#  APCTP Pre2020-033 Spontaneous CP violation by modulus $\tau$
in $A_{4}$ model of lepton flavors

Hiroshi Okada a,b111E-mail address<EMAIL_ADDRESS>and Morimitsu
Tanimoto c222E-mail address<EMAIL_ADDRESS>

aAsia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea
bDepartment of Physics, Pohang University of Science and Technology, Pohang
37673,
Republic of Korea cDepartment of Physics, Niigata University, Niigata
950-2181, Japan

( Abstract
We discuss the modular $A_{4}$ invariant model of leptons combining with the
generalized CP symmetry. In our model, both CP and modular symmetries are
broken spontaneously by the vacuum expectation value of the modulus $\tau$.
The source of the CP violation is a non-trivial value of ${\rm Re}[\tau]$
while other parameters of the model are real. The allowed region of $\tau$ is
in very narrow one close to the fixed point $\tau=i$ for both normal hierarchy
(NH) and inverted ones (IH) of neutrino masses. The CP violating Dirac phase
$\delta_{CP}$ is predicted clearly in $[98^{\circ},\,110^{\circ}]$ and
$[250^{\circ},\,262^{\circ}]$ for NH at $3\,\sigma$ confidence level. On the
other hand, $\delta_{CP}$ is in $[95^{\circ},100^{\circ}]$ and
$[260^{\circ},\,265^{\circ}]$ for IH at $5\,\sigma$ confidence level. The
predicted $\sum m_{i}$ is in $[82,\,102]$ meV for NH and $\sum
m_{i}=[134,\,180]$ meV for IH. The effective mass $\langle m_{ee}\rangle$ for
the $0\nu\beta\beta$ decay is predicted in $[12.5,\,20.5]$ meV and $[54,\,67]$
meV for NH and IH, respectively. )

##  1 Introduction

The non-Abelian discrete symmetries are attractive ones to understand flavors
of quarks and leptons. The $S_{3}$ flavor symmetry was a pioneer for the quark
flavor mixing [2, 1]. It was also discussed to understand the large mixing
angle [3] in the oscillation of atmospheric neutrinos [4]. For the last twenty
years, the non-Abelian discrete symmetries of flavors have been developed,
that is motivated by the precise observation of flavor mixing angles of
leptons [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Among them, the $A_{4}$ flavor
model is an attractive one because the $A_{4}$ group is the minimal one
including a triplet irreducible representation, which allows for a natural
explanation of the existence of three families of quarks and leptons [15, 16,
17, 18, 19, 20, 21]. However, it is difficult to obtain clear predictions of
the $A_{4}$ flavor symmetry because of a lot of free parameters associated
with scalar flavon fields.

Recently, a new approach to the lepton flavor problem has been put forward
based on the invariance under the modular transformation [22], where the model
of the finite modular group $\Gamma_{3}\simeq A_{4}$ has been presented. In
this approach, fermion matrices are written in terms of modular forms which
are holomorphic functions of the modulus $\tau$. This work inspired further
studies of the modular invariance approach to the lepton flavor problem.

The finite groups $S_{3}$, $A_{4}$, $S_{4}$, and $A_{5}$ are realized in
modular groups [23]. Modular invariant flavor models have been also proposed
on the $\Gamma_{2}\simeq S_{3}$ [24], $\Gamma_{4}\simeq S_{4}$ [25] and
$\Gamma_{5}\simeq A_{5}$ [26]. Phenomenological discussions of the neutrino
flavor mixing have been done based on $\rm A_{4}$ [27, 28, 29], $\rm S_{4}$
[30, 31, 32] and $\rm A_{5}$ [33]. A clear prediction of the neutrino mixing
angles and the CP violating phase was given in the simple lepton mass matrices
with the $\rm A_{4}$ modular symmetry [28]. On the other hand, the Double
Covering groups $\rm T^{\prime}$ [34, 35] and $\rm S_{4}^{\prime}$ [36, 37]
were realized in the modular symmetry. Furthermore, modular forms for
$\Delta(96)$ and $\Delta(384)$ were constructed [38], and the extension of the
traditional flavor group was discussed with modular symmetries [39]. The level
$7$ finite modular group $\rm\Gamma_{7}\simeq PSL(2,Z_{7})$ was also presented
for the lepton mixing [40]. Based on those works, phenomenological studies
have been developed in many works [48, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51,
52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70,
71, 72, 73, 74, 75, 76, 77, 78, 79, 80] while theoretical investigations have
been also proceeded [81, 83, 82, 85, 84, 86].

In order to test the modular symmetry of flavors, the prediction of the CP
violating Dirac phase is important. The CP transformation is non-trivial if
the non-Abelian discrete flavor symmetry is set in the Yukawa sector of a
Lagrangian. Then, we should discuss so called the generalized CP symmetry in
the flavor space [87, 88, 89, 90, 91]. It can predict the CP violating phase
[92]. The modular invariance has been also studied combining with the
generalized CP symmetry in flavor theories [93, 94]. It provides a powerful
framework to predict CP violating phases of quarks and leptons.

In our work, we present the modular $A_{4}$ invariant model with the
generalized CP symmetry. Both CP and modular symmetries are broken
spontaneously by the vacuum expectation value (VEV) of the modulus $\tau$. We
discuss the phenomenological implication of this model, that is the
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing angles [95, 96] and the CP
violating Dirac phase of leptons, which is expected to be observed at T2K and
NO$\nu$A experiments [97, 98].

The paper is organized as follows. In section 2, we give a brief review on the
generalized CP transformation in the modular symmetry. In section 3, we
present the CP invariant lepton mass matrix in the $A_{4}$ modular symmetry.
In section 4, we show the phenomenological implication of our model. Section 5
is devoted to the summary. In Appendix A, we present the tensor product of the
$A_{4}$ group. In Appendix B, we show the modular forms for weight $2$ and
$4$. In Appendix C, we show how to determine the coupling coefficients of the
charged lepton sector. In Appendix D , we present how to obtain the Dirac $CP$
phase, the Majorana phases and the effective mass of the $0\nu\beta\beta$
decay.

##  2 Generalized CP transformation in modular symmetry

###  2.1 Generalized CP symmetry

Let us start with discussing the generalised $CP$ symmetry [92, 99]. The CP
transformation is non-trivial if the non-Abelian discrete flavor symmetry $G$
is set in the Yukawa sector of a Lagrangian. Let us consider the chiral
superfields. The CP is a discrete symmetry which involves both Hermitian
conjugation of a chiral superfield $\psi(x)$ and inversion of spatial
coordinates,

$\psi(x)\rightarrow{\bf X}_{\bf r}\overline{\psi}(x_{P})\ ,$ (1)

where $x_{P}=(t,-{\bf x})$ and ${\bf X_{r}}$ is a unitary transformations of
$\psi(x)$ in the irreducible representation $\bf r$ of the discrete flavor
symmetry $G$. If ${\bf X_{r}}$ is the unit matrix, the $CP$ transformation is
the trivial one. This is the case for the continuous flavor symmetry [99].
However, in the framework of the non-Abelian discrete family symmetry, non-
trivial choices of ${\bf X_{r}}$ are possible. The unbroken $CP$
transformations of ${\bf X_{r}}$ form the group $H_{CP}$. Then, ${\bf X_{r}}$
must be consistent with the flavor symmetry transformation,

$\psi(x)\rightarrow{\rho}_{\bf r}(g)\psi(x)\ ,\quad g\in G\ ,$ (2)

where ${\rho}_{\bf{r}}(g)$ is the representation matrix for $g$ in the
irreducible representation $\bf{r}$.

The consistent condition is obtained as follows. At first, perform a $CP$
transformation $\psi(x)\rightarrow{\bf X}_{\bf r}\overline{\psi}(x_{P})$, then
apply a flavor symmetry transformation,
$\overline{\psi}(x_{P})\rightarrow{\rho}_{\bf
r}^{*}(g)\overline{\psi}(x_{P})$, and finally perform an inverse CP
transformation. The whole transformation is written as $\psi(x)\rightarrow{\bf
X}_{\bf r}\rho^{*}(g){\bf X}^{-1}_{\bf r}\psi(x)$, which must be equivalent to
some flavor symmetry $\psi(x)\rightarrow{\rho}_{\bf r}(g^{\prime})\psi(x)$.
Thus, one obtains [100]

${\bf X}_{\bf r}\rho_{\bf r}^{*}(g){\bf X}^{-1}_{\bf r}={\rho}_{\bf
r}(g^{\prime})\ ,\qquad g,\,g^{\prime}\in G\ .$ (3)

This equation defines the consistency condition, which has to be respected for
consistent implementation of a generalized CP symmetry along with a flavor
symmetry [102, 101]. This chain $CP\rightarrow g\rightarrow CP^{-1}$ maps the
group element $g$ onto $g^{\prime}$ and preserves the flavor symmetry group
structure. That is a homomorphism $v(g)=g^{\prime}$ of $G$. Assuming the
presence of faithful representations $\bf r$, Eq. (3) defines a unique mapping
of $G$ to itself. In this case, $v(g)$ is an automorphism of $G$ [101].

It has been also shown that the full symmetry group is isomorphic to a semi-
direct product of $G$ and $H_{CP}$, that is $G\rtimes H_{CP}$, where
$H_{CP}\simeq\mathbb{Z}_{2}^{CP}$, is the group generated by the generalised
$CP$ transformation under the assumption of $\bf X_{r}$ being a symmetric
matrix [102].

###  2.2 Modular symmetry

The modular group $\bar{\Gamma}$ is the group of linear fractional
transformations $\gamma$ acting on the modulus $\tau$, belonging to the upper-
half complex plane as:

$\tau\longrightarrow\gamma\tau=\frac{a\tau+b}{c\tau+d}\ ,~{}~{}{\rm
where}~{}~{}a,b,c,d\in\mathbb{Z}~{}~{}{\rm and}~{}~{}ad-bc=1,~{}~{}{\rm
Im}[\tau]>0~{},$ (4)

which is isomorphic to $PSL(2,\mathbb{Z})=SL(2,\mathbb{Z})/\\{\rm I,-I\\}$
transformation. This modular transformation is generated by $S$ and $T$,

$\displaystyle S:\tau\longrightarrow-\frac{1}{\tau}\ ,\qquad\qquad
T:\tau\longrightarrow\tau+1\ ,$ (5)

which satisfy the following algebraic relations,

$S^{2}=\mathbb{1}\ ,\qquad(ST)^{3}=\mathbb{1}\ .$ (6)

We introduce the series of groups $\Gamma(N)$, called principal congruence
subgroups, where $N$ is the level $1,2,3,\dots$. These groups are defined by

$\displaystyle\begin{aligned} \Gamma(N)=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in SL(2,\mathbb{Z})~{},~{}~{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}~{}~{}({\rm
mod}N)\right\\}\end{aligned}.$ (7)

For $N=2$, we define $\bar{\Gamma}(2)\equiv\Gamma(2)/\\{\rm I,-I\\}$. Since
the element $\rm-I$ does not belong to $\Gamma(N)$ for $N>2$, we have
$\bar{\Gamma}(N)=\Gamma(N)$. The quotient groups defined as
$\Gamma_{N}\equiv\bar{\Gamma}/\bar{\Gamma}(N)$ are finite modular groups. In
these finite groups $\Gamma_{N}$, $T^{N}=\mathbb{1}$ is imposed. The groups
$\Gamma_{N}$ with $N=2,3,4,5$ are isomorphic to $S_{3}$, $A_{4}$, $S_{4}$ and
$A_{5}$, respectively [23].

Modular forms $f_{i}(\tau)$ of weight $k$ are the holomorphic functions of
$\tau$ and transform as

$f_{i}(\tau)\longrightarrow(c\tau+d)^{k}\rho(\gamma)_{ij}f_{j}(\tau)\,,\quad\gamma\in
G\,,$ (8)

under the modular symmetry, where $\rho(\gamma)_{ij}$ is a unitary matrix
under $\Gamma_{N}$.

Superstring theory on the torus $T^{2}$ or orbifold $T^{2}/Z_{N}$ has the
modular symmetry [103, 104, 105, 106, 107, 108]. Its low energy effective
field theory is described in terms of supergravity theory, and string-derived
supergravity theory has also the modular symmetry. Under the modular
transformation of Eq. (4), chiral superfields $\psi_{i}$ ($i$ denotes flavors)
transform as [109],

$\psi_{i}\longrightarrow(c\tau+d)^{-k_{I}}\rho(\gamma)_{ij}\psi_{j}\,.$ (9)

We study global supersymmetric models, e.g., minimal supersymmetric extensions
of the Standard Model (MSSM). The superpotential which is built from matter
fields and modular forms is assumed to be modular invariant, i.e., to have a
vanishing modular weight. For given modular forms this can be achieved by
assigning appropriate weights to the matter superfields.

The kinetic terms are derived from a Kähler potential. The Kähler potential of
chiral matter fields $\psi_{i}$ with the modular weight $-k$ is given simply
by

$K^{\rm matter}=\frac{1}{[i(\bar{\tau}-\tau)]^{k}}\sum_{i}|\psi_{i}|^{2},$
(10)

where the superfield and its scalar component are denoted by the same letter,
and $\bar{\tau}=\tau^{*}$ after taking VEV of $\tau$. Therefore, the canonical
form of the kinetic terms is obtained by changing the normalization of
parameters [28]. The general Kähler potential consistent with the modular
symmetry possibly contains additional terms [110]. However, we consider only
the simplest form of the Kähler potential.

For $\Gamma_{3}\simeq A_{4}$, the dimension of the linear space ${\cal
M}_{k}(\Gamma{(3)})$ of modular forms of weight $k$ is $k+1$ [111, 112, 113],
i.e., there are three linearly independent modular forms of the lowest non-
trivial weight $2$, which form a triplet of the $A_{4}$ group, ${\bf Y^{(\rm
2)}_{\bf 3}}(\tau)=(Y_{1}(\tau),\,Y_{2}(\tau),\,Y_{3}(\tau))^{T}$. As shown in
Appendix A, these modular forms have been explicitly obtained [22] in the
symmetric base of the $A_{4}$ generators $S$ and $T$ for the triplet
representation:

$\displaystyle\begin{aligned} S=\frac{1}{3}\begin{pmatrix}-1&2&2\\\ 2&-1&2\\\
2&2&-1\end{pmatrix},\end{aligned}\qquad\qquad\begin{aligned}
T=\begin{pmatrix}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{2}\end{pmatrix},\end{aligned}$ (11)

where $\omega=\exp(i\frac{2}{3}\pi)$ .

###  2.3 CP transformation of the modulus $\tau$

The CP transformation in the modular symmetry was given by using the
generalized CP symmetry [93]. We summarize the discussion in Ref.[93] briefly.
Consider the CP and modular transformation $\gamma$ of the chiral superfield
$\psi(x)$ assigned to an irreducible unitary representation $\bf r$ of
$\Gamma_{N}$. The chain $CP\rightarrow\gamma\rightarrow
CP^{-1}=\gamma^{\prime}\in\bar{\Gamma}$ is expressed as:

$\displaystyle\psi(x)$ $\displaystyle\xrightarrow{\,CP\,}{\bf X}_{\bf
r}\overline{\psi}(x_{P})\xrightarrow{\ \gamma\ }(c\tau^{*}+d)^{-k}{\bf X}_{\bf
r}\,{\rho}_{\bf r}^{*}(\gamma)\overline{\psi}(x_{P})$
$\displaystyle\xrightarrow{\,CP^{-1}\,}(c\tau^{*}_{CP^{-1}}+d)^{-k}{\bf
X}_{\bf r}\,{\rho}_{\bf r}^{*}(\gamma){\bf X}_{\bf r}^{-1}\psi(x)\,,$ (12)

where $\tau_{CP^{-1}}$ is the operation of $CP^{-1}$ on $\tau$. The result of
this chain transformation should be equivalent to a modular transformation
$\gamma^{\prime}$ which maps $\psi(x)$ to
$(c^{\prime}\tau+d^{\prime})^{-k}{\rho}_{\bf r}(\gamma^{\prime})\psi(x)$.
Therefore, one obtains

$\displaystyle{\bf X}_{\bf r}\rho_{\bf r}^{*}(\gamma){\bf X}^{-1}_{\bf
r}=\left(\frac{c^{\prime}\tau+d^{\prime}}{c\tau^{*}_{CP^{-1}}+d}\right)^{-k}{\rho}_{\bf
r}(\gamma^{\prime})\,.$ (13)

Since ${\bf X}_{\bf r}$, $\rho_{\bf r}$ and $\rho_{\bf r^{\prime}}$ are
independent of $\tau$, the overall coefficient on the right-hand side of Eq.
(13) has to be a constant (complex) for non-zero weight $k$:

$\displaystyle\frac{c^{\prime}\tau+d^{\prime}}{c\tau^{*}_{CP^{-1}}+d}=\frac{1}{\lambda^{*}}\,,$
(14)

where $|\lambda|=1$ due to the unitarity of $\rho_{\bf r}$ and $\rho_{\bf
r^{\prime}}$. The values of $\lambda$, $c^{\prime}$ and $d^{\prime}$ depend on
$\gamma$.

Taking $\gamma=S$ ($c=1$, $d=0$) , and denoting $c^{\prime}(S)=C$,
$d^{\prime}(S)=D$ while keeping $\lambda(S)=\lambda$, we find
$\tau=(\lambda\tau^{*}_{CP^{-1}}-D)/C$ from Eq. (14), and consequently,

$\displaystyle\tau\xrightarrow{\,CP^{-1}\,}\tau_{CP^{-1}}=\lambda(C\tau^{*}+D)\,,\qquad\tau\xrightarrow{\,CP\,}\tau_{CP}=\frac{1}{C}(\lambda\tau^{*}-D)\,.$
(15)

Let us act with chain $CP\rightarrow T\rightarrow CP^{-1}$ on the mudular
$\tau$ itself:

$\displaystyle\tau\xrightarrow{\,CP\,}\tau_{CP}=\frac{1}{C}(\lambda\tau^{*}-D)\xrightarrow{\
T\
}\frac{1}{C}(\lambda(\tau^{*}+1)-D)\xrightarrow{\,CP^{-1}\,}\tau+\frac{\lambda}{C}\,.$
(16)

The resulting transformation has to be a modular transformation, therefore
$\lambda/C$ is an integer. Since $|\lambda|=1$, we find $|C|=1$ and
$\lambda=\pm 1$. After choosing the sign of $C$ as $C=\mp 1$ so that ${\rm
Im}[\tau_{CP}]>0$, the CP transformation of Eq. (15) turns to

$\displaystyle\tau\xrightarrow{\,CP\,}n-\tau^{*}\,,$ (17)

where $n$ is an integer. The chain $CP\rightarrow S\rightarrow
CP^{-1}=\gamma^{\prime}(S)$ imposes no furher restrictions on $\tau_{CP}$. It
is always possible to redefine the CP transformation in such a way that $n=0$
by using the freedom of $T$ transformation. Therefore, we define that the
modulus $\tau$ transforms under CP as

$\displaystyle\tau\xrightarrow{\,CP\,}-\tau^{*}\,,$ (18)

without loss of generality.

The same transformation of $\tau$ was also derived from the higher dimensional
theories [94]. The four-dimensional CP symmetry can be embedded into $(4+d)$
dimensions as higher dimensional proper Lorentz symmetry with positive
determinant. That is, one can combine the four-dimensional CP transformation
and $d$-dimensional transformation with negative determinant so as to obtain
$(4+d)$ dimensional proper Lorentz transformation. For example in six-
dimensional theory, we denote the two extra coordinates by a complex
coordinate $z$. The four-dimensional CP symmetry with $z\rightarrow z^{*}$ or
$z\rightarrow-z^{*}$ is a six-dimensional proper Lorentz symmetry. Note that
$z=x+\tau y$, where $x$ and $y$ are real coordinates. The latter
transformation $z\rightarrow-z^{*}$ maps the upper half plane ${\rm
Im}[\tau]>0$ to the same half plane. Hence, we consider the transformation
$z\rightarrow-z^{*}$ $(\tau\rightarrow-\tau^{*})$ as the CP symmetry.

###  2.4 CP transformation of modular multiplets

Chiral superfields and modular forms transform in Eqs. (8) and (9),
respectively, under a modular transformation. Chiral superfields also
transform in Eq. (1) under the CP transformation. The CP transformation of
modular forms were given in Ref.[93] as follows. Define a modular multiplet of
the irreducible representation $\bf r$ of $\Gamma_{N}$ with weight $k$ as $\bf
Y^{\rm(k)}_{\bf r}(\tau)$, which is transformed as:

$\displaystyle\bf Y^{\rm(k)}_{\bf r}(\tau)\xrightarrow{\,{\rm
CP}\,}Y^{\rm(k)}_{\bf r}(-\tau^{*})\,,$ (19)

under the CP transformation. The complex conjugated CP transformed modular
forms $\bf Y^{\rm(k)*}_{\bf r}(-\tau^{*})$ transform almost like the original
multiplets $\bf Y^{\rm(k)}_{\bf r}(\tau)$ under a modular transformation,
namely:

$\displaystyle\bf Y^{\rm(k)*}_{\bf r}(-\tau^{*})\xrightarrow{\ \gamma\
}Y^{\rm(k)*}_{\bf
r}(-(\gamma\tau)^{*})={\rm(c\tau+d)^{k}}\rho_{\bf{r}}^{*}({\rm
u}(\gamma))Y^{\rm(k)*}_{\bf r}(-\tau^{*})\,,$ (20)

where $u(\gamma)\equiv CP\gamma CP^{-1}$. Using the consistency condition of
Eq. (3), we obtain

$\displaystyle\bf X_{r}^{T}Y^{\rm(k)*}_{\bf r}(-\tau^{*})\xrightarrow{\
\gamma\ }{\rm(c\tau+d)^{k}}\rho_{\bf{r}}(\gamma)X_{r}^{T}Y^{\rm(k)*}_{\bf
r}(-\tau^{*})\,.$ (21)

Therefore, if there exist a unique modular multiplet at a level $N$, weight
$k$ and representation $\bf r$, which is satisfied for $N=2$–$5$ with weight
$2$, we can express the modular form $\bf Y^{\rm(k)}_{\bf r}(\tau)$ as:

$\displaystyle\bf Y^{\rm(k)}_{\bf
r}(\tau)={\rm\kappa}X_{r}^{T}Y^{\rm(k)*}_{\bf r}(-\tau^{*})\,,$ (22)

where $\kappa$ is a proportional coefficient. Since $\bf Y^{\rm(k)}_{\bf
r}(-(-\tau^{*})^{*})=\bf Y^{\rm(k)}_{\bf r}(\tau)$, Eq. (22) gives $\bf
X_{r}^{*}X_{r}={\rm|\kappa|^{2}}\mathbb{1}_{r}$. Therefore, the matrix $\bf
X_{r}$ is symmetric one, and $\kappa=e^{i\phi}$ is a phase, which can be
absorbed in the normalization of modular forms. In conclusion, the CP
transformation of modular forms is given as:

$\displaystyle\bf Y^{\rm(k)}_{\bf r}(\tau)\xrightarrow{\,{\rm
CP}\,}Y^{\rm(k)}_{\bf r}(-\tau^{*})=X_{r}Y^{\rm(k)*}_{\bf r}(\tau)\,.$ (23)

It is also emphasized that $\bf X_{r}=\mathbb{1}_{r}$ satisfies the
consistency condition Eq. (3) in a basis that generators of $S$ and $T$ of
$\Gamma_{N}$ are represented by symmetric matrices because of
$\rho^{*}_{\bf{r}}(S)=\rho^{\dagger}_{\bf{r}}(S)=\rho_{\bf{r}}(S^{-1})=\rho_{\bf{r}}(S)$
and $\rho^{*}_{\bf{r}}(T)=\rho^{\dagger}_{\bf{r}}(T)=\rho_{\bf{r}}(T^{-1})$.

The CP transformations of chiral superfields and modular multiplets are
summalized as follows:

$\displaystyle\tau\xrightarrow{\,{\rm
CP}\,}-\tau^{*}\,,\qquad\psi(x)\xrightarrow{\,{\rm
CP}\,}X_{r}\overline{\psi}(x_{P})\,,\qquad\bf Y^{\rm(k)}_{\bf
r}(\tau)\xrightarrow{\,{\rm CP}\,}Y^{\rm(k)}_{\bf
r}(-\tau^{*})=X_{r}Y^{\rm(k)*}_{\bf r}(\tau)\,,$ (24)

where $\bf X_{r}=\mathbb{1}_{r}$ can be taken in the base of symmetric
generators of $S$ and $T$. We use this CP transformation of modular forms to
construct the CP invariant mass matrices in the next section.

##  3 CP invariant mass matrix in $A_{4}$ modular symmetry

Let us discuss the CP invariant lepton mass matrix in the framework of the
$A_{4}$ modular symmetry. We assign the $A_{4}$ representation and weight for
superfields of leptons in Table 1, where the three left-handed lepton doublets
compose a $A_{4}$ triplet $L$, and the right-handed charged leptons $e^{c}$,
$\mu^{c}$ and $\tau^{c}$ are $A_{4}$ singlets. The weights of the superfields
of left-handed leptons and right-handed charged leptons are $-2$ and $0$,
respectively. Then, the simple lepton mass matrices for charged leptons and
neutrinos are obtained [75].

| $L$ | $(e^{c},\mu^{c},\tau^{c})$ | $H_{u}$ | $H_{d}$ | $\bf Y_{r}^{\rm(2)},\ \ Y_{r}^{\rm(4)}$
---|---|---|---|---|---
$SU(2)$ | $\bf 2$ | $\bf 1$ | $\bf 2$ | $\bf 2$ | $\bf 1$
$A_{4}$ | $\bf 3$ | (1, 1′′, 1′) | $\bf 1$ | $\bf 1$ | $\bf 3,\ \\{3,1,1^{\prime}\\}$
$k$ | $-2$ | $(0,\ 0,\ 0)$ | 0 | 0 | $2,\qquad 4$
Table 1:  Representations and weights $k$ for MSSM fields and modular forms of
weight $2$ and $4$.

The superpotential of the charged lepton mass term is given in terms of
modular forms of weight $2$, $\bf Y^{\rm(2)}_{3}$. It is given as:

$\displaystyle w_{E}$ $\displaystyle=\alpha_{e}e^{c}H_{d}{\bf
Y^{\rm(2)}_{3}}L+\beta_{e}\mu^{c}H_{d}{\bf
Y^{\rm(2)}_{3}}L+\gamma_{e}\tau^{c}H_{d}{\bf Y^{\rm(2)}_{3}}L~{},$ (25)

where $L$ is the left-handed $A_{4}$ triplet leptons. We can take real for
$\alpha_{e}$, $\beta_{e}$ and $\gamma_{e}$. Under CP, the superfields
transform as:

$\displaystyle e^{c}\xrightarrow{\,CP\,}\,X_{\bf
1}^{*}\,\overline{e}^{c}\,,\quad\mu^{c}\xrightarrow{\,CP\,}X_{\bf
1^{\prime\prime}}^{*}\,\overline{\mu}^{c}\,,\quad\tau^{c}\xrightarrow{\,CP\,}\,X_{\bf
1^{\prime}}^{*}\,\overline{\tau}^{c}\,,\quad L\xrightarrow{\,CP\,}\,X_{\bf
3}\overline{L}\,,\quad
H_{d}\xrightarrow{\,CP\,}\,\eta_{d}\,\overline{H}_{d}\,,$ (26)

and we can take $\eta_{d}=1$ without loss of generality. Since the
representations of $S$ and $T$ are symmetric as seen in Eq. (11), we can
choose $X_{\bf 3}=\mathbb{1}$ and $X_{\bf 1}=X_{\bf 1^{\prime}}=X_{\bf
1^{\prime\prime}}=\mathbb{1}$.

Taking $(e_{L},\mu_{L},\tau_{L})$ in the flavor base, the charged lepton mass
matrix $M_{E}$ is simply written as:

$\displaystyle\begin{aligned}
M_{E}(\tau)=v_{d}\begin{pmatrix}\alpha_{e}&0&0\\\ 0&\beta_{e}&0\\\
0&0&\gamma_{e}\end{pmatrix}\begin{pmatrix}Y_{1}(\tau)&Y_{3}(\tau)&Y_{2}(\tau)\\\
Y_{2}(\tau)&Y_{1}(\tau)&Y_{3}(\tau)\\\
Y_{3}(\tau)&Y_{2}(\tau)&Y_{1}(\tau)\end{pmatrix}_{RL}\ ,\end{aligned}$ (27)

where $v_{d}$ is VEV of the neutral component of $H_{d}$, and coefficients
$\alpha_{e}$, $\beta_{e}$ and $\gamma_{e}$ are taken to be real without loss
of generality. Under CP transformation, the mass matrix $M_{E}$ is transformed
following from Eq. (24) as:

$\displaystyle\begin{aligned}
M_{E}(\tau)\xrightarrow{\,CP\,}M_{E}(-\tau^{*})=M_{E}^{*}(\tau)=v_{d}\begin{pmatrix}\alpha_{e}&0&0\\\
0&\beta_{e}&0\\\
0&0&\gamma_{e}\end{pmatrix}\begin{pmatrix}Y_{1}(\tau)^{*}&Y_{3}(\tau)^{*}&Y_{2}(\tau)^{*}\\\
Y_{2}(\tau)^{*}&Y_{1}(\tau)^{*}&Y_{3}(\tau)^{*}\\\
Y_{3}(\tau)^{*}&Y_{2}(\tau)^{*}&Y_{1}(\tau)^{*}\end{pmatrix}_{RL}\
.\end{aligned}$ (28)

Let us discuss the neutrino mass matrix. Suppose neutrinos to be Majorana
particles. By using the Weinberg operator, the superpotential of the neutrino
mass term, $w_{\nu}$ is given as:

$\displaystyle w_{\nu}$ $\displaystyle=-\frac{1}{\Lambda}(H_{u}H_{u}LL{\bf
Y_{r}^{\rm(4)}})_{\bf 1}~{},$ (29)

where $\Lambda$ is a relevant cutoff scale. Since the left-handed lepton
doublet has weight $-2$, the superpotential is given in terms of modular forms
of weight $4$, ${\bf Y_{3}^{\rm(4)}}$, ${\bf Y_{1}^{\rm(4)}}$ and ${\bf
Y_{1^{\prime}}^{\rm(4)}}$.

By putting $v_{u}$ for VEV of the neutral component of $H_{u}$ and using the
tensor products of $A_{4}$ in Appendix A, we have

$\displaystyle w_{\nu}$
$\displaystyle=\frac{v_{u}^{2}}{\Lambda}\left[\begin{pmatrix}2\nu_{e}\nu_{e}-\nu_{\mu}\nu_{\tau}-\nu_{\tau}\nu_{\mu}\\\
2\nu_{\tau}\nu_{\tau}-\nu_{e}\nu_{\mu}-\nu_{\mu}\nu_{\tau}\\\
2\nu_{\mu}\nu_{\mu}-\nu_{\tau}\nu_{e}-\nu_{e}\nu_{\tau}\end{pmatrix}\otimes{\bf
Y_{3}^{\rm(4)}}\right.$ $\displaystyle\left.+\
(\nu_{e}\nu_{e}+\nu_{\mu}\nu_{\tau}+\nu_{\tau}\nu_{\mu})\otimes
g^{\nu}_{1}{\bf
Y_{1}^{\rm(4)}}+(\nu_{e}\nu_{\tau}+\nu_{\mu}\nu_{\mu}+\nu_{\tau}\nu_{e})\otimes
g^{\nu}_{2}{\bf Y_{1^{\prime}}^{\rm(4)}}\right]$ $\displaystyle=$
$\displaystyle\frac{v_{u}^{2}}{\Lambda}\left[(2\nu_{e}\nu_{e}-\nu_{\mu}\nu_{\tau}-\nu_{\tau}\nu_{\mu})Y_{1}^{(4)}+(2\nu_{\tau}\nu_{\tau}-\nu_{e}\nu_{\mu}-\nu_{\mu}\nu_{e})Y_{3}^{(4)}+(2\nu_{\mu}\nu_{\mu}-\nu_{\tau}\nu_{e}-\nu_{e}\nu_{\tau})Y_{2}^{(4)}\right.$
$\displaystyle\left.+\
(\nu_{e}\nu_{e}+\nu_{\mu}\nu_{\tau}+\nu_{\tau}\nu_{\mu})g^{\nu}_{1}{\bf
Y_{1}^{\rm(4)}}+(\nu_{e}\nu_{\tau}+\nu_{\mu}\nu_{\mu}+\nu_{\tau}\nu_{e})g^{\nu}_{2}{\bf
Y_{1^{\prime}}^{\rm(4)}}\right]\ ,$ (30)

where ${\bf Y_{3}^{\rm(4)}}$, ${\bf Y_{1}^{\rm(4)}}$ and ${\bf
Y_{1^{\prime}}^{\rm(4)}}$ are given in Eq. (LABEL:weight4) of Appendix B, and
$g^{\nu}_{1}$, $g^{\nu}_{2}$ are complex parameters in general. The neutrino
mass matrix is written as follows:

$\displaystyle
M_{\nu}(\tau)=\frac{v_{u}^{2}}{\Lambda}\left[\begin{pmatrix}2Y_{1}^{(4)}(\tau)&-Y_{3}^{(4)}(\tau)&-Y_{2}^{(4)}(\tau)\\\
-Y_{3}^{(4)}(\tau)&2Y_{2}^{(4)}(\tau)&-Y_{1}^{(4)}(\tau)\\\
-Y_{2}^{(4)}(\tau)&-Y_{1}^{(4)}(\tau)&2Y_{3}^{(4)}(\tau)\end{pmatrix}+g^{\nu}_{1}{\bf
Y_{1}^{\rm(4)}(\tau)}\begin{pmatrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix}+g^{\nu}_{2}{\bf
Y_{1^{\prime}}^{\rm(4)}(\tau)}\begin{pmatrix}0&0&1\\\ 0&1&0\\\
1&0&0\end{pmatrix}\right]\,,$ (31)

which is the same one in Ref.[75]. Under CP transformation, the mass matrix
$M_{\nu}$ is transformed following from Eq. (24) as:

$\displaystyle
M_{\nu}(\tau)\xrightarrow{\,CP\,}M_{\nu}(-\tau^{*})=M^{*}_{\nu}(\tau)$
$\displaystyle=\frac{v_{u}^{2}}{\Lambda}\left[\begin{pmatrix}2Y_{1}^{(4)*}(\tau)&-Y_{3}^{(4)*}(\tau)&-Y_{2}^{(4)*}(\tau)\\\
-Y_{3}^{(4)*}(\tau)&2Y_{2}^{(4)*}(\tau)&-Y_{1}^{(4)*}(\tau)\\\
-Y_{2}^{(4)*}(\tau)&-Y_{1}^{(4)*}(\tau)&2Y_{3}^{(4)*}(\tau)\end{pmatrix}+g^{\nu*}_{1}{\bf
Y_{1}^{\rm(4)*}(\tau)}\begin{pmatrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix}+g^{\nu*}_{2}{\bf
Y_{1^{\prime}}^{\rm(4)*}(\tau)}\begin{pmatrix}0&0&1\\\ 0&1&0\\\
1&0&0\end{pmatrix}\right].$ (32)

In a CP conserving modular invariant theory, both CP and modular symmetries
are broken spontaneously by VEV of the modulus $\tau$. However, there exist
certain values of $\tau$ which conserve CP while breaking the modular
symmetry. Obviously, this is the case if $\tau$ is left invariant by CP, i.e.

$\displaystyle\tau\xrightarrow{\,CP\,}-\tau^{*}=\tau\,\,,$ (33)

which indicates $\tau$ lies on the imaginary axis, ${\rm Re}[\tau]=0$. In
addition to ${\rm Re}[\tau]=0$, CP is conserved at the boundary of the
fundamental domain. Then, one has

$\displaystyle M_{E}(\tau)=M_{E}^{*}(\tau)\,,\qquad\qquad
M_{\nu}(\tau)=M_{\nu}^{*}(\tau)\,,$ (34)

which leads to $g^{\nu}_{1}$ and $g^{\nu}_{2}$ being real. Since parameters
$\alpha_{e}$, $\beta_{e}$, $\gamma_{e}$ are also real, the source of the CP
violation is only non-trivial ${\rm Re}[\tau]$ after breaking the modular
symmetry. In the next section, we present numerical analysis of the CP
violation by investigating the value of the modulus $\tau$.

##  4 Numerical results of leptonic CP violation

We have presented the CP invariant lepton mass matrices in the $A_{4}$ modular
symmetry. These mass matrices are the same ones in Ref.[75] except for
parameters $g_{1}^{\nu}$ and $g_{2}^{\nu}$ being real. If the CP violation
will be confirmed at the experiments of neutrino oscillations, the CP symmetry
should be broken spontaneously by VEV of the modulus $\tau$. Thus, VEV of
$\tau$ breaks the CP symmetry as well as the modular invariance. The source of
the CP violation is only the real part of $\tau$. This situation is different
from the previous work in Ref.[75], where imaginary parts of $g_{1}^{\nu}$ and
$g_{2}^{\nu}$ also break the CP symmetry explicitly. Our phenomenological
concern is whether the spontaneous CP violation is realized due to the value
of $\tau$, which is consistent with observed lepton mixing angles and neutrino
masses. If this is the case, the CP violating Dirac phase and Majorana phases
are predicted clearly under the fixed value of $\tau$.

Parameter ratios $\alpha_{e}/\gamma_{e}$ and $\beta_{e}/\gamma_{e}$ are given
in terms of charged lepton masses and $\tau$ as shown in Appendix C.
Therefore, the lepton mixing angles, the Dirac phase and Majorana phases are
given by our model parameters $g^{\nu}_{1}$ and $g^{\nu}_{2}$ in addition to
the value of $\tau$.

As the input charged lepton masses, we take Yukawa couplings of charged
leptons at the GUT scale $2\times 10^{16}$ GeV, where $\tan\beta=5$ is taken
as a bench mark [114, 115]:

$\displaystyle y_{e}=(1.97\pm 0.024)\times 10^{-6},\quad y_{\mu}=(4.16\pm
0.050)\times 10^{-4},\quad y_{\tau}=(7.07\pm 0.073)\times 10^{-3},$ (35)

where lepton masses are given by $m_{\ell}=y_{\ell}v_{H}$ with $v_{H}=174$
GeV.

observable | best fit $\pm 1\,\sigma$ for NH | best fit $\pm 1\,\sigma$ for IH
---|---|---
$\sin^{2}\theta_{12}$ | $0.304^{+0.012}_{-0.012}$ | $0.304^{+0.013}_{-0.012}$
$\sin^{2}\theta_{23}$ | $0.573^{+0.016}_{-0.020}$ | $0.575^{+0.016}_{-0.019}$
$\sin^{2}\theta_{13}$ | $0.02219^{+0.00062}_{-0.00063}$ | $0.02238^{+0.00063}_{-0.00062}$
$\Delta m_{\rm sol}^{2}$ | $7.42^{+0.21}_{-0.20}\times 10^{-5}{\rm eV}^{2}$ | $7.42^{+0.21}_{-0.20}\times 10^{-5}{\rm eV}^{2}$
$\Delta m_{\rm atm}^{2}$ |  $2.517^{+0.026}_{-0.028}\times 10^{-3}{\rm eV}^{2}$ | $-2.498^{+0.028}_{-0.028}\times 10^{-3}{\rm eV}^{2}$
Table 2: The best fit $\pm 1\,\sigma$ of neutrino parameters from NuFIT 5.0
for NH and IH [116].

We also input the lepton mixing angles and neutrino mass parameters which are
given by NuFit 5.0 in Table 2 [116]. In our analysis, $\delta_{CP}$ is output
because its observed range is too wide at $3\,\sigma$ confidence level. We
investigate two possible cases of neutrino masses $m_{i}$, which are the
normal hierarchy (NH), $m_{3}>m_{2}>m_{1}$, and the inverted hierarchy (IH),
$m_{2}>m_{1}>m_{3}$. Neutrino masses and the Pontecorvo-Maki-Nakagawa-Sakata
(PMNS) matrix $U_{\rm PMNS}$ [95, 96] are obtained by diagonalizing
$M_{E}^{\dagger}M_{E}$ and $M_{\nu}^{\dagger}M_{\nu}$. We also investigate the
effective mass for the $0\nu\beta\beta$ decay, $\langle m_{ee}\rangle$ (see
Appendix D) and the sum of three neutrino masses $\sum m_{i}$ since it is
constrained by the recent cosmological data, which is the upper-bound $\sum
m_{i}\leq 120$ meV obtained at the 95% confidence level [117, 118].

###  4.1 Case of normal hierarchy of neutrino masses

Let us discuss numerical results for NH of neutrino masses. The ratios
$\alpha_{e}/\gamma_{e}$ and $\beta_{e}/\gamma_{e}$ are given after fixing
charged lepton masses and $\tau$ as shown in Appendix C. However, in practice,
we scan $\alpha_{e}/\gamma_{e}$ and $\beta_{e}/\gamma_{e}$ to obtain the
observed charged lepton mass ratio and include them in $\chi^{2}$ fit as well
as three mixing angles and $\Delta m_{\rm atm}^{2}/\Delta m_{\rm sol}^{2}$.

We have already studied the lepton mass matrices in Eqs. (27) and (31)
phenomenologically at the nearby fixed points of the modulus because the
spontaneous CP violation in Type IIB string theory is possibly realized at
nearby fixed points, where the moduli stabilization is performed in a
controlled way [119, 120]. There are two fixed points in the fundamental
domain of $PSL(2,\mathbb{Z})$, $\tau=i$ and $\tau=\omega$. Indeed, the viable
$\tau$ of our lepton mass matrices is found around $\tau=i$ [75].

Based on this result of Ref. [75], we scan $\tau$ around $i$ while neutrino
couplings $g^{\nu}_{1}$ and $g^{\nu}_{2}$ are scanned in the real space of
$[-10,\,10]$. As a measure of good-fit, we adopt the sum of one-dimensional
$\chi^{2}$ function for four accurately known dimensionless observables
$\Delta m_{\rm atm}^{2}/\Delta m_{\rm sol}^{2}$, $\sin^{2}\theta_{12}$,
$\sin^{2}\theta_{23}$ and $\sin^{2}\theta_{13}$ in NuFit 5.0 [116]. In
addition, we employ Gaussian approximations for fitting $m_{e}/m_{\tau}$ and
$m_{\mu}/m_{\tau}$ by using the data of PDG [121].

In Fig. 1 we show the allowed region on the ${\rm Re}\,[\tau]$ – ${\rm
Im}\,[\tau]$ plane, where three mixing angles and $\Delta m_{\rm
atm}^{2}/\Delta m_{\rm sol}^{2}$ are consistent with observed ones. The green,
yellow and red regions correspond to $2\sigma$, $3\sigma$ and $5\sigma$
confidence levels, respectively.

Figure 1: Allowed regions of $\tau$ for NH. Green, yellow and red correspond
to $2\sigma$, $3\sigma$, $5\sigma$ confidence levels, respectively. The solid
curve is the boundary of the fundamental domain, $|\tau|=1$.

Figure 2: The allowed region of $g^{\nu}_{1}$ and $g^{\nu}_{2}$, which are
real parameters, for NH. Colors denote same ones in Fig. 1.

The allowed region of $\tau$ is restricted in the narrow regions. This result
is contrast to the previous one in Ref. [75], where non-trivial phases of
$g^{\nu}_{1}$ and $g^{\nu}_{2}$ enlarged the allowed region of $\tau$. The
predicted range of $\tau$ is in ${\rm Re}\,[\tau]=\pm[0.073,0.083]$ and ${\rm
Im}\,[\tau]=[1.006,1.014]$ at $3\,\sigma$ confidence level (yellow), which are
close to the fixed point $\tau=i$.

The allowed region of $g^{\nu}_{1}$ and $g^{\nu}_{2}$ is also shown in Fig. 2,
where $g^{\nu}_{1}$ is in the rather wide region of $[-0.18,0.18]$ while
$g^{\nu}_{2}$ is restricted in $[-0.87,-0.79]$ at $3\,\sigma$ confidence level
(yellow).

Due to restricted ${\rm Re}\,[\tau]$, the CP violating Dirac phase
$\delta_{CP}$, which is defined in Appendix D, is predicted clearly. In Fig.
3, we show prediction of $\delta_{CP}$ versus the sum of neutrino masses $\sum
m_{i}$. It is remarked that $\delta_{CP}$ is almost independent of $\sum
m_{i}$. The predicted ranges of $\delta_{CP}$ are narrow such as
$[98^{\circ},110^{\circ}]$ and $[250^{\circ},262^{\circ}]$ at $3\,\sigma$
confidence level (yellow). The predicted ranges $[98^{\circ},110^{\circ}]$ and
$[250^{\circ},262^{\circ}]$ correspond to ${\rm Re}\,[\tau]=(0.073$–$0.083)$
and ${\rm Re}\,[\tau]=-(0.073$–$0.083)$, respectively. The predicted $\sum
m_{i}$ is in $[82,\,102]$ meV for $3\,\sigma$ confidence level (yellow). The
minimal cosmological model, ${\rm\Lambda CDM}+\sum m_{i}$, provides the upper-
bound $\sum m_{i}<120$ meV [117, 118]. Thus, our predicted sum of neutrino
masses is consistent with the cosmological bound $120$ meV.

In Fig. 4, we show the allowed region on the $\sin^{2}\theta_{23}$ – $\sum
m_{i}$ plane. Since $\sum m_{i}$ depends on the value of $\sin^{2}\theta_{23}$
significantly, the crucial test of our prediction will be available in the
near future.

Figure 3: The prediction of $\delta_{CP}$ versus $\sum m_{i}$ for NH. Colors
denote same ones in Fig. 1.

Figure 4: The allowed region on $\sin^{2}\theta_{23}$–$\sum m_{i}$ plane for
NH. Colors denote same ones in Fig. 1.

In Fig. 5, we show the prediction of Majorana phases $\alpha_{21}$ and
$\alpha_{31}$, which are defined by Appendix D. The predicted
$[\alpha_{21},\,\alpha_{31}]$ are around $[30^{\circ},20^{\circ}]$ and
$[330^{\circ},340^{\circ}]$ since the source of the CP violation, ${\rm
Re}\,[\tau]$ is in the narrow range ${\rm Re}\,[\tau]=\pm[0.073,0.083]$.

We can calculate the effective mass $\langle m_{ee}\rangle$ for the
$0\nu\beta\beta$ decay by using the Dirac phase and Majorana phases as seen in
Appendix D. We show the predicted value of $\langle m_{ee}\rangle$ versus
$\sin^{2}\theta_{23}$ as seen in Fig. 6. The predicted $\langle m_{ee}\rangle$
is in $[12.5,\,20.5]$ meV for $3\,\sigma$ confidence level (yellow). The
prediction of $\langle m_{ee}\rangle\simeq 20$ meV will be testable in the
future experiments of the neutrinoless double beta decay.

Figure 5: Predicted Majorana phases $\alpha_{21}$ and $\alpha_{31}$ for NH.
Colors denote same ones in Fig. 1.

Figure 6:  The predicted $\langle m_{ee}\rangle$ versus $\sin^{2}\theta_{23}$
for NH. Colors denote same ones in Fig. 1.

It is important to understand the difference between the results in the
present paper and the previous ones in Ref.[75], where imaginary parts of
$g_{1}^{\nu}$ and $g_{2}^{\nu}$ also break the CP symmetry explicitly. The
modulus $\tau$ is severely restricted around ${\rm Re}\,[\tau]=\pm 0.08$ and
${\rm Im}\,[\tau]=1.01$ in this work while it is allowed in rather wide region
in the previous work. Indeed, the samller ${\rm Re}\,[\tau]$ and the larger
${\rm Im}\,[\tau]$ are allowed such as ${\rm Re}\,[\tau]\simeq\pm 0.03$ and
${\rm Im}\,[\tau]\simeq 1.1$ in the previous results. Due to this restricted
$\tau$ in this work, $\delta_{CP}$ and the sum of neutrino masses $\sum m_{i}$
are predicted clearly. On the other hand, the CP conservation is still allowed
and $\sum m_{i}$ could be larger than $120$ meV in the previous work.
Moreover, the Dirac phase $\delta_{CP}$ depends on $\sum m_{i}$.

###  4.2 Case of inverted hierarchy of neutrino masses

We discuss the case of IH of neutrino masses. In Fig. 7, we show the allowed
region on the ${\rm Re}\,[\tau]$ – ${\rm Im}\,[\tau]$ plane, where the red
region corresponds to $5\,\sigma$ confidence level like in Fig. 1. However,
there are no green and yellow regions of $2\,\sigma$ and $3\,\sigma$
confidence levels.

Figure 7: Allowed regions of $\tau$ for IH. Red corresponds to $5\,\sigma$
confidence level.

Figure 8: The allowed region of $g^{\nu}_{1}$ and $g^{\nu}_{2}$, which are
real parameters, for IH.

The range of $\tau$ is in ${\rm Re}\,[\tau]=\pm[0.009,\,0.012]$ and ${\rm
Im}\,[\tau]=[1.076,\,1.087]$ at $5\,\sigma$ confidence level, which are close
to $\tau=i$.

The allowed region of $g^{\nu}_{1}$ and $g^{\nu}_{2}$ is also shown in Fig. 8,
where $g^{\nu}_{1}$ is restricted in the narrow range of $[-1.20,\,-1.15]$
while $g^{\nu}_{2}$ is rather large as in $[4.8,\,9.6]$ for $5\,\sigma$.

In Fig. 9, we show prediction of $\delta_{CP}$ versus $\sum m_{i}$. It is
remarked that $\delta_{CP}$ is almost independent of $\sum m_{i}$. The
predicted range of $\delta_{CP}$ is in $[95^{\circ},100^{\circ}]$ and
$[260^{\circ},265^{\circ}]$ at $5\,\sigma$ confidence level while the sum of
neutrino masses are in the range of $[134,\,180]$ meV. In our numerical
result, there is no region of the sum of neutrino masses less than $120$ meV.
The upper-bound of the minimal cosmological model, ${\rm\Lambda CDM}+\sum
m_{i}$, is $\sum m_{i}<120$ meV [117, 118], however, it becomes weaker when
the data are analysed in the context of extended cosmological models [121].
The predicted sum of neutrino masses of IH may be still consistent with the
cosmological bound.

Figure 9: The prediction of $\delta_{CP}$ versus $\sum m_{i}$ for IH.

Figure 10: The allowed region on $\sin^{2}\theta_{23}$–$\sum m_{i}$ plane for
IH.

We show the allowed region on the $\sum m_{i}$ – $\sin^{2}\theta_{23}$ plane
in Fig. 10. The precise measurement of $\sin^{2}\theta_{23}$ will provide a
severe test for our prediction since $\sin^{2}\theta_{23}>0.55$ is obtained
for IH.

In Fig. 11, we show the prediction of Majorana phases $\alpha_{21}$ and
$\alpha_{31}$. The predicted $[\alpha_{21},\,\alpha_{31}]$ are restricted
around $[3^{\circ},182^{\circ}]$ and $[356^{\circ},178^{\circ}]$. We also show
the predicted value of $\langle m_{ee}\rangle$ versus $\sin^{2}\theta_{23}$ as
seen in Fig. 12. The predicted $\langle m_{ee}\rangle$ is in $[54,\,67]$ meV
for $5\,\sigma$ confidence level.

As well as the case of NH, we comment on the difference between the results in
the present paper and the previous ones in Ref.[75], where $g_{1}^{\nu}$ and
$g_{2}^{\nu}$ are complex. Our results are obtained at more than $3\,\sigma$
confidence level, on the other hand, the previous ones are at less than
$3\,\sigma$ confidence level. The modulus $\tau$ is also severely restricted
in this work while it is allowed in rather wide region in the previous work.
The sum of neutrino masses $\sum m_{i}$ is lager than $120$ meV in this work,
on the other hand, it is allowed to be smaller than $120$ meV in the previous
work. For example, it could be $90$ meV, and the Dirac phase $\delta_{CP}$
depends on $\sum m_{i}$.

Figure 11: Predicted Majorana phases $\alpha_{21}$ and $\alpha_{31}$ for IH.

Figure 12:  The predicted $\langle m_{ee}\rangle$ versus $\sin^{2}\theta_{23}$
for IH.

###  4.3 Parameter samples of NH and IH

We show the numerical result of two samples for NH and IH, respectively. In
Table 3, parameters and outputs of our calculations are presented for both NH
and IH.

| NH | IH
---|---|---
$\tau$ | $-0.0796+1.0065\,i$ | $0.0103+1.0812\,i$
$g^{\nu}_{1}$ | $0.124$ | -1.17
$g^{\nu}_{2}$ | $-0.802$ | 6.79
$\alpha_{e}/\gamma_{e}$ | $6.82\times 10^{-2}$ | $6.76\times 10^{-2}$
$\beta_{e}/\gamma_{e}$ | $1.02\times 10^{-3}$ | $1.02\times 10^{-3}$
$\sin^{2}\theta_{12}$ | $0.290$ | $0.291$
$\sin^{2}\theta_{23}$ | $0.564$ | $0.579$
$\sin^{2}\theta_{13}$ | $0.0225$ | $0.0219$
$\delta_{CP}^{\ell}$ | $258^{\circ}$ | $262^{\circ}$
$[\alpha_{21},\,\alpha_{31}]$ | $[330^{\circ},\,338^{\circ}]$ | $[3.24^{\circ},\,182^{\circ}]$
$\sum m_{i}$ |  $97.9$ meV |  $153$ meV
$\langle m_{ee}\rangle$ |  $19.2$ meV |  $59.1$ meV
$\chi^{2}$ | $1.98$ | $4.12$
Table 3: Numerical values of parameters and observables at the sample points
of NH and IH.

We also present the mixing matrices of charged leptons $U_{E}$ and neutrinos
$U_{\nu}$ for the samples of Table 3. For NH, those are:

$\displaystyle\begin{aligned}
U_{E}&\approx\begin{pmatrix}0.983&-0.020+0.158\,i&-0.011+0.092\,i\\\
0.016+0.130\,i&0.958&-0.255+0.001\,i\\\
0.016+0.129\,i&0.239+0.001\,i&0.962\end{pmatrix}\ ,\\\
U_{\nu}&\approx\begin{pmatrix}0.838&-0.541+0.068\,i&-0.008+0.031\,i\\\
0.450+0.076\,i&0.688&0.564-0.0008\,i\\\
-0.299-0.021\,i&-0.478-0.020\,i&0.825\end{pmatrix}\ ,\end{aligned}$ (36)

which are given in the diagonal base of the generator $S$ in order to see the
hierarchical structure of flavor mixing [75]. The PMNS mixing matrix is given
as $U_{\rm PMNS}=U_{E}^{\dagger}\,U_{\nu}$. The diagonal base of $S$ is
obtained by using the following unitary matrix:

$\displaystyle\begin{aligned} V_{S}\equiv\begin{pmatrix}-\frac{1}{\sqrt{6}}&\
\frac{2}{\sqrt{6}}&-\frac{1}{\sqrt{6}}\\\ \
\,\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\ \frac{1}{\sqrt{3}}\\\
-\frac{1}{\sqrt{2}}&0&\ \frac{1}{\sqrt{2}}\end{pmatrix},\end{aligned}$ (37)

which leads to $V_{S}\,S\,V_{S}^{\dagger}={\rm diag}\,(1,\,-1,\,-1)$ [75].
Then, the charged lepton and neutrino mass matrices are transformed as
$V_{S}M_{f}^{\dagger}M_{f}V_{S}^{\dagger}\,(f=E,\,\nu)$.

For IH, the mixing matrices are:

$\displaystyle\begin{aligned}
U_{E}&\approx\begin{pmatrix}0.983&0.155+0.019\,i&0.091+0.011\,i\\\
0.127+0.015\,i&0.956&-0.264-0.001\,i\\\
-0.128+0.016\,&0.248-0.001\,i&0.960\end{pmatrix}\ ,\\\
U_{\nu}&\approx\begin{pmatrix}0.840&0.0007+0.542\,i&0.032-0.001\,i\\\
-0.022+0.445\,i&0.691&0.570-0.002\,i\\\
-0.016-0.310\,i&-0.478-0.023\,i&0.821\end{pmatrix}\ ,\end{aligned}$ (38)

which are also given in the diagonal base of the generator $S$.

For both NH and IH, the mixing matrix of charged leptons $U_{E}$ is
hierarchical one, on the other hand, two large mixing angles of 1–2 and 2–3
flavors appear in the neutrino mixing matrix $U_{\nu}$.

In our numerical calculations, we have not included the RGE effects in the
lepton mixing angles and neutrino mass ratio $\Delta m_{\rm sol}^{2}/\Delta
m_{\rm atm}^{2}$. We suppose that those corrections are very small between the
electroweak and GUT scales. This assumption is justified well in the case of
$\tan\beta\leq 5$ unless neutrino masses are almost degenerate [27].

##  5 Summary and discussions

The modular invariant $A_{4}$ model of lepton flavors has been studied
combining with the generalized CP symmetry. In our model, both CP and modular
symmetries are broken spontaneously by VEV of the modulus $\tau$. The source
of the CP violation is a non-trivial value of ${\rm Re}[\tau]$ while
parameters of neutrinos $g^{\nu}_{1}$ and $g^{\nu}_{2}$ are real.

We have found allowed region of $\tau$ close to the fixed point $\tau=i$,
which is consistent with the observed lepton mixing angles and lepton masses
for NH at $2\,\sigma$ confidence level. The CP violating Dirac phase
$\delta_{CP}$ is predicted clearly in $[98^{\circ},110^{\circ}]$ and
$[250^{\circ},262^{\circ}]$ at $3\,\sigma$ confidence level. The predicted
$\sum m_{i}$ is in $[82,\,102]$ meV with $3\,\sigma$ confidence level.

There is also allowed region of $\tau$ close to the fixed point $\tau=i$ for
IH at $5\,\sigma$ confidence level. The predicted $\delta_{CP}$ is in
$[95^{\circ},100^{\circ}]$ and $[260^{\circ},265^{\circ}]$ at $5\,\sigma$
confidence level. The sum of neutrino masses is predicted in $\sum
m_{i}=[134,\,180]$ meV.

By using the predicted Dirac phase and the Majorana phases, we have obtained
the effective mass $\langle m_{ee}\rangle$ for the $0\nu\beta\beta$ decay,
which are in $[12.5,\,20.5]$ meV for NH at $3\,\sigma$ confidence level and in
$[54,\,67]$ meV for IH at $5\,\sigma$ confidence level. Since KamLAND-Zen
experiment [122] presented the upper bound on the effective Majorana mass as
$\langle m_{ee}\rangle<61$–$165$ meV by using a variety of nuclear matrix
element calculations, the prediction of $[54,\,67]$ meV for IH will be tested
in the near future. Furthermore, the prediction of $\langle
m_{ee}\rangle\simeq 20$ meV for NH will be also testable in the future
experiments of the neutrinoless double beta decay.

Since the CP symmetry is conserved at the boundary of the fundamental domain,
one may expect the size of CP violation to be small at the nearby fixed point
of $\tau=i$. In order to estimate of the size of CP violation, we can
calculate the rephasing invariant CP violating measure of leptons, $J_{CP}$
[123, 124] from mass matrices directly [125]. By using aproximate forms of
lepton mass matrices at nearby fixed points in Ref.[75], we have obtained the
relation between the magnitude of $J_{CP}$ and the deviation from $\tau=i$
semi-quantitatively. In order to reproduce the almost maximal size
$|J_{CP}|=0.03$, it is enough to take $\epsilon=\pm{\cal O}(0.05)$ where
$\epsilon$ is supposed to be real in the definition of $\tau=i+\epsilon$.
Since it is important to study CP violation at nearby fixed points
complehensively, we will present appropriate forms in another paper.

In our model, the modulus $\tau$ dominates the CP violation. Therefore, the
determination of $\tau$ is the most important work. Although we have
constrained $\tau$ by observables of leptons phenomenologically, one also
should pay attention to the recent theoretical work of the moduli
stabilization from the viewpoint of modular flavor symmetries [126]. The study
of modulus $\tau$ is interesting to reveal the flavor theory in both
theoretical and phenomenological aspects.

## Acknowledgments

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.

## Appendix

##  Appendix A Tensor product of $A_{4}$ group

We take the generators of $A_{4}$ group for the triplet as follows:

$\displaystyle\begin{aligned} S=\frac{1}{3}\begin{pmatrix}-1&2&2\\\ 2&-1&2\\\
2&2&-1\end{pmatrix},\end{aligned}\qquad\begin{aligned}
T=\begin{pmatrix}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{2}\end{pmatrix},\end{aligned}$ (39)

where $\omega=e^{i\frac{2}{3}\pi}$ for a triplet. In this base, the
multiplication rule is

$\displaystyle\begin{pmatrix}a_{1}\\\ a_{2}\\\ a_{3}\end{pmatrix}_{\bf
3}\otimes\begin{pmatrix}b_{1}\\\ b_{2}\\\ b_{3}\end{pmatrix}_{\bf 3}$
$\displaystyle=\left(a_{1}b_{1}+a_{2}b_{3}+a_{3}b_{2}\right)_{\bf
1}\oplus\left(a_{3}b_{3}+a_{1}b_{2}+a_{2}b_{1}\right)_{{\bf 1}^{\prime}}$
$\displaystyle\oplus\left(a_{2}b_{2}+a_{1}b_{3}+a_{3}b_{1}\right)_{{\bf
1}^{\prime\prime}}$
$\displaystyle\oplus\frac{1}{3}\begin{pmatrix}2a_{1}b_{1}-a_{2}b_{3}-a_{3}b_{2}\\\
2a_{3}b_{3}-a_{1}b_{2}-a_{2}b_{1}\\\
2a_{2}b_{2}-a_{1}b_{3}-a_{3}b_{1}\end{pmatrix}_{{\bf
3}}\oplus\frac{1}{2}\begin{pmatrix}a_{2}b_{3}-a_{3}b_{2}\\\
a_{1}b_{2}-a_{2}b_{1}\\\ a_{3}b_{1}-a_{1}b_{3}\end{pmatrix}_{{\bf 3}\ }\ ,$
$\displaystyle{\bf 1}\otimes{\bf 1}={\bf 1}\ ,\qquad$ $\displaystyle{\bf
1^{\prime}}\otimes{\bf 1^{\prime}}={\bf 1^{\prime\prime}}\ ,\qquad{\bf
1^{\prime\prime}}\otimes{\bf 1^{\prime\prime}}={\bf 1^{\prime}}\ ,\qquad{\bf
1^{\prime}}\otimes{\bf 1^{\prime\prime}}={\bf 1}\ ,$ (40)

where

$\displaystyle T({\bf 1^{\prime})}=\omega\,,\qquad T({\bf
1^{\prime\prime}})=\omega^{2}.$ (41)

More details are shown in the review [6, 7].

##  Appendix B Modular forms in $A_{4}$ symmetry

For $\Gamma_{3}\simeq A_{4}$, the dimension of the linear space ${\cal
M}_{k}(\Gamma{(3)})$ of modular forms of weight $k$ is $k+1$ [111, 112, 113],
i.e., there are three linearly independent modular forms of the lowest non-
trivial weight $2$. These forms have been explicitly obtained [22] in terms of
the Dedekind eta-function $\eta(\tau)$:

$\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}(1-q^{n})~{},\quad\qquad q=\exp\
(i2\pi\tau)~{},$ (42)

where $\eta(\tau)$ is a so called modular form of weight $1/2$. In what
follows we will use the following base of the $A_{4}$ generators $S$ and $T$
in the triplet representation:

$\displaystyle\begin{aligned} S=\frac{1}{3}\begin{pmatrix}-1&2&2\\\ 2&-1&2\\\
2&2&-1\end{pmatrix},\end{aligned}\qquad\qquad\begin{aligned}
T=\begin{pmatrix}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{2}\end{pmatrix},\end{aligned}$ (43)

where $\omega=\exp(i\frac{2}{3}\pi)$ . The modular forms of weight 2 $(k=2)$
transforming as a triplet of $A_{4}$, ${\bf
Y^{\rm(2)}_{3}}(\tau)=(Y_{1}(\tau)\,Y_{2}(\tau),Y_{3}(\tau))^{T}$, can be
written in terms of $\eta(\tau)$ and its derivative [22]:

$\displaystyle Y_{1}(\tau)$ $\displaystyle=$
$\displaystyle\frac{i}{2\pi}\left(\frac{\eta^{\prime}(\tau/3)}{\eta(\tau/3)}+\frac{\eta^{\prime}((\tau+1)/3)}{\eta((\tau+1)/3)}+\frac{\eta^{\prime}((\tau+2)/3)}{\eta((\tau+2)/3)}-\frac{27\eta^{\prime}(3\tau)}{\eta(3\tau)}\right),$
$\displaystyle Y_{2}(\tau)$ $\displaystyle=$
$\displaystyle\frac{-i}{\pi}\left(\frac{\eta^{\prime}(\tau/3)}{\eta(\tau/3)}+\omega^{2}\frac{\eta^{\prime}((\tau+1)/3)}{\eta((\tau+1)/3)}+\omega\frac{\eta^{\prime}((\tau+2)/3)}{\eta((\tau+2)/3)}\right),$
(44) $\displaystyle Y_{3}(\tau)$ $\displaystyle=$
$\displaystyle\frac{-i}{\pi}\left(\frac{\eta^{\prime}(\tau/3)}{\eta(\tau/3)}+\omega\frac{\eta^{\prime}((\tau+1)/3)}{\eta((\tau+1)/3)}+\omega^{2}\frac{\eta^{\prime}((\tau+2)/3)}{\eta((\tau+2)/3)}\right)\,.$

The overall coefficient in Eq. (44) is one possible choice. It cannot be
uniquely determined. The triplet modular forms of weight $2$ have the
following $q$-expansions:

$\displaystyle{\bf Y^{\rm(2)}_{3}}(\tau)=\begin{pmatrix}Y_{1}(\tau)\\\
Y_{2}(\tau)\\\
Y_{3}(\tau)\end{pmatrix}=\begin{pmatrix}1+12q+36q^{2}+12q^{3}+\dots\\\
-6q^{1/3}(1+7q+8q^{2}+\dots)\\\ -18q^{2/3}(1+2q+5q^{2}+\dots)\end{pmatrix}.$
(45)

They satisfy also the constraint [22]:

$\displaystyle Y_{2}(\tau)^{2}+2Y_{1}(\tau)Y_{3}(\tau)=0~{}.$ (46)

The modular forms of the higher weight, $k$, can be obtained by the $A_{4}$
tensor products of the modular forms with weight 2, ${\bf
Y^{\rm(2)}_{3}}(\tau)$, as given in Appendix A. For weight 4, that is $k=4$,
there are five modular forms by the tensor product of $\bf 3\otimes 3$ as:

where ${\bf Y^{\rm(4)}_{1^{\prime\prime}}}(\tau)$ vanishes due to the
constraint of Eq. (46).

##  Appendix C Determination of $\alpha_{e}/\gamma_{e}$ and
$\beta_{e}/\gamma_{e}$

The coefficients $\alpha_{e}$, $\beta_{e}$, and $\gamma_{e}$ in Eq.(27) are
taken to be real positive without loss of generality. We show these parameters
are described in terms of the modular parameter $\tau$ and the charged lepton
masses. We rewrite the mass matrix of Eq. (27) as

$\displaystyle\begin{aligned}
M_{E}=v_{d}\gamma_{e}\begin{pmatrix}\hat{\alpha}_{e}&0&0\\\
0&\hat{\beta}_{e}&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}Y_{1}(\tau)&Y_{3}(\tau)&Y_{2}(\tau)\\\
Y_{2}(\tau)&Y_{1}(\tau)&Y_{3}(\tau)\\\
Y_{3}(\tau)&Y_{2}(\tau)&Y_{1}(\tau)\end{pmatrix}\,,\end{aligned}$ (47)

where $\hat{\alpha}_{e}\equiv\alpha_{e}/\gamma_{e}$ and
$\hat{\beta}_{e}\equiv\beta_{e}/\gamma_{e}$. Denoting charged lepton masses
$m_{1}=m_{e}$, $m_{2}=m_{\mu}$ and $m_{3}=m_{\tau}$, we have three equations
as:

$\displaystyle{\sum_{i=1}^{3}m_{i}^{2}}={\rm Tr}[M_{E}^{\dagger}M_{E}]$
$\displaystyle=v_{d}^{2}\gamma_{e}^{2}\
(1+\hat{\alpha}_{e}^{2}+\hat{\beta}_{e}^{2})\ C^{e}_{1}\,,$ (48)
$\displaystyle{\prod_{i=1}^{3}m_{i}^{2}}={\rm Det}[M_{E}^{{\dagger}}M_{E}]$
$\displaystyle=v_{d}^{6}\gamma_{e}^{6}\
\hat{\alpha}^{2}_{e}\hat{\beta}^{2}_{e}\ C^{e}_{2}\,,$ (49)
$\displaystyle\chi=\frac{{\rm Tr}[M_{E}^{{\dagger}}M_{E}]^{2}-{\rm
Tr}[(M_{E}^{{\dagger}}M_{E})^{2}]}{2}$ $\displaystyle=v_{d}^{4}\gamma^{4}_{e}\
(\hat{\alpha}_{e}^{2}+\hat{\alpha}^{2}_{e}\hat{\beta}_{e}^{2}+\hat{\beta}_{e}^{2})~{}C^{e}_{3}\,,$
(50)

where $\chi\equiv m_{1}^{2}m_{2}^{2}+m_{2}^{2}m_{3}^{2}+m_{3}^{2}m_{1}^{2}$.
The coefficients $C^{e}_{1}$, $C^{e}_{2}$ and $C^{e}_{3}$ depend only on
$Y_{i}(\tau)$’s, where $Y_{i}(\tau)$’s are determined if the value of modulus
$\tau$ is fixed. Those are given explicitly as follows:

$\displaystyle\begin{aligned}
C^{e}_{1}&=|Y_{1}(\tau)|^{2}+|Y_{2}(\tau)|^{2}+|Y_{3}(\tau)|^{2}\,,\\\
C^{e}_{2}&=|Y_{1}(\tau)^{3}+Y_{2}(\tau)^{3}+Y_{3}(\tau)^{3}-3Y_{1}(\tau)Y_{2}(\tau)Y_{3}(\tau)|^{2}\,,\\\
C^{e}_{3}&=|Y_{1}(\tau)|^{4}+|Y_{2}(\tau)|^{4}+|Y_{3}(\tau)|^{4}+|Y_{1}(\tau)Y_{2}(\tau)|^{2}+|Y_{2}(\tau)Y_{3}(\tau)|^{2}+|Y_{1}(\tau)Y_{3}(\tau)|^{2}\\\
&-2{\rm
Re}\left[Y_{1}^{*}(\tau)Y_{2}^{*}(\tau)Y_{3}^{2}(\tau)+Y_{1}^{2}(\tau)Y_{2}^{*}(\tau)Y_{3}^{*}(\tau)+Y_{1}^{*}(\tau)Y_{2}^{2}(\tau)Y_{3}^{*}(\tau)\right]\,.\end{aligned}$

Then, we obtain two equations which describe $\hat{\alpha}_{e}$ and
$\hat{\beta}_{e}$ in terms of masses and $\tau$:

$\displaystyle\begin{aligned} \frac{(1+s)(s+t)}{t}&=\frac{(\sum
m_{i}^{2}/C^{e}_{1})(\chi/C^{e}_{3})}{\prod
m_{i}^{2}/C^{e}_{2}}~{},\quad\qquad\frac{(1+s)^{2}}{s+t}&=\frac{(\sum
m_{i}^{2}/C^{e}_{1})^{2}}{\chi/C^{e}_{3}}~{},\end{aligned}$ (51)

where we redefine the parameters $\hat{\alpha}_{e}^{2}+\hat{\beta}_{e}^{2}=s$
and $\hat{\alpha}_{e}^{2}\hat{\beta}_{e}^{2}=t$. After fixing charged lepton
masses and $\tau$, we obtain $s$ and $t$ numerically. They are related as
follows:

$\displaystyle\hat{\alpha}_{e}^{2}=\frac{s\pm\sqrt{s^{2}-4t}}{2}\,,\quad\quad\hat{\beta}_{e}^{2}=\frac{s\mp\sqrt{s^{2}-4t}}{2}\,.$
(52)

##  Appendix D Majorana and Dirac phases and $\langle m_{ee}\rangle$ in
$0\nu\beta\beta$ decay

Supposing neutrinos to be Majorana particles, the PMNS matrix
$U_{\text{PMNS}}$ [95, 96] is parametrized in terms of the three mixing angles
$\theta_{ij}$ $(i,j=1,2,3;~{}i<j)$, one CP violating Dirac phase
$\delta_{\text{CP}}$ and two Majorana phases $\alpha_{21}$, $\alpha_{31}$ as
follows:

$\displaystyle
U_{\text{PMNS}}=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{\text{CP}}}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{\text{CP}}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{\text{CP}}}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{\text{CP}}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{\text{CP}}}&c_{23}c_{13}\end{pmatrix}\begin{pmatrix}1&0&0\\\
0&e^{i\frac{\alpha_{21}}{2}}&0\\\
0&0&e^{i\frac{\alpha_{31}}{2}}\end{pmatrix},$ (53)

where $c_{ij}$ and $s_{ij}$ denote $\cos\theta_{ij}$ and $\sin\theta_{ij}$,
respectively.

The rephasing invariant CP violating measure of leptons [123, 124] is defined
by the PMNS matrix elements $U_{\alpha i}$. It is written in terms of the
mixing angles and the CP violating phase as:

$J_{CP}=\text{Im}\left[U_{e1}U_{\mu 2}U_{e2}^{\ast}U_{\mu
1}^{\ast}\right]=s_{23}c_{23}s_{12}c_{12}s_{13}c_{13}^{2}\sin\delta_{\text{CP}}\,,$
(54)

where $U_{\alpha i}$ denotes the each component of the PMNS matrix.

There are also other invariants $I_{1}$ and $I_{2}$ associated with Majorana
phases

$I_{1}=\text{Im}\left[U_{e1}^{\ast}U_{e2}\right]=c_{12}s_{12}c_{13}^{2}\sin\left(\frac{\alpha_{21}}{2}\right)\,,\quad
I_{2}=\text{Im}\left[U_{e1}^{\ast}U_{e3}\right]=c_{12}s_{13}c_{13}\sin\left(\frac{\alpha_{31}}{2}-\delta_{\text{CP}}\right)\,.$
(55)

We can calculate $\delta_{\text{CP}}$, $\alpha_{21}$ and $\alpha_{31}$ with
these relations by taking account of

$\displaystyle\cos\delta_{CP}=\frac{|U_{\tau
1}|^{2}-s_{12}^{2}s_{23}^{2}-c_{12}^{2}c_{23}^{2}s_{13}^{2}}{2c_{12}s_{12}c_{23}s_{23}s_{13}}\,,$
$\displaystyle\text{Re}\left[U_{e1}^{\ast}U_{e2}\right]=c_{12}s_{12}c_{13}^{2}\cos\left(\frac{\alpha_{21}}{2}\right)\,,\qquad\text{Re}\left[U_{e1}^{\ast}U_{e3}\right]=c_{12}s_{13}c_{13}\cos\left(\frac{\alpha_{31}}{2}-\delta_{\text{CP}}\right)\,.$
(56)

In terms of this parametrization, the effective mass for the $0\nu\beta\beta$
decay is given as follows:

$\displaystyle\langle
m_{ee}\rangle=\left|m_{1}c_{12}^{2}c_{13}^{2}+m_{2}s_{12}^{2}c_{13}^{2}e^{i\alpha_{21}}+m_{3}s_{13}^{2}e^{i(\alpha_{31}-2\delta_{CP})}\right|\,.$
(57)

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