# A radiatively induced inverse seesaw model
with hidden $U(1)$ gauge symmetry

Takaaki Nomura<EMAIL_ADDRESS>College of Physics, Sichuan University,
Chengdu 610065, China  Hiroshi Okada<EMAIL_ADDRESS>Asia Pacific
Center for Theoretical Physics (APCTP) - Headquarters San 31, Hyoja-dong, Nam-
gu, Pohang 790-784, Korea  Prasenjit Sanyal<EMAIL_ADDRESS>Asia
Pacific Center for Theoretical Physics (APCTP) - Headquarters San 31, Hyoja-
dong, Nam-gu, Pohang 790-784, Korea

###### Abstract

We propose an inverse seesaw scenario under hidden $U(1)$ gauge symmetry,
having rather natural hierarchies among neutral fermion mass scales. The
hierarchies are derived from theory and experimental constraints. The theory
requires Majorana exotic masses have to be induced at one-loop level. The
experimental side requests that the Dirac mass terms has to be highly
suppressed to satisfy the constraints from the lepton flavor violations(LFVs)
such as $\mu\to e\gamma$. In order to induce such a small Majorana exotic
masses we introduce exotic fermions and bosons that also provide us additional
intriguing phenomenologies such as muon anomalous magnetic moment, dark matter
candidate as well as LFVs and deviations in leptonic $Z$ decays. We analyze
these phenomenologies including neutrino oscillation data numerically, and
show allowed region. Finally, we discuss the DM candidate in both the cases of
fermion and scalar boson, where we focus on rather lighter range that is equal
or less than 10 GeV. The dominant contributions originate from interactions of
hidden gauge sector, and we show allowed ranges for both cases.

††preprint: CTP-SCU/2021009, APCTP Pre2021-005

##  I Introduction

Understanding electrically neutral fields is our common issues to be resolved
in both theoretical and experimental view points. Typically, it is considered
in the framework of physics beyond the standard model (SM) since the active
neutrino masses as well as dark matter(DM) candidate cannot be explained in
the SM unless new scale or fields are supplied. Even though several
experiments have partially discovered nature of the active neutrinos measuring
three mixing angles and two squared mass differences, we only have few ideas
about CP phases etc.; Dirac CP phase, Majorana phases, neutrino mass
eigenvalues, Majorana or Dirac mass type, and so on. This is because the
mechanism to generate neutrino masses is not uniquely determined yet. On the
other hand there are almost null direct information on nature of DM in spite
of huge efforts by experimentalists on direct searches such as XENON1T
Aprile:2018dbl  and LUX Akerib:2016vxi , indirect searches such as Fermi-LAT
FermiLAT:2011ab , AMS-02 Aguilar:2013qda , and CALET Adriani:2017efm , and
collider searches at LHC Khachatryan:2014rra . These experiments especially
focus on weakly interacting massive particles (WIMP) whose typical mass would
be the order 100 GeV to 1000 GeV. Although it does not suggest WIMP has
totally been excluded, it would be worthwhile to consider a lighter DM
candidate below e.g., 10 GeV. It would be more natural in view of relaxed
hierarchies among neutral fields, since the active neutrino masses are
minuscule that is the order $10^{-9}$ GeV. In order to realize such a tiny
mass of neutrinos, we propose an improved inverse seesaw model with a hidden
gauged $U(1)$ symmetry that plays crucial roles in carrying out our neutrino
model and assuring the stability of our DM due to remnant $Z_{2}$ symmetry
after spontaneous symmetry breaking. Hidden $U(1)$ symmetry has an advantage
that the mass of hidden gauge boson $(m_{Z^{\prime}})$ and its gauge coupling
$(g^{\prime\prime})$ are almost taken to be free, since the $Z^{\prime}$ boson
does not directly couple to the SM fields and there are almost no bounds by
LEP Barate:2003sz  and LHC Tanabashi:2018oca . 111Even though a bound arises
from kinetic mixing, one can arbitrary neglect this effect. It is unlikely to
flavorful gauge symmetries such as $U(1)_{B-L}$. This is why this kind of
symmetry is frequently applied to lighter DM scenarios Zhang:2009dd ;
Chiang:2013kqa ; Chen:2015nea ; Chen:2015dea ; Gross:2015cwa ; Hambye:2008bq ;
Boehm:2014bia ; Baek:2013dwa ; Khoze:2014woa ; Daido:2019tbm ; Karam:2015jta ;
Davoudiasl:2013jma ; Ko:2020qlt ; Nomura:2020zlm ; Cai:2018upp ;
Nomura:2017wxf .

The original inverse seesaw mass matrix in basis of
$[\nu_{L},N_{R}^{C},N_{L}]^{T}$ Mohapatra:1986bd ; Wyler:1982dd  can be
sketched as follows:

$\displaystyle\left(\begin{array}[]{ccc}0&M_{D}&0\\\ M_{D}&0&M_{N}\\\
0&M_{N}&\mu\end{array}\right),$ (4)

then the active neutrino mass matrix is given by
$\mu\left(\frac{M_{D}}{M_{N}}\right)^{2}$ after the block-diagonalization.
Notice here that it requires mass hierarchies $\mu<M_{D}<M_{N}$ at least in
order to generate the appropriate neutrino mass scale. Usually, these mass
hierarchies are imposed by hand.

To achieve the mass hierarchies, one of the interesting scenario is generation
of Majorana $\mu$ mass term at loop level; here we call it as radiative
inverse seesaw. In ref. Ma:2009gu , a radiative inverse seesaw model is
proposed applying local $U(1)_{\chi}$ motivated by $SO(10)$ and adding some
scalar fields; a model with local $U(1)_{B-L}$ is found in ref. Mandal:2019oth
. Moreover a model with lepton triplet Law:2012mj  is considered where $\mu$
mass term at tree level is forbidden by the nature of gauge symmetry. $Z_{4}$
symmetry can be also used to forbid tree level $\mu$ mass term Ahriche:2016acx
. A radiative inverse seesaw model with a new gauge symmetry is
phenomenologically interesting since it provides new phenomena at experiments
222Tree level inverse seesaw models with local/global $U(1)$ symmetries are
also discussed by the current authors Nomura:2018mwr ; Dey:2019cts ;
Cai:2018upp . Phenomenological constraints are also discussed in ref.
Das:2019pua  for general $U(1)$ gauge symmetry where the SM fermions are
charged under it.. Here we are interested in a scenario with hidden $U(1)$
gauge symmetry where the SM fields are not charged under it. A $Z^{\prime}$
boson from hidden $U(1)$ would provide different phenomenology from non-hidden
$U(1)$ symmetry such as $U(1)_{B-L}$. Remarkably $Z^{\prime}$ can have
interactions with lepton flavor violation(LFV) when the SM lepton and exotic
fermions with hidden $U(1)$ charge mixes. It is thus interesting to consider a
radiative inverse seesaw model with hidden $U(1)$ gauge symmetry and discuss
various lepton flavor physics.

In this paper we propose a radiative inverse seesaw model with hidden $U(1)$
gauge symmetry. In our model, we explain the hierarchies among
$\\{\mu,M_{D},M_{N}\\}$ through theoretical and experimental manners; $\mu$ is
suppressed by generating it at one-loop level 333Similar models are found in
e.g. refs. Das:2017ski ; Pilaftsis:1991ug ; Dev:2012sg ; Dev:2012bd , while
$M_{D}/M_{N}$ has to be tiny enough to satisfy lepton flavor violations such
as $\mu\to e\gamma$ process, as can be seen in our analysis. Here $M_{N}$ is
introduced as a mass of vector-like lepton doublet $L^{\prime}$ and it is not
suppressed. Majorana mass $\mu$ is forbidden at tree level due to the nature
of gauge symmetry. We realize it at one loop level where we introduce singlet
fermion $N$, inert scalar doublet $\eta$ and singlet $\chi$ with nonzero
hidden charges, in order to induce relevant loop diagrams. Here
$\\{N,\eta,\chi\\}$ fields will have remnant $Z_{2}$ odd parity after
spontaneous symmetry breaking of hidden $U(1)$ and the lightest neutral
particle among them can be our DM candidate. Thanks to these additional
fields, we can also discuss flavor physics such as muon anomalous magnetic
moment(muon $g-2$), LFVs and deviations in leptonic $Z$ boson decays as well
as DM phenomenologies.

This paper is organized as follows. In Sec. II, we review our model and
formulate the Higgs sector, and gauge sector. In Sect. III, we discuss the
neutrino masses, LFVs, muon $g-2$, leptonic $Z$ boson decays, and show
numerical analysis satisfying all the constraints except DM. In Sect. IV, we
study two types of our DM; bosonic and fermionic one, and demonstrate the
allowed region to satisfy the relic density for each candidate. Notice here
that the dominant processes are via new gauge interactions that would not be
independent of neutrino and flavor physics. In Sec. V, we summarize and
conclude.

##  II model

Fields |  $Q_{L}$ |  $U_{R}$ |  $d_{R}$ |  $L_{L}$ |  $e_{R}$ |  $L^{\prime}_{L,R}$ |  $N_{L,R}$ |  $\Phi$ |  $\eta$ |  $\varphi$ |  $\chi$
---|---|---|---|---|---|---|---|---|---|---|---
$SU(3)_{C}$ | $\bm{3}$ | $\bm{3}$ | $\bm{3}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$
$SU(2)_{L}$ | $\bm{2}$ | $\bm{1}$ | $\bm{1}$ | $\bm{2}$ | $\bm{1}$ | $\bm{2}$ | $\bm{1}$ | $\bm{2}$ | $\bm{2}$ | $\bm{1}$ | $\bm{1}$
$U(1)_{Y}$ | $\frac{1}{6}$ | $\frac{2}{3}$ | $-\frac{1}{3}$ | $-\frac{1}{2}$ | $-1$ | $-\frac{1}{2}$ | $0$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $0$ | $0$
$U(1)_{H}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-x$ | $\frac{1}{2}x$ | $0$ | $\frac{3}{2}x$ | $x$ | $\frac{1}{2}x$
Table 1: Field contents and charge assignment in the model.

In this model we consider hidden $U(1)_{H}$ gauge symmetry in addition to the
SM gauge symmetry. Here we introduce the vector like extra lepton doublet
$L^{\prime}=(N^{\prime},E^{\prime})$ and SM singlet fermions $N_{L,R}$ which
have $U(1)_{H}$ charge $x$ and $x/2$ respectively. In scalar sector,
$SU(2)_{L}$ doublet $\eta$, singlet $\varphi$ and singlet $\chi$ are
introduced whose $U(1)_{H}$ charges are $x/2$, $x$ and $x/2$. The field
contents and their charge assignments are summarized in Table. 1. In this
model, $\varphi$ and the SM Higgs field $\Phi$ develop VEVs spontaneously
breaking $U(1)_{H}$ and electroweak symmetry. The scalar fields are written by

$\displaystyle\Phi=\begin{pmatrix}G^{+}\\\
\frac{1}{\sqrt{2}}(v+\tilde{h}+iG_{Z})\end{pmatrix},\quad\eta=\begin{pmatrix}\eta^{+}\\\
\frac{1}{\sqrt{2}}(\eta_{R}+i\eta_{I})\end{pmatrix}$
$\displaystyle\varphi=\frac{1}{\sqrt{2}}(v_{\varphi}+\varphi_{R}+iG_{Z^{\prime}}),\quad\chi=\frac{1}{\sqrt{2}}(\chi_{R}+i\chi_{I}),$
(5)

where $v$ and $v_{\varphi}$ are VEVs of $\Phi$ and $\varphi$, and $G^{\pm}$,
$G_{Z}$ and $G_{Z^{\prime}}$ are Nambu-Goldstone(NG) bosons absorbed by
$W^{\pm}$, $Z$ and $Z^{\prime}$ gauge bosons. Note that $Z_{2}$ symmetry
remains after $U(1)_{H}$ symmetry breaking where $N_{L/R}$, $\eta$ and $\chi$
are parity odd under the $Z_{2}$ while the other particles are parity even.
Thus the lightest $Z_{2}$ odd particle can be our DM candidate.

In this model, Yukawa interactions for leptons 444Yukawa interactions for
quarks are the same as the SM one and we do not discuss in the paper. and
scalar potential are given by

$\displaystyle\mathcal{L}\supset\ $ $\displaystyle y_{\ell}\bar{L}_{L}\Phi
e_{R}+M_{L^{\prime}}\bar{L}^{\prime}_{L}L^{\prime}_{R}+M_{N}\bar{N}_{R}N_{L}+y_{D}\bar{L}_{L}L^{\prime}_{R}\varphi+y_{R}\bar{L}^{\prime}_{L}\tilde{\eta}N_{R}+y_{L}\bar{L}^{\prime}_{R}\tilde{\eta}N_{L}$
$\displaystyle+y_{\varphi_{R}}\bar{N}^{c}_{R}N_{R}\varphi^{*}+y_{\varphi_{L}}\bar{N}^{c}_{L}N_{L}\varphi^{*}+h.c.\,,$
(6) $\displaystyle V=\ $ $\displaystyle
m_{\Phi}^{2}\Phi^{\dagger}\Phi+m_{\eta}^{2}\eta^{\dagger}\eta+m_{\varphi}^{2}\varphi^{*}\varphi+m_{\chi}^{2}\chi^{*}\chi+\mu(\chi^{2}\varphi^{*}+h.c.)$
$\displaystyle+\lambda_{0}((\Phi^{\dagger}\eta)\chi^{*}\varphi^{*}+h.c.)+\lambda_{\Phi}(\Phi^{\dagger}\Phi)^{2}+\lambda_{\eta}(\eta^{\dagger}\eta)^{2}+\lambda_{\chi}(\chi^{*}\chi)^{2}+\lambda_{\varphi}(\varphi^{*}\varphi)^{2}$
$\displaystyle+\lambda_{\Phi\eta}(\Phi^{\dagger}\Phi)(\eta^{\dagger}\eta)+\tilde{\lambda}_{\Phi\eta}(\Phi^{\dagger}\eta)(\eta^{\dagger}\Phi)+\lambda_{\Phi\varphi}(\Phi^{\dagger}\Phi)(\varphi^{*}\varphi)+\lambda_{\Phi\chi}(\Phi^{\dagger}\Phi)(\chi^{*}\chi)$
$\displaystyle+\lambda_{\eta\varphi}(\eta^{\dagger}\eta)(\varphi^{*}\varphi)+\lambda_{\eta\chi}(\eta^{\dagger}\eta)(\chi^{*}\chi)+\lambda_{\varphi\chi}(\varphi^{*}\varphi)(\chi^{*}\chi)\,.$
(7)

###  II.1 Scalar sector

Here we consider scalar sector in the model and formulate mass spectrum and
corresponding mass eigenvalues. Firstly the VEVs of $\Phi$ and $\varphi$ are
calculated from scalar potential Eq. (7) by solving the condition $\partial
V/\partial v=\partial V/\partial v_{\varphi}=0$. Then we obtain the conditions

$\displaystyle
m_{\Phi}^{2}+\lambda_{\Phi}v^{2}+\frac{1}{2}\lambda_{\Phi\varphi}v_{\varphi}^{2}=0,$
(8) $\displaystyle
m_{\varphi}^{2}+\lambda_{\varphi}v_{\varphi}^{2}+\frac{1}{2}\lambda_{\Phi\varphi}v^{2}=0.$
(9)

Note that the conditions $m_{\Phi}^{2},m_{\varphi}^{2}<0$ are required to make
square of VEVs positive definite. In addition we assume
$m_{\chi}^{2},m_{\eta}^{2}>0$ so that $\chi$ and $\eta$ do not develop non-
zero VEVs.

After spontaneous symmetry breaking, we obtain mass matrix for CP-even neutral
scalar fields with $Z_{2}$-even parity such that

$\mathcal{L}\supset\frac{1}{2}\begin{pmatrix}\tilde{h}\\\
\varphi_{R}\end{pmatrix}^{T}\begin{pmatrix}2\lambda_{\Phi}v^{2}&\lambda_{\Phi\varphi}vv_{\varphi}\\\
\lambda_{\Phi\varphi}vv_{\varphi}&2\lambda_{\varphi}v_{\varphi}^{2}\end{pmatrix}.\begin{pmatrix}\tilde{h}\\\
\varphi_{R}\end{pmatrix}$ (10)

Diagonalizing the mass matrix, the mass eigenstates are given by

$\displaystyle
m_{h,H}^{2}=\lambda_{\Phi}v^{2}+\lambda_{\varphi}v_{\varphi}^{2}\pm\sqrt{(\lambda_{\Phi}v^{2}-\lambda_{\varphi}v_{\varphi}^{2})^{2}+\lambda_{\Phi\varphi}^{2}v^{2}v_{\varphi}^{2}},$
(11)

where we identify $m_{h}=125$ GeV as the SM Higgs mass. The mass eigenstates
and mixing are written as

$\displaystyle\begin{pmatrix}\tilde{h}\\\
\varphi_{R}\end{pmatrix}=\begin{pmatrix}\cos\beta&-\sin\beta\\\
\sin\beta&\cos\beta\end{pmatrix}\begin{pmatrix}h\\\ H\end{pmatrix},$
$\displaystyle\tan
2\beta=\frac{vv_{\varphi}\lambda_{\Phi\varphi}}{v^{2}\lambda_{\Phi}-v^{2}_{\varphi}\lambda_{\varphi}},$
(12)

where $h$ is identified as the SM Higgs boson.

The mass for $Z_{2}$-odd charged scalar field is given by

$\displaystyle
m_{\eta^{\pm}}^{2}=m_{\eta}^{2}+\frac{1}{2}\lambda_{\Phi\eta}v^{2}+\frac{1}{2}\lambda_{\eta\varphi}v_{\varphi}^{2},$
(13)

where the corresponding mass eigenstate is $\eta^{\pm}$ given in Eq. (5). We
also obtain mass matrix for $Z_{2}$-odd CP-odd scalar fields such as

$\displaystyle\mathcal{L}\supset\frac{1}{2}\begin{pmatrix}\eta_{I}\\\
\chi_{I}\end{pmatrix}^{T}\begin{pmatrix}m_{\eta^{\pm}}^{2}+\frac{1}{2}\tilde{\lambda}_{\Phi\eta}v^{2}&-\frac{1}{2}\lambda_{0}vv_{\varphi}\\\
-\frac{1}{2}\lambda_{0}vv_{\varphi}&m_{\chi^{\prime}}^{2}\end{pmatrix}\begin{pmatrix}\eta_{I}\\\
\chi_{I}\end{pmatrix},$ $\displaystyle m^{2}_{\chi^{\prime}}\equiv
m_{\chi}^{2}+\frac{1}{2}\lambda_{\Phi\chi}v^{2}+\frac{1}{2}\lambda_{\varphi\chi}v_{\varphi}^{2}-\sqrt{2}\mu
v_{\varphi}.$ (14)

Diagonalizing the mass matrix, the mass eigenstates are given by

$\displaystyle m_{A_{1},A_{2}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{[}2m_{\eta^{\pm}}^{2}+2m_{\chi^{\prime}}^{2}+v^{2}\tilde{\lambda}_{\Phi\eta}$
$\displaystyle\pm$
$\displaystyle\sqrt{(2m_{\eta^{\pm}}^{2}+2m_{\chi^{\prime}}^{2}+v^{2}\tilde{\lambda}_{\Phi\eta})^{2}-4(4m_{\eta^{\pm}}^{2}m_{\chi^{\prime}}^{2}-v^{2}v_{\varphi}^{2}\lambda_{0}^{2}+2m_{\chi^{\prime}}^{2}v^{2}\tilde{\lambda}_{\Phi\eta})}\Big{]},$

The mass eigenstates and mixing are written as

$\displaystyle\begin{pmatrix}\eta_{I}\\\
\chi_{I}\end{pmatrix}=\begin{pmatrix}\cos\gamma&-\sin\gamma\\\
\sin\gamma&\cos\gamma\end{pmatrix}\begin{pmatrix}A_{1}\\\ A_{2}\end{pmatrix},$
$\displaystyle\tan
2\gamma=\frac{vv_{\varphi}\lambda_{0}}{m^{2}_{\chi^{\prime}}-m^{2}_{\eta^{\pm}}-\frac{1}{2}v^{2}\tilde{\lambda}_{\Phi\eta}}.$
(16)

Furthermore mass matrix for $Z_{2}$-odd CP-even scalar fields are obtained as
follows

$\displaystyle\mathcal{L}\supset\frac{1}{2}\begin{pmatrix}\eta_{R}\\\
\chi_{R}\end{pmatrix}^{T}\begin{pmatrix}m_{\eta^{\pm}}^{2}+\frac{1}{2}\tilde{\lambda}_{\Phi\eta}v^{2}&\frac{1}{2}\lambda_{0}vv_{\varphi}\\\
\frac{1}{2}\lambda_{0}vv_{\varphi}&m_{\chi^{\prime\prime}}^{2}\end{pmatrix}\begin{pmatrix}\eta_{R}\\\
\chi_{R}\end{pmatrix},$ $\displaystyle m^{2}_{\chi^{\prime\prime}}\equiv
m_{\chi}^{2}+\frac{1}{2}\lambda_{\Phi\chi}v^{2}+\frac{1}{2}\lambda_{\varphi\chi}v_{\varphi}^{2}+\sqrt{2}\mu
v_{\varphi}.$ (17)

Diagonalizing the mass matrix, the mass eigenstates are given by

$\displaystyle m_{H_{1},H_{2}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{[}2m_{\eta^{\pm}}^{2}+2m_{\chi^{\prime\prime}}^{2}+v^{2}\tilde{\lambda}_{\Phi\eta}$
$\displaystyle\pm$
$\displaystyle\sqrt{(2m_{\eta^{\pm}}^{2}+2m_{\chi^{\prime\prime}}^{2}+v^{2}\tilde{\lambda}_{\Phi\eta})^{2}-4(4m_{\eta^{\pm}}^{2}m_{\chi^{\prime\prime}}^{2}-v^{2}v_{\varphi}^{2}\lambda_{0}^{2}+2m_{\chi^{\prime\prime}}^{2}v^{2}\tilde{\lambda}_{\Phi\eta})}\Big{]},$

The mass eigenstates and mixing are written as

$\displaystyle\begin{pmatrix}\eta_{R}\\\
\chi_{R}\end{pmatrix}=\begin{pmatrix}\cos\alpha&-\sin\alpha\\\
\sin\alpha&\cos\alpha\end{pmatrix}\begin{pmatrix}H_{1}\\\ H_{2}\end{pmatrix},$
(19) $\displaystyle\tan
2\alpha=\frac{vv_{\varphi}\lambda_{0}}{m^{2}_{\eta^{\pm}}-m^{2}_{\chi^{\prime\prime}}+\frac{1}{2}v^{2}\tilde{\lambda}_{\Phi\eta}}.$
(20)

We find that the mixing $\sin\alpha$ and $\sin\gamma$ can be written in terms
of mass eigenvalues such that

$\displaystyle\cos^{2}\alpha$ $\displaystyle=$
$\displaystyle\frac{(m^{2}_{A_{2}}-m^{2}_{H_{2}})(m^{2}_{H_{2}}-m^{2}_{A_{1}})}{(m^{2}_{A_{2}}+m^{2}_{A_{1}}-m^{2}_{H_{2}}-m^{2}_{H_{1}})(m^{2}_{H_{2}}-m^{2}_{H_{1}})}$
$\displaystyle\cos^{2}\gamma$ $\displaystyle=$
$\displaystyle\frac{(m^{2}_{A_{2}}-m^{2}_{H_{2}})(m_{A_{2}}^{2}-m^{2}_{H_{1}})}{(m^{2}_{A_{2}}+m^{2}_{A_{1}}-m^{2}_{H_{2}}-m^{2}_{H_{1}})(m^{2}_{A_{2}}-m^{2}_{A_{1}})}.$
(21)

Note that we can write some parameters in the scalar potential by VEVs, scalar
masses and mixings as summarized in Appendix where the independent parameters
are
$\\{m_{h},m_{H},m_{H_{1}},m_{H_{2}},m_{A_{1}},m_{A_{2}},m_{\eta^{\pm}},v,v_{\varphi},\cos\beta$,
$\lambda_{\Phi\chi},\lambda_{\varphi\chi},\lambda_{\Phi\eta},\lambda_{\eta\varphi},\lambda_{\eta\chi},\lambda_{\eta},\lambda_{\chi}\\}$.

###  II.2 Gauge sector

The most general $U(1)$ gauge Lagrangian including the kinetic mixing term is

$\mathcal{L}_{\text{gauge}}=-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}B^{\prime}_{\mu\nu}B^{\prime\mu\nu}-\frac{1}{2}\epsilon
B_{\mu\nu}B^{\prime\mu\nu}$ (22)

where $B_{\mu\nu}$ and $B^{\prime}_{\mu\nu}$ are the field strength tensors of
$U(1)_{Y}$ and $U(1)_{H}$ gauge symmetries. We can diagonalize Eq. (22) by the
following transformation

$\displaystyle\left(\begin{array}[]{c}\tilde{B}^{\prime}_{\mu}\\\
\tilde{B}_{\mu}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\sqrt{1-\epsilon^{2}}&0\\\
\epsilon&1\\\ \end{array}\right)\left(\begin{array}[]{c}B^{\prime}_{\mu}\\\
B_{\mu}\\\ \end{array}\right)$ (29)

where $\epsilon$ is a dimensionless quantity ($\epsilon<<1$) and we
parameterize $\rho=-\frac{\epsilon}{\sqrt{1-\epsilon^{2}}}$. Under the
transformation Eq. (29), the gauge Lagrangian can be written as

$\displaystyle\mathcal{L}_{\text{gauge}}=-\frac{1}{4}\tilde{B}_{\mu\nu}\tilde{B}^{\mu\nu}-\frac{1}{4}\tilde{B}^{\prime}_{\mu\nu}\tilde{B}^{\prime\mu\nu}$
(30)

where
$\tilde{B}_{\mu\nu}=\partial_{\mu}\tilde{B}_{\nu}-\partial_{\nu}\tilde{B}_{\mu}$
and
$\tilde{B}^{\prime}_{\mu\nu}=\partial_{\mu}\tilde{B}^{\prime}_{\nu}-\partial_{\nu}\tilde{B}^{\prime}_{\mu}$.

The kinetic term of the scalar fields are

$\displaystyle\mathcal{L}_{\text{kin}}=(D_{\mu}\Phi)^{\dagger}(D^{\mu}\Phi)+(D_{\mu}\eta)^{\dagger}(D^{\mu}\eta)+(D_{\mu}\varphi)^{\dagger}(D^{\mu}\varphi)+(D_{\mu}\chi)^{\dagger}(D^{\mu}\chi).$
(31)

The covariant derivatives of scalar fields are written by

$\displaystyle D_{\mu}\Phi$ $\displaystyle=$
$\displaystyle\Big{(}\partial_{\mu}+ig\frac{\tau^{a}}{2}W_{\mu}^{a}+i\frac{g^{\prime}}{2}\tilde{B}_{\mu}+i\frac{g^{\prime}}{2}\rho\tilde{B}^{\prime}_{\mu}\Big{)}\Phi,$
$\displaystyle D_{\mu}\eta$ $\displaystyle=$
$\displaystyle\Big{(}\partial_{\mu}+ig\frac{\tau^{a}}{2}W_{\mu}^{a}+i\frac{g^{\prime}}{2}\tilde{B}_{\mu}+\Big{(}i\frac{g^{\prime}}{2}\rho-
ig^{\prime\prime}\frac{3x}{2}\frac{\rho}{\epsilon}\Big{)}\tilde{B}^{\prime}_{\mu}\Big{)}\eta,$
$\displaystyle D_{\mu}\varphi$ $\displaystyle=$
$\displaystyle\Big{(}\partial_{\mu}-ig^{\prime\prime}x\frac{\rho}{\epsilon}\tilde{B}^{\prime}_{\mu}\Big{)}\varphi,$
$\displaystyle D_{\mu}\chi$ $\displaystyle=$
$\displaystyle\Big{(}\partial_{\mu}-i\frac{g^{\prime\prime}}{2}x\frac{\rho}{\epsilon}\tilde{B}^{\prime}_{m}u\Big{)}\chi,$
(32)

where $g$, $g^{\prime}$ and $g^{\prime\prime}$ are 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.

The scalar fields $H$ and $\varphi$ get VEVs whereas other fields do not
develop VEVs to preserve the $Z_{2}$ symmetry. The masses of the gauge bosons
come from the first and third term of Eq. (31). The mass term of the neutral
gauge bosons written in the basis of neutral gauge fields
$(W_{\mu}^{3},\tilde{B}_{\mu},\tilde{B}^{\prime}_{\mu})$ is

$\displaystyle\mathcal{L}_{\text{gauge}}^{\text{mass}}=\frac{1}{2}\left(\begin{array}[]{c}W_{\mu}^{3}\\\
\tilde{B}_{\mu}\\\
\tilde{B}^{\prime}_{\mu}\end{array}\right)^{T}M^{2}_{\text{gauge}}\left(\begin{array}[]{c}W_{\mu}^{3}\\\
\tilde{B}_{\mu}\\\ \tilde{B}^{\prime}_{\mu}\end{array}\right)$ (39)

where

$\displaystyle
M^{2}_{\text{gauge}}=\frac{1}{4}\left(\begin{array}[]{ccc}g^{2}v^{2}&-gg^{\prime}v^{2}&-gg^{\prime}v^{2}\rho\\\
-gg^{\prime}v^{2}&g^{\prime 2}v^{2}&g^{\prime 2}v^{2}\rho\\\
-gg^{\prime}v^{2}\rho&g^{\prime 2}v^{2}\rho&g^{\prime
2}v^{2}\rho^{2}+4g^{\prime\prime
2}v^{2}_{\varphi}x^{2}\frac{\rho^{2}}{\epsilon^{2}}\\\ \end{array}\right).$
(43)

Here we parameterize

$\displaystyle M_{Z^{\prime}}^{2}=\frac{1}{4}g^{\prime
2}v^{2}\rho^{2}+g^{\prime\prime
2}v_{\varphi}^{2}x^{2}\frac{\rho^{2}}{\epsilon^{2}}.$ (44)

We rotate the fields $(W_{\mu}^{3},\tilde{B}_{\mu})$ by Weinberg angle
$\theta_{W}$ to obtain the massless photon field $A_{\mu}$

$\displaystyle\left(\begin{array}[]{c}W^{3}_{\mu}\\\ \tilde{B}_{\mu}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta_{W}&\sin\theta_{W}\\\
-\sin\theta_{W}&\cos\theta_{W}\\\
\end{array}\right)\left(\begin{array}[]{c}\tilde{Z}_{\mu}\\\ A_{\mu}\\\
\end{array}\right).$ (51)

and the mass matrix for the massive neutral gauge bosons

$\displaystyle\mathcal{L}^{\text{mass}}_{\text{gauge}}=\frac{1}{2}\left(\begin{array}[]{c}\tilde{Z}_{\mu}\\\
\tilde{B}^{\prime}_{\mu}\\\
\end{array}\right)^{T}\left(\begin{array}[]{cc}M_{Z,SM}^{2}&-\Delta^{2}\\\
-\Delta^{2}&M_{Z^{\prime}}^{2}\end{array}\right)\left(\begin{array}[]{c}\tilde{Z}_{\mu}\\\
\tilde{B}^{\prime}_{\mu}\\\ \end{array}\right)$ (58)

where $\Delta^{2}=\frac{1}{4}g^{\prime}v^{2}\rho\sqrt{g^{2}+g^{\prime 2}}$ and
$M_{Z,SM}^{2}=\frac{1}{4}v^{2}(g^{2}+g^{\prime 2})$. The physical masses of
the neutral gauge bosons are

$\displaystyle m_{Z}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}M_{Z,SM}^{2}+M_{Z}^{\prime
2}+\sqrt{(M_{Z,SM}^{2}-M_{Z^{\prime}}^{2})^{2}+4\Delta^{4}}\Big{]}$
$\displaystyle m_{Z^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}M_{Z,SM}^{2}+M_{Z}^{\prime
2}-\sqrt{(M_{Z,SM}^{2}-M_{Z^{\prime}}^{2})^{2}+4\Delta^{4}}\Big{]}.$ (59)

In the limit $\epsilon\to 0$, we have $m_{Z}\approx M_{Z,SM}$ and
$m_{Z^{\prime}}\approx M_{Z^{\prime}}$. The mass matrix in Eq. (58) can be
diagonalized by rotation matrix

$\displaystyle\left(\begin{array}[]{c}\tilde{Z}_{\mu}\\\
\tilde{B}^{\prime}_{\mu}\\\ \end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\\\ \end{array}\right)\left(\begin{array}[]{c}Z_{\mu}\\\
Z^{\prime}_{\mu}\\\ \end{array}\right)$ (66) $\displaystyle\tan 2\theta$
$\displaystyle=$
$\displaystyle\frac{2\Delta^{2}}{M_{Z,SM}^{2}-M_{Z^{\prime}}^{2}}$ (67)

where $Z_{\mu}$ and $Z^{\prime}_{\mu}$ are the two physical gauge bosons
correspond to the SM $Z$ boson and extra gauge boson. The $Z-Z^{\prime}$
mixing would disappear in the limit $\epsilon\rightarrow 0$. In summary, the
original unphysical gauge fields are transformed into mass eigenstates by

$\displaystyle\left(\begin{array}[]{c}W_{\mu}^{3}\\\ B_{\mu}\\\
B^{\prime}_{\mu}\\\ \end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}R^{w}_{11}&R^{w}_{12}&R^{w}_{13}\\\
R^{w}_{21}+R^{w}_{31}\rho&R^{w}_{22}+R^{w}_{32}\rho&R^{w}_{23}+R^{w}_{33}\rho\\\
-R^{w}_{31}\frac{\rho}{\epsilon}&-R^{w}_{32}\frac{\rho}{\epsilon}&-R^{w}_{33}\frac{\rho}{\epsilon}\\\
\end{array}\right)\left(\begin{array}[]{c}Z_{\mu}\\\ A_{\mu}\\\
Z^{\prime}_{\mu}\\\ \end{array}\right)$ (77) $\displaystyle R^{w}_{ab}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}\cos{\theta_{w}}\cos{\theta}&\sin{\theta_{w}}&\cos{\theta_{w}}\sin{\theta}\\\
-\sin{\theta_{w}}\cos{\theta}&\cos{\theta_{w}}&-\sin{\theta_{w}}\sin{\theta}\\\
-\sin{\theta}&0&\cos{\theta}\end{pmatrix}_{ab},$ (78)

where $a(b)=1,2,3$.

###  II.3 Constraints in boson sector

Here we consider constraints in boson sector such as perturbativity,
unitarity, vacuum stability, $T$-parameter and Higgs invisible decay. We
mainly focus on two Higgs doublet $\\{\Phi,\eta\\}$ and $\varphi$ since
parameters associated with $\chi$ are completely free and phenomenological
constraints can be easily satisfied. The constraints from unitarity and
perturbativity are given by Bian:2017xzg ; Muhlleitner:2016mzt

$\displaystyle|\lambda_{\Phi,\eta,\varphi,\Phi\eta}|\leq
4\pi,\quad|\lambda_{\Phi\varphi,\Phi\eta}|\leq
8\pi,\quad|\lambda_{\Phi\eta}\pm\tilde{\lambda}_{\Phi\eta}|\leq
8\pi,\quad|\lambda_{\Phi\eta}+2\tilde{\lambda}_{\Phi\eta}|\leq 8\pi,$
$\displaystyle\sqrt{|\lambda_{\Phi\eta}(\lambda_{\Phi\eta}+2\tilde{\lambda}_{\Phi\eta})|}\leq
8\pi,\quad\left|\lambda_{\Phi}+\lambda_{\eta}\pm\sqrt{(\lambda_{\Phi}-\lambda_{\eta})^{2}+\tilde{\lambda}^{2}_{\Phi\eta}}\right|\leq
8\pi,$
$\displaystyle\left|\lambda_{\Phi}+\lambda_{\eta}\pm\sqrt{(\lambda_{\Phi}-\lambda_{\eta})^{2}}\right|\leq
8\pi,\quad a_{1,2,3}\leq 8\pi,$ (79)

where $a_{1,2,3}$ are obtained as the solutions of the following equation

$\displaystyle
x^{3}-2x^{2}(3\lambda_{\Phi}+3\lambda_{\eta}+2\lambda_{\varphi})$
$\displaystyle-x(2\lambda_{\Phi\varphi}^{2}+2\lambda_{\eta\varphi}^{2}-36\lambda_{\Phi}\lambda_{\eta}-24\lambda_{\Phi}\lambda_{\Phi\varphi}-24\lambda_{1}\lambda_{\Phi\varphi}+4\lambda_{\Phi\eta}^{2}+4\lambda_{\Phi\eta}\tilde{\lambda}_{\Phi\eta}+\tilde{\lambda}_{\Phi\eta}^{2})$
$\displaystyle+4(3\lambda_{\Phi\varphi}^{2}\lambda_{\eta}-\lambda_{\Phi\varphi}\lambda_{\eta\varphi}(2\lambda_{\Phi\eta}+\tilde{\lambda}_{\Phi\eta})+3\lambda_{\eta\varphi}^{2}\lambda_{\Phi}+\lambda_{\varphi}((2\lambda_{\Phi\eta}+\tilde{\lambda}_{\Phi\eta})^{2}-36\lambda_{\Phi}\lambda_{\eta}))=0.$
(80)

We can also writhe the conditions to guarantee vacuum stability such that
Muhlleitner:2016mzt :

$\displaystyle\Omega_{1}\cup\Omega_{2}$ (81)
$\displaystyle\Omega_{1}=\biggl{\\{}\lambda_{\Phi,\eta,\varphi}>0,\
2\sqrt{\lambda_{\Phi}\lambda_{\varphi}}+\lambda_{\Phi\varphi}>0,\
2\sqrt{\lambda_{\eta}\lambda_{\varphi}}+\lambda_{\eta\varphi}>0,$
$\displaystyle\qquad\qquad
2\sqrt{\lambda_{\Phi}\lambda_{\eta}}+\lambda_{\Phi\eta}>0,\
\lambda_{\Phi\varphi}+\sqrt{\frac{\lambda_{\Phi}}{\lambda_{\eta}}}\lambda_{\Phi\varphi}\geq
0\biggr{\\}}$ (82)
$\displaystyle\Omega_{2}=\biggl{\\{}\lambda_{\Phi,\eta,\varphi}>0,\
2\sqrt{\lambda_{\Phi}\lambda_{\varphi}}\geq\lambda_{\eta\varphi}>-2\sqrt{\lambda_{\eta}\lambda_{\varphi}},\
2\sqrt{\lambda_{\Phi}\lambda_{\varphi}}>-\lambda_{\Phi\varphi}\geq\sqrt{\frac{\lambda_{\Phi}}{\lambda_{\eta}}}\lambda_{\eta\varphi},$
$\displaystyle\qquad\qquad\sqrt{(\lambda_{\Phi\varphi}^{2}-4\lambda_{\Phi}\lambda_{\varphi})(\lambda_{\eta\varphi}^{2}-4\lambda_{\eta}\lambda_{\varphi})}>\lambda_{\Phi\varphi}\lambda_{\eta\varphi}-2\lambda_{\Phi\eta}\lambda_{\varphi}\biggr{\\}}.$
(83)

In addition we impose inert condition for $\eta$ requiring the potential be
bounded below Nomura:2019yft

$-\frac{2}{\sqrt{21}}\sqrt{\lambda_{\Phi}\lambda_{\eta}}<\lambda_{\Phi\eta}+\tilde{\lambda}_{\Phi\eta},\quad-\sqrt{6}\lambda^{\prime}_{\Phi\eta}<|\lambda_{\Phi\eta}+\tilde{\lambda}_{\Phi\eta}|,\quad-\frac{2}{\sqrt{21}}\sqrt{\lambda_{\eta}\lambda_{\varphi}}<\lambda_{\eta\varphi}.$
(84)

The contribution to $T$-parameter from scalar loops can be written as
Barbieri:2006dq

$\displaystyle\Delta T=\frac{1}{32\alpha
v^{2}}\left(G[m_{\eta^{\pm}}^{2},m_{A_{1}}^{2}]\cos^{2}\gamma+G[m_{\eta^{\pm}}^{2},m_{A_{2}}^{2}]\sin^{2}\gamma\right.$
$\displaystyle\qquad\qquad\qquad\left.+G[m_{\eta^{\pm}}^{2},m_{H_{1}}^{2}]\cos^{2}\alpha+G[m_{\eta^{\pm}}^{2},m_{H_{2}}^{2}]\sin^{2}\alpha\right),$
(85) $\displaystyle
G[m_{1}^{2},m_{2}^{2}]\equiv\frac{m_{1}^{2}+m_{2}^{2}}{2}-\frac{m_{1}^{2}m_{2}^{2}}{m_{1}^{2}-m_{2}^{2}}\ln\left[\frac{m_{1}^{2}}{m_{2}^{2}}\right],$
(86)

where $\alpha$ is the fine structure constant. Here we adopt the constraint
Tanabashi:2018oca

$-0.01<\Delta T<0.11,$ (87)

where vanishing $U$-parameter is assumed.

In our analysis below, we consider relatively small $v_{\varphi}(\ll v)$ to
satisfy constraints from $L^{\prime}$–$L$ mixing induced by
$\bar{L}_{L}L^{\prime}_{R}\varphi$ Yukawa interaction. Thus $Z^{\prime}$ and
$H$ masses tend to be light and we need to consider decay processes of $h\to
Z^{\prime}Z^{\prime}$ and $h\to HH$. The decay widths are given by

$\displaystyle\Gamma(h\to
Z^{\prime}Z^{\prime})=\frac{1}{8\pi}\frac{m_{Z^{\prime}}^{4}}{v_{\varphi}^{2}m_{h}}s^{2}_{\beta}\left(2+\frac{m_{h}^{4}}{4m_{Z^{\prime}}^{4}}\left(1-\frac{2m_{Z^{\prime}}^{2}}{m_{h}^{2}}\right)^{2}\right)\sqrt{1-\frac{4m_{Z^{\prime}}^{2}}{m_{h}^{2}}},$
(88) $\displaystyle\Gamma(h\to HH)=\frac{C_{hHH}^{2}}{16\pi
m_{h}}\sqrt{1-\frac{4m_{H}^{2}}{m_{h}^{2}}},$ (89) $\displaystyle
C_{hHH}\equiv\frac{\lambda_{\Phi\varphi}}{2}\left(v(c^{2}_{\beta}-2c_{\beta}s^{2}_{\beta})+v_{\varphi}(s^{3}_{\beta}-2c^{2}_{\beta}s_{\beta})\right)+\frac{9}{2}s_{\beta}c_{\beta}(\lambda_{\Phi}s_{\beta}v+\lambda_{\varphi}c_{\beta}v_{\varphi}),$
(90)

where $c_{\beta}=\cos\beta(s_{\beta}=\sin\beta)$. Here $H$ mainly decays into
$Z^{\prime}Z^{\prime}$ mode and $Z^{\prime}$ decays into the SM particles via
kinetic mixing or Lepton mixing effect. In our scenario we consider kinetic
mixing is tiny and $Z^{\prime}$ is long lived to escape detector at the LHC
experiments; $Z^{\prime}$ could decay into leptons via $L^{\prime}$–$L$ mixing
discussed below and it is also considered to be tiny. Then we apply constrains
from invisible decay of $h$ for these decay branching ratio (BR) where the
upper bound is taken to be $BR(h\to inv)<0.19$ Aaboud:2019rtt ;
Sirunyan:2018owy . We obtain BRs of $h\to Z^{\prime}Z^{\prime}$ and $h\to HH$
modes in the case of $m_{Z^{\prime}H}^{2}\ll m_{h}^{2}$ as follows

$\displaystyle BR(h\to Z^{\prime}Z^{\prime})\simeq 4.6\times
10^{-4}\left(\frac{s_{\beta}}{10^{-5}}\right)^{2}\left(\frac{1\ {\rm
GeV}}{v_{\varphi}}\right)^{2},$ (91) $\displaystyle BR(h\to HH)\simeq
9.3\times 10^{-4}\left(\frac{s_{\beta}}{10^{-5}}\right)^{2}\left(\frac{1\ {\rm
GeV}}{v_{\varphi}}\right)^{2},$ (92)

where we used the SM Higgs decay width $\Gamma_{h}\simeq 4.2$ MeV. Thus very
small $s_{\beta}$ is required to avoid constraints from the SM Higgs decay.

Finally we discuss allowed region from the above constraints where we assume
$v_{\varphi}\in[1,5]$ GeV, $g^{\prime\prime}\in[0.001,1]$, $m_{H}\in[0.1,50]$
GeV, $m_{H_{1}}\in[150,1500]$ GeV, $m_{H_{2}}\in[0.1,1500]$ GeV,
$m_{A_{1}}\in[150,1500]$ GeV and $m_{A_{2}}\in[0.1,1500]$ GeV. We also assume
small mixing among inert scalar bosons as $\sin\alpha<0.1$ and
$\sin\gamma<0.1$. We find that mass of $H$ is light as $m_{H}\lesssim 20$ GeV
due to small $v_{\varphi}$ and unitarity condition as shown in left plot of
Fig. 1. Also $\Delta T$ constraint requires $|m_{H_{1}}-m_{A_{1}}|\lesssim 4$
GeV and $|m_{\eta^{\pm}}-m_{H_{1}}|\lesssim 120$ GeV. The mixing between the
SM Higgs and $H$ should be suppressed to avoid constraint from invisible decay
as shown in right plot of Fig. 1. The other scalar mass values are not
constrained.

Figure 1: Allowed region on $\\{v_{\varphi},m_{H}\\}$ and
$\\{v_{\varphi},\sin\beta\\}$ planes.

##  III Neutrino mass and lepton flavor violation

In this section we formulate neutrino mass matrix, LFV processes and muon
$g-2$. We then carry out numerical analysis to search for allowed parameter
ranges taking into account experimental constraints.

###  III.1 Neutrino mass generation

We obtain neutral fermion mass terms at tree level, after scalar fields
developing VEVs, such that

$\displaystyle
L\supset\bar{\nu}_{L}M_{D}N^{\prime}_{R}+\bar{N}^{\prime}_{L}M_{L^{\prime}}N^{\prime}_{R}+\bar{N}_{R}M_{N}N_{L}+\bar{N}^{c}_{R}M_{R}N_{R}+\bar{N}^{c}_{L}M_{L}N_{L}+h.c.,$
(93)

where $M_{D}\equiv y_{D}v_{\varphi}/\sqrt{2}$ and $M_{L/R}\equiv
y_{\varphi_{L/R}}v_{\varphi}/\sqrt{2}$. Then neutrino mass matrices at tree
level are obtained as

$L_{M_{\nu}}^{\text{tree
}}=\frac{1}{2}\left(\begin{array}[]{c}\bar{\nu}^{c}_{L}\\\
\bar{N}^{\prime}_{R}\\\ \bar{N}^{\prime
c}_{L}\end{array}\right)^{T}\left(\begin{array}[]{ccc}0&M^{*}_{D}&0\\\
M_{D}^{\dagger}&0&M_{L^{\prime}}^{\dagger}\\\
0&M_{L^{\prime}}^{*}&0\end{array}\right)\left(\begin{array}[]{c}\nu_{L}\\\
N^{\prime c}_{R}\\\ N^{\prime}_{L}\end{array}\right)+h.c.$ (94) $\displaystyle
L_{M_{N}}^{\text{tree}}=\frac{1}{2}\left(\begin{array}[]{c}\bar{N}^{c}_{L}\\\
\bar{N}_{R}\end{array}\right)^{T}\left(\begin{array}[]{cc}2M_{L}&M^{T}_{N}\\\
M_{N}&2M^{\dagger}_{R}\\\ \end{array}\right)\left(\begin{array}[]{c}N_{L}\\\
N^{c}_{R}\end{array}\right)+h.c.$ (101)

The $22$ and $33$ components of the mass matrix in Eq. (94) are generated at
one-loop level by diagrams in Fig. 2. Calculating the diagrams, we obtain one-
loop contributions

$\displaystyle\delta\mu_{22(33)}$ $\displaystyle=$
$\displaystyle-\sum_{\alpha}\frac{{Y_{R}}_{2(3)\alpha}{Y^{T}_{R}}_{\alpha
2(3)}}{32\pi^{2}}M_{N_{\alpha}}\int_{0}^{1}dx\Big{[}\cos^{2}\alpha\log
Q^{2}_{H_{1}}+\sin^{2}\alpha\log Q^{2}_{H_{2}}$ (102) $\displaystyle-$
$\displaystyle\cos^{2}\gamma\log Q^{2}_{A_{1}}-\sin^{2}\gamma\log
Q^{2}_{A_{2}}\Big{]}$

where $Q_{\phi}^{2}=xM_{N_{\alpha}}^{2}+(1-x)m_{\phi}^{2}$ and
$M_{N_{\alpha}}$ is the mass obtained after diagonalizing Eq. [101].

Then the inverse seesaw mass matrix becomes

$L_{M_{\nu}}^{\text{tree +
1-loop}}=\frac{1}{2}\left(\begin{array}[]{c}\bar{\nu}^{c}_{L}\\\
\bar{N}^{\prime}_{R}\\\ \bar{N}^{\prime
c}_{L}\end{array}\right)^{T}\left(\begin{array}[]{ccc}0&M^{*}_{D}&0\\\
M_{D}^{\dagger}&\delta\mu_{22}&M_{L^{\prime}}^{\dagger}\\\
0&M_{L^{\prime}}^{*}&\delta\mu_{33}\end{array}\right)\left(\begin{array}[]{c}\nu_{L}\\\
N^{\prime c}_{R}\\\ N^{\prime}_{L}\end{array}\right)+h.c.$ (103)

Since our model has the following hierarchies
$\delta\mu_{22(33)},M_{D}<<M_{L^{\prime}}$, 555We will see
$M_{D}/M_{L^{\prime}}<<1$ is demanded by LFVs in the subsection III B.
$\delta\mu_{22(33)}<<M_{L^{\prime}}$ would be rather natural, since
$\mu_{22(33)}$ is induced at one-loop level while $M_{L^{\prime}}$ is bare
mass.  the inverse seesaw mass matrix can be block diagonalized by unitary
matrix $U_{\nu}$ to obtain the mass matrix of the light (active) neutrinos and
heavy neutrinos as

$\displaystyle M_{\nu}$ $\displaystyle=$
$\displaystyle-\left(\begin{array}[]{cc}M^{*}_{D}&0\\\
\end{array}\right)\left(\begin{array}[]{cc}\delta\mu_{22}&M^{\dagger}_{L^{\prime}}\\\
M^{*}_{L^{\prime}}&\delta\mu_{33}\\\
\end{array}\right)^{-1}\left(\begin{array}[]{c}M_{D}^{\dagger}\\\ 0\\\
\end{array}\right)$ $\displaystyle=$ $\displaystyle
M_{D}^{*}{M^{*}_{L^{\prime}}}^{-1}\delta
m_{33}{M^{\dagger}_{L^{\prime}}}^{-1}M_{D}^{\dagger}$

and

$\displaystyle
M_{N^{\prime}}=\left(\begin{array}[]{cc}\delta\mu_{22}&M^{\dagger}_{L^{\prime}}\\\
M^{*}_{L^{\prime}}&\delta\mu_{33}\\\ \end{array}\right)$ (112)

such that

$\displaystyle U_{\nu}^{T}\left(\begin{array}[]{ccc}0&M^{*}_{D}&0\\\
M_{D}^{\dagger}&\delta\mu_{22}&M_{L^{\prime}}^{\dagger}\\\
0&M_{L^{\prime}}^{*}&\delta\mu_{33}\end{array}\right)U_{\nu}=\left(\begin{array}[]{cc}M_{\nu}&0\\\
0&M_{N^{\prime}}\\\ \end{array}\right)$ (118)

where the unitary matrix $U_{\nu}$ is given by

$\displaystyle U_{\nu}=\left[\begin{array}[]{cc}1_{3\times
3}&\left(\begin{array}[]{cc}M_{D}^{*}&0\end{array}\right)\left(\begin{array}[]{cc}\delta\mu_{22}&M^{\dagger}_{L^{\prime}}\\\
M^{*}_{L}&\delta\mu_{33}\\\ \end{array}\right)^{-1}\\\
-\left(\begin{array}[]{cc}\delta\mu_{22}&M^{\dagger}_{L^{\prime}}\\\
M^{*}_{L}&\delta\mu_{33}\\\
\end{array}\right)^{-1}\left(\begin{array}[]{c}M_{D}^{\dagger}\\\ 0\\\
\end{array}\right)&1_{6\times 6}\end{array}\right]$ (128)
$\displaystyle=\left[\begin{array}[]{cc}1_{3\times
3}&\begin{array}[]{cc}-M_{D}^{*}(M_{L^{\prime}}^{*})^{-1}\delta\mu_{33}(M_{L^{\prime}}^{\dagger})^{-1}&M_{D}^{*}(M_{L^{\prime}}^{*})^{-1}\end{array}\\\
\begin{array}[]{c}(M_{L^{\prime}}^{*})^{-1}\delta\mu_{33}(M_{L^{\prime}}^{\dagger})^{-1}M_{D}^{\dagger}\\\
-(M_{L^{\prime}}^{\dagger})^{-1}M_{D}^{\dagger}\end{array}&1_{6\times
6}\end{array}\right]$ (134)

Using the Cholesky factorization procedure, the symmetric matrix
$N={M^{*}_{L^{\prime}}}^{-1}\delta m_{33}{M^{\dagger}_{L^{\prime}}}^{-1}$ can
be written as $N=RR^{T}$ where $R$ is an upper-right triangle matrix. The
light neutrino mass matrix $M_{\nu}$ can be diagonalized by Pontecorvo-Maki-
Nakagawa-Sakata (PMNS) matrix $U_{PMNS}$ to obtain the physical masses of the
light neutrinos $D_{\nu}={\rm diag}(m_{\nu_{1}},m_{\nu_{2}},m_{\nu_{3}})$

$\displaystyle D_{\nu}=U_{PMNS}^{T}M_{\nu}U_{PMNS}$ (135)

The standard parameterization of $U_{PMNS}$ in terms of mixing angles
$\theta_{12},~{}\theta_{13},~{}\theta_{23}$, Dirac $CP$ phase $\delta$, and
Majorana phases $\alpha_{1},~{}\alpha_{2}$ is

$\displaystyle
U_{PMNS}=\left(\begin{array}[]{ccc}c_{12}c_{13}e^{i\alpha_{1}/2}&s_{12}c_{13}e^{i\alpha_{2}/2}&s_{13}e^{-i\delta}\\\
(-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta})e^{i\alpha_{1}/2}&(c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta})e^{i\alpha_{2}/2}&s_{23}c_{13}\\\
(s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta})e^{i\alpha_{1}/2}&(-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta})e^{i\alpha_{2}/2}&c_{23}c_{13}\\\
\end{array}\right)$ (139)

where $s(c)_{ij}=\sin(\cos)\theta_{ij}$ and without any loss of generality we
can consider $\theta_{ij}\in[0,\pi/2]$. Following Casas-Ibarra
parameterization, we get

$\displaystyle M^{*}_{D}$ $\displaystyle=$ $\displaystyle
U^{*}_{PMNS}D_{\nu}^{1/2}\mathcal{O}R^{-1}$ $\displaystyle\text{which
implies}~{}~{}y^{*}_{D}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{v_{\varphi}}U^{*}_{PMNS}D_{\nu}^{1/2}\mathcal{O}R^{-1}$
(140)

where $\mathcal{O}$ is an orthogonal matrix with three independent complex
angles. In our numerical analysis we will consider the perturbative limit
$|y_{D}|\leq\sqrt{4\pi}$.

Figure 2: The one-loop diagram generating Majorana mass terms of
$N^{\prime}_{L(R)}$.

The mass matrix for the heavy neutrino $M_{N^{\prime}}$ is diagonalized
perturbatively by an unitary matrix $U_{N^{\prime}}$ to obtain the masses
$D_{N^{\prime}}={\rm
diag}(m_{N^{\prime}_{1}},m_{N^{\prime}_{2}},m_{N^{\prime}_{3}},m_{N^{\prime}_{4}},m_{N^{\prime}_{5}},m_{N^{\prime}_{6}})$
of the six heavy majorana neutrinos

$\displaystyle
D_{N^{\prime}}=U_{N^{\prime}}^{T}M_{N^{\prime}}U_{N^{\prime}}=\left(\begin{array}[]{cccccc}m_{E^{\prime}_{i}}-\delta{\mu_{22}}^{ii}-\delta\mu_{33}^{ii}&0\\\
0&m_{E^{\prime}_{i}}+\delta{\mu_{22}}^{ii}+\delta\mu_{33}^{ii}\\\
\end{array}\right)$ (143)

In the limit $\delta_{\mu_{22(33)}}\to 0$ we can consider the the heavy
neutrinos to be Dirac type fermions. In our case we identify $N^{\prime}$
flavour state as three heavy Dirac neutrinos.

###  III.2 Lepton flavor violation and muon $g-2$

The charged lepton mass term obtained from the Yukawa interaction is

$\displaystyle
L^{\text{mass}}_{\text{lepton}}=\bar{e}_{L}M_{l^{\prime}}e_{R}+\bar{E^{\prime}}_{L}M_{L^{\prime}}E^{\prime}_{R}+\bar{e}_{L}M_{D}E^{\prime}_{R}+h.c.$
(144)

which can be written in matrix form as

$\displaystyle
L^{\text{mass}}_{\text{lepton}}=\left(\begin{array}[]{cc}\bar{e}_{L}&\bar{E^{\prime}}_{L}\\\
\end{array}\right)\left(\begin{array}[]{cc}M_{l^{\prime}}&M_{D}\\\
0&M_{L^{\prime}}\\\ \end{array}\right)\left(\begin{array}[]{c}e_{R}\\\
E^{\prime}_{R}\\\ \end{array}\right)+h.c.$ (150)

where $M_{l^{\prime}}=y_{l}v/\sqrt{2}$. The mass matrix can be diagonalized by
bi-unitary transformation

$\displaystyle
M^{\text{diag}}=U_{L}^{\dagger}\left(\begin{array}[]{cc}M_{l^{\prime}}&M_{D}\\\
0&M_{L^{\prime}}\\\ \end{array}\right)U_{R}$ (153)

where $U_{L}$ and $U_{R}$ are $6\times 6$ unitary matrices such that the field
transformations are

$\displaystyle\left(\begin{array}[]{c}e_{L}\\\
E^{\prime}_{L}\end{array}\right)=U_{L}~{}l_{L},\quad\left(\begin{array}[]{c}e_{R}\\\
E^{\prime}_{R}\end{array}\right)=U_{R}~{}l_{R}.$ (158)

Here we consider the elements of the submatrix $M_{D}$ to be very small in
comparison to diagonal heavy lepton mass matrix $M_{L^{\prime}}$; $M_{D}\ll
M_{L^{\prime}}$. The smallness of the submatrix $M_{D}$ is required by the
constraints from lepton flavor violating (LFV) processes like $BR(l_{i}\to
l_{j}\gamma)$. In such a case, unitary matrices $U_{L}$ and $U_{R}$ are
approximated as

$U_{L}\simeq\begin{pmatrix}\bm{1}&\bm{0}\\\
-\left(M_{D}M_{L^{\prime}}^{-1}\right)^{\dagger}&\bm{1}\end{pmatrix}\equiv\begin{pmatrix}\bm{1}&\bm{0}\\\
-\delta U&\bm{1}\end{pmatrix},\quad U_{R}\simeq\begin{pmatrix}\bm{1}&\bm{0}\\\
\bm{0}&\bm{1}\end{pmatrix},$ (159)

where $\bm{0}$ and $\bm{1}$ are $3\times 3$ zero and unity matrices
respectively.

To estimate LFV BRs the relevant Yukawa interaction terms are

$\displaystyle L\supset$
$\displaystyle\bar{\nu}_{i}\frac{U^{*}_{PMNS_{ji}}y_{D}^{j\alpha}}{\sqrt{2}}N^{\prime}_{\alpha}(\sin\beta
h+\cos\beta
H)+\bar{N^{\prime}_{\alpha}}\frac{y_{D}^{*j\alpha}U_{PMNS_{ji}}}{\sqrt{2}}P_{L}\nu_{i}(\sin\beta
h+\cos\beta H)$ (160) $\displaystyle+$
$\displaystyle\bar{e}_{i}\frac{y_{D}^{i\alpha}}{\sqrt{2}}E^{\prime}_{\alpha}(\sin\beta
h+\cos\beta
H)+\bar{E^{\prime}_{\alpha}}\frac{y_{D}^{*i\alpha}}{\sqrt{2}}P_{L}e_{i}(\sin\beta
h+\cos\beta H),$ $\displaystyle+$
$\displaystyle\frac{i}{\sqrt{2}}y_{D}^{i\alpha}\bar{e}_{i}P_{R}E^{\prime}_{\alpha}G_{Z^{\prime}}+h.c.,$

where summation over repeated index is implied. In this Lagrangian the light
neutrinos $\nu_{i}$ are three Majorana neutrinos and $N^{\prime}_{\alpha}$ are
three heavy Dirac neutrinos. Note that we write interaction among $e^{i}$,
$E^{\alpha}$ and NG boson $G_{Z^{\prime}}$ to calculate muon $g-2$ in Feynman
gauge. Flavor violating $Z^{\prime}$ interaction terms are also given by

$\displaystyle L\supset-xg^{\prime\prime}\left[\delta U_{\alpha
i}\bar{E}^{\prime}_{\alpha}\gamma_{\mu}P_{L}e_{i}+(\delta
U^{\dagger})_{i\alpha}\bar{e}_{i}\gamma_{\mu}P_{L}E^{\prime}_{\alpha}\right]Z^{\prime\mu}+xg^{\prime\prime}(\delta
U^{\dagger}\delta U)_{ij}\bar{e}_{i}\gamma_{\mu}P_{L}e_{j}Z^{\prime\mu},$
(161)

where $\delta U$ is $3\times 3$ matrix given in Eq. (159). From the
interaction we obtain $Z^{\prime}\to\ell_{i}\bar{\ell}_{j}$ decay width for
vanishing kinetic mixing such that

$\displaystyle\Gamma_{Z^{\prime}\to\ell_{i}\bar{\ell}_{j}}=$
$\displaystyle\frac{|xg^{\prime\prime}(\delta U^{\dagger}\delta
U)_{ij}|^{2}}{24\pi}m_{Z^{\prime}}\lambda^{\frac{1}{2}}\left(1,\frac{m_{\ell_{1}}^{2}}{m_{Z^{\prime}}^{2}},\frac{m_{\ell_{2}}^{2}}{m_{Z^{\prime}}^{2}}\right)$
(162)
$\displaystyle\times\left[1-\frac{m_{\ell_{1}}^{2}}{m_{Z^{\prime}}^{2}}-\frac{m_{\ell_{2}}^{2}}{m_{Z^{\prime}}^{2}}+\sqrt{\left(\lambda\left(1,\frac{m_{\ell_{1}}^{2}}{m_{Z^{\prime}}^{2}},\frac{m_{\ell_{2}}^{2}}{m_{Z^{\prime}}^{2}}\right)+\frac{4m_{\ell_{1}}^{2}}{m_{Z^{\prime}}^{2}}\right)\left(\lambda\left(1,\frac{m_{\ell_{1}}^{2}}{m_{Z^{\prime}}^{2}},\frac{m_{\ell_{2}}^{2}}{m_{Z^{\prime}}^{2}}\right)+\frac{4m_{\ell_{2}}^{2}}{m_{Z^{\prime}}^{2}}\right)}\right],$

where $\lambda(1,x,y)\equiv 1+x^{2}+y^{2}-2x-2y-2xy$. If light $Z^{\prime}$
can only decay through this lepton mixing effect its lifetime will be long;
for example $\tau_{Z^{\prime}}\sim 5\times 10^{-4}s$ for $|(\delta
U^{\dagger}\delta U)_{22}|=10^{-8}$, $g^{\prime\prime}\sim 0.1$ and
$m_{Z^{\prime}}=1$ GeV.

Considering the one-loop diagram with scalars $h,~{}H$ and leptons
$E^{\prime}_{\alpha}$ running in the loop, we obtain Lindner:2016bgg ;
Baek:2016kud

$\displaystyle BR(l_{i}\to
l_{j}\gamma)=\frac{3(4\pi)^{3}\alpha_{em}}{4G_{F}^{2}}\Big{[}|A^{M}_{ji}|^{2}+|A^{E}_{ji}|^{2}\Big{]}BR(\l_{i}\to
l_{j}\nu_{i}\bar{\nu}_{j})$ (163)

where

$\displaystyle A^{M}_{ji}$ $\displaystyle=$
$\displaystyle\frac{m_{i}+m_{j}}{16\pi^{2}m_{i}}\sin^{2}\beta\sum_{\alpha}F(m_{E^{\prime}_{\alpha}},m_{h})y^{j\alpha}_{D}y^{*i\alpha}_{D}$
$\displaystyle+$
$\displaystyle\frac{m_{i}+m_{j}}{16\pi^{2}m_{i}}\cos^{2}\beta\sum_{\alpha}F(m_{E^{\prime}_{\alpha}},m_{H})y^{j\alpha}_{D}y^{*i\alpha}_{D}$
$\displaystyle-$
$\displaystyle\frac{m_{i}+m_{j}}{8\pi^{2}m_{i}}x^{2}g^{\prime\prime
2}\sum_{\alpha}\hat{F}(m_{E^{\prime}_{\alpha}},m_{Z^{\prime}})\delta U_{\alpha
j}(\delta U^{\dagger})_{i\alpha}$ $\displaystyle+$
$\displaystyle\frac{m_{i}+m_{j}}{16\pi^{2}m_{i}}\sum_{\alpha}F(m_{E^{\prime}_{\alpha}},m_{Z^{\prime}})y^{j\alpha}_{D}y^{*i\alpha}_{D}$
$\displaystyle A^{E}_{ji}$ $\displaystyle=$
$\displaystyle\frac{m_{i}-m_{j}}{16\pi^{2}m_{i}}\sin^{2}\beta\sum_{\alpha}F(m_{E^{\prime}_{\alpha}},m_{h})y^{j\alpha}_{D}y^{*i\alpha}_{D}$
(164) $\displaystyle+$
$\displaystyle\frac{m_{i}-m_{j}}{16\pi^{2}m_{i}}\cos^{2}\beta\sum_{\alpha}F(m_{E^{\prime}_{\alpha}},m_{H})y^{j\alpha}_{D}y^{*i\alpha}_{D}$
$\displaystyle-$
$\displaystyle\frac{m_{i}-m_{j}}{8\pi^{2}m_{i}}x^{2}g^{\prime\prime
2}\sum_{\alpha}\hat{F}(m_{E^{\prime}_{\alpha}},m_{Z^{\prime}})\delta U_{\alpha
j}(\delta U^{\dagger})_{i\alpha}$ $\displaystyle+$
$\displaystyle\frac{m_{i}-m_{j}}{16\pi^{2}m_{i}}\sum_{\alpha}F(m_{E^{\prime}_{\alpha}},m_{Z^{\prime}})y^{j\alpha}_{D}y^{*i\alpha}_{D},$

where the last two terms in the RHS are respectively contributions from
$Z^{\prime}$ (transverse mode) and NG boson, and loop integration factor is

$\displaystyle F(m_{a},m_{b})$ $\displaystyle=$
$\displaystyle\frac{m_{a}^{6}+2m_{b}^{6}-6m_{a}^{4}m_{b}^{2}+3m_{a}^{2}m_{b}^{4}+12m_{a}^{2}m_{b}^{4}\log(\frac{m_{a}}{m_{b}})}{12(m_{a}^{2}-m_{b}^{2})^{4}},$
$\displaystyle\hat{F}(m_{a},m_{b})$ $\displaystyle=$
$\displaystyle\frac{5m_{a}^{6}+4m_{b}^{6}-9m_{a}^{2}m_{b}^{4}-12m_{a}^{2}m_{b}^{2}(2m_{a}^{2}-m_{b}^{2})\log(\frac{m_{a}}{m_{b}})}{12(m_{a}^{2}-m_{b}^{2})^{4}},$
(165)

Note that the second term of RHS in Eq. (161) also contributes to
$\ell_{i}\to\ell_{j}\gamma$ process through $Z^{\prime}$ loop. However this
effect is suppressed since we take $\delta
U=-(M_{D}/M_{L^{\prime}})^{\dagger}$ to be very small. In our numerical
calculation, we use explicit form of Eq. (163) to impose constraints from the
LFV decay BRs.

In addition we consider $\mu$-$e$ conversion through LFV interaction
associated with $Z$ boson induced by the lepton mixing effect
Crivellin:2020ebi . The effective interaction for $\mu$-$e$ conversion is
written as

$\displaystyle
L_{eff}=\sum_{q=u,d}(C^{LL}_{q}O_{q}^{LL}+C^{LR}_{q}O_{q}^{LR})+(L\leftrightarrow
R)+h.c.\ ,$ (166) $\displaystyle
O_{q}^{XX}=(\bar{e}\gamma^{\mu}P_{X}\mu)(\bar{q}\gamma_{\mu}P_{X}q),\quad
C_{q}^{XX}=\Gamma^{X}_{e\mu}\frac{1}{m_{Z}^{2}}\Gamma_{qq}^{X},$ (167)

where $X=L,R$ and $\Gamma^{X}_{qq(e\mu)}$ is $Z$ couplings given by

$\displaystyle\Gamma_{uu}^{L}=-\frac{g}{c_{W}}\left(\frac{1}{2}-\frac{2}{3}s_{W}^{2}\right),\quad\Gamma_{uu}^{R}=\frac{2}{3}\frac{gs_{W}^{2}}{c_{W}},$
(168)
$\displaystyle\Gamma_{dd}^{L}=-\frac{g}{c_{W}}\left(-\frac{1}{2}+\frac{1}{3}s_{W}^{2}\right),\quad\Gamma_{dd}^{R}=-\frac{1}{3}\frac{gs_{W}^{2}}{c_{W}},$
(169)
$\displaystyle\Gamma_{e\mu}^{L}\simeq-\frac{g}{c_{W}}\left(-\frac{1}{2}+s_{W}^{2}\right)\frac{M_{D_{ea}}M^{\dagger}_{D_{a\mu}}}{M^{2}_{L^{\prime}_{a}}},\quad\Gamma_{e\mu}^{R}\simeq
0,$ (170)

with $s_{W}(c_{W})=\sin\theta_{W}(\cos\theta_{W})$. Then $\mu$–$e$ conversion
ratio is given by Crivellin:2017rmk ; Cirigliano:2009bz ; Kitano:2002mt

$\Gamma^{N}_{\mu\to
e}=4m_{\mu}^{5}\left|\sum_{q=u,d}\left(C_{q}^{RL}+C_{q}^{RR}\right)\left(f_{p}^{q}V_{N}^{p}+f_{n}^{q}V_{N}^{n}\right)\right|^{2}+(L\leftrightarrow
R),$ (171)

where $f_{p}^{u}=2$, $f_{n}^{u}=1$, $f_{p}^{d}=1$, $f_{n}^{d}=2$ and
$V_{N}^{p(n)}$ is the overlap integrals for nuclear $N$. For gold, we obtain
$V_{Au}^{p(n)}=0.0974(0.146)$ Kitano:2002mt . We need to normalize the
conversion ratio by capture rate $\Gamma_{Au}^{\rm capt}=8.7\times 10^{-18}$
GeV Suzuki:1987jf , and the constraint is given by Bertl:2006up

$\frac{\Gamma^{Au}_{\mu\to e}}{\Gamma_{Au}^{\rm capt}}<7.0\times 10^{-13}.$
(172)

We then obtain the constraint on lepton mixing such that

$\frac{M_{D_{ea}}M^{\dagger}_{D_{a\mu}}}{M^{2}_{L^{\prime}_{a}}}\lesssim
2.8\times 10^{-7}$ (173)

The constraint is weaker than $\mu\to e\gamma$ but it will be tested in future
experiments with more precision. Hence the lepton mixing should be very small,
and for a good approximation we can consider no lepton mixing when we consider
SM lepton interactions.

Muon anomalous magnetic moment
It is known as hint of physics beyond the SM since observed value is deviated
from the SM prediction. The deviation from the SM prediction $\Delta a_{\mu}$
is about 3.7$\sigma$ Bennett:2006fi ; Aoyama:2020ynm ;

$\displaystyle\Delta a_{\mu}=(2.706\pm 0.726)\times 10^{-9}.~{}$ (174)

It suggests that new physics would be needed. Our new contribution to the muon
$g-2$ is obtained to be

$\displaystyle\Delta a_{\mu}$ $\displaystyle=$
$\displaystyle\frac{m_{\mu}^{2}}{8\pi^{2}}\sin^{2}\beta\sum_{\alpha}F(m_{E^{\prime}_{\alpha}},m_{h})y_{D}^{2\alpha}y_{D}^{*2\alpha}$
(175) $\displaystyle+$
$\displaystyle\frac{m_{\mu}^{2}}{8\pi^{2}}\cos^{2}\beta\sum_{\alpha}F(m_{E^{\prime}_{\alpha}},m_{H})y_{D}^{2\alpha}y_{D}^{*2\alpha}$
$\displaystyle-$
$\displaystyle\frac{m_{\mu}^{2}}{4\pi^{2}}x^{2}g^{\prime\prime
2}\sum_{\alpha}\hat{F}(m_{E^{\prime}_{\alpha}},m_{Z^{\prime}})\delta U_{\alpha
2}(\delta U^{\dagger})_{2\alpha}$ $\displaystyle+$
$\displaystyle\frac{m_{\mu}^{2}}{8\pi^{2}}\sum_{\alpha}F(m_{E^{\prime}_{\alpha}},m_{Z^{\prime}})y_{D}^{2\alpha}y_{D}^{*2\alpha},$

where the last two terms in the RHS are respectively contributions from
$Z^{\prime}$ (transverse mode) and NG boson.

Flavor violating $\ell_{i}\to\ell_{j}Z^{\prime}$ decay
We also have this kind of LFV process induced by the second term of Eq. (161).
For $m_{Z^{\prime}}<m_{i}-m_{j}$, decay width is given by

$\displaystyle\Gamma(\ell_{i}\to\ell_{j}Z^{\prime})=x^{2}g^{\prime\prime
2}(\delta U^{\dagger}\delta U)^{2}_{ij}\frac{k}{8\pi
m_{i}}\left(\sqrt{k^{2}+m_{j}^{2}}+\frac{1}{m_{Z^{\prime}}^{2}}(m_{i}^{2}-m_{j}^{2}-m^{2}_{Z^{\prime}})\sqrt{k^{2}+m^{2}_{Z^{\prime}}}\right),$
(176) $\displaystyle
k=\frac{m_{i}}{2}\lambda^{\frac{1}{2}}\left(1,\frac{m_{j}^{2}}{m^{2}_{i}},\frac{m^{2}_{Z^{\prime}}}{m^{2}_{i}}\right).$

Then $Z^{\prime}$ decays into leptons through the mixing effect giving three
charged lepton final states. Thus when these modes are kinematically allowed
there are strong constraint for $\delta U=M_{D}^{\dagger}/M_{L^{\prime}}$. For
$m_{Z^{\prime}}<m_{\mu}$, it is constrained by $BR(\mu\to 3e)\lesssim
10^{-12}$ Bellgardt1988  which results in

$\left|g^{\prime\prime}\frac{M_{D_{ea}}M^{\dagger}_{D_{a\mu}}}{M^{2}_{L^{\prime}_{a}}}\right|\lesssim
9\times 10^{-15}$ (177)

where we used $m_{Z^{\prime}}=0.05$ GeV as a reference value. For
$m_{\mu}<m_{Z^{\prime}}<m_{\tau}$, it is constrained by , for example
$BR(\tau\to 3\mu)\lesssim 2\times 10^{-8}$ Hayasaka2010 , which results in

$\left|g^{\prime\prime}\frac{M_{D_{\mu
a}}M^{\dagger}_{D_{a\tau}}}{M^{2}_{L^{\prime}_{a}}}\right|\lesssim 4\times
10^{-10}$ (178)

where we used $m_{Z^{\prime}}=0.5$ GeV as a reference value. Thus we find that
strong constraints are imposed from $\ell_{i}\to\ell_{j}Z^{\prime}$ when these
decay modes are kinematically allowed. In our analysis, we consider
$Z^{\prime}>m_{\tau}$ to avoid these strong LFV constraints.

###  III.3 Leptonic $Z$ boson decays

The $Z_{\mu}$ to the leptonic (charged and uncharged) interaction terms in the
limit $\epsilon\to 0$ i.e. $(\sin\theta\to 0)$ is given by Chiang:2017tai ;
Kumar:2020web ; Nomura:2019btk

$\displaystyle L$ $\displaystyle\supset$
$\displaystyle\frac{g}{c_{W}}\Big{[}\bar{l}\gamma_{\mu}\Big{(}\frac{P_{L}}{2}-s_{W}^{2}\Big{)}l+\bar{E^{\prime}}\gamma_{\mu}\Big{(}\frac{1}{2}-s_{W}^{2}\Big{)}E^{\prime}\Big{]}Z^{\mu}$
(179) $\displaystyle-$
$\displaystyle\frac{g}{c_{W}}\Big{[}\bar{\nu}\gamma_{\mu}\frac{P_{L}}{2}\nu+\bar{N^{\prime}}\gamma_{\mu}\frac{1}{2}N^{\prime}\Big{]}Z^{\mu}$

where the sine of the Weinberg angle is $s_{W}\sim 0.23$. The equation is
written in the mass basis where $\nu$ are the three light Majorana neutrinos
and $N^{\prime}$ are three heavy Dirac neutrinos. The deviation of $Z$ boson
branching ratios from SM prediction up to 1-loop level is given by

$\displaystyle\Delta{\rm BR}(Z\to f_{i}\bar{f_{j}})$
$\displaystyle=\frac{\Gamma(Z\to f_{i}\bar{f_{j}})_{\rm SM+New}-\Gamma(Z\to
f_{i}\bar{f_{j}})_{\rm SM}}{\Gamma^{tot}_{Z}}$
$\displaystyle\approx\left\\{~{}\begin{array}[]{cc}\frac{\Gamma(Z\to
f_{i}\bar{f}_{j})_{\rm SM+New}-\Gamma(Z\to f_{i}\bar{f}_{j})_{\rm
SM}}{\Gamma^{\text{tot}}_{Z}}&(i=j)\\\ \frac{\Gamma(Z\to
f_{i}\bar{f}_{j})_{\rm New}}{\Gamma^{\text{tot}}_{Z}}&(i\neq j)\\\
\end{array}\right.,$ (182)

where $\Gamma_{Z}^{\rm{tot}}=2.4952\pm 0.0023$ Tanabashi:2018oca . Then our
new contributions for charged-lepton and neutrino final modes are respectively
given by

$\displaystyle\Delta{\rm BR}(Z\to l_{i}\bar{l_{j}})$ $\displaystyle=$
$\displaystyle\frac{m_{Z}g^{2}}{12\pi\Gamma_{Z}^{\rm{tot}}c^{2}_{W}}\Big{[}\frac{|B^{l}_{ij}|^{2}}{2}-s^{2}_{W}\delta_{ij}Re[B^{l*}_{ij}]-\Big{(}-\frac{s_{W}^{2}}{2}+\frac{1}{8}\Big{)}\delta_{ij}\Big{]},$
$\displaystyle\Delta{\rm BR}(Z\to\nu_{i}\bar{\nu_{j}})$ $\displaystyle=$
$\displaystyle\frac{m_{Z}g^{2}}{24\pi\Gamma_{Z}^{\rm{tot}}c^{2}_{W}}\Big{[}|B^{\nu}_{ij}|^{2}-\frac{\delta_{ij}}{4}\Big{]},$
(183)

with

$\displaystyle
B^{l}_{ij}=\frac{\delta_{ij}}{2}-\frac{1}{2}\sum_{\alpha}\frac{y_{D}^{i\alpha}y_{D}^{*j\alpha}}{(4\pi)^{2}}G^{l}(m_{E^{\prime}_{\alpha}},m_{h})\sin^{2}\beta-\frac{1}{2}\sum_{\alpha}\frac{y_{D}^{i\alpha}y_{D}^{*j\alpha}}{(4\pi)^{2}}G^{l}(m_{E^{\prime}_{\alpha}},m_{H})\cos^{2}\beta,$
$\displaystyle
B^{\nu}_{ij}=\frac{\delta_{ij}}{2}+\frac{1}{2}\sum_{\alpha,k,l}\frac{U^{*}_{PMNS_{ki}}U_{PMNS_{lj}}y_{D}^{k\alpha}y_{D}^{*l\alpha}}{(4\pi)^{2}}G^{\nu}(m_{N^{\prime}_{\alpha}},m_{h})\sin^{2}\beta$
$\displaystyle\hskip
49.79231pt+\frac{1}{2}\sum_{\alpha,k,l}\frac{U^{*}_{PMNS_{ki}}U_{PMNS_{lj}}y_{D}^{k\alpha}y_{D}^{*l\alpha}}{(4\pi)^{2}}G^{\nu}(m_{N^{\prime}_{\alpha}},m_{H})\cos^{2}\beta,$
$\displaystyle
G^{l}(m_{a},m_{b})=-\Big{(}-\frac{1}{2}+s^{2}_{W}\Big{)}^{2}H_{1}(m_{a},m_{b})+\Big{(}-\frac{1}{2}+s^{2}_{W}\Big{)}H_{2}(m_{a},m_{b})$
$\displaystyle
G^{\nu}(m_{a},m_{b})=-\frac{1}{2}H_{1}(m_{a},m_{b})-\frac{1}{2}H_{2}(m_{a},m_{b}),$
$\displaystyle
H_{1}(m_{a},m_{b})=\frac{m_{b}^{4}-4m_{a}^{2}m_{b}^{2}+3m_{a}^{4}-4m_{b}^{2}(m_{b}^{2}-2m_{a}^{2})\log(m_{b})-4m_{a}^{4}\log(m_{a})}{4(m_{a}^{2}-m_{b}^{2})^{2}},$
$\displaystyle
H_{2}(m_{a},m_{b})=m_{a}^{2}\Big{[}\frac{m_{a}^{2}-m_{b}^{2}+2m_{b}^{2}\log(\frac{m_{b}}{m_{a}})}{(m_{a}^{2}+m_{b}^{2})^{2}}\Big{]}.$
(184)

###  III.4 Numerical Analysis

In our numerical analysis we consider the Yukawa couplings
$|y_{D}^{1\alpha}|\in[10^{-8},10^{-3}]$, $|y_{D}^{2\alpha}|\in[10^{-3},1.0]$
and $|y_{D}^{3\alpha}|\in[10^{-6},10^{-1}]$, $m_{H}\in[1,10]$ GeV,
$m_{E^{\prime}_{\alpha}}\in[100,1500]$ GeV,
$m_{N^{\prime}_{\alpha}}\in[100,1500]$ GeV and $\sin\beta\sim 10^{-4}$. The
allowed parameter spaces are obtained which satisfy the neutrino data of
recent global fit by NuFIT 5.0 Esteban:2020cvm . For simplicity we consider
normal ordering and the Majorana phases as free parameters in the range
$[0,2\pi]$.

$\displaystyle
s^{2}_{12}\in[0.269,0.343],~{}~{}~{}s^{2}_{23}\in[0.415,0.616],~{}~{}~{}s^{2}_{13}\in[0.02032,0.02410],~{}~{}\delta\in\left[\frac{2}{3}\pi,2\pi\right],$
$\displaystyle\Delta m_{21}^{2}\in[6.82,8.04]\times 10^{-5}{\rm
eV}~{}~{}\text{and}~{}~{}\Delta m_{31}^{2}\in[2.435,2.598]\times 10^{-3}{\rm
eV}.$ (185)

Figure 3: Allowed region of $|y_{D}^{21}|$, $|y_{D}^{22}|$ and $|y_{D}^{23}|$

Under these parameter space, we have randomly selected and show plots to
satisfy all the constraints as discussed above. Fig. 3 shows the allowed
spaces for relevant components of the Dirac Yukawa couplings $|y^{{21}}_{D}|$,
$|y^{{22}}_{D}|$ and $|y^{{23}}_{D}|$, which are characterized by Casas-Ibarra
parametrization and restricted by leptonic $Z$ boson decays as well as LFVs.
It implies that these components are 0.6 at most that would be reasonable in
our model.

Figure 4: Allowed region for LFVs and the correlation to muon $g-2$. The left
one shows muon $g-2$ in terms of BR($\mu\to e\gamma)$, while the right one
BR($\tau\to e\gamma)$ in terms of BR($\tau\to\mu\gamma)$.

Fig. 4 shows the possibly allowed regions for muon $g-2$, BR($\mu\to e\gamma)$
in the left figure, and BR($\tau\to e\gamma)$ and BR($\tau\to\mu\gamma)$ in
the right one. In the left one, new contribution to the muon $g-2$ reaches at
$3\times 10^{-10}$ at around the the upper bound on BR($\mu\to
e\gamma)=4.2\times 10^{-13}$. Even though the scale of muon $g-2$ is smaller
than the expected value in Eq. (174) by 1 magnitude, it will still be
verifiable to be tested in near future. In the right one, BR($\tau\to
e\gamma)$ is much smaller than the current upper bound while
BR($\tau\to\mu\gamma)$ reaches the current upper bound. Therefore,
BR($\tau\to\mu\gamma)$ could also be tested in near future.

##  IV Dark matter physics

For the dark matter (DM) analysis we consider two scenarios, fermionic DM
scenario and the scalar DM scenario. In the fermionic DM scenario, we consider
the lightest of the $Z_{2}$ odd Majorana fermions $N_{1,..,6}$ obtained by
diagonalizing Eq.(101) as the DM candidate. For simplicity we consider the
mass ordering $m_{N_{1}}<m_{N_{2}}<m_{N_{3}}<m_{N_{4}}<m_{N_{5}}<m_{N_{6}}$
and therefore the lightest DM candidate is $N_{1}$. For numerical analysis we
consider $m_{N_{1}}\in[1,10]$ GeV, $m_{Z^{\prime}}\in[2,10]$ GeV, the gauge
kinetic mixing parameter $\epsilon\sim 10^{-4}$ and the $U(1)_{X}$ charge,
$x=1$, $m_{H}\in[1,10]$ GeV, $v_{\varphi}\in[1,5]$ GeV and $\sin\beta\sim
10^{-4}$. The relic density is computed using the public code Micromegas-5.2.4
Belanger:2014vza  implementing our model. The left panel of Fig. 5 shows the
allowed region of $m_{Z^{\prime}}-m_{N_{1}}$ space with $g^{\prime\prime}$ in
colour bar satisfying the observed relic density; $0.11<\Omega h^{2}<0.13$.
Above the dashed line, the dominant contribution to $\langle\sigma v\rangle$
comes from the $N_{1}N_{1}\to Z^{\prime}Z^{\prime}$ channel. Below the dashed
line, the dominant contribution to $\langle\sigma v\rangle$ comes from
$N_{1}N_{1}\to Z^{\prime}\to f_{SM}\bar{f}_{SM}$ processes. Along the dashed
line slightly below it, the cross sections are enhanced due to resonance which
leads to a sharp fall in the relic density Nayak:2017dwg .

For the scalar DM scenario, we have two candidates; $H_{1}$ or $H_{2}$, where
one can select the lighter one between CP even and CP odd bosons. When the
mixing is not so large, $H_{1}$ is $\eta_{R}$ dominant and $H_{2}$ is
$\chi_{R}$ one. In case of $H_{1}$, the detailed behavior of DM has been known
unless the Yukawa interactions $y_{L,R}$ are dominant. Below 100 GeV, a
solution has at the pole of half of DM Higgs mass $\sim 63$ GeV. Above 100
GeV, another solution is at around 534 GeV that is induced by gauge
interactions in kinetic terms. In details, see, e.g., ref. Hambye:2009pw .
However since our focus region is that the DM mass is less than 1 GeV, these
solution is not valid. Even though we have new channels via
$H_{1}-A_{1}-Z^{\prime}$ interactions, the DM mass wouldn’t be below 1 GeV
because of the electroweak precision test. Thus, we consider $H_{2}$ as the
lightest $Z_{2}$ odd particle. For numerical analysis we consider
$m_{H_{2}}\in[1,10]$ GeV, $m_{Z^{\prime}}\in[2,10]$ GeV and the remaining
parameters same as above. The right panel of Fig. 5 shows the allowed region
of $m_{Z^{\prime}}-m_{H_{2}}$ space with $g^{\prime\prime}$ in color bar
satisfying the observed relic density. The dominant contribution to the relic
density comes from $H_{2}H_{2}\to Z^{\prime}\to f_{SM}\bar{f}_{SM}$ channel.

Figure 5: Allowed regions of parameter space satisfying the observed DM relic
density constraint. Left: The DM candidate is $N_{1}$ and Right: The DM
candidate is $H_{2}$.

##  V Summary and discussion

We have proposed an inverse seesaw scenario under hidden $U(1)$ gauge
symmetry, having natural hierarchies among mass scales of neutral fermions.
The hierarchies are derived from theory and experimental constraints. The
theoretical aspect tells us $\delta\mu_{22,33}<<M_{L^{\prime}}$ since
$\delta\mu_{22,33}$ is induces at one-loop level, while $M_{L^{\prime}}$ is
bare mass. The experimental side requests that $M_{D}<<M_{L^{\prime}}$ since
$M_{D}$ is highly suppressed by the LFVs such as $\mu\to e\gamma$. In order to
induce small $\delta\mu_{22,33}$, we introduce exotic fermions and bosons that
provide us additional intriguing phenomenologies such as muon anomalous
magnetic moment, dark matter candidate as well as LFVs and leptonic $Z$
decays. We have analyzed these phenomenologies including neutrino oscillation
data numerically, and shown allowed region. Here, Dirac Yukawa coupling
$y_{D}$ possesses the information of neutrino data derived by modified Casas-
Ibarra parametrization. We have found that our new contribution to the muon
$g-2$ is about $3\times 10^{-10}$ which is smaller than 1 order magnitude, but
it would still be verifiable in near future experiments. BR$(\mu\to e\gamma)$
and BR$(\tau\to\mu\gamma)$ would be tested soon since these upper bounds
reaches at the current experimental bounds. Finally, we have discussed the DM
candidate in both the cases of fermion and boson, where we have focussed on
rather lighter range that is equal or less than 1 GeV. The dominant
contributions originate from interactions of hidden gauge sector, and we have
found allowed ranges for both the case as can be seen in Fig. 5.

In our scenario, we consider light $Z^{\prime}$ and extra scalar $H$ since it
is preferred to provide sizable muon $g-2$ relaxing tension between
experimental data and theoretical prediction. In such a case the SM Higgs
boson can decay into $HH$ and $Z^{\prime}Z^{\prime}$ mode where $H$ further
decays into $Z^{\prime}Z^{\prime}$ and $Z^{\prime}$ decays into charged
leptons via lepton mixing effect. Thus we would have multi-lepton signals from
rare Higgs decay as $h\to Z^{\prime}Z^{\prime}\to 4\ell$ and/or $h\to HH\to
4Z^{\prime}\to 8\ell$ which can be distinguishable signature of our scenario.
Interestingly, leptonic $Z^{\prime}$ decays includes LFV modes. We need
detailed simulation analysis to estimate discovery potential of these signals
since $H$ and $Z^{\prime}$ are light and some final state particles are
collimated. Therefore we leave the analysis in future work.

Before closing our paper, we will briefly mention collider phenomenologies. In
our scenario, extra charged particles $\eta^{\pm}$ and $E^{\pm}$ can be
produced by electroweak interactions at the LHC. Inert charged scalar
$\eta^{\pm}$ dominantly decays into $W^{\pm(*)}H_{1,2}$ assuming $E^{\pm}$ has
heavier mass; $H_{1,2}$ would be DM or it decays into the states including DM
with other particles. If $\eta^{\pm}$ decays into $W^{\pm(*)}DM$ the situation
is the same as inert Higgs doublet scenario, and it is well studied, e.g. in
ref. Belyaev:2018ext . Exotic lepton $E^{\pm}$ can decay into $\ell^{\pm}H$
and $\eta^{\pm}N_{i}$ where $H$ decays into $Z^{\prime}Z^{\prime}$ and
$\eta^{\pm}$ decays as described above. Thus new signals from these exotic
charged particles provide us complicated final states with many particles and
analysis of them is beyond the scope of this paper. Analysis of collider
signals would be given in future work.

###### Acknowledgements.

The work of H.O. and P.S. was supported by the Junior Research Group (JRG)
Program at the Asia-Pacific Center for Theoretical Physics (APCTP) through the
Science and Technology Promotion Fund and Lottery Fund of the Korean
Government and was supported by the Korean Local Governments-Gyeongsangbuk-do
Province and Pohang City. The authors would like to thank Dr. Arindam Das for
his fruitful discussions. H.O. is sincerely grateful for all the KIAS members.

##  Appendix A Couplings in scalar potential

The parameters in the scalar potential can be written by scalar masses and
mixings:

$\displaystyle\lambda_{\Phi}$ $\displaystyle=$
$\displaystyle\frac{m^{2}_{h}\cos^{2}\beta+m_{H}^{2}\sin^{2}\beta}{2v^{2}}$
$\displaystyle\lambda_{\varphi}$ $\displaystyle=$
$\displaystyle\frac{m_{h}^{2}\sin^{2}\beta+m_{H}^{2}\cos^{2}\beta}{2v_{\varphi}^{2}}$
$\displaystyle\lambda_{\Phi\varphi}$ $\displaystyle=$
$\displaystyle-\frac{(m_{H}^{2}-m_{h}^{2})\cos\beta\sin\beta}{vv_{\varphi}}$
$\displaystyle\lambda_{0}$ $\displaystyle=$
$\displaystyle-\frac{2(m^{2}_{H_{2}}-m^{2}_{H_{1}})\cos\alpha\sin\alpha}{vv_{\varphi}}$
$\displaystyle=$
$\displaystyle\frac{2(m^{2}_{A_{2}}-m^{2}_{A_{1}})\cos\gamma\sin\gamma}{vv_{\varphi}}$
$\displaystyle\tilde{\lambda}_{\Phi\eta}$ $\displaystyle=$
$\displaystyle-\frac{2(m_{\eta^{\pm}}^{2}-m_{H_{1}}^{2}\cos^{2}\alpha-
m_{H_{2}}^{2}\sin^{2}\alpha)}{v^{2}}$ $\displaystyle=$
$\displaystyle-\frac{2(m_{\eta^{\pm}}^{2}-m_{A_{1}}^{2}\cos^{2}\gamma-
m_{A_{2}}^{2}\sin^{2}\gamma)}{v^{2}}$ $\displaystyle m_{\chi^{\prime}}^{2}$
$\displaystyle=$ $\displaystyle
m_{A_{1}}^{2}\sin^{2}\gamma+m_{A_{2}}^{2}\cos^{2}\gamma$ $\displaystyle
m_{\chi^{\prime\prime}}^{2}$ $\displaystyle=$ $\displaystyle
m_{H_{1}}^{2}\sin^{2}\alpha+m_{H_{2}}^{2}\cos^{2}\alpha$ $\displaystyle\mu$
$\displaystyle=$
$\displaystyle\frac{m_{\chi^{\prime\prime}}^{2}-m_{\chi^{\prime}}^{2}}{2\sqrt{2}v_{\varphi}}$
$\displaystyle m_{\chi}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}m_{\chi^{\prime}}^{2}+m_{\chi^{\prime\prime}}^{2}-v^{2}\lambda_{H\chi}-v_{\varphi}^{2}\lambda_{\varphi\chi}\Big{]}.$
(186)

Thus the independent parameters are
$m_{h},m_{H},m_{H_{1}},m_{H_{2}},m_{A_{1}},m_{A_{2}},m_{\eta^{\pm}},v,v_{\varphi},\cos\beta,$
$\lambda_{\Phi\chi},\lambda_{\varphi\chi},\lambda_{\Phi\eta},\lambda_{\eta\varphi},\lambda_{\eta\chi},\lambda_{\eta},\lambda_{\chi}$.

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