right)<4.2\times
10^{-8}$ and $\mathcal{B}\\!\left(\tau\to e\gamma\right)<3.3\times 10^{-8}$
[26, 27], we obtain the upper-bound of the Wilson coefficient as:

$\displaystyle\frac{1}{\Lambda^{2}}|\mathcal{C}_{\underset{\mu\tau(\tau\mu)}{e\gamma}}^{\prime}|<2.65\times
10^{-6}\,\mathrm{TeV}^{-2}\,,\qquad\frac{1}{\Lambda^{2}}|\mathcal{C}_{\underset{e\tau(\tau
e)}{e\gamma}}^{\prime}|<2.35\times 10^{-6}\,\mathrm{TeV}^{-2}\,,$ (63)

respectively.

##  Appendix B Explicit forms of $3\times 3$ matrices for $YY^{\dagger}$ and
$Y^{\dagger}Y$

Here, we present the $3\times 3$ matrices for $YY^{\dagger}$ and
$Y^{\dagger}Y$ by using Eq. (24) explicitly. We can obtain the left
(right)-handed mixing angles and phases by using these Hermitian matrices. The
left-handed mixing is obtained by diagonalizing $YY^{\dagger}$, which is:

$\displaystyle YY^{\dagger}=$
$\displaystyle\begin{pmatrix}{C^{y}_{\Delta}}^{2}(c_{e}^{2}{\delta^{\prime}_{e}}^{2}+s_{e}^{2}\delta_{e}^{2})&C^{y}_{\Delta}(C^{y\,*}_{\Delta}+C^{y\,*}_{VV\Delta}\epsilon^{2}_{\ell})({\delta^{\prime}_{e}}^{2}-{\delta_{e}}^{2})c_{e}s_{e}&[3,1]^{*}\\\
[1.2]^{*}&|(C^{y}_{\Delta}+C^{y}_{VV\Delta}\epsilon^{2}_{\ell})|^{2}(s_{e}^{2}{\delta^{\prime}_{e}}^{2}+c_{e}^{2}\delta_{e}^{2})+|C^{y}_{V}|^{2}\epsilon^{2}_{\ell}&[3,2]^{*}\\\
C^{y\,*}_{\Delta}C^{y}_{V\Delta}({\delta^{\prime}_{e}}^{2}-{\delta_{e}}^{2})c_{e}s_{e}\epsilon_{\ell}&[C^{y\,*}_{V\Delta}(C^{y\,*}_{\Delta}+C^{y\,*}_{VV\Delta}\epsilon^{2}_{\ell})(s_{e}^{2}{\delta^{\prime}_{e}}^{2}+c_{e}^{2}\delta_{e}^{2})+C^{y\,*}_{V}C^{y}]\,\epsilon_{\ell}&|C^{y}|^{2}+(s_{e}^{2}{\delta^{\prime}_{e}}^{2}+c_{e}^{2}\delta_{e}^{2})|C^{y}_{V\Delta}|^{2}\end{pmatrix}.$
(64)

On the other hand, the right-handed mixing is obtained by diagonalizing
$Y^{\dagger}Y$, which is:

$\displaystyle Y^{\dagger}Y=$
$\displaystyle\begin{pmatrix}[{C^{y}_{\Delta}}^{2}c_{e}^{2}+(|C^{y}_{\Delta}+C^{y}_{VV\Delta}\epsilon^{2}_{\ell}|^{2}+|C^{y}_{V\Delta}|^{2}\epsilon^{2}_{\ell})s_{e}^{2}]{\delta^{\prime}_{e}}^{2}&(2{\rm
Re}{C^{y}_{\Delta}}+|C^{y}_{VV\Delta}|^{2}\epsilon^{2}_{\ell}+|C^{y}_{V\Delta}|^{2})\delta^{\prime}_{e}\delta_{e}\epsilon^{2}_{\ell}c_{e}s_{e}&[3,1]^{*}\\\
[1,2]^{*}&[{C^{y}_{\Delta}}^{2}s_{e}^{2}+(|C^{y}_{\Delta}+C^{y}_{VV\Delta}\epsilon^{2}_{\ell}|^{2}+|C^{y}_{V\Delta}|^{2}\epsilon^{2}_{\ell})c_{e}^{2}]{\delta_{e}}^{2}&[3,2]^{*}\\\
(C^{y}_{\Delta}C^{y\,*}_{V}+C^{y}_{V}C^{y\,*}+C^{y}_{VV\Delta}C^{y\,*}_{V}\epsilon^{2}_{\ell})\delta^{\prime}_{e}\epsilon_{\ell}s_{e}&(C^{y}_{\Delta}C^{y\,*}_{V}+C^{y}_{V\Delta}C^{y\,*}+C^{y}_{VV\Delta}C^{y\,*}_{V}\epsilon^{2}_{\ell})\delta_{e}\epsilon_{\ell}c_{e}&|C^{y}|^{2}+|C^{y}_{V}|^{2}\epsilon^{2}_{\ell}\end{pmatrix}.$
(65)

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