# Radiative inverse seesaw model with hidden $U(1)$ gauge symmetry enhancing
lepton $g-2$

Takaaki Nomura<EMAIL_ADDRESS>College of Physics, Sichuan University,
Chengdu 610065, China  Hiroshi Okada<EMAIL_ADDRESS>Department of Physics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka,
819-0395, Japan

###### Abstract

We propose a new inverse seesaw model based on hidden local $U(1)$ symmetry
framework where inverse seesaw mechanism is induced at one loop level. A
Majorana mass term of singlet fermion is forbidden by the $U(1)$ symmetry and
it is generated at one-loop level by introducing relevant particle contents to
get loop diagram, inducing inverse seesaw mechanism. The same particle
contents also contribute to lepton magnetic(electric) dipole moment and lepton
flavor violating decays without chiral suppression. We can then obtain sizable
muon anomalous magnetic dipole moment that accommodate with deviation from the
standard model prediction. The constraints from lepton flavor violating decays
and electron magnetic(electric) dipole moment are also discussed to explore
testability of the model.

##  I Introduction

A mechanism to generate tiny neutrino masses is one of the important issue
that definitely requires physics beyond the standard model (SM). A well known
scenario is the type-I seesaw mechanism Yanagida (1979); Minkowski (1977);
Mohapatra and Senjanovic (1980) that introduces heavy right-handed neutrinos
$(N_{R})$ in realizing smallness of active neutrino masses. Typically required
new physics scale is very high such as around Grand Unified Theory (GUT) one
($\mathcal{O}(10^{16})$ GeV) which is also motivated to work the leptogensis
mechanism well Fukugita and Yanagida (1986). In this framework it is thus
difficult to see effects of new physics at low energy scale through, for
example, lepton flavor violations (LFVs) and electric(magnetic) dipole moments
of charged leptons.

An inverse seesaw (IS) Mohapatra and Valle (1986); Wyler and Wolfenstein
(1983) mechanism is an alternative scenario to generate the neutrino mass that
tends to require lower mass scale compared to type-I seesaw enabling us to
obtain higher verifiability. In this case we introduce SM singlet left-handed
chiral fermions $(N_{L})$ in addition to right-handed neutrinos and active
neutrino mass is given by $m_{\nu}\sim\mu\left(\frac{M_{D}}{M_{N}}\right)^{2}$
where $M_{D}$ is Dirac mass between active and right-handed neutrinos, $M_{N}$
is Dirac mass between $N_{R}$ and $N_{L}$, and $\mu$ is Majorana mass
parameter for $\bar{N}^{c}_{L}N_{L}$ term violating lepton number. Then
neutrino mass can be tiny without much high energy scale if $\mu$ is very
suppressed. Basically one can assume $\mu$ is small as it is related to lepton
number violation, but we can also consider a mechanism to get its smallness
where a Majorana term with $\mu$ is generated at loop level. In fact we can
realize loop induced inverse seesaw introducing hidden $U(1)$ gauge symmetry
where SM particles are not charged under it Nomura et al. (2022). Two singlets
$N_{R}$ and $N_{L}$ are originated from one vector-like fermion with hidden
$U(1)$ charge where it should be vector-like to cancel gauge anomaly. Then
this symmetry plays a role of forbidding Majorana mass term at tree level 111
Although we do not discuss in this paper, a hidden $U(1)$ gauge symmetry is
also motivated for dark matter (DM) physics providing its stability and
mediator of interactions with SM particles. Zhang et al. (2010); Chiang et al.
(2014); Chen and Nomura (2015, 2016); Gross et al. (2015); Hambye (2009);
Boehm et al. (2014); Baek et al. (2014); Khoze and Ro (2014); Daido et al.
(2020); Karam and Tamvakis (2015); Davoudiasl and Lewis (2014); Ko et al.
(2021); Nomura et al. (2021); Cai et al. (2019); Nomura and Okada (2018);
Matsui et al. (2023); Li et al. (2023); Nagao et al. (2023) . After
spontaneous symmetry breaking, a Majorana mass term of $N_{L}$ can be induced
at loop level by introducing relevant particle contents to make a loop
diagram. Interestingly these new particle contents also induce LFV and
contribute to electric(magnetic) dipole moments.

A test of new physics by lepton flavor physics is achieving high precision and
we expect further improvements in near future. In particular muon anomalous
magnetic dipole moment (muon $g-2$) would indicate necessity of beyond the SM
where the current discrepancy with the SM prediction is 5.1$\sigma$ level Abi
et al. (2021); Aguillard et al. (2023); Bennett et al. (2006); Aoyama et al.
(2012, 2019); Czarnecki et al. (2003); Gnendiger et al. (2013); Davier et al.
(2017); Keshavarzi et al. (2018); Colangelo et al. (2019); Hoferichter et al.
(2019); Davier et al. (2020); Keshavarzi et al. (2020a); Kurz et al. (2014);
Melnikov and Vainshtein (2004); Masjuan and Sanchez-Puertas (2017); Colangelo
et al. (2017); Hoferichter et al. (2018); Gérardin et al. (2019); Bijnens et
al. (2019); Colangelo et al. (2020); Blum et al. (2020); Colangelo et al.
(2014); Hagiwara et al. (2011) combining new results from the E989
collaboration at Fermilab Abi et al. (2021); Aguillard et al. (2023) and the
previous result from BNL experiment Bennett et al. (2006) as

$\displaystyle\Delta a^{\rm new}_{\mu}=(24.9\pm 4.9)\times 10^{-10}.$ (1)

Although we would have uncertainty on the hadron vacuum polarization (HVP),
estimated by recent lattice calculations Borsanyi et al. (2021); Alexandrou et
al. (2023); Cè et al. (2022), it is shown in refs. Crivellin et al. (2020); de
Rafael (2020); Keshavarzi et al. (2020b) 222The effect in modifying HVP for
muon $g-2$ and electroweak precision test is also discussed previously in ref.
Passera et al. (2008). that the lattice results imply new tensions with the
HVP extracted from $e^{+}e^{-}$ data and the global fits to the electroweak
precision observables. We expect more precise test of muon $g-2$ can be a key
to understand new physics. It is thus interesting to consider new physics that
can have sizable muon $g-2$. To obtain the sizable muon $g-2$ with natural
manner by Yukawa couplings we would need one-loop contributions with chiral
flip by heavy fermion mass inside a loop diagram Lindner et al. (2018); Athron
et al. (2021); Guedes and Olgoso (2022); Crivellin and Hoferichter (2021) or
the Yukawa couplings would exceed perturbation limit. In this work we explore
the possibility to get such a mechanism to enhance muon $g-2$ in a neutrino
mass model, especially inverse seesaw mechanism. In addition to muon $g-2$,
this enhancement affects electron $g-2$, LFVs, electric dipole moment(EDM) of
electron/muon that are also tested with high precision Afanaciev et al.
(2024); Baldini et al. (2016); Aubert et al. (2010); Renga (2018); Parker et
al. (2018); Morel et al. (2020); Roussy et al. (2023); Bennett et al. (2009)
and we check these constraints.

In this paper, we propose a new model with hidden $U(1)$ gauge symmetry where
inverse seesaw mechanism is induced at one-loop level by introducing some new
field contents. It is found that our field contents can also enhance lepton
($g-2$) without chiral suppression compared to the previous model Nomura et
al. (2022) and we can explain muon $g-2$ anomaly. The same enhancements appear
in LFV decay amplitude and electron/muon EDMs and the model would be testable
in future experiments. We formulate active neutrino mass, LFV and
electric(magnetic) dipole moments in the model, and perform numerical analysis
searching for parameter region that is consistent with current experimental
data of neutrino masses/mixings and lepton flavor constraints. Then we
investigate flavor observables such as LFV decay ratios under the condition
where we can obtain sizable muon $g-2$ to show testability of the model.

This paper is organized as follows. In Section II, we present the model with
mass spectrum, relevant interactions and formulas for phenomenology of our
interest. In Section III, we perform numerical analysis to search for allowed
parameter region and predicted flavor observables. Finally we devote Section
IV to the summary and conclusion.

##  II A model

| $~{}~{}N~{}~{}$ | $~{}~{}E~{}~{}$ | $~{}~{}H_{1}~{}~{}$ | $~{}~{}H_{2}~{}~{}$ | $~{}~{}\varphi~{}~{}$ | $~{}~{}s^{+}_{1}~{}~{}$ | $~{}~{}s^{+}_{3}~{}~{}$
---|---|---|---|---|---|---|---
$~{}~{}SU(2)_{L}~{}~{}$ | $\bf{1}$ | $\bf{1}$ | $\bf{2}$ | $\bf{2}$ | $\bf{1}$ | $\bf{1}$ | $\bf{1}$
$~{}~{}U(1)_{Y}~{}~{}$ | $0$ | $-1$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $0$ | $1$ | $1$
$~{}~{}U(1)_{H}~{}~{}$ | $1$ | $-2$ | $1$ | $0$ | $-1$ | $1$ | $3$
Table 1: Charge assignment for new vector-like lepton and scalar fields
including SM-like Higgs $H_{2}$.

We consider a model that is based on hidden $U(1)_{H}$ gauge symmetry. In the
model we introduce vector-like $SU(2)_{L}$ singlet neutral and charged leptons
$N$ and $E$, two Higgs doublets $H_{1}$ and $H_{2}$, singlet scalar $\varphi$,
and $SU(2)_{L}$ singlet charged scalar fields $s_{1}^{+}$ and $s_{3}^{+}$.
Here $H_{2}$ is the SM-like Higgs. The $U(1)_{H}$ charges of
$\\{N,E,H_{1},\varphi,s_{1}^{+},s^{+}_{3}\\}$ are chosen as
$\\{1,-2,1,-1,1,3\\}$ while other fields including the SM fermions are neutral
under the hidden charge. The charge assignment in the model is summarized in
TABLE 1. In our scenario scalar fields $\\{H_{1},H_{2},\varphi\\}$ will
develop vacuum expectation values(VEVs) to break $U(1)_{H}$ and electroweak
symmetry.

The Yukawa interactions of our model are given by

$\displaystyle\mathcal{L}_{\rm Yukawa}=$ $\displaystyle
y_{\ell}\overline{L_{L}}e_{R}H_{2}+y_{\nu}\overline{L_{L}}\tilde{H}_{1}N_{R}+y_{1}\overline{N^{c}_{R}}E_{R}s_{1}^{+}+M_{N}\overline{N_{L}}N_{R}$
$\displaystyle+y_{2}\overline{E_{R}}N_{L}s^{-}_{3}+y_{3}\overline{N^{c}_{L}}E_{L}s^{+}_{1}+y_{4}\overline{E_{L}}N_{R}s^{-}_{3}+y_{5}\overline{N_{L}}e_{R}s^{+}_{1}+h.c.\
,$ (2)

where we omitted flavor indices and quark Yukawa sector, and define
$\tilde{H}_{1}=i\sigma_{2}H^{*}_{1}$ and $s^{-}_{1,3}=(s^{+}_{1,3})^{*}$.
Scalar potential is also written by

$\displaystyle V=$
$\displaystyle\mu_{1}^{2}H_{1}^{\dagger}H_{1}+\mu_{2}^{2}H_{2}^{\dagger}H_{2}+\mu_{\varphi}^{2}\varphi^{*}\varphi+\mu_{s_{1}}^{2}s^{+}_{1}s^{-}_{1}+\mu_{s_{3}}^{2}s^{+}_{3}s^{-}_{3}+\mu_{12}(H_{2}^{\dagger}H_{1}\varphi+h.c.)$
$\displaystyle+\mu_{s}(H_{1}^{T}i\sigma_{2}H_{2}s_{1}^{-}+h.c.)+\frac{1}{2}\lambda_{1}(H_{1}^{\dagger}H_{1})^{2}+\frac{1}{2}\lambda_{2}(H_{2}^{\dagger}H_{2})^{2}+\lambda_{3}(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{2})$
$\displaystyle+\lambda_{4}(H_{1}^{\dagger}H_{2})(H_{2}^{\dagger}H_{1})+\lambda_{s_{1}}(s^{+}_{1}s^{-}_{1})^{2}+\lambda_{s_{3}}(s^{+}_{3}s^{-}_{3})^{2}+\lambda_{H_{1}\varphi}(H_{1}^{\dagger}H_{1})(\varphi^{*}\varphi)$
$\displaystyle+\lambda_{H_{2}\varphi}(H_{2}^{\dagger}H_{2})(\varphi^{*}\varphi)+\lambda_{H_{1}s_{1}}(H_{1}^{\dagger}H_{1})(s_{1}^{+}s_{1}^{-})+\lambda_{H_{1}s_{3}}(H_{1}^{\dagger}H_{1})(s_{3}^{+}s_{3}^{-})$
$\displaystyle+\lambda_{H_{2}s_{1}}(H_{2}^{\dagger}H_{2})(s_{1}^{+}s_{1}^{-})+\lambda_{H_{2}s_{3}}(H_{2}^{\dagger}H_{2})(s_{3}^{+}s_{3}^{-})+\lambda_{s_{1}s_{3}}(s_{1}^{+}s_{1}^{-})(s_{3}^{+}s_{3}^{-})$
$\displaystyle+(\lambda_{ss}s_{3}^{+}s_{1}^{-}\varphi\varphi+c.c.),$ (3)

where coupling constants are taken to be real.

###  II.1 Scalar sector

In our scenario scalar fields $\\{H_{1},H_{2},\varphi\\}$ develop VEVs and
they are written by

$H_{i}=\begin{pmatrix}w^{+}_{i}\\\
\frac{1}{\sqrt{2}}(v_{i}+h_{i}+iz_{i})\end{pmatrix},\quad\varphi=\frac{1}{\sqrt{2}}(v_{\varphi}+\phi^{0}+iz^{\prime}),$
(4)

where $v_{i}$ and $v_{\varphi}$ are the VEVs of $H_{i}$ and $\varphi$.

For charged scalar sector, we rewrite $w^{+}_{1,2}$ by Higgs basis as

$\begin{pmatrix}w_{1}^{+}\\\
w_{2}^{+}\end{pmatrix}=\begin{pmatrix}\cos\beta&-\sin\beta\\\
\sin\beta&\cos\beta\end{pmatrix}\begin{pmatrix}w^{+}\\\ H^{+}\end{pmatrix},$
(5)

where $\tan\beta=v_{1}/v_{2}$ and $w^{+}$ is identified as NG boson absorbed
by $W^{+}$ boson. Then we obtain mass terms for charged scalar fields as
follows

$\displaystyle\mathcal{L_{C}}=$ $\displaystyle\
m^{2}_{H^{\pm}}H^{+}H^{-}+\mu^{2}_{s_{1}}s_{1}^{+}s_{1}^{-}+\mu^{2}_{s_{3}}s^{+}_{3}s^{-}_{3}$
$\displaystyle+m^{2}_{13}(s^{+}_{3}s^{-}_{1}+s^{-}_{3}s^{+}_{1})+m^{2}_{Hs_{1}}(H^{+}s_{1}^{-}+H^{-}s^{+}_{1}),$
(6)

where

$\displaystyle
m^{2}_{H^{\pm}}=\frac{\mu_{12}v_{\varphi}}{\sin\beta\cos\beta}-\frac{v^{2}\lambda_{4}}{2},\quad
m^{2}_{13}=\frac{v^{2}_{\varphi}\lambda_{ss}}{2},\quad
m^{2}_{Hs_{1}}=-\frac{v\mu_{s}}{\sqrt{2}}.$ (7)

Thus we have 3 by 3 mass matrix in the basis of
$\\{H^{+},s_{1}^{+},s_{3}^{+}\\}$. In our analysis, we require mixing between
$s_{1}^{\pm}$ and $s_{3}^{\pm}$ is small by choosing parameter in the
potential; it is motivated to obtain small Majorana mass terms of $N_{L(R)}$
in realizing inverse seesaw mechanism. In this case we can approximately
obtain mass eigenstate $\\{H_{1}^{+},H_{2}^{+},H_{3}^{+}\\}$ as

$\displaystyle\begin{pmatrix}H^{+}\\\ s^{+}_{1}\\\ s^{+}_{3}\end{pmatrix}$
$\displaystyle\simeq\begin{pmatrix}1&0&0\\\ 0&c_{A}&-s_{A}\\\
0&s_{A}&c_{A}\end{pmatrix}\begin{pmatrix}c_{B}&-s_{B}&0\\\ s_{B}&c_{B}&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}H_{1}^{+}\\\ H_{2}^{+}\\\
H_{3}^{+}\end{pmatrix},$ (8)

where $c_{A[B]}(s_{A[B]})=\cos A[B](\sin A[B])$ with $A$ and $B$ being mixing
angles ($A\ll B$ by our assumption). The mass eigenvalues are denoted as
$m_{H_{i}^{\pm}}$ for $H_{i}^{\pm}$.

We have three degrees of freedom for CP-even neutral scalar boson sector that
is the same situation as two Higgs doublet with one singlet scalar case. There
is also one CP-odd neutral scalar boson that is similar to the case of two
Higgs doublet model. In this paper we do not discuss phenomenology related to
neutral scalar bosons and we just assume mass spectrum and parameters in the
potential satisfying current experimental/theoretical constraints.

###  II.2 Gauge sector

The gauge sector associated with $U(1)_{Y}$ and $U(1)_{H}$ is written by

$\mathcal{L}_{\rm
gauge}=-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}B^{\prime}_{\mu\nu}B^{\prime\mu\nu}-\frac{1}{2}\epsilon
B_{\mu\nu}B^{\prime\mu\nu}$ (9)

where $B_{\mu\nu}$ and $B^{\prime}_{\mu\nu}$ are the gauge field strength
tensors regarding $U(1)_{Y}$ and $U(1)_{H}$ respectively, and the last term is
kinetic mixing one with kinetic mixing parameter $\epsilon$. The kinetic terms
can be diagonalized by the following transformation

$\left(\begin{array}[]{c}\tilde{B}^{\prime}_{\mu}\\\ \tilde{B}_{\mu}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\sqrt{1-\epsilon^{2}}&0\\\
\epsilon&1\\\ \end{array}\right)\left(\begin{array}[]{c}B^{\prime}_{\mu}\\\
B_{\mu}\\\ \end{array}\right).$ (10)

The gauge boson masses are generated from kinetic terms of VEV developing
scalar fields

$\displaystyle\mathcal{L}_{\text{kin}}=(D_{\mu}H_{1})^{\dagger}(D^{\mu}H_{1})+(D_{\mu}H_{2})^{\dagger}(D^{\mu}H_{2})+(D_{\mu}\varphi)^{\dagger}(D^{\mu}\varphi).$
(11)

We write the covariant derivatives as follows

$\displaystyle D_{\mu}H_{1}$ $\displaystyle=$
$\displaystyle\Big{(}\partial_{\mu}+ig_{2}\frac{\tau^{a}}{2}W_{\mu}^{a}+i\frac{g_{1}}{2}\tilde{B}_{\mu}+\Big{(}i\frac{g_{1}}{2}\rho-
ig_{X}\frac{\rho}{\epsilon}\Big{)}\tilde{B}^{\prime}_{\mu}\Big{)}H_{1},$
$\displaystyle D_{\mu}H_{2}$ $\displaystyle=$
$\displaystyle\Big{(}\partial_{\mu}+ig_{2}\frac{\tau^{a}}{2}W_{\mu}^{a}+i\frac{g_{1}}{2}\tilde{B}_{\mu}+i\frac{g_{1}}{2}\rho\tilde{B}^{\prime}_{\mu}\Big{)}H_{2},$
$\displaystyle D_{\mu}\varphi$ $\displaystyle=$
$\displaystyle\Big{(}\partial_{\mu}+ig_{X}\frac{\rho}{\epsilon}\tilde{B}^{\prime}_{\mu}\Big{)}\varphi,$
(12)

where we parameterize $\rho=-\frac{\epsilon}{\sqrt{1-\epsilon^{2}}}$, $g_{2}$,
$g_{1}$ and $g_{X}$ are respectively gauge couplings of $SU(2)_{L}$,
$U(1)_{Y}$ and $U(1)_{H}$, $W^{a}$ is the $SU(2)_{L}$ gauge field, and
$\tau^{a}$ is the Pauli matrix. After symmetry breaking, we obtain $W$ boson
mass as in the SM and neutral gauge boson mass terms such that

$\displaystyle\mathcal{L}_{ZZ^{\prime}}=\frac{1}{2}m_{Z_{\rm
SM}}^{2}\tilde{Z}_{\mu}\tilde{Z}^{\mu}+\Delta
M^{2}\tilde{Z}_{\mu}\tilde{B}^{\prime\mu}+\frac{1}{2}m_{B^{\prime}}^{2}\tilde{B}^{\prime}_{\mu}\tilde{B}^{\prime\mu},$
(13) $\displaystyle m_{Z_{\rm
SM}}^{2}=\frac{1}{4}(g_{1}^{2}+g_{2}^{2})v^{2},\quad
m_{B^{\prime}}^{2}=\left(\frac{1}{2}g_{1}\rho-
g_{X}\frac{\rho}{\epsilon}\right)^{2}v_{1}^{2}+\frac{1}{4}g_{1}^{2}\rho^{2}v_{2}^{2}+\frac{1}{2}g_{X}^{2}\frac{\rho^{2}}{\epsilon^{2}}v_{\varphi}^{2},$
$\displaystyle\Delta
M^{2}=\frac{1}{2}\sqrt{g_{1}^{2}+g_{2}^{2}}\left(\frac{1}{2}g_{1}\rho-
g_{X}\frac{\rho}{\epsilon}\right)v_{1}^{2}+\frac{1}{4}\sqrt{g_{1}^{2}+g_{2}^{2}}g_{1}\rho
v_{2}^{2},$ (14)

where
$\tilde{Z}_{\mu}\equiv-\cos\theta_{W}W^{3}_{\mu}+\sin\theta_{W}\tilde{B}_{\mu}$
with $\theta_{W}$ being the Weinberg angle. By diagonalizing the mass terms
the physical masses of the neutral gauge bosons are obtained as

$\displaystyle m_{Z}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}M_{Z,SM}^{2}+M_{Z}^{\prime
2}+\sqrt{(M_{Z,SM}^{2}-M_{Z^{\prime}}^{2})^{2}+4\Delta^{4}}\Big{]}$
$\displaystyle m_{Z^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}M_{Z,SM}^{2}+M_{Z}^{\prime
2}-\sqrt{(M_{Z,SM}^{2}-M_{Z^{\prime}}^{2})^{2}+4\Delta^{4}}\Big{]}.$ (15)

To avoid constraint from the $\rho$-parameter, we require $\epsilon\ll 1$ and
$m_{Z}^{2}\ll g_{X}^{2}v_{\varphi}^{2}$ so that SM $Z$ boson mass is
$m_{Z}\simeq m_{Z_{\rm SM}}$ and $m_{Z^{\prime}}\simeq m_{B^{\prime}}$. The
rotation matrix diagonalizing neutral gauge boson sector is given by

$\displaystyle\left(\begin{array}[]{c}\tilde{Z}_{\mu}\\\
\tilde{B}^{\prime}_{\mu}\\\ \end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\\\ \end{array}\right)\left(\begin{array}[]{c}Z_{\mu}\\\
Z^{\prime}_{\mu}\\\ \end{array}\right)$ (22) $\displaystyle\tan 2\theta$
$\displaystyle=\frac{2\Delta^{2}}{M_{Z,SM}^{2}-M_{Z^{\prime}}^{2}}$ (23)

where $Z_{\mu}$ and $Z^{\prime}_{\mu}$ are the physical gauge bosons that are
the SM $Z$ boson and extra $Z^{\prime}$ gauge boson. For neutrino mass and
flavor physics $Z^{\prime}$ boson does not contribute and we just assume it is
heavy enough and $Z$-$Z^{\prime}$ mixing is small to avoid phenomenological
constraints.

###  II.3 Neutrino mass

After scalar fields developing VEVs mass terms of neutral fermions at tree
level are written by

$\displaystyle\mathcal{L}_{M}=M_{N}\overline{N_{L}}N_{R}+m_{D}\overline{\nu_{L}}N_{R}+h.c.,$
(24)

where $m_{D}\equiv y_{\nu}v_{1}/\sqrt{2}$. Thus structure of tree level mass
matrix is

$M_{\nu}=\left(\begin{array}[]{ccc}0&m_{D}^{*}&0\\\
m_{D}^{T}&0&M_{N}^{\dagger}\\\ 0&M_{N}^{*}&0\end{array}\right)$ (25)

in the $(\nu_{L}~{}N_{R}^{c}~{}N_{L})$ basis. The 22 and 33 block components
of $M_{\nu}$ can be generated at loop level. The relevant interactions to
generate them are

$\displaystyle\mathcal{L}\supset$ $\displaystyle\
y_{1}\overline{N_{R}^{c}}E_{R}(c_{A}s_{B}H_{1}^{+}+c_{A}c_{B}H_{2}^{+}-s_{A}H_{3}^{+})+y_{2}\overline{E_{R}}N_{L}(s_{A}s_{B}H_{1}^{-}+s_{A}c_{B}H_{2}^{-}+c_{A}H_{3}^{-})$
$\displaystyle+y_{3}\overline{N^{c}_{L}}E_{L}(c_{A}s_{B}H_{1}^{+}+c_{A}c_{B}H_{2}^{+}-s_{A}H_{3}^{+})+y_{4}\overline{E_{L}}N_{R}(s_{A}s_{B}H_{1}^{-}+s_{A}c_{B}H_{2}^{-}+c_{A}H_{3}^{-})$
$\displaystyle+h.c.\,,$ (26)

where we used Eq. (8) to write mass eigenstates for charged scalars. The 22
and 33 elements of $M_{\nu}$ are generated by the diagram in Fig. 1. We then
calculate one-loop diagrams and find

$\displaystyle(\delta m_{22})_{ij}$
$\displaystyle=\frac{s_{A}c_{A}}{(4\pi)^{2}}\sum_{\alpha=1,2,3}(y_{4}^{\dagger})_{j\alpha}(y_{1}^{\dagger})_{\alpha
i}M_{E_{\alpha}}I(M^{2}_{E_{\alpha}},m^{2}_{H^{+}_{1,2,3}}),$ (27)
$\displaystyle(\delta m_{33})_{ij}$
$\displaystyle=\frac{s_{A}c_{A}}{(4\pi)^{2}}\sum_{\alpha=1,2,3}(y_{3})_{i\alpha}(y_{2})_{\alpha
j}M_{E_{\alpha}}I(M^{2}_{E_{\alpha}},m^{2}_{H^{+}_{1,2,3}}),$ (28)

where $I(M^{2}_{E_{\alpha}},m^{2}_{H^{+}_{a}})$ is function of extra charged
lepton/Higgs masses from loop integration. The explicit form of
$I(M^{2}_{E_{\alpha}},m^{2}_{H^{+}_{1,2,3}})$ is given by

$\displaystyle I(M^{2}_{E_{\alpha}},m^{2}_{H^{+}_{1,2,3}})$
$\displaystyle=\int[dX]_{4}\left[2\frac{s^{2}_{B}m^{2}_{H_{1}^{+}}+c^{2}_{B}m^{2}_{H_{2}^{+}}-m^{2}_{H_{3}^{+}}}{M^{2}_{E_{\alpha}}x+m^{2}_{H_{1}^{+}}y+m^{2}_{H_{2}^{+}}z+m^{2}_{H_{3}^{+}}w}\right.$
$\displaystyle\left.\qquad\qquad\qquad+\frac{s^{2}_{B}m^{2}_{H_{2}^{+}}m^{2}_{H_{3}^{+}}+c^{2}_{B}m^{2}_{H_{1}^{+}}m^{2}_{H_{3}^{+}}-m^{2}_{H_{1}^{+}}m^{2}_{H_{2}^{+}}}{(M^{2}_{E_{\alpha}}x+m^{2}_{H_{1}^{+}}y+m^{2}_{H_{2}^{+}}z+m^{2}_{H_{3}^{+}}w)^{2}}\right],$
(29)

where $\int[dX]_{4}=\int_{0}^{1}dxdydzdw\delta(1-x-y-z-w)$.

Figure 1: One-loop diagram that generates Majorana mass term of $N_{L(R)}$.

The loop induced elements $\delta m_{22}$ and $\delta m_{33}$ are naturally
taken to be small and we assume $\delta m_{22(33)}\ll m_{D}\ll M_{N}$. Then we
find the active neutrino mass matrix via IS mechanism such as

$m_{\nu}\simeq(m_{D}M^{-1}_{N})\delta m_{33}(m_{D}M^{-1}_{N})^{T}.$ (30)

Note that $\delta m_{22}$ element does not contribute to the active neutrino
mass matrix at the leading order. To simplify our analysis we factor out scale
of mass scale in our new physics sector such that

$\\{M_{N_{\alpha}},M_{E_{\alpha}},m_{H^{\pm}_{i}}\\}=\mu_{\rm
NP}\times\\{\tilde{M}_{N_{\alpha}},\tilde{M}_{E_{\alpha}},\tilde{m}_{H^{\pm}_{i}}\\},$
(31)

where $\mu_{\rm NP}$ has mass scale and parameters in RHS with ”$\sim$” are
dimensionless ones representing the ratio of mass parameters. We then write
active neutrino mass matrix by

$\displaystyle m_{\nu}$
$\displaystyle\simeq\frac{v^{2}s_{\beta}^{2}}{2\mu_{\rm
NP}}(y_{\nu}\tilde{M}^{-1}_{N})\delta\tilde{m}_{33}(y_{\nu}\tilde{M}^{-1}_{N})^{T}$
$\displaystyle\equiv\kappa(y_{\nu}\tilde{M}^{-1}_{N})\delta\tilde{m}_{33}(y_{\nu}\tilde{M}^{-1}_{N})^{T}$
$\displaystyle\equiv\kappa\tilde{m}_{\nu},$ (32)

where $\delta m_{33}=\mu_{\rm NP}\delta\tilde{m}_{33}$. The neutrino mass
matrix $m_{\nu}$ is diagonalized by a unitary matrix $V_{\nu}$ by
$D_{\nu}=|\kappa|\tilde{D}_{\nu}=V_{\nu}^{T}m_{\nu}V_{\nu}=|\kappa|V_{\nu}^{T}\tilde{m}_{\nu}V_{\nu}$.
We then determine $|\kappa|$ using squared mass differences observed in
neutrino oscillation experiments such that

$\displaystyle(\mathrm{NO}):\ |\kappa|^{2}=\frac{|\Delta m_{\rm
atm}^{2}|}{\tilde{D}_{\nu_{3}}^{2}-\tilde{D}_{\nu_{1}}^{2}},\quad(\mathrm{IO}):\
|\kappa|^{2}=\frac{|\Delta m_{\rm
atm}^{2}|}{\tilde{D}_{\nu_{2}}^{2}-\tilde{D}_{\nu_{3}}^{2}},$ (33)

where $\Delta m_{\rm atm}^{2}$ corresponds to the atmospheric neutrino mass-
squared splitting, and NO and IO respectively indicate the normal and the
inverted ordering of neutrino mass. Using the determined $|\kappa|$, the solar
mass squared splitting can be given by

$\displaystyle\Delta m_{\rm
sol}^{2}=|\kappa|^{2}({\tilde{D}_{\nu_{2}}^{2}-\tilde{D}_{\nu_{1}}^{2}}),$
(34)

which is compared with observed value in numerical analysis. The observed
neutrino mixing matrix is defined by $U=V_{\nu}$ Maki et al. (1962) since we
assume the charged-lepton mass matrix to be diagonal, where elements of the
matrix are parametrized by three mixing angles $\theta_{ij}(i,j=1,2,3;i<j)$,
one CP violating Dirac phase $\delta_{CP}$, and two Majorana phases
$\alpha_{21},\alpha_{31}$. The mixing matrix is thus explicitly written by

$U=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{CP}}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{CP}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{CP}}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{CP}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{CP}}&c_{23}c_{13}\end{pmatrix}\begin{pmatrix}1&0&0\\\
0&e^{i\frac{\alpha_{21}}{2}}&0\\\
0&0&e^{i\frac{\alpha_{31}}{2}}\end{pmatrix}.$ (35)

where $c_{ij}$ and $s_{ij}$ stand for $\cos\theta_{ij}$ and $\sin\theta_{ij}$
($i,j=1-3$), respectively. Then, each of the mixing angle is obtained in terms
of the component of $U$ such that

$\displaystyle\sin^{2}\theta_{13}=|U_{e3}|^{2},\quad\sin^{2}\theta_{23}=\frac{|U_{\mu
3}|^{2}}{1-|U_{e3}|^{2}},\quad\sin^{2}\theta_{12}=\frac{|U_{e2}|^{2}}{1-|U_{e3}|^{2}}.$
(36)

The Dirac phase $\delta_{CP}$ is also given by computing the Jarlskog
invariant as

$\displaystyle\sin\delta_{CP}$ $\displaystyle=\frac{\text{Im}[U_{e1}U_{\mu
2}U_{e2}^{*}U_{\mu
1}^{*}]}{s_{23}c_{23}s_{12}c_{12}s_{13}c^{2}_{13}},\quad\cos\delta_{CP}=-\frac{|U_{\tau
1}|^{2}-s^{2}_{12}s^{2}_{23}-c^{2}_{12}c^{2}_{23}s^{2}_{13}}{2c_{12}s_{12}c_{23}s_{23}s_{13}},$
(37)

where $\delta_{CP}$ be subtracted from $\pi$ when $\cos\delta_{CP}$ is
negative. Also Majorana phase $\alpha_{21},\ \alpha_{31}$ are found as follows

$\displaystyle\sin\left(\frac{\alpha_{21}}{2}\right)=\frac{\text{Im}[U^{*}_{e1}U_{e2}]}{c_{12}s_{12}c_{13}^{2}},\quad\cos\left(\frac{\alpha_{21}}{2}\right)=\frac{\text{Re}[U^{*}_{e1}U_{e2}]}{c_{12}s_{12}c_{13}^{2}},\
,$ (38)
$\displaystyle\sin\left(\frac{\alpha_{31}}{2}-\delta_{CP}\right)=\frac{\text{Im}[U^{*}_{e1}U_{e3}]}{c_{12}s_{13}c_{13}},\quad\cos\left(\frac{\alpha_{31}}{2}-\delta_{CP}\right)=\frac{\text{Re}[U^{*}_{e1}U_{e3}]}{c_{12}s_{13}c_{13}},$
(39)

where $\alpha_{21}/2,\ \alpha_{31}/2-\delta_{CP}$ are subtracted from $\pi$ if
$\cos\left(\frac{\alpha_{21}}{2}\right),\
\cos\left(\frac{\alpha_{31}}{2}-\delta_{CP}\right)$ are negative. Finally the
effective mass for the neutrinoless double beta decay is written by

$\displaystyle\langle
m_{ee}\rangle=|\kappa||\tilde{D}_{\nu_{1}}\cos^{2}\theta_{12}\cos^{2}\theta_{13}+\tilde{D}_{\nu_{2}}\sin^{2}\theta_{12}\cos^{2}\theta_{13}e^{i\alpha_{21}}+\tilde{D}_{\nu_{3}}\sin^{2}\theta_{13}e^{-2i\delta_{CP}}|,$
(40)

where its observed value could be measured by KamLAND-Zen in future Gando et
al. (2016).

In addition to the neutrino mass data, we need to consider a constraint from
non-unitarity that should be taken into account for IS model. Here the non-
unitarity matrix is denoted by $U^{\prime}$ and it represents the deviation
from the unitarity parametrized by the following form

$\displaystyle U^{\prime}\equiv\left(1-\frac{1}{2}FF^{\dagger}\right)U,$ (41)

where $F\equiv(M^{*}_{N})^{-1}m^{T}_{D}$ is a hermitian matrix given by
elements of neutral fermion matrix $M_{\nu}$. The global constraints are found
combining several experimental results such as the the effective Weinberg
angle, SM $W$ boson mass, several ratios of $Z$ boson fermionic decays,
invisible decay of $Z$, electroweak universality, measurements of Cabbibo-
Kobayashi-Maskawa matrix, and lepton flavor violations Fernandez-Martinez et
al. (2016). These results provide the following constraints for
$|FF^{\dagger}|$ Agostinho et al. (2018); Das et al. (2017)

$\displaystyle|FF^{\dagger}|\leq\left[\begin{array}[]{ccc}2.5\times
10^{-3}&2.4\times 10^{-5}&2.7\times 10^{-3}\\\ 2.4\times 10^{-5}&4.0\times
10^{-4}&1.2\times 10^{-3}\\\ 2.7\times 10^{-3}&1.2\times 10^{-3}&5.6\times
10^{-3}\\\ \end{array}\right].$ (45)

In numerical analyses below we impose the above constraints.

###  II.4 Lepton flavor violation and muon $g-2$

Figure 2: One-loop diagram contributing lepton $g-2$, EDM and LFV decay
amplitude.

In the model muon $g-2$ and LFV decay amplitudes can be induced at one loop
level without chiral suppression. The relevant Lagrangian terms are given by

$\displaystyle\mathcal{L}_{LFV}=-y_{\nu}s_{\beta}\overline{\ell_{L}}N_{R}(c_{B}H_{1}^{-}-s_{B}H_{2}^{-})+y_{5}\overline{N_{L}}e_{R}(c_{A}s_{B}H_{1}^{+}+c_{A}c_{B}H_{2}^{+}-s_{A}H^{+}_{3})+h.c.\,.$
(46)

The dominant one loop diagram is given in Fig. 2. Calculating one loop
diagram, we obtain new physics contribution to electron and muon $g-2$ as

$\Delta a_{e,\mu}=-m_{\mu}(a_{R}+a_{L})_{11,22},$ (47)

where the amplitudes are given by

$\displaystyle(a_{R(L)})_{ij}$
$\displaystyle=(a_{R_{1(2)}})_{ij}+(a_{L_{1(2)}})_{ij},$
$\displaystyle(a_{R_{1(2)}})_{ij}$
$\displaystyle\simeq\frac{y^{jk}_{\nu}y^{ki}_{5}}{16\pi^{2}}s_{\beta}c_{A}c_{B}s_{B}M_{N_{k}}\int_{0}^{1}dx\frac{x}{xM^{2}_{N_{k}}+(1-x)m^{2}_{H_{1(2)}^{+}}},$
$\displaystyle(a_{L_{1(2)}})_{ij}$
$\displaystyle\simeq\frac{(y_{\nu}^{\dagger})^{ki}(y^{\dagger}_{5})^{jk}}{16\pi^{2}}s_{\beta}c_{A}c_{B}s_{B}M_{N_{k}}\int_{0}^{1}dx\frac{x}{xM^{2}_{N_{k}}+(1-x)m^{2}_{H_{1(2)}^{+}}},$
(48)

where we ignored charged lepton masses and $N_{k}$ is treated as Dirac fermion
approximately.

For electron $g-2$, the SM prediction depends on the value of the fine-
structure constant $\alpha_{em}$ where the latest measurements based on Cesium
and Rubidium show different values Parker et al. (2018); Morel et al. (2020).
We then have difference between the SM prediction and observed value of
electron $g-2$ such that

$\displaystyle\Delta a_{e}({\rm Cesium})=(-8.8\pm 3.6)\times 10^{-13},$ (49)
$\displaystyle\Delta a_{e}({\rm Rubidium})=(4.8\pm 3.0)\times 10^{-13}.$ (50)

In numerical analysis below, our theoretical value of $\Delta a_{e}$ is
compared with these observations.

In addition to electron and muon $g-2$, we have contribution to EDM from the
same loop diagram. Electron and muon EDMs are given by

$d_{e,\mu}=\frac{e}{2}{\rm Im}[(a_{R}-a_{L})_{11,22}],$ (51)

where $e$ is the electromagnetic coupling. We adopt the latest experimental
constraints on $|d_{e,\mu}|$ Roussy et al. (2023); Bennett et al. (2009);

$|d_{e}|<6.3\times 10^{-17}{\rm GeV}^{-1},\quad|d_{\mu}|<2.76\times
10^{-6}{\rm GeV}^{-1}.$ (52)

The same one-loop diagram also induces LFV decay processes of charged leptons
$\ell_{i}\to\ell_{j}\gamma$. The decay BRs of the processes are given by

$BR(\ell_{i}\to\ell_{j}\gamma)=\frac{48\pi^{3}C_{ij}\alpha_{\rm
em}}{G_{F}^{2}m_{i}^{2}}m_{i}^{3}(|(a_{R})_{ij}|^{2}+|(a_{L})_{ij}|^{2}),$
(53)

where $(C_{21},C_{31},C_{32})=(1,0.1784,0.1736)$. The current experimental
upper limits for these BRs are given by Afanaciev et al. (2024); Baldini et
al. (2016); Aubert et al. (2010); Renga (2018)

$\displaystyle B(\mu\rightarrow e\gamma)$ $\displaystyle\leq 3.1\times
10^{-13},\quad B(\tau\rightarrow\mu\gamma)\leq 4.4\times 10^{-8},\quad
B(\tau\rightarrow e\gamma)\leq 3.3\times 10^{-8}~{}.$ (54)

We impose these constraints in numerical analysis.

###  II.5 One-loop modification in leptonic $Z$ boson decays

Figure 3: Diagrams for modifying interactions between $Z$ and SM charged
leptons where dashed line indicate charged scalar bosons.

We consider the one-loop contributions to $Z$ boson decay into two charged
leptons that are induced by the Yukawa interactions in Eq. (46). The relevant
diagrams are shown in Fig. 3 where the dashed line includes three charged
scalar bosons and we consider all the possible combinations of Yukawa
couplings in Eq. (46) for the diagrams. By these diagrams, interactions
between $Z$ boson and charged leptons are modified as

$\mathcal{L}^{\rm
SM+NP}_{Z\ell\bar{\ell}}=\frac{g_{2}}{c_{W}}\overline{\ell_{i}}\gamma^{\mu}\left[\left(-\frac{1}{2}+s_{W}^{2}\right)\left(\delta_{ij}+(\delta^{\rm
NP}_{L})_{ij}\right)P_{L}+s^{2}_{W}\left(\delta_{ij}+(\delta^{\rm
NP}_{R})_{ij}\right)P_{R}\right]\ell_{j}$ (55)

where $g_{2}$ is the $SU(2)_{L}$ gauge coupling constant. The new physics
contribution is written by

$(\delta^{\rm NP}_{R(L)})_{ij}=(\delta A_{R(L)})_{ij}+(\delta
B_{R(L)})_{ij}+(\delta C_{R(L)})_{ij}+(\delta D_{R(L)}),$ (56)

where $\delta A_{R(L)}$, $\delta B_{R(L)}$, $\delta C_{R(L)}$ and $\delta
D_{R(L)}$ are obtained by calculating loop diagrams in Fig. 3. Here the
contribution $\delta A_{R(L)}$ and $\delta B_{R(L)}$ are obtained from diagram
(C) in the figure with coupling combination of $y_{\nu}$ and $y_{5}$ for loop.
The contribution $\delta C_{R(L)}$ is obtained from summation of diagrams (A)
and (B) with coupling combination of $y_{\nu}(y_{\nu}^{\dagger})$ and
$y_{5}(y_{5}^{\dagger})$ in the loops. Also the contribution $\delta D_{R(L)}$
is obtained from summation of diagrams (A), (B) and (C) with coupling
combination of $y_{\nu}(y_{5})$ and $y_{\nu}^{\dagger}(y_{5}^{\dagger})$ for
loops. In our parameter region of interest, $\delta D_{R(L)}$ is dominant one
and the approximated expression for dominant contribution is given by

$\displaystyle(\delta D_{R})_{ij}$
$\displaystyle\simeq-\frac{(y_{5}^{\dagger})_{i\alpha}(y_{5})_{\alpha
j}}{(4\pi)^{2}}\int[dX]_{3}\frac{z(1-z+x)m_{Z}^{2}}{xM^{2}_{E_{\alpha}}+(1-x)m^{2}_{H_{2}^{+}}},$
(57) $\displaystyle(\delta D_{L})_{ij}$
$\displaystyle\simeq-\frac{(y_{\nu})_{i\alpha}(y_{\nu}^{\dagger})_{\alpha
j}}{(4\pi)^{2}}\int[dX]_{3}\frac{z(1-z+x)m_{Z}^{2}}{xM^{2}_{E_{\alpha}}+(1-x)m^{2}_{H_{1}^{+}}},$
(58)

where $\int[dX]_{3}=\int_{0}^{1}dxdydz\delta(1-x-y-z)$ and we approximated the
formula under
$m_{i}^{2}\ll\\{M_{E_{\alpha}}^{2},m^{2}_{H_{1}^{\pm}},m^{2}_{H_{2}^{\pm}}\\}$.
The full expressions of subdominant contributions $\delta A_{R(L)}$, $\delta
B_{R(L)}$ and $\delta C_{R(L)}$ are summarized in the Appendix. The widths of
$Z$ boson into charged leptons are modified by new physics contribution such
that

$\Gamma^{\rm SM+NP}_{Z\to\ell_{i}\overline{\ell_{j}}}=\frac{g_{2}^{2}}{24\pi
c_{W}^{2}}m_{Z}\left[\left(-\frac{1}{2}+s^{2}_{W}\right)\left|\delta_{ij}+(\delta_{L}^{NP})_{ij}\right|^{2}+S_{W}^{4}\left|\delta_{ij}+(\delta_{R}^{NP})_{ij}\right|^{2}\right].$
(59)

Note that we have flavor non-conserving $Z$ boson decays from new physics
contributions. Then we impose the constraints Zyla et al. (2020):

$\displaystyle{\rm BR}(Z\to e^{\pm}\mu^{\mp})\leq 7.5\times 10^{-7},\quad{\rm
BR}(Z\to e^{\pm}\tau^{\mp})\leq 9.8\times 10^{-6},$ $\displaystyle{\rm
BR}(Z\to\mu^{\pm}\tau^{\mp})\leq 1.2\times 10^{-5}.$ (60)

For flavor conserving modes, we define deviation from the SM prediction by

$\Delta{\rm BR}(Z\to\ell_{i}\overline{\ell_{i}})=\frac{\Gamma^{\rm
SM+NP}(Z\to\ell_{i}\overline{\ell_{i}})-\Gamma^{\rm
SM}(Z\to\ell_{i}\overline{\ell_{i}})}{\Gamma_{Z}^{\rm tot}},$ (61)

where $\Gamma_{Z}^{\rm tot}=2.4952\pm 0.0023$ GeV is the total decay width of
$Z$ boson. We impose the current bounds on the deviation for the lepton flavor
conserving $Z$ boson decays which are given by Zyla et al. (2020)

$\displaystyle|\Delta{\rm BR}(Z\to e^{+}e^{-})|\leq 4.2\times
10^{-5},\quad|\Delta{\rm BR}(Z\to\mu^{+}\mu^{-})|\leq 6.6\times 10^{-5},$
$\displaystyle|\Delta{\rm BR}(Z\to\tau^{+}\tau^{-})|\leq 1.2\times 10^{-5}.$
(62)

Notice that we do not consider $Z\to\nu\bar{\nu}$ decay mode since new physics
effect is smaller and the experimental constraints are also less significant.

##  III Numerical analysis and phenomenology

In this section we perform numerical analysis searching for allowed parameter
points that satisfy neutrino data and constraints from non-unitarity and LFVs.
Firstly we fix the ratio of mass parameters such that

$\displaystyle\\{\tilde{m}_{H^{\pm}_{1}},\tilde{m}_{H^{\pm}_{2}},\tilde{m}_{H^{\pm}_{3}},\tilde{M}_{E_{1}},\tilde{M}_{E_{2}},\tilde{M}_{E_{3}},\tilde{M}_{N_{1}},\tilde{M}_{N_{2}},\tilde{M}_{N_{3}}\\}$
$\displaystyle=\\{1.0,\ 1.5,\ 2.0,\ 1.2,\ 1.7,\ 2.2,\ 1.1,\ 1.6,\ 2.1\\}.$
(63)

Notice that neutrino mass matrix and LFV amplitudes are not much sensitive to
the mass parameters and other choices do not change tendency of the results
unless there is large hierarchy among them. We also fix
$s_{\beta}=1/\sqrt{2}$, $s_{A(B)}=0.01(0.1)$, and $\mu_{NP}$ is determined by
$|\kappa|$. Then, as relevant parameters, we adopt
$\\{(y_{\nu})_{ij},(y_{5})_{ij},(\delta\tilde{m}_{33})_{ij}\\}$ where we
consider $(\delta\tilde{m}_{33})_{ij}$ as free parameter although it is given
by other parameters as Eq. (28) for simplicity. Then, free parameters are
globally scanned in the following region:

$\displaystyle|(y_{\nu})_{ij}|\in[10^{-5},1.0],\quad|(y_{5})_{ij}|\in[10^{-5},1.0],$
$\displaystyle|(\delta\tilde{m}_{33})_{ij}|\in\left[10^{-10},\frac{s_{A}c_{A}}{(4\pi)^{2}}{\rm
Max}[M_{E_{\alpha}}I(M^{2}_{E_{\alpha}},m^{2}_{H^{+}_{1,2,3}})]\right],$ (64)

where the upper bound of $|(\delta\tilde{m}_{33})_{ij}|$ is fixed based on Eq.
(28).

For neutrino sector, we require $\\{s_{12},s_{13},s_{23},\Delta m^{2}_{\rm
sol}\\}$ satisfy $3\sigma$ range of observed value in NuFIT 5.2 Esteban et al.
(2020) where observed value of $\Delta m^{2}_{\rm atm}$ is used as input
parameter choosing within 3$\sigma$ range randomly. We then search for allowed
parameter sets of $(y_{\nu})_{ij}$ satisfying the neutrino observation
conditions and imposing the constraints in Eq. (45). The new physics scale
$\mu_{NP}$ is also obtained as output from relation in Eq. (II.3).

Performing numerical analysis for neutrino sector, we obtain allowed values of
Yukawa coupling $(y_{\nu})_{ij}$ and $\mu_{NP}$. Here we do not show
observables related to neutrinos since any pattern is possible by changing
$\delta\tilde{m}_{33}$ and there is no prediction. The neutrino data is used
to constrain structure of $(y_{\nu})_{ij}$ and possible value of $\mu_{NP}$.
Then we calculate lepton $g-2$, deviations in $Z$ boson decays, EDM and LFVs
using the allowed Yukawa coupling $(y_{\nu})_{ij}$ and $\mu_{NP}$ with
$(y_{5})_{ij}$ scanned within the range in Eq. (64) and search for the
parameters providing sizable muon $g-2$ satisfying the other constraints. It
is found that deviations in $Z$ boson decays are much smaller than current
constraints in the model and we do not show the plot for them.

Figure 4: $\Delta a_{\mu}$ and $BR(\ell\to\ell^{\prime}\gamma)$ for allowed
parameter sets where horizontal dot-dashed line shows current constraint from
$BR(\mu\to e\gamma)$, and region between vertical solid, dashed and dotted
lines indicate $1\sigma$, $2\sigma$ and $3\sigma$ range of muon $g-2$. The
left(right) plot corresponds to NO(IO) case.

Figure 5: $\mu_{NP}$ and $\Delta a_{\mu}$ for allowed parameter sets where
region between horizontal solid, dashed and dotted lines indicate $1\sigma$,
$2\sigma$ and $3\sigma$ range of muon $g-2$. The left(right) plot corresponds
to NO(IO) case.

In left(right) panel of Fig. 4, we show the scatter plots on $\\{\Delta
a_{\mu},{\rm BR(\ell_{i}\to\ell_{j}\gamma)}\\}$ in NO(IO) case obtained from
allowed parameter sets requiring $\Delta a_{\mu}>5.3\times 10^{-10}$ which is
$4\sigma$ below the central value of $\Delta a_{\mu}^{\rm exp}$. The
horizontal dot-dashed line shows current constraint from $BR(\mu\to e\gamma)$,
and region between vertical solid, dashed and dotted lines indicate $1\sigma$,
$2\sigma$ and $3\sigma$ range of muon $g-2$. The obtained values of
$BR(\tau\to e(\mu)\gamma)$ are sufficiently smaller than current limits and we
do not show lines corresponding to the limits. We find that IO case tends to
give larger $BR(\mu\to e\gamma)$ and $BR(\tau\to e\gamma)$ values compared to
NO case. Also we have less points to provide sizable muon $g-2$ in IO case
since constraints from $BR(\mu\to e\gamma)$ exclude many points in this case.

In left(right) panel of Fig. 5, we show the scatter plot on $\\{\mu_{\rm
NP},\Delta a_{\mu}\\}$ in NO(IP) case for our allowed parameter sets where
region between horizontal solid, dashed and dotted lines indicate $1\sigma$,
$2\sigma$ and $3\sigma$ range of muon $g-2$. We find that preferred new
physics scale $\mu_{\rm NP}$ is less than $\mathcal{O}(100)(\mathcal{O}(1))$
TeV to realize sizable muon $g-2$ within $3\sigma$ of Eq. (1) in NO(IO) case.
In particular smaller scale is preferred to get sizable muon $g-2$ in IO case.

Figure 6: $\Delta a_{\mu}$ and $|\Delta a_{e}|$ for allowed parameter sets
where region between vertical solid, dashed and dotted lines indicate
$1\sigma$, $2\sigma$ and $3\sigma$ range of muon $g-2$. The left(right) plot
corresponds to NO(IO) case.

Figure 7: $|d_{\mu}|$ and $|d_{e}|$ for allowed parameter sets where the gray
region is excluded by experimental constraint on $|d_{e}|$. The left(right)
plot corresponds to NO(IO) case.

In left(right) panel of Fig. 6, we show the scatter plot on $\\{\Delta
a_{\mu},|\Delta a_{e}|\\}$ in NO(IO) case for our allowed parameter sets where
the vertical lines are the same as Fig. 4. We find that the magnitude of
$\Delta a_{e}$ is smaller than current sensitivity of it shown in Eqs. (49)
and (50) where we have same amount of points for both positive and negative
sign of $\Delta a_{e}$. The magnitude of electron $g-2$ tends to be small
since Yukawa couplings associated with electron should be small to satisfy
constraint from $BR(\mu\to e\gamma)$ when we require muon $g-2$ to be
$\mathcal{O}(10^{-9})$. Also we find magnitude of electron $g-2$ tends to be
larger in IO case.

In left(right) panel of Fig. 7, we show the scatter plot on
$\\{|d_{\mu}|,|d_{e}|\\}$ in NO(IO) case for our allowed parameter sets where
the gray region is excluded by experimental constraint on $|d_{e}|$. We can
not find a clear correlation of them. The value of $|d_{e}|$ is concentrated
around current limit and we expect more parameter region can be tested in
future experiments. We also find that obtained $|d_{\mu}|$ is much smaller
than the current upper limit.

##  IV Summary

In this work we have proposed a new inverse seesaw model based on hidden local
$U(1)$ symmetry framework where inverse seesaw mechanism is induced at one
loop level. Two types of SM singlet chiral fermions for inverse seesaw
mechanism are naturally obtained from one vector-like fermion where vector-
like nature requires anomaly cancellation associated with hidden $U(1)$ and
Majorana mass terms of $N_{L(R)}$ are forbidden at tree level. But the
Majorana mass terms are generated at one-loop level by introducing relevant
particle contents to get loop diagram. Thus, inverse seesaw mechanism still
works well.

The same particle contents also induce contributions to lepton $g-2$/EDM and
LFV decays at one loop level. Interestingly we have obtained enhancement of
them since chirality flip appears inside loop diagram picking up new heavy
fermion masses. As a result, we have obtained sizable muon $g-2$ that
accommodates with deviation from the SM prediction. We have shown LFV decay
ratios, electron $g-2$ and electron/muon EDM for parameters that realizes
neutrino data and sizable muon $g-2$ by performing numerical analysis. It has
been found that new physics scale is preferred to be less than
$\mathcal{O}(100)(\mathcal{O}(1))$ TeV in NO(IO) case to realize muon $g-2$
within $3\sigma$. The magnitude of electron $g-2$ is smaller than current
central value since Yukawa coupling associated with electron should be small
to avoid $BR(\mu\to e\gamma)$ when we require muon $g-2$ is more than
$\mathcal{O}(10^{-10})$. Also LFV decays including electrons in final state
tend to be large in IO case compared to NO case. We expect that future
improvement of precision regarding LFV decays and electron(muon) $g-2$/EDM
measurements can test the model.

## Acknowledgments

The work was supported by the Fundamental Research Funds for the Central
Universities (T. N.).

##  Appendix A Expressions of subdominant one-loop contribution for
$Z\to\ell_{i}\bar{\ell}_{j}$

Here we summarize expressions for $\delta A_{R(L)}$, $\delta B_{R(L)}$ and
$\delta C_{R(L)}$ as follows:

$\displaystyle(\delta A_{R})_{ij}\simeq$
$\displaystyle-\frac{(y_{5}^{\dagger})_{i\alpha}(y^{\dagger}_{\nu})_{\alpha
j}}{(4\pi)^{2}s_{W}^{2}}M_{E_{\alpha}}m_{i}s_{\beta}\int[dX](1+2x-2z)$
$\displaystyle\times\left[\frac{c_{A}s_{B}c_{B}}{\Delta_{H_{11}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)c_{B}^{2}-s_{W}^{2}c_{A}^{2}s_{B}^{2}\right)+\frac{c_{A}s_{B}^{2}}{\Delta_{H_{12}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)c_{B}s_{B}+s_{W}^{2}c_{A}^{2}s_{B}c_{B}\right)\right.$
$\displaystyle\left.\quad-\frac{c_{A}c_{B}^{2}}{\Delta_{H_{21}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)c_{B}s_{B}+s_{W}^{2}c_{A}^{2}s_{B}c_{B}\right)-\frac{c_{A}s_{B}c_{B}}{\Delta_{H_{22}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)s_{B}^{2}-s_{W}^{2}c_{A}^{2}c_{B}^{2}\right)\right],$
(65) $\displaystyle(\delta A_{L})_{ij}\simeq$
$\displaystyle\frac{(y_{5}^{\dagger})_{i\alpha}(y^{\dagger}_{\nu})_{\alpha
j}}{(4\pi)^{2}}\left(-\frac{1}{2}+s^{2}_{W}\right)^{-1}M_{E_{\alpha}}m_{j}s_{\beta}\int[dX](1-2z)$
$\displaystyle\times\left[\frac{c_{A}s_{B}c_{B}}{\Delta_{H_{11}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)c_{B}^{2}-s_{W}^{2}c_{A}^{2}s_{B}^{2}\right)+\frac{c_{A}s_{B}^{2}}{\Delta_{H_{12}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)c_{B}s_{B}+s_{W}^{2}c_{A}^{2}s_{B}c_{B}\right)\right.$
$\displaystyle\left.\quad-\frac{c_{A}c_{B}^{2}}{\Delta_{H_{21}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)c_{B}s_{B}+s_{W}^{2}c_{A}^{2}s_{B}c_{B}\right)-\frac{c_{A}s_{B}c_{B}}{\Delta_{H_{22}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)s_{B}^{2}-s_{W}^{2}c_{A}^{2}c_{B}^{2}\right)\right],$
(66)

where
$\Delta_{H_{ab}}\simeq-z(1-z+x)m_{Z}^{2}+xM^{2}_{E_{\alpha}}+ym^{2}_{H_{a}^{+}}+zm^{2}_{H_{b}^{+}}$
ignoring charged lepton mass.

$\displaystyle(\delta B_{R})_{ij}\simeq$
$\displaystyle\frac{(y_{\nu})_{i\alpha}(y_{5})_{\alpha
j}}{(4\pi)^{2}s_{W}^{2}}M_{E_{\alpha}}m_{j}s_{\beta}\int[dX](1-2z)$
$\displaystyle\times\left[\frac{c_{A}s_{B}c_{B}}{\Delta_{H_{11}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)c_{B}^{2}-s_{W}^{2}c_{A}^{2}s_{B}^{2}\right)+\frac{c_{A}s_{B}^{2}}{\Delta_{H_{12}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)c_{B}s_{B}+s_{W}^{2}c_{A}^{2}s_{B}c_{B}\right)\right.$
$\displaystyle\left.\quad-\frac{c_{A}c_{B}^{2}}{\Delta_{H_{21}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)c_{B}s_{B}+s_{W}^{2}c_{A}^{2}s_{B}c_{B}\right)-\frac{c_{A}s_{B}c_{B}}{\Delta_{H_{22}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)s_{B}^{2}-s_{W}^{2}c_{A}^{2}c_{B}^{2}\right)\right],$
(67) $\displaystyle(\delta B_{L})_{ij}\simeq$
$\displaystyle-\frac{(y_{\nu})_{i\alpha}(y_{5})_{\alpha
j}}{(4\pi)^{2}}\left(-\frac{1}{2}+s^{2}_{W}\right)^{-1}M_{E_{\alpha}}m_{i}s_{\beta}\int[dX](1+2x-2z)$
$\displaystyle\times\left[\frac{c_{A}s_{B}c_{B}}{\Delta_{H_{11}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)c_{B}^{2}-s_{W}^{2}c_{A}^{2}s_{B}^{2}\right)+\frac{c_{A}s_{B}^{2}}{\Delta_{H_{12}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)c_{B}s_{B}+s_{W}^{2}c_{A}^{2}s_{B}c_{B}\right)\right.$
$\displaystyle\left.\quad-\frac{c_{A}c_{B}^{2}}{\Delta_{H_{21}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)c_{B}s_{B}+s_{W}^{2}c_{A}^{2}s_{B}c_{B}\right)-\frac{c_{A}s_{B}c_{B}}{\Delta_{H_{22}}}\left(\left(\frac{1}{2}-s_{W}^{2}\right)s_{B}^{2}-s_{W}^{2}c_{A}^{2}c_{B}^{2}\right)\right],$
(68) $\displaystyle(\delta C_{R})_{ij}\simeq$ $\displaystyle\
-\frac{c_{A}c_{B}s_{B}s_{\beta}}{(4\pi)^{2}}\left[m_{i}(y_{\nu})_{i\alpha}(y_{5})_{\alpha
j}+m_{j}y_{5}^{\dagger})_{i\alpha}(y^{\dagger}_{\nu})_{\alpha j}\right]$
$\displaystyle\times
M_{E_{\alpha}}\int[dX]_{2}y(1-y)\left[\frac{1}{M^{2}_{E_{\alpha}}x+m^{2}_{H^{+}_{1}}y}-\frac{1}{M^{2}_{E_{\alpha}}x+m^{2}_{H^{+}_{2}}y}\right],$
$\displaystyle(\delta C_{L})_{ij}\simeq$ $\displaystyle\
-\frac{c_{A}c_{B}s_{B}s_{\beta}}{(4\pi)^{2}}\left[m_{j}(y_{\nu})_{i\alpha}(y_{5})_{\alpha
j}+m_{i}y_{5}^{\dagger})_{i\alpha}(y^{\dagger}_{\nu})_{\alpha j}\right]$
$\displaystyle\times
M_{E_{\alpha}}\int[dX]_{2}y(1-y)\left[\frac{1}{M^{2}_{E_{\alpha}}x+m^{2}_{H^{+}_{1}}y}-\frac{1}{M^{2}_{E_{\alpha}}x+m^{2}_{H^{+}_{2}}y}\right],$
(69)

where $\int[dX]_{2}=\int_{0}^{1}dxdy\delta(1-x-y)$.

## References

  * Yanagida (1979) T. Yanagida, Phys. Rev. D 20, 2986 (1979).
  * Minkowski (1977) P. Minkowski, Phys. Lett. B 67, 421 (1977).
  * Mohapatra and Senjanovic (1980) R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980).
  * Fukugita and Yanagida (1986) M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 (1986).
  * Mohapatra and Valle (1986) R. N. Mohapatra and J. W. F. Valle, Phys. Rev. D 34, 1642 (1986).
  * Wyler and Wolfenstein (1983) D. Wyler and L. Wolfenstein, Nucl. Phys. B 218, 205 (1983).
  * Nomura et al. (2022) T. Nomura, H. Okada, and P. Sanyal, Eur. Phys. J. C 82, 697 (2022), eprint 2103.09494.
  * Zhang et al. (2010) H. Zhang, C. S. Li, Q.-H. Cao, and Z. Li, Phys. Rev. D 82, 075003 (2010), eprint 0910.2831.
  * Chiang et al. (2014) C.-W. Chiang, T. Nomura, and J. Tandean, JHEP 01, 183 (2014), eprint 1306.0882.
  * Chen and Nomura (2015) C.-H. Chen and T. Nomura, Phys. Lett. B 746, 351 (2015), eprint 1501.07413.
  * Chen and Nomura (2016) C.-H. Chen and T. Nomura, Phys. Rev. D 93, 074019 (2016), eprint 1507.00886.
  * Gross et al. (2015) C. Gross, O. Lebedev, and Y. Mambrini, JHEP 08, 158 (2015), eprint 1505.07480.
  * Hambye (2009) T. Hambye, JHEP 01, 028 (2009), eprint 0811.0172.
  * Boehm et al. (2014) C. Boehm, M. J. Dolan, and C. McCabe, Phys. Rev. D 90, 023531 (2014), eprint 1404.4977.
  * Baek et al. (2014) S. Baek, P. Ko, and W.-I. Park, JCAP 10, 067 (2014), eprint 1311.1035.
  * Khoze and Ro (2014) V. V. Khoze and G. Ro, JHEP 10, 061 (2014), eprint 1406.2291.
  * Daido et al. (2020) R. Daido, S.-Y. Ho, and F. Takahashi, JHEP 01, 185 (2020), eprint 1909.03627.
  * Karam and Tamvakis (2015) A. Karam and K. Tamvakis, Phys. Rev. D 92, 075010 (2015), eprint 1508.03031.
  * Davoudiasl and Lewis (2014) H. Davoudiasl and I. M. Lewis, Phys. Rev. D 89, 055026 (2014), eprint 1309.6640.
  * Ko et al. (2021) P. Ko, T. Nomura, and H. Okada, Phys. Rev. D 103, 095011 (2021), eprint 2007.08153.
  * Nomura et al. (2021) T. Nomura, H. Okada, and S. Yun, JHEP 06, 122 (2021), eprint 2012.11377.
  * Cai et al. (2019) H. Cai, T. Nomura, and H. Okada, Nucl. Phys. B 949, 114802 (2019), eprint 1812.01240.
  * Nomura and Okada (2018) T. Nomura and H. Okada, Phys. Rev. D 97, 075038 (2018), eprint 1709.06406.
  * Matsui et al. (2023) T. Matsui, T. Nomura, and K. Yagyu, Phys. Rev. D 108, L011301 (2023), eprint 2303.00417.
  * Li et al. (2023) J. Li, T. Nomura, J. Pei, X. Yin, and C. Zhang, Phys. Rev. D 108, 035021 (2023), eprint 2302.09839.
  * Nagao et al. (2023) K. I. Nagao, T. Nomura, H. Okada, and T. Shimomura, Phys. Rev. D 108, 055032 (2023), eprint 2212.14528.
  * Abi et al. (2021) B. Abi et al. (Muon g-2), Phys. Rev. Lett. 126, 141801 (2021), eprint 2104.03281.
  * Aguillard et al. (2023) D. P. Aguillard et al. (Muon g-2), Phys. Rev. Lett. 131, 161802 (2023), eprint 2308.06230.
  * Bennett et al. (2006) G. W. Bennett et al. (Muon g-2), Phys. Rev. D 73, 072003 (2006), eprint hep-ex/0602035.
  * Aoyama et al. (2012) T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, Phys. Rev. Lett. 109, 111808 (2012), eprint 1205.5370.
  * Aoyama et al. (2019) T. Aoyama, T. Kinoshita, and M. Nio, Atoms 7, 28 (2019).
  * Czarnecki et al. (2003) A. Czarnecki, W. J. Marciano, and A. Vainshtein, Phys. Rev. D 67, 073006 (2003), [Erratum: Phys.Rev.D 73, 119901 (2006)], eprint hep-ph/0212229.
  * Gnendiger et al. (2013) C. Gnendiger, D. Stöckinger, and H. Stöckinger-Kim, Phys. Rev. D 88, 053005 (2013), eprint 1306.5546.
  * Davier et al. (2017) M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, Eur. Phys. J. C 77, 827 (2017), eprint 1706.09436.
  * Keshavarzi et al. (2018) A. Keshavarzi, D. Nomura, and T. Teubner, Phys. Rev. D 97, 114025 (2018), eprint 1802.02995.
  * Colangelo et al. (2019) G. Colangelo, M. Hoferichter, and P. Stoffer, JHEP 02, 006 (2019), eprint 1810.00007.
  * Hoferichter et al. (2019) M. Hoferichter, B.-L. Hoid, and B. Kubis, JHEP 08, 137 (2019), eprint 1907.01556.
  * Davier et al. (2020) M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, Eur. Phys. J. C 80, 241 (2020), [Erratum: Eur.Phys.J.C 80, 410 (2020)], eprint 1908.00921.
  * Keshavarzi et al. (2020a) A. Keshavarzi, D. Nomura, and T. Teubner, Phys. Rev. D 101, 014029 (2020a), eprint 1911.00367.
  * Kurz et al. (2014) A. Kurz, T. Liu, P. Marquard, and M. Steinhauser, Phys. Lett. B 734, 144 (2014), eprint 1403.6400.
  * Melnikov and Vainshtein (2004) K. Melnikov and A. Vainshtein, Phys. Rev. D 70, 113006 (2004), eprint hep-ph/0312226.
  * Masjuan and Sanchez-Puertas (2017) P. Masjuan and P. Sanchez-Puertas, Phys. Rev. D 95, 054026 (2017), eprint 1701.05829.
  * Colangelo et al. (2017) G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, JHEP 04, 161 (2017), eprint 1702.07347.
  * Hoferichter et al. (2018) M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schneider, JHEP 10, 141 (2018), eprint 1808.04823.
  * Gérardin et al. (2019) A. Gérardin, H. B. Meyer, and A. Nyffeler, Phys. Rev. D 100, 034520 (2019), eprint 1903.09471.
  * Bijnens et al. (2019) J. Bijnens, N. Hermansson-Truedsson, and A. Rodríguez-Sánchez, Phys. Lett. B 798, 134994 (2019), eprint 1908.03331.
  * Colangelo et al. (2020) G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub, and P. Stoffer, JHEP 03, 101 (2020), eprint 1910.13432.
  * Blum et al. (2020) T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin, C. Jung, and C. Lehner, Phys. Rev. Lett. 124, 132002 (2020), eprint 1911.08123.
  * Colangelo et al. (2014) G. Colangelo, M. Hoferichter, A. Nyffeler, M. Passera, and P. Stoffer, Phys. Lett. B 735, 90 (2014), eprint 1403.7512.
  * Hagiwara et al. (2011) K. Hagiwara, R. Liao, A. D. Martin, D. Nomura, and T. Teubner, J. Phys. G 38, 085003 (2011), eprint 1105.3149.
  * Borsanyi et al. (2021) S. Borsanyi et al., Nature 593, 51 (2021), eprint 2002.12347.
  * Alexandrou et al. (2023) C. Alexandrou et al. (Extended Twisted Mass), Phys. Rev. D 107, 074506 (2023), eprint 2206.15084.
  * Cè et al. (2022) M. Cè et al., Phys. Rev. D 106, 114502 (2022), eprint 2206.06582.
  * Crivellin et al. (2020) A. Crivellin, M. Hoferichter, C. A. Manzari, and M. Montull, Phys. Rev. Lett. 125, 091801 (2020), eprint 2003.04886.
  * de Rafael (2020) E. de Rafael, Phys. Rev. D 102, 056025 (2020), eprint 2006.13880.
  * Keshavarzi et al. (2020b) A. Keshavarzi, W. J. Marciano, M. Passera, and A. Sirlin, Phys. Rev. D 102, 033002 (2020b), eprint 2006.12666.
  * Passera et al. (2008) M. Passera, W. J. Marciano, and A. Sirlin, Phys. Rev. D 78, 013009 (2008), eprint 0804.1142.
  * Lindner et al. (2018) M. Lindner, M. Platscher, and F. S. Queiroz, Phys. Rept. 731, 1 (2018), eprint 1610.06587.
  * Athron et al. (2021) P. Athron, C. Balázs, D. H. J. Jacob, W. Kotlarski, D. Stöckinger, and H. Stöckinger-Kim, JHEP 09, 080 (2021), eprint 2104.03691.
  * Guedes and Olgoso (2022) G. Guedes and P. Olgoso, JHEP 09, 181 (2022), eprint 2205.04480.
  * Crivellin and Hoferichter (2021) A. Crivellin and M. Hoferichter, JHEP 07, 135 (2021), [Erratum: JHEP 10, 030 (2022)], eprint 2104.03202.
  * Afanaciev et al. (2024) K. Afanaciev et al. (MEG II), Eur. Phys. J. C 84, 216 (2024), eprint 2310.12614.
  * Baldini et al. (2016) A. M. Baldini et al. (MEG), Eur. Phys. J. C 76, 434 (2016), eprint 1605.05081.
  * Aubert et al. (2010) B. Aubert et al. (BaBar), Phys. Rev. Lett. 104, 021802 (2010), eprint 0908.2381.
  * Renga (2018) F. Renga (MEG), Hyperfine Interact. 239, 58 (2018), eprint 1811.05921.
  * Parker et al. (2018) R. H. Parker, C. Yu, W. Zhong, B. Estey, and H. Müller, Science 360, 191 (2018), eprint 1812.04130.
  * Morel et al. (2020) L. Morel, Z. Yao, P. Cladé, and S. Guellati-Khélifa, Nature 588, 61 (2020).
  * Roussy et al. (2023) T. S. Roussy et al., Science 381, adg4084 (2023), eprint 2212.11841.
  * Bennett et al. (2009) G. W. Bennett et al. (Muon (g-2)), Phys. Rev. D 80, 052008 (2009), eprint 0811.1207.
  * Maki et al. (1962) Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. 28, 870 (1962).
  * Gando et al. (2016) A. Gando et al. (KamLAND-Zen), Phys. Rev. Lett. 117, 082503 (2016), [Addendum: Phys.Rev.Lett. 117, 109903 (2016)], eprint 1605.02889.
  * Fernandez-Martinez et al. (2016) E. Fernandez-Martinez, J. Hernandez-Garcia, and J. Lopez-Pavon, JHEP 08, 033 (2016), eprint 1605.08774.
  * Agostinho et al. (2018) N. R. Agostinho, G. C. Branco, P. M. F. Pereira, M. N. Rebelo, and J. I. Silva-Marcos, Eur. Phys. J. C 78, 895 (2018), eprint 1711.06229.
  * Das et al. (2017) A. Das, T. Nomura, H. Okada, and S. Roy, Phys. Rev. D 96, 075001 (2017), eprint 1704.02078.
  * Zyla et al. (2020) P. A. Zyla et al. (Particle Data Group), PTEP 2020, 083C01 (2020).
  * Esteban et al. (2020) I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz, and A. Zhou, JHEP 09, 178 (2020), eprint 2007.14792.