EPHOU-23-017

KYUSHU-HET-270

# Modular flavor models with positive modular weights:
a new lepton model building

Tatsuo Kobayashi<EMAIL_ADDRESS>Department of Physics,
Hokkaido University, Sapporo 060-0810, Japan  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  Hajime
Otsuka<EMAIL_ADDRESS>Department of Physics, Kyushu
University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

###### Abstract

We propose an interesting assignment of positive modular wights for fields in
a modular non-Abelian discrete flavor symmetry. By this assignment, we can
construct inverse seesaw and linear seesaw models without any additional
symmetries which possess good testability in current experiments. At first, we
discuss probabilities for positive modular wights from a theoretical point of
view. Then we show concrete examples of inverse seesaw and linear seesaw
scenarios applying modular $A_{4}$ symmetry as examples and demonstrate some
predictions as well as being consistent with experimental results such as
their masses and mixings.

##  I Introduction

Neutrinos may be Majorana type particles if we minimally extend the Standard
Model (SM) by introducing right-handed (heavier) neutrinos. This is called
canonical seesaw scenario Yanagida (1979); Minkowski (1977); Mohapatra and
Senjanovic (1980). This scenario does not require any additional symmetry.
Also, it leads us to an origin of Baryon number asymmetry via leptogenesis
Fukugita and Yanagida (1986). However this scenario typically requires a high
energy scale for right-handed neutrino masses such as Grand Unified Theory
(GUT) one ($\mathcal{O}(10^{16})$ GeV); this scale is also motivated to work
the leptogensis well. Thus it would be difficult to verify the canonical
seesaw scenario directly by current/future experiments. It is then important
and interesting to explore other scenarios that could be tested.

Inverse seesaw (IS) Mohapatra and Valle (1986); Wyler and Wolfenstein (1983)
and linear seesaw (LS) Wyler and Wolfenstein (1983); Akhmedov et al. (1996a,
b) mechanisms have been proposed as other possibilities of neutrino mass
generation that enable us to obtain higher verifiability since these scenarios
can induce the active neutrino masses with lower mass scale compared to
canonical seesaw one, like TeV scale. However these models require an
additional symmetry such as $U(1)_{B-L}$ to discriminate two types of heavier
Majorana fermions. Also, there would not be any simple IS/LS models with
predictive flavor structure which are nowadays tremendously important to
verify them at collider experiments. This difficulty is because flavor
structures cannot be constrained through such a new symmetry like
$U(1)_{B-L}$. Thus we typically need multiple symmetries including flavor
specific ones in realizing predictive IS/LS models. Phenomenologically, IS or
LS model would be more attractive but a realization of predictive models is
rather complicated than the canonical seesaw model because of the above
reasons.

In this paper, we show an intriguing fact that one can construct minimal IS
and LS models by introducing a modular $A_{4}$ symmetry only. Moreover, due to
determining the flavor structure for the lepton matter superfields by the
symmetry, these models automatically possess high testability. However, in
realizing minimal models, we need to impose an unusual condition that we have
to allow assignments with both plus and minus signs of the modular weight for
four-dimensional (4D) superfields. 111When we only apply negative modular
weights in a standard manner, we also need to introduce one more symmetry at
least as we explain in Sec.III. Notice here that only the minus sign
assignments for 4D fields are theoretically confirmed so far. 222Modular non-
Abelian discrete flavor symmetries have recently been proposed in refs
Feruglio (2019); Criado and Feruglio (2018). After that, a lot of models and
their applications to flavor physics were studied. Here, we refer to models
with modular $A_{4}$ symmetries. Kobayashi et al. (2018a); Okada and Tanimoto
(2019a); Nomura and Okada (2019a); Okada and Tanimoto (2019b); de Anda et al.
(2020); Novichkov et al. (2019); Nomura and Okada (2019b); Okada and Orikasa
(2019); Ding et al. (2019a); Nomura et al. (2020); Kobayashi et al. (2019);
Asaka et al. (2020a); Zhang (2020); Ding et al. (2019b); Kobayashi et al.
(2020); Nomura et al. (2019); Wang (2020); Okada and Shoji (2020); Okada and
Tanimoto (2020a); Behera et al. (2020a, b); Nomura and Okada (2020a, b); Asaka
et al. (2020b); Okada and Tanimoto (2020b); Nagao and Okada (2020); Okada and
Tanimoto (2020c); Kang et al. (2022); Nomura et al. (2023). See for more
references Ref. Kobayashi and Tanimoto (2023). We thus need to investigate the
theoretical verification of positive modular weights. There exist some
articles on quark and lepton masses and mixing assigning plus modular weights
for fields under the $A_{4}$ symmetry. See for examples Refs. Mishra et al.
(2023); Liu and Ding (2022).

Here, we will discuss some theoretical possibilities on how to realize the
positive signs of modular weight for 4D superfields. Then, we demonstrate two
examples of neutrino models for IS and LS scenarios and display some
predictions via numerical analyses. Our realization of these models indicates
that the use of positive modular weights provides an extended possibility in
constructing phenomenological models with modular flavor symmetry.

This paper is organized as follows. In Section II, we present theoretical
possibilities for positive modular weights for 4D fields. In Section III, we
show two examples of IS and LS scenarios on how to realize these models and
demonstrate predictions via numerical analyses. Finally we devote Section IV
to the summary and conclusion of our results.

##  II Realizations of positive modular weights for chiral superfields from
theoretical backgrounds

In this section, we discuss the possibility of 4D positive modular weights in
the context of higher-dimensional theory including the string theory.333See,
Ref. Kikuchi et al. (2022a), for the discussion of conventional 4D negative
modular weights in the context of modular flavor models from a higher-
dimensional perspective. The 4D effective action can be obtained by using the
Kaluza-Klein reduction on compact extra-dimensional spaces. For instance,
higher-dimensional scalar fields $\Phi(x,y)$ and spinor fields $\Psi(x,y)$,
with $x$ and $y$ being 4D and $d$-dimensional internal coordinates, can be
decomposed as

$\displaystyle\Phi(x,y)=\sum_{n}\phi^{(n)}(x)\varphi^{(n)}(y),$
$\displaystyle\Psi(x,y)=\sum_{n}\psi^{(n)}(x)\chi^{(n)}(y),$ (1)

where $n$ labels Kaluza-Klein momentum in the internal space. Since the mass
scale of Kaluza-Klein modes is typically a compactification scale, only
massless modes with $n=0$ are relevant in the low-energy physics. Note that
internal background sources lead to degenerate massless modes
$\phi^{(0)}_{i}$, $\psi^{(0)}_{i}$ whose index $i$ will be identified with the
generation number of quarks, leptons and/or Higgs. It indicates that the
flavor structure of quarks and leptons will be determined by the structure of
matter wavefunctions in extra-dimensional spaces $\varphi^{(0)}_{i}$,
$\chi^{(0)}_{i}$. Hereafter, we omit the Kaluza-Klein index $n$ and flavor
index $i$ unless we specify them.

To see the flavor structure in more detail, let us discuss three-point
couplings of matter wavefunctions. When higher-dimensional matter fields
propagate in extra-dimensional spaces, 4D Yukawa couplings are given by an
overlap integral of internal matter wavefunction such as $\varphi$ and $\chi$.
For instance, the 4D Yukawa coupling term in the lepton sector is written by

$\displaystyle W=Y\bar{N}H_{u}L,$ (2)

in the superfield language, where $\bar{N}$, $L$ and $H_{u}$ denote right-
handed neutral fermions, left-handed leptons and up-sector Higgs field,
respectively. Here, the 4D Yukawa coupling $Y$ is given by an overlap integral
of corresponding matter wavefunctions in the extra-dimensional space:

$\displaystyle Y=g\int
d^{d}y\chi_{\bar{N}}(y)\chi_{L}(y)\varphi^{\ast}_{H_{u}}(y),$ (3)

with $g$ being a higher-dimensional coupling. Remarkably, on certain toroidal
backgrounds, such wavefunctions transform under finite modular groups
$\Gamma_{N}=PSL(2,\mathbb{Z})/\Gamma(N)$ where $\Gamma(N)$ is the principal
congruence subgroup:

$\displaystyle\begin{aligned} \Gamma(N)=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in SL(2,\mathbb{Z})~{},~{}~{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}~{}~{}({\rm
mod}\,N)\right\\}\end{aligned}.$ (4)

Specifically, the modular transformation of internal wavefunction $\chi_{i}$
in its flavor space is given by

$\displaystyle\chi_{i}(y)\rightarrow\rho(\gamma)_{ij}(c\tau+d)^{k}\chi_{j}(y),$
(5)

where $\rho(\gamma)$ represents a reducible representation of $\Gamma_{N}$ and
$k$ is the so-called modular weight. Here, $\tau$ denotes the modulus of the
torus. Similarly, the wavefunction $\varphi_{i}(y)$ transforms non-trivially.
It is notable that the higher-dimensional fields $\Phi(x,y)$ and $\Psi(x,y)$
themselves are invariant under the modular symmetry, indicating that the 4D
matter fields $\phi_{i}(x)$ and $\psi_{i}(x)$ (or chiral superfields in the
superfield language) enjoy opposite modular transformations in comparison with
the corresponding wavefunctions $\varphi_{i}(y)$ and $\chi_{i}(y)$ in the
extra-dimensional space, respectively. Thus, the 4D matters, $\phi_{i}(x)$ and
$\psi_{i}(x)$, and internal fields, $\varphi_{i}(y)$ and $\chi_{i}(y)$, have
the modular weight $-k_{i}$ and $k_{i}$, respectively. The 4D superpotential
(2) is indeed modular invariant because the 4D Yukawa coupling has the modular
weight $k_{Y}=k_{\bar{N}}+k_{L}-k_{H_{d}}$.

The matter wavefunctions as well as the modular weights are explicitly
calculated in toroidal backgrounds as well as its orbifolds. For instance, on
$T^{2}$ background with magnetic fluxes, the modulus $\tau$ identified with
the complex structure modulus determines the flavor structure of quarks and
leptons. The massless wavefunction in the extra-dimensional space has the
modular weight 1/2, and the corresponding Yukawa couplings are described by
the modular form with weight 1/2 Kobayashi et al. (2018b); Kobayashi and Tamba
(2019); Ohki et al. (2020); Kikuchi et al. (2020a, b); Kikuchi et al. (2021);
Almumin et al. (2021). Thus, the 4D matter fields have the modular weight
$-1/2$. (See for the modular symmetry and modular weights in heterotic
orbifold models Refs. Lauer et al. (1989, 1991); Ferrara et al. (1989); Dixon
et al. (1990); Baur et al. (2019); Nilles et al. (2020, 2021).) Such a
negative modular weight of 4D matter fields is natural in the 4D effective
action in the following sense.

  * •

Property of modular forms from the viewpoint of higher-dimensional theories

Firstly, suppose that the massless wavefunction in the extra-dimensional space
$\varphi(y)$ is a holomorphic function of the modulus at the origin $y=0$.
Then, when the wavefunction has the positive modular weight $k>0$, it is a
modular form. Even if we incorporate the radiative corrections, the sign of
modular weight will be unchanged as discussed in Ref. Kikuchi et al. (2023).
Indeed, in the opposite case $k<0$, $\varphi(0)$ will not be the modular form
but rather a singular function.

  * •

Control of matter Kähler potential from the viewpoint of 4D effective theories

Second, let us see the modular invariant Kähler potential of 4D matter fields:

$\displaystyle K=\frac{1}{(2{\rm Im}(\tau))^{k}}|\phi(x)|^{2},$ (6)

corresponding to the normalization of internal wavefunction:

$\displaystyle\int d^{d}y\,\sqrt{g}\,|\varphi(y)|^{2}=\frac{1}{(2{\rm
Im}(\tau))^{k}}.$ (7)

The positivity of modular weight $k$ controls higher-order corrections in the
Kähler potential, for instance $|\phi(x)|^{4}/(2{\rm Im}(\tau))^{2k}$,
corresponding to the volume expansion of the torus. Hence, the negative
modular weight will be out of control.

Note that the above viewpoints are different from each other. The former point
is to construct the 4D effective theories from fully known higher-dimensional
theories by Kaluza-Klein reduction. On the other hand, the latter point is the
controllability of the 4D effective theories, which may be derived, e.g. by
string perturbation theory of 4D modes.

In the following, let us present two possibilities of realizing 4D positive
modular weights from a top-down approach.

  1. 1.

Localized modes

For the query of modular form, localized modes in extra dimensions may have a
chance to have the positive modular weight for 4D chiral matters. Suppose that
they are localized at $y\neq 0$. Then, their wavefunctions vanish at $y=0$.
These modes may be free from the discussion on modular forms. For example, the
heterotic string theory on toroidal orbifolds has untwisted and twisted modes.
The twisted modes are localized modes, while the untwisted modes are bulk
modes. The ground states of twisted modes on $T^{2}/Z_{N}$ have the modular
weights $-1/N$ for 4D matter fields. On top of that, the oscillators shift the
modular weight by $\pm 1$, where the sign depends on the property of
oscillators. Thus, 4D matter fields have a chance of the positive modular
weights. (See, Refs. Dixon et al. (1990); Ibanez and Lust (1992); Kawabe et
al. (1995), for an explicit construction.) The matter Kähler potential would
be controlled in the stringy calculations or the following multi moduli
scenario.

  2. 2.

Bulk modes in a multi moduli scenario

For the query of controlling the Kähler potential, multi moduli scenario will
be crucial. For instance, in the two moduli case, the matter Kähler potential
will be expanded in the large volume regime:

$\displaystyle K=\frac{1}{(2{\rm Im}(T))^{n}(2{\rm Im}(t))^{k}}|\phi(x)|^{2},$
(8)

where $n$ and $k$ are modular weights associated with the modulus $T$ and $t$,
respectively. Let us suppose that the modular flavor symmetry is originated
from the modulus $t$, but the overall volume is determined by another modulus
$T$. In this case, the negative modular weight $k<0$ will be controllable. For
illustrative purposes, we deal with the $E_{8}\times E_{8}$ heterotic string
with standard embedding, where the matter Kähler potential will be extracted
from the moduli Kähler potential. Specifically, on $T^{6}/\mathbb{Z}_{3}$
orbifold, the Kähler potential is known up to the second order in the chiral
matter field $A_{\alpha}$ which is a ${\bf 27}$ fundamental representation of
$E_{6}$ Ferrara et al. (1986); Cvetic et al. (1988):444A hierarchical
structure of holomorphic Yukawa couplings was discussed on this background
Ishiguro et al. (2021a).

$\displaystyle K$ $\displaystyle=-\ln\det(2{\rm
Im}(T)-A^{\alpha}A^{\bar{\alpha}})^{a\bar{b}}=-\ln\det(2{\rm
Im}(T))^{a\bar{b}}+(2{\rm
Im}(T))^{-1}_{a\bar{b}}A^{a}_{\alpha}A^{\bar{b}}_{\bar{\alpha}},$ (9)

with

$\displaystyle T=\begin{pmatrix}T^{1}&T^{4}&T^{5}\\\ T^{7}&T^{2}&T^{6}\\\
T^{8}&T^{9}&T^{3}\\\ \end{pmatrix}.$ (10)

Here, $a$ and $\alpha$ denote the index of moduli and the index of
$SU(3)\subset E_{8}$, respectively. Note that we focus on untwisted Kähler
moduli and corresponding untwisted matters. When we restrict ourselves on the
locus $T:=T^{1}=T^{2}=T^{3}$ and $t:=T^{4}=T^{5}=T^{6}=T^{7}=T^{8}=T^{9}$, an
explicit form of matter Kähler metric $K_{a\bar{b}}^{(27)}=(2{\rm
Im}(T))^{-1}_{a\bar{b}}$ is

$\displaystyle 2({\rm Im}(T)-{\rm Im}(t))^{2}({\rm Im}(T)+2{\rm
Im}(t))K_{a\bar{b}}^{(27)}$ $\displaystyle=\begin{pmatrix}{\rm Im}(T)^{2}-{\rm
Im}(t)^{2}&({\rm Im}(t)-{\rm Im}(T)){\rm Im}(t)&({\rm Im}(t)-{\rm Im}(T)){\rm
Im}(t)\\\ ({\rm Im}(t)-{\rm Im}(T)){\rm Im}(t)&{\rm Im}(T)^{2}-{\rm
Im}(t)^{2}&({\rm Im}(t)-{\rm Im}(T)){\rm Im}(t)\\\ ({\rm Im}(t)-{\rm
Im}(T)){\rm Im}(t)&({\rm Im}(t)-{\rm Im}(T)){\rm Im}(t)&{\rm Im}(T)^{2}-{\rm
Im}(t)^{2}\\\ \end{pmatrix},$ (11)

which can be diagonalized as

$\displaystyle\Lambda_{\hat{a}\bar{\hat{b}}}=$
$\displaystyle\begin{pmatrix}(2{\rm Im}(T)-2{\rm Im}(t))^{-1}&0&0\\\ 0&(2{\rm
Im}(T)-2{\rm Im}(t))^{-1}&0\\\ 0&0&(2{\rm Im}(T)+4{\rm Im}(t))^{-1}\\\
\end{pmatrix}.$ (12)

Thus, when we focus on the $t$-dependent Kähler metric in the diagonal basis,
one can arrive at the matter Kähler metric with 4D positive modular weight:

$\displaystyle K$ $\displaystyle\propto\frac{2{\rm Im}(t)}{2{\rm
Im}(T)}|\phi(x)|^{2},$ (13)

where it is notable that the overall volume is now controlled by ${\rm Im}(T)$
rather than ${\rm Im}(t)$, and both of them are considered to be larger than
the string length. It results in the control of matter Kähler potential
against higher-order corrections. A moderate hierarchy between the overall
volume and the volume of local cycle plays an important role in the
realization of positive modular weights. Since the modular flavor symmetry is
realized in the local cycle of underlying manifolds, the matter wavefunctions
would be localized in the extra-dimensional space. Thus, the query of modular
form would be resolved, but it is required to check the property of
wavefunctions themselves, which left for future work. This is just a specific
example of realizing the 4D positive modular weight for the untwisted sector
on toroidal orbifolds, but we expect similar phenomena in a more generic multi
moduli scenario. Indeed, on the so-called “Swiss-cheese like Calabi-Yau
manifolds” as employed in the Large Volume Scenario Balasubramanian et al.
(2005); Conlon et al. (2005), one can derive a similar structure of moduli
Kähler potential due to the hierarchy between the overall volume and the local
divisor volume. Following a general procedure for constructing modular flavor
models on Calabi-Yau manifolds Ishiguro et al. (2021b); Ishiguro et al.
(2022), we would realize the 4D positive modular weights whose comprehensive
study will be left for future work.

##  III Two examples of lepton models

In this section, we show two examples of lepton models: “Inverse Seesaw (IS)”
and “Linear Seesaw (LS)”, introducing positive modular weights. Thanks to the
extension of degrees of freedom for signs of modular weights, we find that
these two scenarios can be constructed by this symmetry itself. 555 Notice
here that we implicitly work on the supersymmetric theory to forbid infinite
terms by making good use of holomorphicity from supersymmetry, and we start
from its effective theory after breaking supersymmetry. See, Refs. Kobayashi
and Otsuka (2022); Kikuchi et al. (2022b), for a realization of modular-
invariant effective field theory after the soft supersymmetry breaking.
Without positive and negative modular weights, these two scenarios can be
constructed only if additional symmetry such as a $U(1)$ symmetry is imposed.
This is (roughly speaking) because we have to differentiate two types of
heavier Majorana fermions denoted by $\bar{N}$ and $S$, and this
discrimination cannot be achieved by applying negative modular weights only.
Below we will show how to realize our models.

Before discussing each of the neutrino mass mechanisms, we review general
formulation on neutrino masses and their mixings. The neutrino mass matrix can
be written in terms of overall mass dimension parameter $\kappa$ and
dimensionless neutrino mass matrix $\tilde{m}_{\nu}$:

$\displaystyle m_{\nu}\equiv\kappa\tilde{m}_{\nu},$ (14)

where $\kappa$ and $\tilde{m}_{\nu}$ are determined if a concrete mechanism is
fixed. $m_{\nu}$ is diagonalized by a unitary matrix $V_{\nu}$;
$D_{\nu}=|\kappa|\tilde{D}_{\nu}=V_{\nu}^{T}m_{\nu}V_{\nu}=|\kappa|V_{\nu}^{T}\tilde{m}_{\nu}V_{\nu}$.
Then $|\kappa|$ is fixed by

$\displaystyle(\mathrm{NH}):\ |\kappa|^{2}=\frac{|\Delta m_{\rm
atm}^{2}|}{\tilde{D}_{\nu_{3}}^{2}-\tilde{D}_{\nu_{1}}^{2}},\quad(\mathrm{IH}):\
|\kappa|^{2}=\frac{|\Delta m_{\rm
atm}^{2}|}{\tilde{D}_{\nu_{2}}^{2}-\tilde{D}_{\nu_{3}}^{2}},$ (15)

where $\Delta m_{\rm atm}^{2}$ is the atmospheric neutrino mass-squared
splitting, and NH and IH indicate the normal hierarchy and the inverted
hierarchy, respectively. Subsequently, the solar mass squared splitting can be
obtained in terms of $|\kappa|$ such that:

$\displaystyle\Delta m_{\rm
sol}^{2}=|\kappa|^{2}({\tilde{D}_{\nu_{2}}^{2}-\tilde{D}_{\nu_{1}}^{2}}),$
(16)

which is compared to the observed value in our numerical analysis. We define
the observed mixing matrix by $U=V^{\dagger}_{eL}V_{\nu}$ Maki et al. (1962)
666Note that if the charged-lepton mass matrix $m_{\ell}$ is not a diagonal
one, we need to diagonalize it as ${\rm
diag}(|m_{e}|^{2},|m_{\mu}|^{2},|m_{\tau}|^{2})\equiv
V_{e_{L}}^{\dagger}m^{\dagger}_{\ell}m_{\ell}V_{e_{L}}$, where $V_{eL}$ is a
unitary matrix., that is 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 $\alpha_{21},\alpha_{31}$. Under the parametrization, we
write the matrix by

$U=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{CP}}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{CP}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{CP}}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{CP}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{CP}}&c_{23}c_{13}\end{pmatrix}\begin{pmatrix}1&0&0\\\
0&e^{i\frac{\alpha_{21}}{2}}&0\\\
0&0&e^{i\frac{\alpha_{31}}{2}}\end{pmatrix},$ (17)

where $c_{ij}$ and $s_{ij}$ stand for $\cos\theta_{ij}$ and $\sin\theta_{ij}$
($i,j=1-3$), respectively. Then, we can write each of the mixing in terms of
the component of $U$ as follows:

$\displaystyle\sin^{2}\theta_{13}=|U_{e3}|^{2},\quad\sin^{2}\theta_{23}=\frac{|U_{\mu
3}|^{2}}{1-|U_{e3}|^{2}},\quad\sin^{2}\theta_{12}=\frac{|U_{e2}|^{2}}{1-|U_{e3}|^{2}}.$
(18)

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

$\displaystyle\sin\delta_{CP}$ $\displaystyle=\frac{\text{Im}[U_{e1}U_{\mu
2}U_{e2}^{*}U_{\mu
1}^{*}]}{s_{23}c_{23}s_{12}c_{12}s_{13}c^{2}_{13}},\quad\cos\delta_{CP}=-\frac{|U_{\tau
1}|^{2}-s^{2}_{12}s^{2}_{23}-c^{2}_{12}c^{2}_{23}s^{2}_{13}}{2c_{12}s_{12}c_{23}s_{23}s_{13}},$
(19)

where $\delta_{CP}$ being subtracted from $\pi$ if $\cos\delta_{CP}$ is
negative. In addtion Majorana phase $\alpha_{21},\ \alpha_{31}$ are found from
following relations:

$\displaystyle\sin\left(\frac{\alpha_{21}}{2}\right)=\frac{\text{Im}[U^{*}_{e1}U_{e2}]}{c_{12}s_{12}c_{13}^{2}},\quad\cos\left(\frac{\alpha_{21}}{2}\right)=\frac{\text{Re}[U^{*}_{e1}U_{e2}]}{c_{12}s_{12}c_{13}^{2}},\
,$ (20)
$\displaystyle\sin\left(\frac{\alpha_{31}}{2}-\delta_{CP}\right)=\frac{\text{Im}[U^{*}_{e1}U_{e3}]}{c_{12}s_{13}c_{13}},\quad\cos\left(\frac{\alpha_{31}}{2}-\delta_{CP}\right)=\frac{\text{Re}[U^{*}_{e1}U_{e3}]}{c_{12}s_{13}c_{13}},$
(21)

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

$\displaystyle\langle
m_{ee}\rangle=|\kappa||\tilde{D}_{\nu_{1}}\cos^{2}\theta_{12}\cos^{2}\theta_{13}+\tilde{D}_{\nu_{2}}\sin^{2}\theta_{12}\cos^{2}\theta_{13}e^{i\alpha_{21}}+\tilde{D}_{\nu_{3}}\sin^{2}\theta_{13}e^{-2i\delta_{CP}}|,$
(22)

where its predicted value could be tested by current/future experiments such
as KamLAND-Zen Gando et al. (2016).

###  III.1 Inverse seesaw

| Leptons | Higgs
---|---|---
|  $(L_{e}$,$L_{\mu}$,$L_{\tau})$ |  ($\bar{e}$,$\bar{\mu}$,$\bar{\tau}$) |  $\bar{N}$ |  $S$ |  $H_{u}$ |  $H_{d}$
$SU(2)_{L}$ | $\bm{2}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{2}$ | $\bm{2}$
$U(1)_{Y}$ | $-\frac{1}{2}$ | $+1$ | $0$ | $0$ | $\frac{1}{2}$ | $-\frac{1}{2}$
$A_{4}$ | $\\{1\\}$ | $\\{\bar{1}\\}$ | $\\{\bar{1}\\}$ | $3$ | $1$ | $1$
$-k$ | $+1$ | $-1$ | $-1$ | $-1$ | $0$ | $0$
Table 1: (IS): Field contents for leptons and Higgs with their charge
assignments under $SU(2)_{L}\otimes U(1)_{Y}\otimes A_{4}$ where $-k$ is the
number of modular weight and
$\\{1\\}\equiv\\{1,1^{\prime},1^{\prime\prime}\\}$ and
$\\{\bar{1}\\}\equiv\\{1,1^{\prime\prime},1^{\prime}\\}$.

The IS scenario requires two types of neutral fermions $\bar{N}$ and $S$ where
we assume each of them has three families. Our model is discriminated by
different charges of flavor symmetry of $A_{4}$; three types of singlets
$\\{\bar{1}\\}\equiv\\{1,1^{\prime\prime},1^{\prime}\\}$ for $\bar{N}$ and
triplet $3$ for $S$ where $-1$ modular weights are imposed for all these
neutral fermions. The SM leptons are assigned under $A_{4}$ as follows:
$\\{\bar{1}\\}$ for three right-handed charged-leptons
$\\{\bar{e},\bar{\mu},\bar{\tau}\\}$ and
$\\{1\\}(\equiv\\{1,1^{\prime},1^{\prime\prime}\\})$ for three left-handed
leptons $\\{L_{e},L_{\mu},L_{\tau}\\}$ where $+1$ and $-1$ modular weights are
imposed respectively. Due to these assignments, we find the charged-lepton
mass matrix is diagonal at the Lagrangian. Therefore, the mass eigenvalues of
charged-leptons are given by $m_{\ell}\equiv y_{\ell}v_{d}/\sqrt{2}$
($\ell\equiv(e,\mu,\tau)$) where $\langle
H_{d}\rangle\equiv[0,v_{d}/\sqrt{2}]^{T}$. It implies that PMNS matrix denoted
by $U$ directly comes from the neutrino sector. Thus, we concentrate on the
neutrino sector below. Moreover, we introduce two Higgs fields to cancel
chiral anomalies as the usual supersymmetric SM. We summarize all the charge
assignments and their field contents in Tab. 1. Under these symmetries,
renormalizable superpotential is found as

$\displaystyle W_{(IS)}$
$\displaystyle=\sum_{i=1}^{3}y_{D_{i}}\bar{N}_{i}H_{u}L_{i}$ (23)
$\displaystyle+m_{NS}\left[\bar{N}_{1}(y_{1}S_{1}+y_{3}S_{2}+y_{2}S_{3})+\tilde{\alpha}_{2}\bar{N}_{2}(y_{2}S_{1}+y_{1}S_{2}+y_{3}S_{3})+\tilde{\alpha}_{3}\bar{N}_{3}(y_{3}S_{1}+y_{2}S_{2}+y_{1}S_{3})\right]$
$\displaystyle+m_{S}\left[y_{1}(2S_{1}S_{1}-S_{2}S_{3}-S_{3}S_{2})+y_{2}(2S_{2}S_{2}-S_{1}S_{3}-S_{3}S_{1})+y_{3}(2S_{3}S_{3}-S_{1}S_{2}-S_{2}S_{1})\right],$

where $y_{1,2,3}$ is Yukawa coupling given by a modular form under $A_{4}$
triplet with modular weight $2$, and all the other terms are forbidden by our
charge assignments. In the lepton sector at least, R-parity is not needed
because of the modular $A_{4}$ symmetry.

Notice here that $\bar{N}H_{u}L$ term is absent if we choose $-1$ modular
weight for $L_{i}$ since $A_{4}$ triplet weight 2 modular form is required
while superfields are singlet. It is indeed possible to have the term
assigning a triplet for $\bar{N}$ but we can also have $LSH_{u}$ and
$\bar{N}\bar{N}$ terms breaking the IS structure. Also if we choose $-(2n+1)$
($n\neq 0$) weight for $L_{i}$, we generally have $LSH_{u}$ term. In general,
we can not obtain the IS structure in a minimal way with negative modular
weights only; there will be more possible invariant terms if we increase the
absolute value of modular weights or assign triplets for other superfields.
Therefore positive modular weights make it possible to realize the IS
structure without imposing any other symmetry.

After spontaneous symmetry breaking, we find nine by nine neutral fermion’s
mass matrix in a basis of $[\nu,\bar{N}^{C},S]^{T}$ as follows

$\displaystyle M_{N}\equiv\begin{pmatrix}0&m_{D}^{T}&0\\\ m_{D}&0&M_{NS}\\\
0&M_{NS}^{T}&M_{S}\end{pmatrix},$ (24)

where $m_{D},M_{S},M_{{NS}}$ are respectively given by

$\displaystyle m_{D}$
$\displaystyle\equiv\frac{y_{D_{1}}v_{u}}{\sqrt{2}}\tilde{m}_{D},\quad\tilde{m}_{D}\equiv\begin{pmatrix}1&0&0\\\
0&\tilde{\mu}_{2}&0\\\ 0&0&\tilde{\mu}_{3}\end{pmatrix},$ (25)
$\displaystyle\mu_{S}$ $\displaystyle\equiv
m_{S}\tilde{\mu}_{S},\quad\tilde{\mu}_{S}\equiv\begin{pmatrix}2y_{1}&-y_{3}&-y_{2}\\\
-y_{3}&2y_{2}&-y_{1}\\\ -y_{2}&-y_{1}&2y_{3}\end{pmatrix},$ (26)
$\displaystyle M_{NS}$ $\displaystyle\equiv
m_{NS}\tilde{M}_{NS},\quad\tilde{M}_{NS}\equiv\begin{pmatrix}1&0&0\\\
0&\tilde{\alpha}_{2}&0\\\
0&0&\tilde{\alpha}_{3}\end{pmatrix}\begin{pmatrix}y_{1}&y_{3}&y_{2}\\\
y_{2}&y_{1}&y_{3}\\\ y_{3}&y_{2}&y_{1}\end{pmatrix},$ (27)

and $\langle H_{u}\rangle\equiv[v_{u}/\sqrt{2},0]^{T}$. Here, all the
dimensionless parameters (except $y_{1,2,3}$);
$\tilde{\mu}_{2,3},\tilde{\alpha}_{2,3}$, can be real after rephasing of
fields. Once we assume the following mass hierarchy $\mu_{S}\ll m_{D}<M_{NS}$,
we find the active neutrino mass matrix in light of the IS mechanism as
follows:

$\displaystyle m_{\nu}\approx(m_{D}M_{ND}^{-1})\mu_{S}(m_{D}M_{ND}^{-1})^{T}.$
(28)

Furthermore, $m_{\nu}$ can be rewritten in terms of dimensionless matrices:

$\displaystyle
m_{\nu}\approx\frac{y_{D_{1}}^{2}v_{u}^{2}m_{S}}{2m^{2}_{NS}}(\tilde{m}_{D}\tilde{M}_{ND}^{-1})\tilde{\mu}_{S}(\tilde{m}_{D}\tilde{M}_{ND}^{-1})^{T}\equiv\kappa\tilde{m}_{\nu},$
(29)

where $\kappa\equiv\frac{y_{D_{1}}^{2}v_{u}^{2}m_{S}}{2m^{2}_{NS}}$ and
$\tilde{m}_{\nu}\equiv(\tilde{m}_{D}\tilde{M}_{ND}^{-1})\tilde{\mu}_{S}(\tilde{m}_{D}\tilde{M}_{ND}^{-1})^{T}$.
Now we have four dimensionless real parameters
$\tilde{\mu}_{2,3},\tilde{\alpha}_{2,3}$ and one complex one $\tau$.
Therefore, we totally have six free parameters.

###  III.2 Linear seesaw

| Leptons | Higgs
---|---|---
|  $(L_{e}$,$L_{\mu}$,$L_{\tau})$ |  ($\bar{e}$,$\bar{\mu}$,$\bar{\tau}$) |  $\bar{N}$ |  $S$ |  $H_{u}$ |  $H_{d}$
$SU(2)_{L}$ | $\bm{2}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{2}$ | $\bm{2}$
$U(1)_{Y}$ | $-\frac{1}{2}$ | $+1$ | $0$ | $0$ | $\frac{1}{2}$ | $-\frac{1}{2}$
$A_{4}$ | $3$ | $\\{\bar{1}\\}$ | $\\{\bar{1}\\}$ | $\\{1\\}$ | $1$ | $1$
$-k$ | $-3$ | $-1$ | $+1$ | $-1$ | $0$ | $0$
Table 2: (LS): Field contents for leptons and Higgs with their charge
assignments under $SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)_{H}\otimes A_{4}$
where $-k$ is the number of modular weight and
$\\{1\\}\equiv\\{1,1^{\prime},1^{\prime\prime}\\}$ and
$\\{\bar{1}\\}\equiv\\{1,1^{\prime\prime},1^{\prime}\\}$.

The LS scenario also requires two types of neutral fermions $\bar{N}$ and $S$
where we assume each of them has three families. Our model is discriminated by
different charges of modular flavor symmetry of $A_{4}$; three types of
singlets $\\{\bar{1}\\}$ with $+1$ modular weight for $\bar{N}$, and singlets
$\\{1\\}$ for three generations of $S$ where $-1$ modular weights are imposed
for them. The SM leptons are assigned under $A_{4}$ as follows:
$\\{\bar{1}\\}$ for three right-handed charged-leptons
$\\{\bar{e},\bar{\mu},\bar{\tau}\\}$ and triplet for three left-handed leptons
$\\{L_{e},L_{\mu},L_{\tau}\\}$ where $-1$ and $-3$ modular weights are imposed
respectively. The Higgs sector is the same as the one in IS case. We summarize
all the charge assignments and their field contents in Tab. 2. Under these
symmetries, renormalizable superpotential is found as

$\displaystyle W_{(LS)}$
$\displaystyle=a_{e}\bar{e}H_{d}(y^{\prime}_{1}L_{1}+y^{\prime}_{3}L_{2}+y^{\prime}_{2}L_{3})+b_{e}\bar{\mu}H_{d}(y^{\prime}_{2}L_{1}+y^{\prime}_{1}L_{2}+y^{\prime}_{3}L_{3})$
$\displaystyle+c_{e}\bar{\tau}H_{d}(y^{\prime}_{3}L_{1}+y^{\prime}_{2}L_{2}+y^{\prime}_{1}L_{3})$
(30)
$\displaystyle+\alpha_{D_{1}}\left[\bar{N}_{1}H_{u}(y_{1}L_{1}+y_{3}L_{2}+y_{2}L_{3})+\tilde{\alpha}_{D_{2}}\bar{N}_{2}H_{u}(y_{2}L_{1}+y_{1}L_{2}+y_{3}L_{3})\right.$
$\displaystyle+\left.\tilde{\alpha}_{D_{3}}\bar{N}_{3}H_{u}(y_{3}L_{1}+y_{2}L_{2}+y_{1}L_{3})\right]$
(31)
$\displaystyle+\alpha_{D^{\prime}_{1}}\left[S_{1}H_{u}(y^{\prime}_{1}L_{1}+y^{\prime}_{3}L_{2}+y^{\prime}_{2}L_{3})+\tilde{\alpha}_{D^{\prime}_{2}}S_{2}H_{u}(y^{\prime}_{3}L_{1}+y^{\prime}_{2}L_{2}+y^{\prime}_{1}L_{3})\right.$
$\displaystyle+\left.\tilde{\alpha}_{D^{\prime}_{3}}S_{3}H_{u}(y^{\prime}_{2}L_{1}+y^{\prime}_{1}L_{2}+y^{\prime}_{3}L_{3})\right]$
(32) $\displaystyle+\sum_{i=1}^{3}m_{NS_{i}}\bar{N}_{i}S_{i},$ (33)

where $y^{\prime}_{1,2,3}$ is Yukawa coupling given by modular form under
$A_{4}$ triplet with modular weight $4$, and all the other terms are forbidden
by our charge assignments.

Notice here that $\bar{N}S$ term is absent if we choose $-1$ modular weight
for $\bar{N}$. It is possible to get the term by considering $-(2n+1)$ ($n\neq
0$) weight for $\bar{N}$ but this weight also allows $\bar{N}\bar{N}^{T}$
term. Therefore positive modular weight is crucial to realize LS structure in
a minimal way as in the IS case.

Charged-lepton mass matrix is given after spontaneous symmetry breaking of
$H_{d}$:

$\displaystyle m_{\ell}=\frac{v_{d}}{\sqrt{2}}\begin{pmatrix}a_{e}&0&0\\\
0&b_{e}&0\\\
0&0&c_{e}\end{pmatrix}\begin{pmatrix}y^{\prime}_{1}&y^{\prime}_{3}&y^{\prime}_{2}\\\
y^{\prime}_{2}&y^{\prime}_{1}&y^{\prime}_{3}\\\
y^{\prime}_{3}&y^{\prime}_{2}&y^{\prime}_{1}\end{pmatrix},$ (34)

where $a_{e},b_{e},c_{e}$ are real without loss of generality. Then the
charged-lepton mass eigenvalues are found as ${\rm
diag}(|m_{e}|^{2},|m_{\mu}|^{2},|m_{\tau}|^{2})\equiv
V_{e_{L}}^{\dagger}m^{\dagger}_{\ell}m_{\ell}V_{e_{L}}$. We fix these three
input parameters by inserting the observed charged-lepton masses and
$V_{e_{L}}$ as follows:

$\displaystyle{\rm
Tr}[m_{\ell}{m_{\ell}}^{\dagger}]=|m_{e}|^{2}+|m_{\mu}|^{2}+|m_{\tau}|^{2},$
(35) $\displaystyle{\rm
Det}[m_{\ell}{m_{\ell}}^{\dagger}]=|m_{e}|^{2}|m_{\mu}|^{2}|m_{\tau}|^{2},$
(36) $\displaystyle({\rm Tr}[m_{\ell}{m_{\ell}}^{\dagger}])^{2}-{\rm
Tr}[(m_{\ell}{m_{\ell}}^{\dagger})^{2}]=2(|m_{e}|^{2}|m_{\mu}|^{2}+|m_{\mu}|^{2}|m_{\tau}|^{2}+|m_{e}|^{2}|m_{\tau}|^{2}).$
(37)

Here, we apply the experimental values summarized in PDG for the charged-
lepton masses Tanabashi et al. (2018).
After spontaneous symmetry breaking, we find nine by nine neutral fermion’s
mass matrix in a basis of $[\nu,\bar{N}^{C},S]^{T}$ as follows

$\displaystyle M_{N}\equiv\begin{pmatrix}0&m_{D}^{T}&m^{\prime T}_{D}\\\
m_{D}&0&M_{NS}\\\
m^{\prime}_{D}&M_{NS}^{T}&0\end{pmatrix}+\begin{pmatrix}0&m_{D}^{T}&m^{\prime
T}_{D}\\\ m_{D}&0&M_{NS}\\\ m^{\prime}_{D}&M_{NS}^{T}&0\end{pmatrix}^{T},$
(38)

where $m_{D},M_{S},M_{{NS}}$ are respectively given by

$\displaystyle m_{D}$
$\displaystyle\equiv\frac{\alpha_{D_{1}}v_{u}}{\sqrt{2}}\tilde{m}_{D},\quad\tilde{m}_{D}\equiv\begin{pmatrix}1&0&0\\\
0&\tilde{\alpha}_{D_{2}}&0\\\
0&0&\tilde{\alpha}_{D_{3}}\end{pmatrix}\begin{pmatrix}y_{1}&y_{3}&y_{2}\\\
y_{2}&y_{1}&y_{3}\\\ y_{3}&y_{2}&y_{1}\end{pmatrix},$ (39) $\displaystyle
m^{\prime}_{D}$
$\displaystyle\equiv\frac{\alpha_{D^{\prime}_{1}}v_{u}}{\sqrt{2}}\tilde{m}^{\prime}_{D},\quad\tilde{m}^{\prime}_{D}\equiv\begin{pmatrix}1&0&0\\\
0&\tilde{\alpha}_{D^{\prime}_{2}}&0\\\
0&0&\tilde{\alpha}_{D^{\prime}_{3}}\end{pmatrix}\begin{pmatrix}y^{\prime}_{1}&y^{\prime}_{3}&y^{\prime}_{2}\\\
y^{\prime}_{3}&y^{\prime}_{2}&y^{\prime}_{1}\\\
y^{\prime}_{2}&y^{\prime}_{1}&y^{\prime}_{3}\end{pmatrix},$ (40)
$\displaystyle M_{NS}$ $\displaystyle\equiv
m_{NS_{1}}\tilde{M}_{NS},\quad\tilde{M}_{NS}\equiv\begin{pmatrix}1&0&0\\\
0&\tilde{\mu}_{NS_{2}}&0\\\ 0&0&\tilde{\mu}_{NS_{3}}\end{pmatrix},$ (41)

$\tilde{\mu}_{NS_{a}}\equiv m_{NS_{a}}/m_{NS_{1}}(a=2,3)$, and $\langle
H_{u}\rangle\equiv[v_{u}/\sqrt{2},0]^{T}$. Here, four dimensionless parameters
$\tilde{\alpha}_{D_{2,3}},\tilde{\alpha}_{D^{\prime}_{2,3}}$ can be real after
rephasing of fields while $\tilde{\mu}_{NS_{2,3}}$ are complex. Once we assume
the following mass hierarchy $m_{D},m^{\prime}_{D}\ll M_{NS}$, we find the
active neutrino mass matrix in light of the LS mechanism as follows:

$\displaystyle m_{\nu}\approx m^{\prime
T}_{D}M_{ND}^{-1}m_{D}^{T}+m_{D}(M^{T}_{ND})^{-1}m^{\prime}_{D}.$ (42)

Furthermore, $m_{\nu}$ can be rewritten in terms of dimensionless matrices:

$\displaystyle
m_{\nu}\approx\frac{\alpha_{D_{1}}\alpha_{D^{\prime}_{1}}v_{u}^{2}}{2m_{NS_{1}}}[\tilde{m}^{\prime
T}_{D}\tilde{M}_{ND}^{-1}\tilde{m}_{D}^{T}+\tilde{m}_{D}(\tilde{M}^{T}_{ND})^{-1}\tilde{m}^{\prime}_{D}]\equiv\kappa\tilde{m}_{\nu},$
(43)

where
$\kappa\equiv\frac{\alpha_{D_{1}}\alpha_{D^{\prime}_{1}}v_{u}^{2}}{2m_{NS_{1}}}$
and $\tilde{m}_{\nu}\equiv\tilde{m}^{\prime
T}_{D}\tilde{M}_{ND}^{-1}\tilde{m}_{D}^{T}+\tilde{m}_{D}(\tilde{M}^{T}_{ND})^{-1}\tilde{m}^{\prime}_{D}$.
Now we have four dimensionless real parameters
$\tilde{\alpha}^{\prime}_{2,3},\tilde{\alpha}_{2,3}$, and three complex ones
$\tau,\tilde{\mu}_{NS_{2,3}}$. Therefore, we totally have ten free parameters.

###  III.3 Numerical analysis

We have numerical analysis for IS and LS models below and show what kind of
correlations are found. Before showing the numerical analysis, we need to
discuss a constraint from non-unitarity that should be taken in account for IS
and LS. The non-unitarity matrix denoted by $U^{\prime}$ that represents the
deviation from the unitarity, and it is typically parametrized by the
following form

$\displaystyle U^{\prime}\equiv\left(1-\frac{1}{2}FF^{\dagger}\right)U,$ (44)

where $F\equiv(M^{*}_{NS})^{-1}m^{T}_{D}$ is a hermitian matrix. $M_{NS}$ and
$m_{D}$ are the same notations as the ones of each model for IS and LS. The
global constraints are found via several experimental results such as the SM
$W$ boson mass, the effective Weinberg angle, several ratios of $Z$ boson
fermionic decays, invisible decay of $Z$, electroweak universality, measured
Cabbibo-Kobayashi-Maskawa, and lepton flavor violations Fernandez-Martinez et
al. (2016). These results give the following constraints for $|FF^{\dagger}|$
Agostinho et al. (2018)

$\displaystyle|FF^{\dagger}|\leq\left[\begin{array}[]{ccc}2.5\times
10^{-3}&2.4\times 10^{-5}&2.7\times 10^{-3}\\\ 2.4\times 10^{-5}&4.0\times
10^{-4}&1.2\times 10^{-3}\\\ 2.7\times 10^{-3}&1.2\times 10^{-3}&5.6\times
10^{-3}\\\ \end{array}\right].$ (48)

In our numerical analyses below, we implicitly impose the above constraints in
addition to the neutrino oscillation data.

####  III.3.1 IS

We randomly select our free parameters within the following range:

$\displaystyle\\{\tilde{\alpha}_{2,3},\tilde{\mu}_{2,3}\\}\in[10^{-3}-10^{3}],$
(49)

and we work on the fundamental region on $\tau$.

Figure 1: Allowed region of real and imaginary part of $\tau$ in the
fundamental region. The blue, green, yellow, and red points respectively
correspond to $\sigma\leq 1$, $1<\sigma\leq 2$, $2<\sigma\leq 3$ and
$3<\sigma\leq 5$ interval in $\Delta\chi^{2}$ analysis..

In Fig. 1, we plot the allowed region of real and imaginary part of $\tau$ in
the fundamental region, where the blue, green, yellow, and red points
respectively correspond to $\sigma\leq 1$, $1<\sigma\leq 2$, $2<\sigma\leq 3$
and $3<\sigma\leq 5$ interval in $\Delta\chi^{2}$ analysis. In the NH case,
the allowed region of real $\tau$ runs over the whole fundamental region while
there is the upper bound on imaginary $\tau$ that is about 4. In the IH case,
the allowed region of real $\tau$ runs over [$-0.4$,0.4] while the range of
imaginary $\tau$ is [2,5].

Figure 2: Allowed region of Majorana phases, where the color legends are the
same as the ones of Fig. 1.

In Fig. 2, we plot the allowed region of Majorana phases, where the color
legends are the same as the ones of Fig. 1. In case of NH, $\alpha_{31}$ runs
the whole range but $\alpha_{21}$ is localized at nearby [$-50^{\circ}$,50∘]
and [130∘,250∘]. In case of IH, $\alpha_{31}$ is [170∘,190∘] while
$\alpha_{21}$ is [$-50^{\circ}$,20∘].

Figure 3: Allowed region of sum of neutrino masses and CP Dirac phase, where
the color legends are the same as the ones of Fig. 1.

In Fig. 3, we plot the allowed region of sum of neutrino masses and CP Dirac
phase, where the color legends are the same as the ones of Fig. 1. In case of
NH, the sum of the active neutrino masses is allowed in the range of
[0.059,0.085] eV while the whole range is allowed for $\delta_{CP}$. In case
of IH, the sum of the active neutrino masses is allowed in the range of
[0.14,1] eV and $\delta_{CP}$ is localized by 90∘ and 270∘.

Figure 4: Allowed region of sum of neutrino masses and the effective
neutrinoless double beta decay, where the color legends are the same as the
ones of Fig. 1.

In Fig. 4, we plot the allowed region of sum of neutrino masses and the
effective neutrinoless double beta decay, where the color legends are the same
as the ones of Fig. 1. In case of NH, the effective neutrinoless double beta
decay is allowed in the range of [0.001,0.016] eV. In case of IH, the
effective neutrinoless double beta decay is allowed in the range of [0.05,0.3]
eV.

Figure 5: Allowed region of input parameters of $\tilde{\alpha}_{2,3}$, where
the color legends are the same as the ones of Fig. 1.

In Fig. 5, we plot the allowed region of $\tilde{\alpha}_{2,3}$, where the
color legends are the same as the ones of Fig. 1. In case of NH, although
$\tilde{\alpha}_{2,3}$ runs the whole range which we set, there is a
correlation between them. In case of IH, $\tilde{\alpha}_{2,3}$ runs
$[0.01,100]$ but their correlation is not so stronger than the case of NH.

Figure 6: Allowed region of input parameters of $\tilde{\mu}_{2,3}$, where the
color legends are the same as the ones of Fig. 1.

In Fig. 6, we plot the allowed region of $\tilde{\mu}_{2,3}$, where the color
legends are the same as the ones of Fig. 1. In case of NH, $\tilde{\mu}_{2}$
runs [0.01,100] and $\tilde{\mu}_{3}$ runs [0.1,1000]. Furthermore, there is a
correlation between them. In case of IH, $\tilde{\mu}_{2,3}$ runs $[0.01,100]$
but their correlation is not so stronger than the case of NH.

####  III.3.2 LS

We randomly select our free parameters within the following range:

$\displaystyle\\{\tilde{\alpha}_{D_{2,3}},\tilde{\alpha}_{D^{\prime}_{2,3}},|\tilde{\mu}_{NS_{2,3}}|\\}\in[10^{-3}-10^{3}],$
(50)

and we work on the fundamental region on $\tau$.

Figure 7: Allowed region of real and imaginary part of $\tau$ in the
fundamental region where the color legends are the same as the ones of Fig. 1.

In Fig. 7, we plot the allowed region of real and imaginary part of $\tau$ in
the fundamental region where the color legends are the same as the ones of
Fig. 1. In the case NH, real $\tau$ runs the whole range but imaginary $\tau$
has the upper bound Im$[\tau]\leq 2$. In the case IH, $\tau$ is localized at
nearby $|{\rm Re}[\tau]|=[0.05,0.07]$ and Im$[\tau]=[1.6,1.8]$.

Figure 8: Allowed region of Majorana phases, where the color legends are the
same as the ones of Fig. 1.

In Fig. 8, we plot the allowed region of Majorana phases, where the color
legends are the same as the ones of Fig. 1. In the case NH, even though the
whole ranges are allowed, $\alpha_{21}$ tends to be localized at nearby 180∘.
In the case IH, $\alpha_{31}$ is localized at nearby
$[90^{\circ},270^{\circ}]$ and $\alpha_{21}$ is also localized at nearby
$[120^{\circ},220^{\circ}]$.

Figure 9: Allowed region of sum of neutrino masses and CP Dirac phase, where
the color legends are the same as the ones of Fig. 1.

In Fig. 9, we plot the allowed region of sum of neutrino masses and CP Dirac
phase, where the color legends are the same as the ones of Fig. 1. In the NH
case, the whole range is allowed for $\delta_{CP}$ and sum of the active
neutrino masses runs over the range of [0.059,0.098] eV. In the IH case, $\sum
m_{i}$ is localized at nearby $[0.12,0.14]$ eV, and $\delta_{CP}$ is localized
at nearby $[-60^{\circ},60^{\circ}]$.

Figure 10: Allowed region of sum of neutrino masses and the effective
neutrinoless double beta decay, where the color legends are the same as the
ones of Fig. 1.

In Fig. 10, we plot the allowed region of sum of neutrino masses and the
effective neutrinoless double beta decay, where the color legends are the same
as the ones of Fig. 1. In the NH case, $\langle m_{ee}\rangle$ runs over
[0,0.013] eV. In the IH case, $\langle m_{ee}\rangle$ is localized at nearby
[0.016,0.024] eV.

Figure 11: Allowed region of input parameters of $\tilde{\alpha}_{D_{2,3}}$,
where the color legends are the same as the ones of Fig. 1.

In Fig. 11(12), we plot the allowed region of
$\tilde{\alpha}_{D_{2,3}}(\tilde{\alpha}_{D^{\prime}_{2,3}})$, where the color
legends are the same as the ones of Fig. 1. In the NH case,
$\tilde{\alpha}_{D_{2,3}}(\tilde{\alpha}_{D^{\prime}_{2,3}})$ runs the whole
ranges and their correlations are weak. In the IH case,
$\tilde{\alpha}_{D_{2}}$ and $\tilde{\alpha}_{D_{3}}$ are respectively
localized at nearby [9,17] and [1.49,2.75]. $\tilde{\alpha}_{D^{\prime}_{2}}$
and $\tilde{\alpha}_{D^{\prime}_{3}}$ are respectively localized at nearby
[0.006,0.012] and [200,400].

Figure 12: Allowed region of input parameters of
$\tilde{\alpha}_{D^{\prime}_{2,3}}$, where the color legends are the same as
the ones of Fig. 1.

Figure 13: Allowed region of input parameters of $\tilde{\mu}_{NS_{2,3}}$,
where the color legends are the same as the ones of Fig. 1. The left figure is
their absolute values while the right one is their arguments.

In Fig. 13, we plot the allowed region of $\tilde{\mu}_{NS_{2,3}}$, where the
color legends are the same as the ones of Fig. 1. The left figure is their
absolute values while the right one is their arguments. In the NH case,
$\tilde{\mu}_{NS_{2,3}}$ runs the whole ranges and there are no correlations
between them. In the IH case, $\tilde{\mu}_{NS_{2,3}}$ are respectively
localized at nearby [130,225] and [170,300].

##  IV Summary and Conclusions

We have proposed an interesting assignment of positive modular weights for 4D
fields that enables us to construct Inverse Seesaw and Linear Seesaw models
without any additional symmetries. These scenarios can be realized at a lower
energy scale that would be attractive in view of verifiability via current
experiments. At first, we have discussed possibilities for the positive
modular weights from a theoretical point of view. Then, we have shown two
examples; explicit constructions of IS and LS models, and demonstrated some
predictions via numerical analyses. We find that the minimal realization of
these models is achieved by positive modular weights for 4D superfields.

Before closing this paper, we would like to emphasize that this possibility of
positive modular assignment would possess more interesting aspects in model
building and vast applications for flavor physics. The positive modular
weights will give new insights not only into the flavor structure of quarks
and leptons but also into higher-dimensional operators in the SM effective
field theory Kobayashi et al. (2022a, b).

## Acknowledgments

The work was supported by the Fundamental Research Funds for the Central
Universities (T. N.), JSPS KAKENHI Grant Numbers JP20K14477 (H. Otsuka),
JP23H04512 (H. Otsuka) and JP23K03375 (T. K.).

## 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).
  * Akhmedov et al. (1996a) E. K. Akhmedov, M. Lindner, E. Schnapka, and J. W. F. Valle, Phys. Lett. B 368, 270 (1996a), eprint hep-ph/9507275.
  * Akhmedov et al. (1996b) E. K. Akhmedov, M. Lindner, E. Schnapka, and J. W. F. Valle, Phys. Rev. D 53, 2752 (1996b), eprint hep-ph/9509255.
  * Feruglio (2019) F. Feruglio, _Are neutrino masses modular forms?_ (2019), pp. 227–266, eprint 1706.08749.
  * Criado and Feruglio (2018) J. C. Criado and F. Feruglio, SciPost Phys. 5, 042 (2018), eprint 1807.01125.
  * Kobayashi et al. (2018a) T. Kobayashi, N. Omoto, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, JHEP 11, 196 (2018a), eprint 1808.03012.
  * Okada and Tanimoto (2019a) H. Okada and M. Tanimoto, Phys. Lett. B 791, 54 (2019a), eprint 1812.09677.
  * Nomura and Okada (2019a) T. Nomura and H. Okada, Phys. Lett. B 797, 134799 (2019a), eprint 1904.03937.
  * Okada and Tanimoto (2019b) H. Okada and M. Tanimoto (2019b), eprint 1905.13421.
  * de Anda et al. (2020) F. J. de Anda, S. F. King, and E. Perdomo, Phys. Rev. D 101, 015028 (2020), eprint 1812.05620.
  * Novichkov et al. (2019) P. Novichkov, S. Petcov, and M. Tanimoto, Phys. Lett. B 793, 247 (2019), eprint 1812.11289.
  * Nomura and Okada (2019b) T. Nomura and H. Okada (2019b), eprint 1906.03927.
  * Okada and Orikasa (2019) H. Okada and Y. Orikasa (2019), eprint 1907.13520.
  * Ding et al. (2019a) G.-J. Ding, S. F. King, and X.-G. Liu, JHEP 09, 074 (2019a), eprint 1907.11714.
  * Nomura et al. (2020) T. Nomura, H. Okada, and O. Popov, Phys. Lett. B 803, 135294 (2020), eprint 1908.07457.
  * Kobayashi et al. (2019) T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, Phys. Rev. D 100, 115045 (2019), [Erratum: Phys.Rev.D 101, 039904 (2020)], eprint 1909.05139.
  * Asaka et al. (2020a) T. Asaka, Y. Heo, T. H. Tatsuishi, and T. Yoshida, JHEP 01, 144 (2020a), eprint 1909.06520.
  * Zhang (2020) D. Zhang, Nucl. Phys. B 952, 114935 (2020), eprint 1910.07869.
  * Ding et al. (2019b) G.-J. Ding, S. F. King, X.-G. Liu, and J.-N. Lu, JHEP 12, 030 (2019b), eprint 1910.03460.
  * Kobayashi et al. (2020) T. Kobayashi, T. Nomura, and T. Shimomura, Phys. Rev. D 102, 035019 (2020), eprint 1912.00637.
  * Nomura et al. (2019) T. Nomura, H. Okada, and S. Patra (2019), eprint 1912.00379.
  * Wang (2020) X. Wang, Nucl. Phys. B 957, 115105 (2020), eprint 1912.13284.
  * Okada and Shoji (2020) H. Okada and Y. Shoji, Nucl. Phys. B 961, 115216 (2020), eprint 2003.13219.
  * Okada and Tanimoto (2020a) H. Okada and M. Tanimoto (2020a), eprint 2005.00775.
  * Behera et al. (2020a) M. K. Behera, S. Singirala, S. Mishra, and R. Mohanta (2020a), eprint 2009.01806.
  * Behera et al. (2020b) M. K. Behera, S. Mishra, S. Singirala, and R. Mohanta (2020b), eprint 2007.00545.
  * Nomura and Okada (2020a) T. Nomura and H. Okada (2020a), eprint 2007.04801.
  * Nomura and Okada (2020b) T. Nomura and H. Okada (2020b), eprint 2007.15459.
  * Asaka et al. (2020b) T. Asaka, Y. Heo, and T. Yoshida, Phys. Lett. B 811, 135956 (2020b), eprint 2009.12120.
  * Okada and Tanimoto (2020b) H. Okada and M. Tanimoto (2020b), eprint 2009.14242.
  * Nagao and Okada (2020) K. I. Nagao and H. Okada (2020), eprint 2010.03348.
  * Okada and Tanimoto (2020c) H. Okada and M. Tanimoto (2020c), eprint 2012.01688.
  * Kang et al. (2022) D. W. Kang, J. Kim, T. Nomura, and H. Okada, JHEP 07, 050 (2022), eprint 2205.08269.
  * Nomura et al. (2023) T. Nomura, H. Okada, and H. Otsuka (2023), eprint 2309.13921.
  * Kobayashi and Tanimoto (2023) T. Kobayashi and M. Tanimoto (2023), eprint 2307.03384.
  * Mishra et al. (2023) P. Mishra, M. K. Behera, P. Panda, M. Ghosh, and R. Mohanta, JHEP 09, 144 (2023), eprint 2305.08576.
  * Liu and Ding (2022) X.-G. Liu and G.-J. Ding, JHEP 03, 123 (2022), eprint 2112.14761.
  * Kikuchi et al. (2022a) S. Kikuchi, T. Kobayashi, H. Otsuka, M. Tanimoto, H. Uchida, and K. Yamamoto, Phys. Rev. D 106, 035001 (2022a), eprint 2201.04505.
  * Kobayashi et al. (2018b) T. Kobayashi, S. Nagamoto, S. Takada, S. Tamba, and T. H. Tatsuishi, Phys. Rev. D 97, 116002 (2018b), eprint 1804.06644.
  * Kobayashi and Tamba (2019) T. Kobayashi and S. Tamba, Phys. Rev. D 99, 046001 (2019), eprint 1811.11384.
  * Ohki et al. (2020) H. Ohki, S. Uemura, and R. Watanabe, Phys. Rev. D 102, 085008 (2020), eprint 2003.04174.
  * Kikuchi et al. (2020a) S. Kikuchi, T. Kobayashi, S. Takada, T. H. Tatsuishi, and H. Uchida, Phys. Rev. D 102, 105010 (2020a), eprint 2005.12642.
  * Kikuchi et al. (2020b) S. Kikuchi, T. Kobayashi, H. Otsuka, S. Takada, and H. Uchida, JHEP 11, 101 (2020b), eprint 2007.06188.
  * Kikuchi et al. (2021) S. Kikuchi, T. Kobayashi, and H. Uchida, Phys. Rev. D 104, 065008 (2021), eprint 2101.00826.
  * Almumin et al. (2021) Y. Almumin, M.-C. Chen, V. Knapp-Pérez, S. Ramos-Sánchez, M. Ratz, and S. Shukla, JHEP 05, 078 (2021), eprint 2102.11286.
  * Lauer et al. (1989) J. Lauer, J. Mas, and H. P. Nilles, Phys. Lett. B 226, 251 (1989).
  * Lauer et al. (1991) J. Lauer, J. Mas, and H. P. Nilles, Nucl. Phys. B 351, 353 (1991).
  * Ferrara et al. (1989) S. Ferrara, . D. Lust, and S. Theisen, Phys. Lett. B 233, 147 (1989).
  * Dixon et al. (1990) L. J. Dixon, V. Kaplunovsky, and J. Louis, Nucl. Phys. B 329, 27 (1990).
  * Baur et al. (2019) A. Baur, H. P. Nilles, A. Trautner, and P. K. S. Vaudrevange, Phys. Lett. B 795, 7 (2019), eprint 1901.03251.
  * Nilles et al. (2020) H. P. Nilles, S. Ramos-Sánchez, and P. K. S. Vaudrevange, JHEP 02, 045 (2020), eprint 2001.01736.
  * Nilles et al. (2021) H. P. Nilles, S. Ramos–Sánchez, and P. K. S. Vaudrevange, Nucl. Phys. B 966, 115367 (2021), eprint 2010.13798.
  * Kikuchi et al. (2023) S. Kikuchi, T. Kobayashi, K. Nasu, H. Otsuka, S. Takada, and H. Uchida, JHEP 04, 003 (2023), eprint 2301.10356.
  * Ibanez and Lust (1992) L. E. Ibanez and D. Lust, Nucl. Phys. B 382, 305 (1992), eprint hep-th/9202046.
  * Kawabe et al. (1995) H. Kawabe, T. Kobayashi, and N. Ohtsubo, Nucl. Phys. B 434, 210 (1995), eprint hep-ph/9405420.
  * Ferrara et al. (1986) S. Ferrara, C. Kounnas, and M. Porrati, Phys. Lett. B 181, 263 (1986).
  * Cvetic et al. (1988) M. Cvetic, J. Louis, and B. A. Ovrut, Phys. Lett. B 206, 227 (1988).
  * Ishiguro et al. (2021a) K. Ishiguro, T. Kobayashi, and H. Otsuka, JHEP 07, 064 (2021a), eprint 2103.10240.
  * Balasubramanian et al. (2005) V. Balasubramanian, P. Berglund, J. P. Conlon, and F. Quevedo, JHEP 03, 007 (2005), eprint hep-th/0502058.
  * Conlon et al. (2005) J. P. Conlon, F. Quevedo, and K. Suruliz, JHEP 08, 007 (2005), eprint hep-th/0505076.
  * Ishiguro et al. (2021b) K. Ishiguro, T. Kobayashi, and H. Otsuka, Nucl. Phys. B 973, 115598 (2021b), eprint 2010.10782.
  * Ishiguro et al. (2022) K. Ishiguro, T. Kobayashi, and H. Otsuka, JHEP 01, 020 (2022), eprint 2107.00487.
  * Kobayashi and Otsuka (2022) T. Kobayashi and H. Otsuka, Eur. Phys. J. C 82, 25 (2022), eprint 2108.02700.
  * Kikuchi et al. (2022b) S. Kikuchi, T. Kobayashi, K. Nasu, H. Otsuka, S. Takada, and H. Uchida, PTEP 2022, 123B02 (2022b), eprint 2203.14667.
  * 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.
  * Tanabashi et al. (2018) M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018).
  * 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.
  * Kobayashi et al. (2022a) T. Kobayashi, H. Otsuka, M. Tanimoto, and K. Yamamoto, Phys. Rev. D 105, 055022 (2022a), eprint 2112.00493.
  * Kobayashi et al. (2022b) T. Kobayashi, H. Otsuka, M. Tanimoto, and K. Yamamoto, JHEP 08, 013 (2022b), eprint 2204.12325.