---
abstract: '[ We propose a radiative seesaw model based on a modular $A_4$ symmetry, which has good predictability in the lepton sector. We execute a numerical analysis to search for parameters that satisfy the experimental constraints such as those from neutrino oscillation data and lepton flavor violations. Then, we present several predictions in our model that originate from the modular symmetry.]{}'
author:
- Hiroshi Okada
- Yutaro Shoji
title: 'A radiative seesaw model in modular $A_4$ symmetry'
---

[APCTP Pre2020 - 005]{}

Introduction
============

One of big mysteries in the standard model (SM) is the origin of the flavor structures. In particular, the flavor structure of the neutrino mass matrix is very important to understand the lepton sector. Historically, models with non-Abelian discrete flavor symmetries have been widely discussed since they not only reproduce the experimental results but also have several model specific predictions. Recently, starting from papers [@Feruglio:2017spp; @deAdelhartToorop:2011re], modular motivated non-Abelian discrete flavor symmetries have attracted attention of many authors to realize more predictable flavor structures in the quark and the lepton sectors. One of their remarkable natures is that any coupling constants as well as fields can also be transformed as non-trivial representations of those symmetries. Thus, we do not need to introduce many scalar fields such as flavons to realize flavor structure. As a result, we obtain a more minimal scenario without assumptions such as vacuum alignments among scalar fields. Here, we list several references where they apply this kind of symmetries to flavor models; the $A_4$ modular group [@Feruglio:2017spp; @Criado:2018thu; @Kobayashi:2018scp; @Okada:2018yrn; @Nomura:2019jxj; @Okada:2019uoy; @deAnda:2018ecu; @Novichkov:2018yse; @Nomura:2019yft; @Okada:2019mjf; @Ding:2019zxk; @Nomura:2019lnr; @Kobayashi:2019xvz; @Asaka:2019vev; @Zhang:2019ngf; @Gui-JunDing:2019wap; @Nomura:2019xsb; @Kobayashi:2019gtp; @Wang:2019xbo; @King:2020qaj; @Abbas:2020qzc], $S_3$ [@Kobayashi:2018vbk; @Kobayashi:2018wkl; @Kobayashi:2019rzp; @Okada:2019xqk], $S_4$ [@Penedo:2018nmg; @Novichkov:2018ovf; @Kobayashi:2019mna; @King:2019vhv; @Okada:2019lzv; @Criado:2019tzk; @Wang:2019ovr], $A_5$ [@Novichkov:2018nkm; @Ding:2019xna; @Criado:2019tzk], larger groups [@Baur:2019kwi], multiple modular symmetries [@deMedeirosVarzielas:2019cyj], and double covering of $A_4$ [@Liu:2019khw], in which masses, mixing, and CP phases for quark and lepton are predicted [^1]. A possible correction from Kähler potential is also discussed in Ref. [@Chen:2019ewa]. Furthermore, a systematic approach to understand the origin of CP transformations is recently discussed in ref. [@Baur:2019iai], and CP violation in models with modular symmetry is discussed in Ref. [@Kobayashi:2019uyt]. Another big mystery in the SM is the lack of a dark matter (DM) candidate. Even though many experiments from different aspects are going on to search for DM signatures, we have not obtained any decisive proofs yet. However, there are a lot of nice scenarios of DM that are connected to other observables. One of interesting models is known as the radiative seesaw model [@Ma:2006km]. This scenario not only explains the neutrino sector and DM at the same time but also provides a lot of new phenomena at a low energy scale such as lepton flavor violations, muon anomalous magnetic moment, collider signatures, etc. Since such a model connects the DM sector and the neutrino sector, the understanding of the neutrino nature leads to the understanding of the DM nature, and vise versa. In this paper, we work on a radiative seesaw scenario with a Dirac DM candidate based on our previous work [@Okada:2020oxh], applying a modular $A_4$ flavor symmetry. Then, we try to find several predictions in the lepton sector.

The manuscript is organized as follows. [In Sec. \[sec:realization\], we give our model setup under the $A_4$ modular symmetry, in which we review the modular $A_4$ symmetry and define relevant interactions needed to formulate the neutrino mass matrix and lepton flavor violations (LFVs). Then, we execute a numerical analysis and give several predictions in the lepton sector in Sec. III. Finally, we give our conclusion and discussion in Sec. \[sec:conclusion\].]{}

Model {#sec:realization}
=====

In this section, we introduce our model, which is based on a modular $A_4$ symmetry. The leptonic fields and the scalar fields of the model, their representations under the $A_4\times Z_3$ symmetry and their modular weights are given in Tab. \[tab:fields\]. We also show the representations of the Yukawa couplings in Tab. \[tab:couplings\]. Under these symmetries, we write the renormalizable Lagrangian for the lepton sector as follows: $$\begin{aligned}
-{\cal L}_L &=
\sum_{\ell=e,\mu,\tau}y_\ell \bar L_{L_\ell} H_{SM} e_{R_\ell}{\nonumber}\\
&\hspace{3ex}+\alpha_\nu \bar L_{L_e} (Y^{(2)}_{\bf 3}\otimes N_{R})_{\bf1}\tilde H_1
+\beta_\nu \bar L_{L_\mu}(Y^{(2)}_{\bf 3}\otimes N_{R})_{\bf1''}\tilde H_1
+\gamma_\nu \bar L_{L_\tau}(Y^{(2)}_{\bf 3}\otimes N_{R})_{\bf1'}\tilde H_1{\nonumber}\\
&\hspace{3ex}+a_\nu (\bar N_{L_e} \otimes Y^{(6)*}_{\bf 3})_{\bf1} {L^C_{L_e}}\tilde H_2
+b_\nu (\bar N_{L_\mu}\otimes Y^{(6)*}_{\bf 3})_{\bf1'} {L^C_{L_\mu}}\tilde H_2
+c_\nu (\bar N_{L_\tau}\otimes Y^{(6)*}_{\bf 3})_{\bf1''} {L^C_{L_\tau}}\tilde H_2{\nonumber}\\
&\hspace{3ex}+a'_\nu (\bar N_{L_e}\otimes Y'^{(6)*}_{\bf 3})_{\bf1} {L^C_{L_e}}\tilde H_2
+b'_\nu (\bar N_{L_\mu}\otimes Y'^{(6)*}_{\bf 3})_{\bf1'} {L^C_{L_\mu}}\tilde H_2
+c'_\nu (\bar N_{L_\tau}\otimes Y'^{(6)*}_{\bf 3})_{\bf1''} {L^C_{L_\tau}}\tilde H_2{\nonumber}\\
&\hspace{3ex}+ {M_D} (\bar N_{L}\otimes N_{R})_{\bf1}
+ {\rm h.c.}, 
\label{eq:lag-lep}\end{aligned}$$ where $\tilde H\equiv i\sigma_2 H^*$ with $\sigma_2$ being the second Pauli matrix and $(A\otimes B)_{\bf R}$ indicates that the representation, $\bf R$, is contracted from $A$ and $B$. Here, $M_D$ includes a modular invariant coefficient, $1/(i\tau-i\tau^*)$, and the charged-lepton matrix is diagonal thanks to the $A_4$ symmetry.

  ----------- --------------------------------------------------- ----------------------------------------- ---------- ------------ ------------ ------------
                $(\bar L_{L_e},\bar L_{L_\mu},\bar L_{L_\tau})$     $(e_{R_e},e_{R_{\mu}},e_{R_{\tau}})$      $N_{}$     $H_{SM}$     $H_1^*$       $H_2$ 
   $SU(2)_L$                       $\bm{2}$                                       $\bm{1}$                   $\bm{1}$    $\bm{2}$     $\bm{2}$     $\bm{2}$
   $U(1)_Y$                        $\frac12$                                        $-1$                       $0$      $\frac12$    $\frac12$    $\frac12$
     $A_4$                      ${(1,1',1'')}$                                 ${(1,1'',1')}$                  $3$         $1$          $1$          $1$
     $-k$                             $0$                                            $0$                       $-1$        $0$          $-1$         $-5$
     $Z_3$                            $1$                                            $1$                     $\omega$      $1$       $\omega^2$   $\omega^2$
  ----------- --------------------------------------------------- ----------------------------------------- ---------- ------------ ------------ ------------

  : Fermionic and bosonic field content of the model and their charge assignments under $SU(2)_L\times U(1)_Y\times A_4$, where $-k$ is the number of modular weight. The quark sector is the same as that in the SM.[]{data-label="tab:fields"}

  ------------ -------------------- -------------------- ------------------- -- -- --
                 $Y^{(4)}_{\bf1}$     $Y^{(2)}_{\bf3}$     $Y^{(6)}_{\bf3}$        
   [ $A_4$]{}        ${\bf1}$             ${\bf3}$            ${\bf3}$             
      $-k$             $4$                  $2$                  $6$               
  ------------ -------------------- -------------------- ------------------- -- -- --

  : Modular weight assignments for Yukawa couplings.[]{data-label="tab:couplings"}

The modular forms of weight 2, [$(y_{1},y_{2},y_{3})$]{}, which transform as a triplet of $A_4$, are written in terms of the Dedekind eta-function, $\eta(\tau)$, and its derivative, $\eta'(\tau)$, as [@Feruglio:2017spp] $$\begin{aligned}
\label{eq:Y-A4}
y_{1}(\tau) &=& \frac{i}{2\pi}\left( \frac{\eta'(\tau/3)}{\eta(\tau/3)}  +\frac{\eta'((\tau +1)/3)}{\eta((\tau+1)/3)}  
+\frac{\eta'((\tau +2)/3)}{\eta((\tau+2)/3)} - \frac{27\eta'(3\tau)}{\eta(3\tau)}  \right), \nonumber \\
y_{2}(\tau) &=& \frac{-i}{\pi}\left( \frac{\eta'(\tau/3)}{\eta(\tau/3)}  +\omega^2\frac{\eta'((\tau +1)/3)}{\eta((\tau+1)/3)}  
+\omega \frac{\eta'((\tau +2)/3)}{\eta((\tau+2)/3)}  \right) , \label{eq:Yi} \\ 
y_{3}(\tau) &=& \frac{-i}{\pi}\left( \frac{\eta'(\tau/3)}{\eta(\tau/3)}  +\omega\frac{\eta'((\tau +1)/3)}{\eta((\tau+1)/3)}  
+\omega^2 \frac{\eta'((\tau +2)/3)}{\eta((\tau+2)/3)}  \right)\,.
\nonumber\end{aligned}$$ Then, any couplings of higher weights are constructed from the multiplication rules of $A_4$. One finds the following expressions: $$\begin{aligned}
&Y^{(4)}_{\bf1}=y^2_1+2y_2y_3,\quad
Y^{(6)}_{\bf3}=
\left[\begin{array}{c}
y^3_1+2y_1y_2y_3 \\ 
y^2_1y_2+2y_2^2y_3 \\ 
y^2_1y_3+2y_3^2y_2 \\ 
\end{array}\right],\quad
Y'^{(6)}_{\bf3}=
\left[\begin{array}{c}
y^3_3 + 2y_1y_2y_3 \\ 
y^2_3y_1+2y_1^2y_2 \\ 
y^2_3y_2+2y_2^2y_1 \\ 
\end{array}\right].\end{aligned}$$ To construct a nonzero neutrino mass matrix, we also need a quartic term in the Higgs potential, [*i.e.*]{} $(H_{SM}^\dag H_1)(H_{SM}^\dag H_2)$, which can be realized as follows: $$\begin{aligned}
a_0\frac{Y^{(4)}_{\bf1}}{i(\tau^*-\tau)}(H_{SM}^\dag H_1)(H_{SM}^\dag H_2)+h.c.
\equiv \lambda_0(H_{SM}^\dag H_1)(H_{SM}^\dag H_2)+h.c.,\end{aligned}$$ where $a_0$ is an arbitrary complex number. Although it mixes the neutral complex Higgs bosons, we assume that the mixing angle is very small and the mass eigenstates are almost the flavor eigenstates, which we denote as $\eta_{1,2}$.

Neutrino mass matrix
--------------------

Due to the modular symmetry, the heavy Dirac neutrino mass matrix is diagonal with the eigenvalue, $M_D$: $$\begin{aligned}
{\cal M}&= {M_D}
\left[\begin{array}{ccc}
1 &0 & 0 \\ 
0 & 1 &0 \\ 
0& 0 & 1 \\ 
\end{array}\right] . \end{aligned}$$

We write down the relevant interactions for the generation of the neutrino mass matrix as $$\begin{aligned}
-{\cal L}_\nu&=\bar\nu_L y_{N_R} N_R \eta_1^* 
+\bar N_Ly_{N_L} \nu^C_L \eta_2^*+\frac{\lambda_0v_H^2}{2}\eta_1\eta_2+h.c.,\end{aligned}$$ where $$\begin{aligned}
y_{N_R} &=
\left[\begin{array}{ccc}
\alpha_\nu &0 & 0 \\ 
0 & \beta_\nu  &0   \\ 
0& 0 & \gamma_\nu  \\ 
\end{array}\right]
\left[\begin{array}{ccc}
y_{1} & y_{3} &y_{2} \\ 
y_{3} & y_{2} &y_{1} \\ 
y_{2} & y_{1} & y_{3} \\ 
\end{array}\right],\\
 y_{N_L} &=
\left[\begin{array}{ccc}
y_{6,1} & y_{6,3} &y_{6,2} \\ 
y_{6,3} & y_{6,2} &y_{6,1} \\ 
y_{6,2} & y_{6,1} & y_{6,3} \\ 
\end{array}\right]^*
\left[\begin{array}{ccc}
a_\nu &0 & 0 \\ 
0 & 0 & c_\nu   \\ 
0& b_\nu & 0  \\ 
\end{array}\right]
+
\left[\begin{array}{ccc}
y'_{6,1} & y'_{6,3} &y'_{6,2} \\ 
y'_{6,3} & y'_{6,2} &y'_{6,1} \\ 
y'_{6,2} & y'_{6,1} & y'_{6,3} \\ 
\end{array}\right]^*
\left[\begin{array}{ccc}
a'_\nu &0 & 0 \\ 
0 & 0  & c'_\nu   \\ 
0& b'_\nu & 0  \\ 
\end{array}\right],
\label{eq:mn}\end{aligned}$$ where ${\bf Y^{(6)}_3}\equiv[ y_{6,1},y_{6,2},y_{6,3}]^T$, ${\bf
Y'^{(6)}_3}\equiv[ y'_{6,1},y'_{6,2},y'_{6,3}]^T$, and we impose the perturbativity limit, ${\rm Max}[y_{N_{R,L}}]\lesssim\sqrt{4\pi}$, in the numerical analysis.

Then, the neutrino mass matrix is calculated as $$\begin{aligned}
&m_{\nu}=\kappa [y_{N_R}  y_{N_L} +(y_{N_R}  y_{N_L})^T]\equiv \kappa \tilde m_\nu,\\
&\kappa=\frac{\lambda_0^* v_{SM}^2}{32\pi^2M_D} F(M_D^2,\eta_1^2,\eta_2^2),\\
& F(a,b,c)=-a\frac{(b-c)a\ln a+(c-a)b\ln b+(a-b)c\ln c}{(a-b)(b-c)(c-a)},\end{aligned}$$ where $v_{SM}\simeq$ 246 GeV is the VEV of the SM Higgs boson. Notice that $\kappa$ does not depend on the flavor structure due to the modular symmetry.

Thus, we determine $\kappa$ by $$\begin{aligned}
(NH):\  \kappa^2= \frac{|\Delta m_{\rm atm}^2|}{\tilde D_{\nu_3}^2-\tilde D_{\nu_1}^2},
\quad
(IH):\  \kappa^2= \frac{|\Delta m_{\rm atm}^2|}{\tilde D_{\nu_2}^2-\tilde D_{\nu_3}^2},
 \end{aligned}$$ where $\tilde m_\nu$ is diagonalized by $V^\dag_\nu (\tilde m_\nu^\dag
\tilde m_\nu)V_\nu=(\tilde D_{\nu_1}^2,\tilde D_{\nu_2}^2,\tilde
D_{\nu_3}^2)$ and $\Delta m_{\rm atm}^2$ is the atmospheric neutrino mass-squared difference. Here, NH and IH stand for the normal hierarchy and the inverted hierarchy, respectively. Subsequently, the solar neutrino mass-squared difference can be written in terms of $\kappa$ as follows: $$\begin{aligned}
\Delta m_{\rm sol}^2= {\kappa^2}({\tilde D_{\nu_2}^2-\tilde D_{\nu_1}^2}),
 \end{aligned}$$ which we compare with the observed value.

Working in the diagonal basis of the charged-leptons, one finds $U_{\rm
PMNS}=V_\nu$, which is parameterized by three mixing angles, $\theta_{ij}
(i,j=1,2,3; i < j)$, one CP violating Dirac phase, $\delta_{CP}$, and two Majorana phases, $\{\alpha_{2}, \alpha_{3}\}$, as $$U_{\rm PMNS} = 
\begin{pmatrix} c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i \delta_{CP}} \\ 
-s_{12} c_{23} - c_{12} s_{23} s_{13} e^{i \delta_{CP}} & c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i \delta_{CP}} & s_{23} c_{13} \\
s_{12} s_{23} - c_{12} c_{23} s_{13} e^{i \delta_{CP}} & -c_{12} s_{23} - s_{12} c_{23} s_{13} e^{i \delta_{CP}} & c_{23} c_{13} 
\end{pmatrix}
\begin{pmatrix} 1 & 0 & 0 \\ 0 & e^{i \frac{\alpha_{2}}{2}} & 0 \\ 0 & 0 & e^{i \frac{\alpha_{3}}{2}} \end{pmatrix},$$ where $c_{ij}$ and $s_{ij}$ stand for $\cos \theta_{ij}$ and $\sin \theta_{ij}$, respectively. Then, the mixing angles are expressed in terms of the components of $U_{\rm PMNS}$ as $$\begin{aligned}
\sin^2\theta_{13}=|(U_{\rm PMNS})_{13}|^2,\quad 
\sin^2\theta_{23}=\frac{|(U_{\rm PMNS})_{23}|^2}{1-|(U_{\rm PMNS})_{13}|^2},\quad 
\sin^2\theta_{12}=\frac{|(U_{\rm PMNS})_{12}|^2}{1-|(U_{\rm PMNS})_{13}|^2}.\end{aligned}$$ 0 Also we compute the Jarlskog invariant, $\delta_{CP}$ derived from PMNS matrix elements $U_{\alpha i}$: $$J_{CP} = \text{Im} [U_{e1} U_{\mu 2} U_{e 2}^* U_{\mu 1}^*] = s_{23} c_{23} s_{12} c_{12} s_{13} c^2_{13} \sin \delta_{CP},$$ and the Majorana phases are also estimated in terms of other invariants $I_1$ and $I_2$: $$I_1 = \text{Im}[U^*_{e1} U_{e2}] = c_{12} s_{12} c_{13}^2 \sin \left( \frac{\alpha_{2}}{2} \right), \
I_2 = \text{Im}[U^*_{e1} U_{e3}] = c_{12} s_{13} c_{13} \sin \left( \frac{\alpha_{3}}{2}  - \delta_{CP} \right).$$ As for the CP phase and the Majorana phases, we have $$\begin{aligned}
 \delta_{\rm CP}&=-{\rm arg}[\det U_{\rm PMNS}]+{\rm arg}[(U_{\rm PMNS})_{11}]+{\rm arg}[(U_{\rm PMNS})_{12}]\nonumber\\
 &\hspace{3ex}-{\rm arg}[(U_{\rm PMNS})_{13}]+{\rm arg}[(U_{\rm PMNS})_{23}]+{\rm arg}[(U_{\rm PMNS})_{33}],\\
 \frac{\alpha_2}{2}&={\rm arg}[(U_{\rm PMNS})_{12}]-{\rm arg}[(U_{\rm PMNS})_{11}],\\
 \frac{\alpha_3}{2}&=-{\rm arg}[\det U_{\rm PMNS}]+{\rm arg}[(U_{\rm PMNS})_{12}]+{\rm arg}[(U_{\rm PMNS})_{23}]+{\rm arg}[(U_{\rm PMNS})_{33}],\end{aligned}$$ where ${\rm arg}[*]$ denotes the argument of $*$.

We will adopt the neutrino experimental data at the 3$\sigma$ interval [@Esteban:2018azc] as follows: $$\begin{aligned}
&{\rm NH}: \Delta m^2_{\rm atm}=[2.431, 2.622]\times 10^{-3}\ {\rm eV}^2,\
\Delta m^2_{\rm sol}=[6.79, 8.01]\times 10^{-5}\ {\rm eV}^2,\label{eq:nuexp_NH}\\
&\sin^2\theta_{13}=[0.02044, 0.02437],\ 
\sin^2\theta_{23}=[0.428, 0.624],\ 
\sin^2\theta_{12}=[0.275, 0.350],{\nonumber}\\
&{\rm IH}: \Delta m^2_{\rm atm}=[2.413, 2.606]\times 10^{-3}\ {\rm eV}^2,\
\Delta m^2_{\rm sol}=[6.79, 8.01]\times 10^{-5}\ {\rm eV}^2,\label{eq:nuexp_IH}\\
&\sin^2\theta_{13}=[0.02067, 0.02461],\ 
\sin^2\theta_{23}=[0.433, 0.623],\ 
\sin^2\theta_{12}=[0.275, 0.350].{\nonumber}$$ Notice that the atmospheric mass-squared difference is considered as an input parameter in our numerical analysis.

Neutrinoless double beta decay
------------------------------

The neutrinoless double beta decay may be observed by KamLAND-Zen in future [@KamLAND-Zen:2016pfg]. The relevant effective mass can be calculated as $$\begin{aligned}
\langle m_{ee}\rangle=\kappa|\tilde D_{\nu_1} \cos^2\theta_{12} \cos^2\theta_{13}+\tilde D_{\nu_2} \sin^2\theta_{12} \cos^2\theta_{13}e^{i\alpha_{2}}+\tilde D_{\nu_3} \sin^2\theta_{13}e^{i(\alpha_{3}-2\delta_{CP})}|.\end{aligned}$$

Lepton flavor violations
------------------------

LFVs are induced via $y_{N_{L,R}}$ and the most stringent constraint arises from $\mu\to e\gamma$ process, which is bounded as ${\rm
BR}(\mu\to e\gamma)\lesssim 4.2\times 10^{-13}$ [@TheMEG:2016wtm; @Renga:2018fpd]. Meanwhile, our theoretical formula is given by $$\begin{aligned}
{\rm BR}(\mu\to e\gamma)&= \frac{3\alpha_{\rm em}}{16\pi{\rm G_F^2}}
\left| G(M_D,m_{\eta_1})\left( \sum_{a=1}^3 y_{N_{R_{1a}}} y^\dag_{N_{R_{a2}}} \right)
+ 
G(M_D,m_{\eta_2}) \left( \sum_{a=1}^3 y_{N_{L_{a1}}} y^\dag_{N_{L_{2a}}}\right)
  \right|^2,\\
  G(m_a,m_b)&\approx
 \frac{2 m_a^6+3 m_a^4 m_b^2-6 m_a^2 m_b^4+m_b^6 +12 m_a^4 m_b^2 \ln\left[\frac{m_b}{m_a}\right]} {12 (m_a^2-m_b^2)^4},\end{aligned}$$ where ${\rm G_F}\approx 1.17\times10^{-5}$ GeV$^{-2}$ is the Fermi constant and $\alpha_{\rm em}\approx1/129$ is the fine structure constant. When the masses of $m_{\eta_{1,2}}, M_D$ are of the order of 100 GeV, the constraints for Yukawa couplings are found as $y_{N_R}\approx y_{N_L}={\cal O}(0.01)$ [^2].

Numerical analysis {#sec:num_analysis}
==================

Here, we search for the allowed regions that satisfy the constraints on the neutrino oscillation data given in Eqs. (\[eq:nuexp\_NH\]) and (\[eq:nuexp\_IH\]) and those on the LFVs. First, we set the ranges of input parameters as follows: $$\begin{aligned}
&\tau=[-0.5+0.1i, 0.5+3i],\quad [\alpha_\nu,\beta_\nu,\gamma_\nu,a_\nu^{(')},b_\nu^{(')},c_\nu^{(')}]=[-1,1],\\
&[M_D,m_{\eta_1},m_{\eta_2}]=[1,10]\ {\rm TeV},\\
&{\rm NH}: \Delta m^2_{\rm atm}=[2.431, 2.622]\times 10^{-3}\ {\rm eV}^2,\
{\rm IH}: \Delta m^2_{\rm atm}=[2.413, 2.606]\times 10^{-3}\ {\rm eV}^2.\end{aligned}$$ Here, the real part of $\tau$ has a periodicity of $1$ and the 3$\sigma$ interval of $\Delta m^2_{\rm atm}$ is used for the range of the scan. Fig. \[fig:1\] shows the allowed region of $\tau$. The left panel is for the NH case and the right one is for the IH case. In both cases, the allowed regions have the similar shape, [*i.e.*]{} having four peaks at around ${\rm Re}[\tau]=\pm0.1,\pm0.5$. A smaller ${\rm Im}[\tau]$ is preferred in the NH case and a larger ${\rm
Im}[\tau]$ is preferred in the IH case. Here, we emphasize that the region around $\tau=i$ is allowed in both cases and the same region is favored in a model of quark sector [@Okada:2019uoy]. It suggests that our model may also provide a viable and predictive quark model with the same value of modulus $\tau$, although it is beyond our scope.

![Allowed region of $\tau$. The left panel is for the NH case and the right one is for the IH case.[]{data-label="fig:1"}](tau-nh.pdf "fig:"){width="80mm"} ![Allowed region of $\tau$. The left panel is for the NH case and the right one is for the IH case.[]{data-label="fig:1"}](tau-ih.pdf "fig:"){width="80mm"}

Fig. \[fig:2\] shows the allowed region of the Majorana phases, $\alpha_2$ and $\alpha_3$. The left panel is for the NH case and the right one is for the IH case. We find characteristic denser regions in both NH and IH.

![Allowed region of $\alpha_2$ and $\alpha_3$. The left panel is for the NH case and the right one is for the IH case.[]{data-label="fig:2"}](alpha-nh.pdf "fig:"){width="80mm"} ![Allowed region of $\alpha_2$ and $\alpha_3$. The left panel is for the NH case and the right one is for the IH case.[]{data-label="fig:2"}](alpha-ih.pdf "fig:"){width="80mm"}

Fig. \[fig:3\] shows the allowed region of $\sin^2\theta_{23}$ and $\delta_{\rm CP}$. The left panel is for the NH case and the right one is for the IH case. There are three denser regions at $\delta_{\rm
CP}=90^\circ,240^\circ, 270^\circ$ for NH, while two denser regions at $\delta_{\rm CP}=90^\circ, 270^\circ$ for IH. Since the current best fit value of the Dirac CP phase is around $270^\circ$, these predictions seem to be consistent. Moreover, we find that a relatively smaller value of $\sin^2\theta_{23}$ tends to be favored from the best fit value of $\delta_{\rm CP}$ for the NH case and a relatively larger value for the IH case.

![Allowed region of $\sin^2\theta_{23}$ and $\delta_{\rm CP}$. The left panel is for the NH case and the right one is for the IH case.[]{data-label="fig:3"}](cp-nh.pdf "fig:"){width="80mm"} ![Allowed region of $\sin^2\theta_{23}$ and $\delta_{\rm CP}$. The left panel is for the NH case and the right one is for the IH case.[]{data-label="fig:3"}](cp-ih.pdf "fig:"){width="80mm"}

Fig. \[fig:4\] shows the allowed region of the lightest neutrino mass, $m_1$ for NH and $m_3$ for IH, and the sum of neutrino masses, $m_1+m_2+m_3$. The left panel is for the NH case and the right one is for the IH case. The horizontal orange dotted line represents the cosmological constraint on the sum of the neutrino masses, where $\sum_i
m_i\lesssim0.146$ eV for NH and $\sum_i m_i\lesssim0.172$ eV for IH [@RoyChoudhury:2019hls; @Aghanim:2018eyx; @Vagnozzi:2017ovm]. If this constraint is considered seriously, we get a stronger constraint on the lightest neutrino mass around $0.04$ eV for both NH and IH.

![Allowed region of the lightest neutrino mass and $m_1+m_2+m_3$. The left panel is for the NH case and the right one is for the IH case.[]{data-label="fig:4"}](sum_m-nh.pdf "fig:"){width="80mm"} ![Allowed region of the lightest neutrino mass and $m_1+m_2+m_3$. The left panel is for the NH case and the right one is for the IH case.[]{data-label="fig:4"}](sum_m-ih.pdf "fig:"){width="80mm"}

Fig. \[fig:5\] shows the allowed region of the lightest neutrino mass and the effective mass for the neutrinoless double beta decay, $m_{ee}$. The left panel is for the NH case and the right one is for the IH case. Considering the cosmological constraint shown in Fig. \[fig:4\], we obtain an upper bound on $m_{ee}$; $m_{ee}\lesssim 0.04$ eV for NH and $m_{ee}\lesssim 0.08$ eV for IH.

![Allowed region of $m_1$ and $m_{ee}$. The left panel is for the NH case and the right one is for the IH case.[]{data-label="fig:5"}](mee-nh.pdf "fig:"){width="80mm"} ![Allowed region of $m_1$ and $m_{ee}$. The left panel is for the NH case and the right one is for the IH case.[]{data-label="fig:5"}](mee-ih.pdf "fig:"){width="80mm"}

All the other oscillation parameters are found over all the 3$\sigma$ intervals of Eq.(\[eq:nuexp\_NH\]) without structures. After the constraint on ${\rm BR}(\mu\to e\gamma)$ is imposed, the typical size of the other LFVs are one or more orders of magnitude smaller than the current constraints of ${\rm BR}(\tau\to e\gamma)\lesssim
3.3\times10^{-8}$ and ${\rm BR}(\tau\to \mu\gamma)\lesssim
4.4\times10^{-8}$.

Conclusion and discussion {#sec:conclusion}
=========================

We have extended the radiative seesaw model proposed in our previous work [@Okada:2020oxh] by applying a modular $A_4$ symmetry in order to increase predictability in the lepton sector. After the formulation of the neutrino sector and LFVs, we have executed a numerical analysis to find parameters that satisfy the constraints from the neutrino oscillation data and LFVs. We have found several preferences in the allowed parameter regions. We highlight some prominent points in the following.

1.  We find characteristic denser allowed regions of the Majorana phases.

2.  There are three denser regions at $\delta_{\rm
    CP}=90^\circ,240^\circ, 270^\circ$ for NH, while two denser regions at $\delta_{\rm
    CP}=90^\circ, 270^\circ$ for IH. Since the current best fit value of the Dirac CP phase is around $270^\circ$, these predictions seem to be consistent. Moreover, one might find that a relatively smaller value of $\sin^2\theta_{23}$ tends to be favored from the best fit value of $\delta_{\rm CP}$ for the NH case and a relatively larger value for the IH case.

3.  We obtain an upper bound on $m_{ee}$; $m_{ee}\lesssim 0.04$ eV for NH and $m_{ee}\lesssim 0.08$ eV for IH.

This research was supported by an appointment to the JRG Program at the APCTP through the Science and Technology Promotion Fund and Lottery Fund of the Korean Government. This was also supported by the Korean Local Governments - Gyeongsangbuk-do Province and Pohang City (H.O.). H.O. is sincerely grateful for the KIAS member. Y.S. is supported by Grant-in-Aid for Scientific research from the Ministry of Education, Science, Sports, and Culture (MEXT), Japan, No. 16H06492.

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[^1]: Some reviews [@Altarelli:2010gt; @Ishimori:2010au; @Ishimori:2012zz; @Hernandez:2012ra; @King:2013eh; @King:2014nza; @King:2017guk; @Petcov:2017ggy] are useful for the understanding of non-Abelian groups and their applications to flavor structures.

[^2]: In our numerical analysis, we also consider the other possible processes such $\tau\to e\gamma$ and $\tau\to \mu\gamma$. These experimental results give the constraints for Yukawa couplings $\lesssim{\cal O}$(0.1).
