# Neutrino Mass Matrices with Generalized CP Symmetries and Texture Zeros

Sanjeev Kumar<EMAIL_ADDRESS>Radha Raman Gautam
<EMAIL_ADDRESS>Department of Physics and Astrophysics, University of
Delhi, Delhi-110007, India  Department of Physics, LBS Govt. Degree College,
Saraswati Nagar, Shimla-171206, India

###### Abstract

We investigate the properties of neutrino mass matrices that incorporate
texture zeros and generalized CP symmetries associated with tribimaximal
mixing. By combining these approaches, we derive predictive neutrino mass
matrices and explore their implications for mass hierarchies, mixing angles,
and CP-violating phases. We find that the three angles defining the
generalized CP symmetries have narrow allowed ranges. We also obtain distinct
correlations between the three mixing angles and the CP-violating phases that
distinguish the various texture patterns from one another. Moreover, we
compute the effective neutrino mass for neutrinoless double beta decay and the
sum of neutrino masses. Our results highlight the predictability and
testability of neutrino mass matrices with generalized CP symmetry.

###### keywords:

Neutrino mass matrix, Generalized CP symmetry

††journal: Nuclear Physics B

##  1 Introduction

The discovery of neutrino oscillations indicates that neutrinos have non-zero
masses and mix with one another. After this discovery, the precision
measurements of lepton masses and mixings and determination of mass hierarchy
and CP violation in the lepton sector are the main focus of forthcoming
neutrino experiments. These discoveries can help us in understanding of
leptogenesis and the matter-antimatter asymmetry of the universe.

The neutrino mixing matrix $U$, also known as the PMNS matrix, can be written
as

$U=\begin{pmatrix}\cos\theta_{12}\cos\theta_{13}&\sin\theta_{12}\cos\theta_{13}&\sin\theta_{13}e^{-i\delta}\\\
-\cos\theta_{23}\sin\theta_{12}-\cos\theta_{12}\sin\theta_{13}\sin\theta_{23}e^{i\delta}&\cos\theta_{12}\cos\theta_{23}-\sin\theta_{12}\sin\theta_{13}\sin\theta_{23}e^{i\delta}&\cos\theta_{13}\sin\theta_{23}\\\
\sin\theta_{12}\sin\theta_{23}-\cos\theta_{12}\sin\theta_{13}\cos\theta_{23}e^{i\delta}&-\cos\theta_{12}\sin\theta_{23}-\sin\theta_{12}\sin\theta_{13}\cos\theta_{23}e^{i\delta}&\cos\theta_{13}\cos\theta_{23}\\\
\end{pmatrix}P$ (1)

in its PDG parametrization [1] after a slight variation in the definition of
the phase matrix $P$:

$P=\begin{pmatrix}e^{-i\phi_{1}}&0&0\\\ 0&e^{-i\phi_{2}}&0\\\
0&0&e^{-i\phi_{3}}\end{pmatrix}.$ (2)

The mixing matrix $U$ is parameterized by three mixing angles ($\theta_{12}$,
$\theta_{23}$, and $\theta_{13}$), one Dirac type CP-violating phase
($\delta$), and two Majorana type CP-violating phases
($\phi_{12}=\phi_{2}-\phi_{1}$ and $\phi_{13}=\phi_{3}-\phi_{1}$).

The lepton mixing matrix $U$ parameterizes the mixing between the three
flavors of neutrinos and diagonalizes the neutrino mass matrix $M_{\nu}$ as

$U^{T}M_{\nu}U=\text{diag}(m_{1},m_{2},m_{3}).$ (3)

The matrix $M_{\nu}$ can be parameterized as a function of three mixing
angles, three neutrino masses, and three CP-violating phases. However,
neutrino oscillation experiments are sensitive only to the mass-squared
differences ($\Delta m^{2}_{21}$ and $\Delta m^{2}_{31}$, where $\Delta
m_{ij}^{2}=m_{i}^{2}-m_{j}^{2}$, $i,j=1,2,3$). We know definitely that the
second neutrino mass eigenstate is heavier than the first mass eigenstate
($m_{2}>m_{1}$). But, we cannot say so about the third mass eigenstate as the
sign of $\Delta m^{2}_{31}$ is still undetermined. The current experiments
allow both normal hierarchy (NH) with $m_{1}<m_{3}$ and inverted hierarchy
(IH) with $m_{1}>m_{3}$ corresponding to the two sign choices of $\Delta
m^{2}_{31}$.

The neutrino oscillation experiments provide information only about the
neutrino mass-squared differences and not on the absolute neutrino mass scale.
The constraints on the absolute neutrino mass come from the experiments on
beta decay, neutrinoless double beta decay, and cosmological experiments. The
current understanding of these neutrino experiments has been reviewed in Ref.
[2].

There exist certain special patterns in the mixing matrix $U$ that can be
attributed to a residual flavor symmetry of $M_{\nu}$. Such residual
symmetries arise when $M_{\nu}$ is invariant under certain non-trivial
transformations $G_{i}$ of the form

$G_{i}=Ud_{i}U^{\dagger}$ (4)

where

$d_{1}=\text{diag}(1,-1,-1),$ (5) $d_{2}=\text{diag}(-1,1,-1),$ (6)

and

$d_{3}=\text{diag}(-1,-1,1).$ (7)

In other words,

$G_{i}^{T}M_{\nu}G_{i}=M_{\nu}\text{ for }i=1,2,3.$ (8)

The concept of residual flavor symmetries has been thoroughly explored in the
literature [3, 4, 5, 6, 7]. One of the most extensively investigated lepton
mixing patterns that has emerged through the use of flavor symmetries is the
tribimaximal (TBM) mixing pattern [8, 9, 10]:

$U_{TBM}=\left(\begin{array}[]{ccc}-\frac{\sqrt{2}}{\sqrt{3}}&\frac{1}{\sqrt{3}}&0\\\
\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\end{array}\right).$
(9)

One of the key predictions of the TBM mixing is that the reactor mixing angle
$\theta_{13}$ should be zero, while the atmospheric mixing angle is expected
to be maximal, i.e., $\theta_{23}=\frac{\pi}{4}$. Additionally, TBM mixing
predicts a specific value for the solar mixing angle,
$\theta_{12}=\sin^{-1}(1/\sqrt{3})$. The transformation matrices for the TBM
mixing are

$G_{1}=\frac{1}{3}\begin{pmatrix}1&-2&2\\\ -2&-2&-1\\\ 2&-1&-2\end{pmatrix},$
(10) $G_{2}=\frac{1}{3}\begin{pmatrix}-1&2&-2\\\ 2&-1&-2\\\
-2&-2&-1\end{pmatrix},$ (11)

and

$G_{3}=\begin{pmatrix}-1&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix}.$ (12)

Even though recent neutrino oscillation experiments have established a non-
zero value for the reactor mixing angle $\theta_{13}$ [11, 12, 13, 14, 15], it
is still possible to explain the experimental data by considering a mixing
matrix that retains only the first or the second column of the TBM mixing
matrix. Such mixing patterns are known as TM1 or TM2 mixing and have been
extensively studied in the literature [16, 17, 18, 19, 20, 21]. Unlike TBM
mixing, the neutrino mass matrix giving rise to TM1 or TM2 mixing has only
$G_{1}$ or $G_{2}$ as its residual flavor symmetry, respectively. The mass
matrix for TM1 or TM2 mixing doesn’t have $G_{3}$ residual symmetry and hence
can accommodate a non-zero $\theta_{13}$.

The study of residual flavor symmetries in the neutrino mass matrix has
inspired the investigation of other types of symmetries as well. One such
symmetry is the generalized remnant CP transformation [22, 23, 24, 25, 26].
Unlike the residual flavor symmetries, these transformations do not leave the
neutrino mass matrix $M_{\nu}$ invariant, but rather transform it to its
complex conjugate $M_{\nu}^{\ast}$. The generalized CP transformations are
denoted by $X_{i}$ and are related to the lepton mixing matrix $U$ through the
relation

$X_{i}=Ud_{i}U^{T}\text{ for }i=1,2,3.$ (13)

The generalized CP symmetry $X_{i}$ is present in $M_{\nu}$ if

$X^{T}_{i}M_{\nu}X_{i}=M_{\nu}^{\ast}.$ (14)

Since, the TBM mixing matrix is real, we note that $G_{i}=X_{i}$
mathematically. Furthermore the three transformations $G_{i}$ or $X_{i}$ along
with the identity transformation form the group $Z_{2}\times Z_{2}$. Starting
form TBM neutrino mixing matrix, one can construct a neutrino mass matrix that
exhibits one of the three independent generalized CP symmetries (X1, X2, or
$X_{3}$). The imposition of any two of these symmetries leads to variants of
TBM mixing possessing a single residual flavor symmetry. In other words,
performing two CP transformations $X_{i}X_{j}^{*}$ is equivalent to a flavor
symmetry transformation $G_{k}$, where $i$, $j$, and $k$ are cyclic
permutations of ${1,2,3}$. This observation is important as it allows to
explain the lepton mixing matrix in terms of the remnant CP symmetries.
Additionally, imposing generalized CP symmetries instead of a flavor symmetry
also puts constraints on the Majorana CP phases, making it an advantageous
approach [27]. Owing to its significance, this approach has been
systematically studied [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
39, 40, 41, 42, 43].

The imposition of a single residual CP-symmetry also allows us to obtain
variants of TBM mixing. Just like the variants TM1 and TM2 of TBM mixing,
these variants can also exhibit non-zero $\theta_{13}$ and CP violation.
Therefore, such variants become relevant for current and forthcoming
experimental data. For TBM mixing, the generalized CP symmetry $X_{3}$ is
known as the generalized $\mu-\tau$ symmetry [44, 45, 46] and has received
more attention than the other two CP symmetries. Nevertheless, the
phenomenology of all three symmetries has been investigated [41].

Apart from the residual flavour symmetries and remnant CP symmetries,
alternative approaches such as imposing texture zeros [47, 48, 49, 50, 51, 52,
53, 54, 55, 56, 57, 58, 59], vanishing cofactors [60, 61, 62, 63, 64, 65, 66,
67, 68, 69, 70], hybrid textures [71, 72, 73, 74, 75, 76], or equalities among
the neutrino mass matrix elements [77] have also been explored to explain the
neutrino masses and mixings. Assuming neutrinos to be Majorana particles and
charged lepton mass matrix to be diagonal, the presence of texture zeros in
the neutrino mass matrix is one of the simplest possibilities consistent with
the current neutrino data. There are fifteen possible patterns of two texture
zeros in the neutrino mass matrix, but only seven of them are compatible with
the current neutrino oscillation data (Table 1). These seven patterns can be
classified into three classes (A, B, and C) depending on the type of the mass
hierarchy [47].

The simultaneous use of both flavor symmetry and texture zeros or vanishing
cofactors in constructing neutrino mass matrices has resulted in highly
predictive patterns. For example, incorporation of texture zeros or vanishing
cofactors along with the mixing patterns TM1 or TM2 are compatible with the
current data and testable with the future data [78, 79, 80, 81, 82].

Another interesting possibility is to impose texture zeros in conjunction with
the generalized CP symmetries. For instance, the effects of texture zeros or
vanishing cofactors in conjunction with the generalized CP symmetry $X_{3}$
corresponding to TBM mixing has been studied in literature [83, 84].

In this work, we aim to investigate the consequences of incorporating texture
zeros into the framework of the generalized CP symmetries X1 and X2
corresponding to TBM mixing. By combining these two approaches together, we
expect to obtain even more predictive neutrino mass matrices. Specifically, we
will explore the possible patterns of texture zeros in the neutrino mass
matrix that are consistent with the symmetries imposed by X1 and X2. Our
analysis will also consider the implications of these patterns for the values
of the neutrino masses, mixing angles, and the CP-violating phases. The form
of neutrino mixing matrix and mass matrix for X1 and X2 CP symmetries is
derived in Section 2 where we also show the way to obtain the predictions for
masses, mixing angles, and CP-violating phases. We study the implications of
imposing the one and two texture zero conditions with X1 and X2 CP symmetries
and study the implications for neutrino parameters in light of the current
experimental data. We summarize the results of the study in Section 3 and
conclusions in Section 4.

##  2 Formalism and Methodology

###  2.1 Generalized CP Symmetry

Neutrino mass matrices invariant under X1 or X2 generalized CP symmetries can
be written as

$M_{X_{1}}=\left(\begin{array}[]{ccc}a+4i(x+y)&2b+2ix&2c+2iy\\\
2b+2ix&4b+d+i(-3x-y)&a-b-c-d+2i(-x-y)\\\ 2c+2iy&a-b-
c-d+2i(-x-y)&4c+d+i(-x-3y)\end{array}\right),$ (15)
$M_{X_{2}}=\left(\begin{array}[]{ccc}a+2i(x+y)&b+ix&c+iy\\\
b+ix&d-2iy&a+c-d+i(-x-y)\\\ c+iy&a+c-d+i(-x-y)&b-c+d-2ix\end{array}\right),$
(16)

where, $a,b,c,d,x$ and $y$ are real parameters. The mass matrix with X1 CP
symmetry ($M_{X_{1}}$) is diagonalized by the lepton mixing matrix [41]

$U_{X_{1}}=U_{TBM}\text{diag}(1,i,i)O_{\nu}Q_{\nu}$ (17)

and the mass matrix with X2 CP symmetry ($M_{X_{2}}$) is diagonalized by the
lepton mixing matrix

$U_{X_{2}}=U_{TBM}\text{diag}(i,1,i)O_{\nu}Q_{\nu}.$ (18)

Here, the matrix $O_{\nu}$ contains the three angles ($\theta_{1}$,
$\theta_{2}$, $\theta_{3}$) that parametrize the mixing matrix $U$ for a
particular generalized CP symmetry and is given as

$O_{\nu}=\begin{pmatrix}1&0&0\\\ 0&\cos\theta_{1}&\sin\theta_{1}\\\
0&-\sin\theta_{1}&\cos\theta_{1}\end{pmatrix}\begin{pmatrix}\cos\theta_{2}&0&\sin\theta_{2}\\\
0&1&0\\\
-\sin\theta_{2}&0&\cos\theta_{2}\end{pmatrix}\begin{pmatrix}\cos\theta_{3}&\sin\theta_{3}&0\\\
-\sin\theta_{3}&\cos\theta_{3}&\\\ 0&0&1\end{pmatrix}.$ (19)

The diagonal matrix of CP parities $Q_{\nu}$ is given by

$Q_{\nu}=\textrm{diag}(e^{ik_{1}\pi/2},e^{ik_{2}\pi/2},e^{ik_{3}\pi/2}),$ (20)

with integer values of $k_{i}$ ($i=1,2,3$). The mixing angles ($\theta_{12}$,
$\theta_{23}$, and $\theta_{13}$) can be calculated from absolute values of
the respective entries of $U$ and the CP-violating phases can be calculated
from the three CP invariants [85, 86, 25] by comparing the values obtained
here with their counterparts in the PDG parametrization [1] [Cf. Eq. (1)]. The
Jarlskog invariant $J_{CP}$ is given as [87]

$J_{CP}=\text{Im}\left[U_{11}U_{33}U^{*}_{13}U^{*}_{31}\right].$ (21)

The invariants $I_{1}$ and $I_{2}$ associated with the Majorana phases are
given as [85, 86, 25]

$I_{1}=\text{Im}\left[U_{12}^{2}U_{11}^{*2}\right]$

and

$I_{2}=\text{Im}\left[U_{13}^{2}U_{11}^{*2}\right].$

###  2.2 Two Texture Zeros

Assuming neutrinos to be Majorana particles and taking the charged lepton mass
matrix as diagonal, there are fifteen potential patterns of two texture zeros
in the neutrino mass matrix. However, only seven of them are compatible with
current neutrino oscillation data. These seven patterns have been summarized
in Table 1.

Type | Constraining Equations
---|---
A1 |  $M_{ee}=0$, $M_{e\mu}=0$
A2 |  $M_{ee}=0$, $M_{e\tau}=0$
B1 |  $M_{e\tau}=0$, $M_{\mu\mu}=0$
B2 |  $M_{e\mu}=0$, $M_{\tau\tau}=0$
B3 |  $M_{e\mu}=0$, $M_{\mu\mu}=0$
B4 |  $M_{e\tau}=0$, $M_{\tau\tau}=0$
C |  $M_{\mu\mu}=0$, $M_{\tau\tau}=0$
Table 1: Viable two texture zero neutrino mass matrices classified into three
classes.

###  2.3 Two Texture Zeros with X1 or X2 Generalized CP Symmetry

Imposing the X1 generalized CP symmetry on the seven neutrino mass matrices
with two texture zeros, we obtain

$M^{A_{1}}_{X_{1}}=\left(\begin{array}[]{ccc}0&0&2c\\\ 0&d&-c-d\\\
2c&-c-d&4c+d\end{array}\right),$ (22)
$M^{A_{2}}_{X_{1}}=\left(\begin{array}[]{ccc}0&2b&0\\\ 2b&4b+d&-b-d\\\
0&-b-d&d\end{array}\right),$ (23)
$M^{B_{1}}_{X_{1}}=\left(\begin{array}[]{ccc}a&2b&0\\\ 2b&0&a+3b\\\
0&a+3b&-4b\end{array}\right),$ (24)
$M^{B_{2}}_{X_{1}}=\left(\begin{array}[]{ccc}a&0&2c\\\ 0&-4c&a+3c\\\
2c&a+3c&0\end{array}\right),$ (25)
$M^{B_{3}}_{X_{1}}=\left(\begin{array}[]{ccc}a&0&2c\\\ 0&0&a-c\\\
2c&a-c&4c\end{array}\right),$ (26)
$M^{B_{4}}_{X_{1}}=\left(\begin{array}[]{ccc}a&2b&0\\\ 2b&4b&a-b\\\
0&a-b&0\end{array}\right),\text{ and}$ (27)
$M^{C}_{X_{1}}=\left(\begin{array}[]{ccc}a&2b&2b\\\ 2b&0&a+2b\\\
2b&a+2b&0\end{array}\right)$ (28)

where, $a,b,c,d$ are real parameters. If we impose the X2 CP symmetry, the
seven possibilities for the neutrino mass matrix are

$M^{A_{1}}_{X_{2}}=\left(\begin{array}[]{ccc}0&0&c\\\ 0&c+d&-d\\\
c&-d&d\end{array}\right)$ (29)
$M^{A_{2}}_{X_{2}}=\left(\begin{array}[]{ccc}0&b&0\\\ b&d&-d\\\
0&-d&b+d\end{array}\right),$ (30)
$M^{B_{1}}_{X_{2}}=\left(\begin{array}[]{ccc}a+2ix&b+ix&0\\\ b+ix&0&a-ix\\\
0&a-ix&b-2ix\end{array}\right),$ (31)
$M^{B_{2}}_{X_{2}}=\left(\begin{array}[]{ccc}a+2iy&0&c+iy\\\ 0&c-2iy&a-iy\\\
c+iy&a-iy&0\end{array}\right),$ (32)
$M^{B_{3}}_{X_{2}}=\left(\begin{array}[]{ccc}a&0&c\\\ 0&0&a+c\\\
c&a+c&-c\end{array}\right),$ (33)
$M^{B_{4}}_{X_{2}}=\left(\begin{array}[]{ccc}a&b&0\\\ b&-b&a+b\\\
0&a+b&0\end{array}\right),\text{ and}$ (34)
$M^{C}_{X_{2}}=\left(\begin{array}[]{ccc}a&b&b\\\ b&0&a+b\\\
b&a+b&0\end{array}\right)$ (35)

where, $a,b,c,d,x$ and $y$ are real parameters.

We observe that for the neutrino mass matrix with X1 CP symmetry and two
texture zeros, all the seven classes become real. However, for the neutrino
mass matrices with X2 CP symmetry and two texture zeros, the classes B1 and
$B_{2}$ are complex while other 5 classes are real. We know that the real
parts of the mass matrices remain invariant under X1 or X2 CP symmetry and are
also invariant under $G_{1}$ or $G_{2}$ flavor symmetry. Hence, they can be
diagonalized by TM1 or TM2 mixing matrices, respectively. Thus, all these
these mass matrices, except those for classes $B_{1},B_{2}$ with X2 CP
symmetry, can be diagonalized either by the TM1 or TM2 mixing matrices. In
other words, all the seven cases of two texture zero mass matrices with X1 CP
symmetry are diagonalized by the TM1 mixing matrix

$U_{TM_{1}}=\left(\begin{array}[]{ccc}\frac{2}{\sqrt{3}}&\frac{1}{\sqrt{3}}\cos\theta&\frac{1}{\sqrt{3}}\sin\theta\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}\cos\theta-\frac{\sin\theta}{\sqrt{2}}&\frac{1}{\sqrt{3}}\sin\theta+\frac{\cos\theta}{\sqrt{2}}\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}\cos\theta+\frac{\sin\theta}{\sqrt{2}}&\frac{1}{\sqrt{3}}\sin\theta-\frac{\cos\theta}{\sqrt{2}}\end{array}\right)P.$
(36)

and the five cases of the two texture zero mass matrices with X2 CP symmetry
(classes A1, A2, B3, B4, and C) are diagonalized by the TM2 mixing matrix

$U_{TM_{2}}=\left(\begin{array}[]{ccc}-\sqrt{\frac{2}{3}}\cos\theta&\frac{1}{\sqrt{3}}&-\sqrt{\frac{2}{3}}\sin\theta\\\
\frac{\cos\theta}{\sqrt{6}}+\frac{\sin\theta}{\sqrt{2}}&\frac{1}{\sqrt{3}}&\frac{\sin\theta}{\sqrt{6}}-\frac{\cos(\theta)}{\sqrt{2}}\\\
\frac{\cos\theta}{\sqrt{6}}-\frac{\sin\theta}{\sqrt{2}}&\frac{1}{\sqrt{3}}&\frac{\cos\theta}{\sqrt{2}}+\frac{\sin(\theta)}{\sqrt{6}}\end{array}\right)P.$
(37)

The mass matrices for classes B1 and B2 with X2 CP symmetry need to be
diagonalized with another method that we will describe later in this section.

The condition of two texture zeros in the neutrino mass matrix at $(p,q)$ and
$(r,s)$ positions, when imposed with X1 CP symmetry, implies:

$(M_{X_{1}})_{pq}=0\ \textrm{and}\ (M_{X_{1}})_{rs}=0.$ (38)

When expressed in terms of the mass eigenvalues and the mixing matrix
elements, these two complex equations become

$m_{1}U_{p1}U_{q1}e^{2i\phi_{1}}+m_{2}U_{p2}U_{q2}e^{2i\phi_{2}}+m_{3}U_{p3}U_{q3}e^{2i\phi_{3}}=0$
(39)

and

$m_{1}U_{r1}U_{s1}e^{2i\phi_{1}}+m_{2}U_{r2}U_{s2}e^{2i\phi_{2}}+m_{3}U_{r3}U_{s3}e^{2i\phi_{3}}=0$
(40)

where $p,q,r$ and $s$ can take the values $1$, $2$, and $3$. These equations
can be simultaneously solved to obtain two complex mass ratios:

$\frac{m_{1}}{m_{2}}e^{2i(\phi_{1}-\phi_{2})}=\frac{U_{r2}U_{s2}U_{p3}U_{q3}-U_{p2}U_{q2}U_{r3}U_{s3}}{U_{p1}U_{q1}U_{r3}U_{s3}-U_{p3}U_{q3}U_{r1}U_{s1}}$
(41)

and

$\frac{m_{1}}{m_{3}}e^{2i(\phi_{1}-\phi_{3})}=\frac{U_{r3}U_{s3}U_{p2}U_{q2}-U_{p3}U_{q3}U_{r2}U_{s2}}{U_{p1}U_{q1}U_{r2}U_{s2}-U_{p2}U_{q2}U_{r1}U_{s1}}.$
(42)

The magnitudes of the two mass ratios in Eqs. (41) and (42), are defined as

$\textrm{r}_{12}=\left|\frac{m_{1}}{m_{2}}e^{2i(\phi_{1}-\phi_{2})}\right|,\ \
\ \ \ \
\textrm{r}_{13}=\left|\frac{m_{1}}{m_{3}}e^{2i(\phi_{1}-\phi_{3})}\right|.$
(43)

The magnitudes of the two mass ratios can be used to obtain the expression for
the parameter $R_{\nu}$ defined as the ratio of mass squared differences:

$R_{\nu}\equiv\frac{\Delta m_{21}^{2}}{\left|\Delta
m_{31}^{2}\right|}=\frac{\left(\frac{1}{\text{r}_{12}}\right)^{2}-1}{\left|\left(\frac{1}{\text{r}_{13}}\right)^{2}-1\right|}$
(44)

where $m_{1}<m_{3}$ for NH ($m_{1}>m_{3}$ for IH). The two CP-violating
Majorana phases $\phi_{12}\equiv(\phi_{1}-\phi_{2})$ and
$\phi_{13}\equiv(\phi_{1}-\phi_{3})$ are given by

$\displaystyle\phi_{12}$
$\displaystyle=-\frac{1}{2}\textrm{Arg}\left(\frac{U_{r2}U_{s2}U_{p3}U_{q3}-U_{p2}U_{q2}U_{r3}U_{s3}}{U_{p1}U_{q1}U_{r3}U_{s3}-U_{p3}U_{q3}U_{r1}U_{s1}}\right),$
(45) $\displaystyle\phi_{13}$
$\displaystyle=-\frac{1}{2}\textrm{Arg}\left(\frac{U_{r3}U_{s3}U_{p2}U_{q2}-U_{p3}U_{q3}U_{r2}U_{s2}}{U_{p1}U_{q1}U_{r2}U_{s2}-U_{p2}U_{q2}U_{r1}U_{s1}}\right).$
(46)

Since the mass matrices in the present case are real, the two Majorana type CP
phases take trivial values of $0$, $\frac{\pi}{2}$ or $\pi$. For the neutrino
mass matrix with two texture zeros and X1 CP symmetry to be compatible with
the present neutrino oscillation data, the values of $R_{\nu}$ obtained from
Eq. (44) should lie within its experimentally allowed range. The mixing matrix
used in the above equations is $U_{TM_{1}}$. If we replace it with
$U_{TM_{2}}$, the same procedure can be used to study the neutrino mass matrix
with two texture zeros and X2 CP symmetry.

The neutrino mass matrices corresponding to classes $B_{1},B_{2}$ with X2 CP
symmetry are complex and follow the symmetry relations

$X_{2}^{T}M^{B_{1}}X_{2}=(M^{B_{1}})^{*}$ (47)

and

$X_{2}^{T}M^{B_{2}}X_{2}=(M^{B_{2}})^{*}.$ (48)

To diagonalize these mass matrices, we first make them real using the
transformation

$M^{B_{j}}_{\textrm{real}}=\textrm{diag}(i,1,i)^{T}U_{TBM}^{T}M^{B_{j}}U_{TBM}\textrm{diag}(i,1,i)$
(49)

where $j=1$ or $2$. The resulting real symmetric mass matrix is diagonalizable
by an orthogonal matrix $O$ as

$\left(M^{B_{j}}\right)_{\textrm{diag}}=O^{T}M^{B_{j}}_{\textrm{real}}O$ (50)

that can be parametrized in terms of three neutrino masses ($m_{1}$, $m_{2}$,
$m_{3}$) using the three invariants

$Tr[M_{\textrm{real}}]=m_{1}+m_{2}+m_{3},$ (51)

,

$Tr[(M_{\textrm{real}})^{2}]=m_{1}^{2}+m_{2}^{2}+m_{3}^{2},$ (52)

and

$Det[M_{\textrm{real}}]=m_{1}m_{2}m_{3}.$ (53)

By solving above three equations, we can obtain the desired orthogonal mixing
matrix $O$ that diagonalizes the neutrino mass matrix of class B1 or B2 with
X2 CP symmetry:

$O=\left(\begin{array}[]{ccc}\frac{\sqrt{3}m_{2}m_{3}-C_{1}}{\left(-m_{1}^{2}+m_{2}^{2}+m_{3}^{2}-m_{2}m_{3}\right)N_{1}}&\frac{\sqrt{3}m_{1}m_{3}-C_{1}}{\left(m_{1}^{2}-m_{3}m_{1}-m_{2}^{2}+m_{3}^{2}\right)N_{2}}&\frac{\sqrt{3}m_{1}m_{2}-C_{1}}{\left(m_{1}^{2}-m_{2}m_{1}+m_{2}^{2}-m_{3}^{2}\right)N_{3}}\\\
\frac{\sqrt{2}C_{2}\left(-m_{1}^{2}+(m_{2}+m_{3})^{2}+\sqrt{3}C_{1}\right)}{(m_{2}+m_{3})\left(-m_{1}^{2}+m_{2}^{2}+m_{3}^{2}-\sqrt{3}C_{1}-4m_{2}m_{3}\right)N_{1}}&\frac{\sqrt{2}C_{2}\left(\sqrt{3}C_{1}+(m_{1}-m_{2}+m_{3})(m_{1}+m_{2}+m_{3})\right)}{(m_{1}+m_{3})\left(m_{1}^{2}-4m_{3}m_{1}-m_{2}^{2}+m_{3}^{2}-\sqrt{3}C_{1}\right)N_{2}}&\frac{\sqrt{2}C_{2}\left(\sqrt{3}C_{1}+(m_{1}+m_{2}-m_{3})(m_{1}+m_{2}+m_{3})\right)}{(m_{1}+m_{2})\left(m_{1}^{2}-4m_{2}m_{1}+m_{2}^{2}-m_{3}^{2}-\sqrt{3}C_{1}\right)N_{3}}\\\
\frac{1}{N_{1}}&\frac{1}{N_{2}}&\frac{1}{N_{3}}\end{array}\right),$ (54)

where

$C_{1}=\sqrt{(m_{1}+m_{2}-m_{3})(m_{1}-m_{2}+m_{3})(-m_{1}+m_{2}+m_{3})(m_{1}+m_{2}+m_{3})},$
(55)

and

$C_{2}=\sqrt{-\frac{(m_{1}+m_{2})(m_{1}+m_{3})(m_{2}+m_{3})}{m_{1}+m_{2}+m_{3}}}.$
(56)

Here, the normalization factors $N_{1},N_{2}$ and $N_{3}$ are given by

$N_{1}=2\sqrt{\frac{(m_{1}-m_{2})(m_{3}-m_{2})(m_{1}+m_{2}+m_{3})}{(m_{2}+m_{3})\left(-\sqrt{3}C_{1}-m_{1}^{2}+m_{2}^{2}-4m_{2}m_{3}+m_{3}^{2}\right)}},$
(57)
$N_{2}=2\sqrt{\frac{(m_{1}-m_{2})(m_{2}-m_{3})(m_{1}+m_{2}+m_{3})}{(m_{1}+m_{3})\left(-\sqrt{3}C_{1}+m_{1}^{2}-4m_{1}m_{3}-m_{2}^{2}+m_{3}^{2}\right)}},$
(58)

and

$N_{3}=2\sqrt{\frac{(m_{1}-m_{3})(m_{3}-m_{2})(m_{1}+m_{2}+m_{3})}{(m_{1}+m_{2})\left(-\sqrt{3}C_{1}+m_{1}^{2}-4m_{1}m_{2}+m_{2}^{2}-m_{3}^{2}\right)}}.$
(59)

Using the above expression of the orthogonal matrix $O$ (Eq. 54), the full
mixing matrix that diagonalizes the neutrino mass matrices $M^{B_{1}}_{X_{2}}$
and $M^{B_{2}}_{X_{2}}$ can be written as

$U=U_{TBM}\text{diag}(i,1,i)OQ_{\nu}.$ (60)

This matrix $U$ depends upon the neutrinos masses $m_{1}$, $m_{2}$, and
$m_{3}$ and can be used to calculate the predictions of the patterns
$M^{B_{1}}_{X_{2}}$ and $M^{B_{2}}_{X_{2}}$ for the mixing angles and CP-
violating phases.

###  2.4 One Texture Zero with X1 or X2 Generalized CP Symmetry

The neutrino mass matrices having one texture zero and X1 CP symmetry can be
obtained from $M_{X_{1}}$ in Eq. (15) by substituting the texture zero
condition at the appropriate place:

$M_{X_{1}}^{I}=\left(\begin{array}[]{ccc}0&2b+2ix&2c-2ix\\\
2b+2ix&4b+d-2ix&-b-c-d\\\ 2c-2ix&-b-c-d&4c+d+2ix\end{array}\right),$ (61)
$M_{X_{1}}^{II}=\left(\begin{array}[]{ccc}a+4iy&0&2c+2iy\\\
0&d-iy&a-c-d-2iy\\\ 2c+2iy&a-c-d-2iy&4c+d-3iy\end{array}\right),$ (62)
$M_{X_{1}}^{III}=\left(\begin{array}[]{ccc}a+4ix&2b+2ix&0\\\
2b+2ix&4b+d-3ix&a-b-d-2ix\\\ 0&a-b-d-2ix&d-ix\end{array}\right),$ (63)
$M_{X_{1}}^{IV}=\left(\begin{array}[]{ccc}a-8ix&2b+2ix&2c-6ix\\\
2b+2ix&0&a+3b-c+4ix\\\ 2c-6ix&a+3b-c+4ix&-4b+4c+8ix\end{array}\right),$ (64)

and

$M_{X_{1}}^{V}=\left(\begin{array}[]{ccc}a&2b+2ix&2c-2ix\\\
2b+2ix&a+3b-c-2ix&0\\\ 2c-2ix&0&a-b+3c+2ix\end{array}\right).$ (65)

$M_{X_{1}}^{VI}=\left(\begin{array}[]{ccc}a+\frac{8ix}{3}&2b+2ix&2c-\frac{2ix}{3}\\\
2b+2ix&4b-4c-\frac{8ix}{3}&a-b+3c-\frac{4ix}{3}\\\
2c-\frac{2ix}{3}&a-b+3c-\frac{4ix}{3}&0\end{array}\right),$ (66)

where $a$, $b$, $c$, $d$, $x$, and $y$ are real parameters. Using the similar
methodology, the neutrino mass matrices having one texture zero and X2 CP
symmetry can be obtained from $M_{X_{2}}$ in Eq. (16) by substituting the
texture zero condition at the appropriate place:

$M_{X_{2}}^{I}=\left(\begin{array}[]{ccc}0&b+ix&c-ix\\\ b+ix&d+2ix&c-d\\\
c-ix&c-d&b-c+d-2ix\end{array}\right),$ (67)
$M_{X_{2}}^{II}=\left(\begin{array}[]{ccc}a+2iy&0&c+iy\\\ 0&d-2iy&a+c-d-iy\\\
c+iy&a+c-d-iy&d-c\end{array}\right),$ (68)
$M_{X_{2}}^{III}=\left(\begin{array}[]{ccc}a+2ix&b+ix&0\\\ b+ix&d&a-d-ix\\\
0&a-d-ix&b+d-2ix\end{array}\right),$ (69)
$M_{X_{2}}^{IV}=\left(\begin{array}[]{ccc}a+2ix&b+ix&c\\\ b+ix&0&a+c-ix\\\
c&a+c-ix&b-c-2ix\end{array}\right),$ (70)
$M_{X_{2}}^{V}=\left(\begin{array}[]{ccc}a&b+ix&c-ix\\\ b+ix&a+c+2ix&0\\\
c-ix&0&a+b-2ix\end{array}\right),$ (71)

and

$M_{X_{2}}^{VI}=\left(\begin{array}[]{ccc}a+2iy&b&c+iy\\\ b&-b+c-2iy&a+b-iy\\\
c+iy&a+b-iy&0\end{array}\right).$ (72)

The condition of one texture zero in the neutrino mass matrix when imposed
along with X1 CP symmetry, implies:

$(M_{X_{1}})_{jk}=0.$ (73)

The above condition yields a complex equation given by

$m_{1}P+m_{2}Q+m_{3}R=0$ (74)

with the tree complex coefficients $P=U_{j1}U_{k1}e^{2i\phi_{1}}$,
$Q=U_{j2}U_{k2}e^{2i\phi_{2}}$, and $R=U_{j3}U_{k3}e^{2i\phi_{3}}$. Here, the
indices $j$ and $k$ can take values $1$, $2$ and $3$. This complex equation
can be used to calculate two mass ratios $\frac{m_{1}}{m_{2}}$ and
$\frac{m_{1}}{m_{3}}$ :

$\frac{m_{1}}{m_{2}}=\frac{\textrm{Re}(R)\textrm{Im}(Q)-\textrm{Re}(Q)\textrm{Im}(R)}{\textrm{Re}(P)\textrm{Im}(R)-\textrm{Re}(R)\textrm{Im}(P)}$
(75)

and

$\frac{m_{1}}{m_{3}}=\frac{\textrm{Re}(R)\textrm{Im}(Q)-\textrm{Re}(Q)\textrm{Im}(R)}{\textrm{Re}(Q)\textrm{Im}(P)-\textrm{Re}(P)\textrm{Im}(Q)}$
(76)

where the symbols Re and Im denote the real and imaginary parts, respectively.

We use the above mass ratios to obtain the expression for the parameter
$R_{\nu}$ defined as

$R_{\nu}\equiv\frac{\Delta m_{21}^{2}}{|\Delta
m_{31}^{2}|}=\frac{(\frac{m_{2}}{m_{1}})^{2}-1}{|(\frac{m_{3}}{m_{1}})^{2}-1|}.$
(77)

For a neutrino mass matrix with one texture zero and X1 CP symmetry to be
compatible with the present neutrino oscillation data, the values of $R_{\nu}$
obtained from Eq. (77) should lie within its experimentally allowed range.
Similarly, one can obtain mass ratios and $R_{\nu}$ for neutrino mass matrices
with one texture zero and X2 CP symmetry and analyze their predictions.

Parameter | Normal Ordering | Inverted Ordering
---|---|---
| best fit $\pm 1\sigma$ $3\sigma$ range | best fit $\pm 1\sigma$ $3\sigma$ range
$\theta_{12}^{\circ}$ |  $34.3^{+1.0}_{-1.0}$ $31.4$ \- $37.4$ |  $34.3^{+1.0}_{-1.0}$ $31.4$ \- $37.4$
$\theta_{23}^{\circ}$ |  $49.26^{+0.79}_{-0.79}$ $41.20$ \- $51.33$ |  $49.46^{+0.60}_{-0.97}$ $41.16$ \- $51.25$
$\theta_{13}^{\circ}$ |  $8.53^{+0.13}_{-0.12}$ $8.13$ \- $8.92$ |  $8.58^{+0.12}_{-0.14}$ $8.17$ \- $8.96$
$\delta_{CP}^{\circ}$ |  $194^{+24}_{-22}$ $128$ \- $359$ |  $284^{+26}_{-28}$ $200$ \- $353$
$\Delta m^{2}_{21}/10^{-5}eV^{2}$ |  $7.50^{+0.22}_{-0.20}$ $6.94$ \- $8.14$ |  $7.50^{+0.22}_{-0.20}$ $6.94$ \- $8.14$
$|\Delta m^{2}_{31}|/10^{-3}eV^{2}$ |  $2.55^{+0.02}_{-0.03}$ $2.47$ \- $2.63$ |  $2.45^{+0.02}_{-0.03}$ $2.37$ \- $2.53$
Table 2: Current neutrino oscillation parameters from global fits [88].

##  3 Analysis, Results, and Discussion

We have 7 patterns of two texture zeros with X1 generalized CP symmetry, 7
patterns of two texture zeros with X2 CP symmetry, 6 patterns of one texture
zeros with X1 CP symmetry, and 6 patterns of one texture zeros with X2 CP
symmetry. Each of these patterns can be studied for NH as well as IH.
Therefore, we have 14 patterns for two texture zeros with one generalized CP
symmetry and 12 patterns for one texture zero with one generalized CP
symmetry. Therefore, we have 26 patterns for NH and 26 patterns for IH.

###  3.1 Two texture zeros with one generalized CP symmetry

When we impose the generalized CP symmetry on the two texture zero patterns of
class C ( $M^{C}_{X_{1}}$ and $M^{C}_{X_{2}}$), the resulting neutrino mass
matrix becomes $\mu-\tau$ symmetric or invariant under the flavor symmetry
$G_{3}$. So, the patterns $M^{C}_{X_{1}}$ and $M^{C}_{X_{2}}$ predict
vanishing $\theta_{13}$ and are disallowed. We study the neutrino mass
matrices of class $A$ and $B$ with one generalized CP symmetry using the
procedure derived in the previous section. When we calculate the predictions
of these patterns for the mixing angles and mass squared differences, we find
that they all are disallowed except for patterns $M^{A_{1}}_{X_{2}}$ and
$M^{A_{2}}_{X_{2}}$. In summary, all the seven cases of two texture zeros with
X1 CP symmetry [Eqs. (22)-(28)] are disallowed by the current data for both
the hierarchies. Out of the seven patterns of neutrino mass matrix with X2 CP
symmetry [Eqs. (29)-(35)], only patterns A1 and A2 with NH are allowed.

The allowed parameter space for these patterns ( $M^{A_{1}}_{X_{2}}$ and
$M^{A_{2}}_{X_{2}}$) is very restricted. Since, the mass matrix is real and
can be diagonalized with TM2 mixing, the mixing angle $\theta_{23}$ takes its
extreme value in the higher octant while the CP-violating phase $\delta=0$.
The Majorana phases can have values $0$, $\frac{\pi}{2}$, and $\pi$. The
correlation plots for various parameters of these patterns are shown in Fig.
1. We observe that they have exactly identical predictions for $\theta_{12}$
and $\theta_{23}$.

###  3.2 One texture zero with one generalized CP symmetry

There are 12 patterns of neutrino mass matrix with one texture zero and one
generalized CP symmetry (6 possibilities for texture zeros and 2 for
generalized CP symmetry) as described in the last section. All these patterns
will have distinct predictions for neutrino masses, mixing angles, and CP
violation.

The neutrino mixing matrix $U$ for any of these patterns depends upon the free
parameters which are three angles ($\theta_{1}$, $\theta_{2}$, and
$\theta_{3}$) defining the orthogonal matrix $O_{\nu}$ and the choices of the
integers ($k_{1}$, $k_{2}$, and $k_{3}$) defining the matrix $Q_{\nu}$. For
each pattern of the neutrino mass matrix, we can calculate the parameter
$R_{\nu}$, the three neutrino mixing angles, and the three CP-violating phases
as described in the last section.

The angles $\theta_{1}$, $\theta_{2}$, and $\theta_{3}$ defining the
generalized CP transformations X1 and X2 are the three free parameters in our
numerical analysis of the neutrino mass matrices with one texture zero and a
generalized CP symmetry. We vary these three angles uniformly in the interval
$[-\frac{\pi}{2},\frac{\pi}{2}]$. We consider all the signs resulting from the
choices of the integers $k_{1}$, $k_{2}$, and $k_{3}$. Then, we calculate the
values of $R_{\nu}$ for these patterns of the neutrino mass matrix using the
relations derived in the last section. Next, we obtain the allowed parameter
space for the angles ($\theta_{1}$, $\theta_{2}$, and $\theta_{3}$) for which
$R_{\nu}$ lies between its experimentally allowed range at 3 standard
deviations. Finally, we calculate the predictions for the three mixing angles
and CP-violating phases as explained in Section 2.1. By varying the smallest
neutrino mass $m_{\circ}$ ($m_{1}$ for NH and $m_{3}$ for IH) and the two
mass-squared differences ($\Delta m_{12}^{2}$ and $\Delta m_{13}^{2}$) in
their experimentally allowed ranges at three standard deviations (Table 2), we
also calculate the predictions for the effective neutrino mass ($|M_{ee}|$)
measurable in neutrinoless beta decay experiments and the sum of neutrino
masses ($\Sigma_{m}$) observable in the cosmological experiments. The results
of this numerical analysis are presented below for one texture zeros with one
generalized CP symmetry.

All the 12 patterns with one texture zero and one CP symmetry are allowed for
NH. However, one of the patterns (texture zero at $M_{11}$) is ruled out for
inverted hierarchy because this condition predicts a too large value of
$\theta_{13}$ for the IH case. The presence of CP symmetries in the mass
matrix doesn’t change this property of $M_{11}=0$ already known in the
previous studies [89, 90, 91]. So, only 10 patterns with one texture zero and
one CP symmetry are allowed for IH. The Figures 2, 3, 4, and 5 depict the
details of the allowed parameter space for all these allowed patterns.

We find that the allowed values for angles $\theta_{1}$, $\theta_{2}$, and
$\theta_{3}$ lie within a narrow range of approximately
$[-15^{\circ},15^{\circ}]$. In certain patterns, we also found a sharp
correlation between $\theta_{1}$ and $\theta_{3}$. The allowed parameter space
depends upon the choice of CP symmetry, hierarchy, and position of zero in the
neutrino mass matrix.

A common feature of various modifications of TBM mixing like TM1 and TM2 is
that they predict a neat correlation between $\theta_{23}$ and $\delta$ and as
$\theta_{23}$ tends to maximal mixing, CP violation also tends to be maximal
($\delta$ tends to $\frac{\pi}{2}$ or $\frac{3\pi}{2}$). We obtain similar
correlations between $\theta_{23}$ and $\delta$ for CP symmetries X1 and X2
when we impose the texture zeros. The correlations are distinct for each
pattern and would act as one of the distinguishing features of these patterns
as experiments improve their precision on these parameters.

An interesting prediction of these patterns that distinguishes them from the
patterns derived from TM1 and TM2 mixing is a correlation between
$\theta_{12}$ and $\theta_{23}$ (and hence with $\delta$). Clearly, the
neutrino mass matrices with generalized CP symmetries and texture zeros are
highly testable and will face a close scrutiny from the precision
measurements. These predictions for correlations between $\theta_{12}$,
$\theta_{23}$, and $\delta$ are depicted in Figures 6, 7, 8, and 9.

Like all neutrino mass matrices with texture zeros, with or without additional
symmetries, there exists a complementarity between predictions for
$\theta_{23}$ and $\delta$ among certain patterns. The pattern $M_{12}=0$ is
complementary to $M_{13}=0$ and the pattern $M_{22}=0$ is complementary to
$M_{33}=0$. This complementarity arises because the expression of $M_{12}$
($M_{22}$) in terms of masses, mixing angles, and phases transforms to
$M_{13}$ ($M_{33}$) under exchange

$\theta_{23}\rightarrow\frac{\pi}{2}-\theta_{23}~{}\text{and}~{}\delta\rightarrow\pi+\delta.$

Hence, the predictions for $\theta_{23}$ for the case $M_{13}=0$ ($M_{33}=0$)
is a mirror reflection of the prediction for $M_{12}=0$ ($M_{22}=0$) around
$\frac{\pi}{4}$. Also, the prediction for $\delta$ for the case $M_{13}=0$
($M_{33}=0$) is a translation of the prediction for $M_{12}=0$ ($M_{22}=0$) by
a constant angle $\pi$. However, the allowed region for the
$\theta_{23}-\delta$ parameter space for the recent experimental results is
not symmetric under above transformation. In the experimental data, the values
of $\theta_{23}$ higher than $\frac{\pi}{4}$ are slightly preferred than the
values less than $\frac{\pi}{4}$ and large portion of the values of $\delta$
in the interval $[0,\pi]$ are ruled out in a statistically significant manner.
This asymmetry in the data makes our predictions for these parameters
asymmetric as well and the complementarity pointed out above will not be
evident in the allowed parameter space after imposing the experimental
constraints on $\theta_{23}$ and $\delta$. However, it is still useful in
understanding many of the predictions for $\theta_{23}$ and $\delta$. One
interesting example is the prediction that $\theta_{23}$ is allowed to be
below or above maximal mixing for certain patterns. For patterns $M_{12}=0$
and $M_{22}=0$ with NH for X2 CP symmetry, $\theta_{23}<\frac{\pi}{4}$ and for
patterns $M_{13}=0$ and $M_{33}=0$ with NH for X2 CP symmetry,
$\theta_{23}>\frac{\pi}{4}$.

The correlations between the CP-violating phases $\delta$, $\phi_{12}$, and
$\phi_{13}$ are almost linear with reflection symmetry around the
$\phi_{12}=\frac{\pi}{2}$ or $\phi_{13}=\frac{\pi}{2}$ lines. In certain
cases, the phases $\phi_{12}$ and $\phi_{13}$ take very narrow ranges. The CP-
violating phase $\delta$ can act as a discriminator among these various cases.
These predictions have been depicted in Figures 10, 11, 12, and 13.

Cosmological observations have put an upper bound on the sum of neutrino
masses ($\Sigma_{m}=m_{1}+m_{2}+m_{3}$). Data from Planck satellite combined
with baryon acoustic oscillations (BAO) measurements limit the sum of neutrino
masses $\Sigma_{m}\leq 0.12$ eV at 95$\%$ Confidence Level (CL) [92]. However,
these limits depend upon certain model specific assumptions. In the present
work, we assume a more conservative limit of $\Sigma_{m}\leq 1$ eV.

For all of the allowed patterns, we calculate the effective neutrino mass for
neutrinoless double beta decay ($|M_{ee}|$) (except for the texture zero at
the element $M_{11}$) and the sum of neutrino masses ($\Sigma_{m}$). The
values of $|M_{ee}|$, $\Sigma_{m}$, and the lowest mass $m_{\circ}$ ($m_{1}$
for NH and $m_{3}$ for IH) for all the allowed cases have been summarized in
Tables 3 and 4. For the patterns $M_{\mu\mu}=0$, $M_{\mu\tau}=0$, and
$M_{\tau\tau}=0$, the predicted values of $\Sigma_{m}$ are always larger than
0.12 eV for all choices of CP symmetries (X1 and X2) and hierarchies (NH and
IH) and tend to violate the Plank data. These patterns will be completely
ruled out as the cosmological data becomes more restrictive on $\Sigma_{m}$ in
future. We also note that there are three patterns of neutrino mass matrix
with IH for which the lowest neutrino mass $m_{3}$ can be zero.

##  4 Conclusions

In this work, we have studied the properties of neutrino mass matrices with
texture zeros and CP symmetries. We find that only two cases of two texture
zeros are allowed in presence of the X1 or X2 CP symmetries. The neutrino mass
matrices in those two patterns are real and predict extreme values of
$\theta_{23}$. For neutrino mass matrices with one texture zero in presence of
one generalized CP symmetry, all the cases except $M_{11}=0$ for inverted
hierarchy are allowed. We find that the angles $\theta_{1}$, $\theta_{2}$, and
$\theta_{3}$ fall within a narrow range of approximately
$[-15^{\circ},15^{\circ}]$ and they are correlated among themselves. We also
observe correlations between $\theta_{12}$, $\theta_{23}$, and $\delta$ for CP
symmetries X1 and X2, which are distinct for each texture and could be used as
their distinguishing features. We note that the complementarity between
predictions for $\theta_{23}$ and $\delta$ for certain pairs of texture zeros
does not hold exactly due to recent experimental results. We also calculate
the effective neutrino mass for neutrinoless double beta decay and the sum of
the neutrino masses for all the relevant cases. Certain patterns of one
texture zero and one CP symmetry are currently disfavored by the cosmological
data and might be completely ruled out as the cosmological data becomes more
restrictive in the future. Our results demonstrate the predictability and
testability of neutrino mass matrices with one CP symmetry and texture zeros.
Their predictions for correlations between mixing angles and CP-violating
phases can be used to distinguish these textures from one another as well as
from the other texture zero schemes proposed in the literature.

|  | Normal Ordering |  |  | Inverted Ordering |
---|---|---|---|---|---|---
|  $|M_{ee}|$ (eV) |  $m_{\circ}$ (eV) |  $\Sigma_{m}$ (eV) |  $|M_{ee}|$ (eV) |  $m_{\circ}$ (eV) |  $\Sigma_{m}$ (eV)
$M_{ee}=0$ | 0 | 0.003-0.004 | 0.058 - 0.077 | - | - | -
|  |  $\oplus$ 0.005- 0.008 |  |  |  |
$M_{e\mu}=0$ | 0.024 - 0.221 | 0.023 - 0.221 | 0.099 - 0.670 | 0.057 - 0.277 | 0 - 0.272 | 0.099 - 0.830
$M_{e\tau}=0$ | 0.0233 - 0.174 | 0.023 - 0.017 | 0.098 - 0.525 | 0.049 - 0.313 | 0.009 - 0.310 | 0.109 - 0.933
$M_{\mu\mu}=0$ | 0.054 - 0.310 | 0.054 - 0.310 | 0.181 - 0.933 | 0.063 - 0.331 | 0.041 - 0.330 | 0.172 - 1.000
$M_{\mu\tau}=0$ | 0.139 - 0.209 | 0.145 - 0.215 | 0.445 - 0.652 | 0.016 - 0.017 | 0.017 - 0.018 | 0.120 - 0.125
$M_{\tau\tau}=0$ | 0.037 - 0.284 | 0.037 - 0.283 | 0.135 - 0.860 | 0.076 - 0.320 | 0.058 - 0.316 | 0.210 - 0.957
Table 3: Allowed ranges of $|M_{ee}|$, $m_{\circ}$, $\Sigma_{m}$ for X1 CP symmetry. |  | Normal Ordering |  |  | Inverted Ordering |
---|---|---|---|---|---|---
|  $|M_{ee}|$ (eV) |  $m_{\circ}$ (eV) |  $\Sigma_{m}$ (eV) |  $|M_{ee}|$ (eV) |  $m_{\circ}$ (eV) |  $\Sigma_{m}$ (eV)
$M_{ee}=0$ | 0 | 0.0021 - 0.0085 | 0.058 - 0.077 | - | - | -
$M_{e\mu}=0$ | 0.0010 - 0.0014 | 0.0067 - 0.0100 | 0.060 - 0.085 | 0.047 - 0.317 | 0 - 0.313 | 0.098 - 0.954
|  $\oplus$ 0.018 - 0.312 |  $\oplus$ 0.018 - 0.300 |  $\oplus$ 0.095 - 0.940 |  |  |
$M_{e\tau}=0$ | 0.0001 - 0.0018 | 0.0026 - 0.0046 | 0.033 - 0.074 | 0.047 - 0.314 | 0 - 0.311 | 0.098 - 0.940
|  $\oplus$ 0.019 - 0.200 |  $\oplus$ 0.018 - 0.200 |  $\oplus$ 0.091 - 0.600 |  |  |
$M_{\mu\mu}=0$ | 0.05 - 0.32 | 0.05 - 0.32 | 0.17 - 0.96 | 0.0172 - 0.0179 | 0.023 - 0.025 | 0.132 - 0.138
|  |  |  |  $\oplus$ 0.066 - 0.270 |  $\oplus$ 0.047 - 0.265 |  $\oplus$ 0.18 - 0.81
$M_{\mu\tau}=0$ | 0.137 - 0.328 | 0.144 - 0.333 | 0.441-1 | 0.0158 - 0.0166 | 0.0193 - 0.0204 | 0.124 - 0.133
|  |  |  |  $\oplus$ 0.0187 - 0.0198 |  $\oplus$ 0.0211 - 0.0224 |
$M_{\tau\tau}=0$ | 0.033-0.324 | 0.034-0.325 | 0.128 - 0.980 | 0.015180 - 0.01593 | 0.00685 - 0.00815 | 0.1056 - 0.1110
|  |  |  |  $\oplus$ 0.076 - 0.243 |  $\oplus$ 0.060 - 0.238 |  $\oplus$ 0.214 - 0.730
Table 4: Allowed ranges of $|M_{ee}|$, $m_{\circ}$, $\Sigma_{m}$ for X2 CP
symmetry.

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(a) $\theta_{12}$ and $\theta_{23}$ correlation for the class A1 with X2 CP
symmetry

(b) $\theta_{12}$ and $\theta_{23}$ correlation for the class A2 with X2 CP
symmetry

(c) $\Sigma_{m}$ and $\theta_{23}$ correlation for the class A1 with X2 CP
symmetry

(d) $\Sigma_{m}$ and $\theta_{23}$ correlation for the class A2 with X2 CP
symmetry

Figure 1: Correlation plots for the classes A1 and A2 with X2 CP symmetry. All
the mixing angles are in degrees and $\Sigma_{m}$ is in eV.

(a) The texture $M_{11}=0$ for X1 CP symmetry in NH

(b) The texture $M_{12}=0$ for X1 CP symmetry in NH

(c) The texture $M_{13}=0$ for X1 CP symmetry in NH

(d) The texture $M_{22}=0$ for X1 CP symmetry in NH

(e) The texture $M_{23}=0$ for X1 CP symmetry in NH

(f) The texture $M_{33}=0$ for X1 CP symmetry in NH

Figure 2: Allowed values of the free parameters ($\theta_{1}$, $\theta_{2}$,
and $\theta_{3}$) for one texture zeros with X1 CP symmetry in NH. All angles
are in degrees.

(a) The texture $M_{12}=0$ for X1 CP symmetry in IH

(b) The texture $M_{13}=0$ for X1 CP symmetry in IH

(c) The texture $M_{22}=0$ for X1 CP symmetry in IH

(d) The texture $M_{23}=0$ for X1 CP symmetry in IH

(e) The texture $M_{33}=0$ for X1 CP symmetry in IH

Figure 3: Allowed values of the free parameters ($\theta_{1}$, $\theta_{2}$,
and $\theta_{3}$) for one texture zeros with X1 CP symmetry in IH. All angles
are in degrees.

(a) $M_{11}=0$ for X2 CP symmetry in NH

(b) $M_{12}=0$ for X2 CP symmetry in NH

(c) $M_{13}=0$ for X2 CP symmetry in NH

(d) $M_{22}=0$ for X2 CP symmetry in NH

(e) $M_{23}=0$ for X2 CP symmetry in NH

(f) $M_{33}=0$ for X2 CP symmetry in NH

Figure 4: Allowed values of the free parameters ($\theta_{1}$, $\theta_{2}$,
and $\theta_{3}$) for one texture zeros with X2 CP symmetry in NH. All angles
are in degrees.

(a) $M_{12}=0$ for X2 CP symmetry in IH

(b) $M_{13}=0$ for X2 CP symmetry in IH

(c) $M_{22}=0$ for X2 CP symmetry in IH

(d) $M_{23}=0$ for X2 CP symmetry in IH

(e) $M_{33}=0$ for X2 CP symmetry in IH

Figure 5: Allowed values of the free parameters ($\theta_{1}$, $\theta_{2}$,
and $\theta_{3}$) for one texture zeros with X2 CP symmetry in IH. All angles
are in degrees.

(a) $M_{11}=0$ for X1 CP symmetry in NH

(b) $M_{12}=0$ for X1 CP symmetry in NH

(c) $M_{13}=0$ for X1 CP symmetry in NH

(d) $M_{22}=0$ for X1 CP symmetry in NH

(e) $M_{23}=0$ for X1 CP symmetry in NH

(f) $M_{33}=0$ for X1 CP symmetry in NH

Figure 6: Predicted values of the mixing angles ($\theta_{12}$, $\theta_{23}$,
and $\theta_{13}$) and CP-violating phase $\delta$ for one texture zeros with
X1 CP symmetry in NH. All angles and phases are in degrees.

(a) $M_{12}=0$ for X1 CP symmetry in IH

(b) $M_{13}=0$ for X1 CP symmetry in IH

(c) $M_{22}=0$ for X1 CP symmetry in IH

(d) $M_{23}=0$ for X1 CP symmetry in IH

(e) $M_{33}=0$ for X1 CP symmetry in IH

Figure 7: Predicted values of the mixing angles ($\theta_{12}$, $\theta_{23}$,
and $\theta_{13}$) and CP-violating phase $\delta$ for one texture zeros with
X1 CP symmetry in IH. All angles and phases are in degrees.

(a) $M_{11}=0$ for X2 CP symmetry in NH

(b) $M_{12}=0$ for X2 CP symmetry in NH

(c) $M_{13}=0$ for X2 CP symmetry in NH

(d) $M_{22}=0$ for X2 CP symmetry in NH

(e) $M_{23}=0$ for X2 CP symmetry in NH

(f) $M_{33}=0$ for X2 CP symmetry in NH

Figure 8: Predicted values of the mixing angles ($\theta_{12}$, $\theta_{23}$,
and $\theta_{13}$) and CP-violating phase $\delta$ for one texture zeros with
X2 CP symmetry in NH. All angles and phases are in degrees.

(a) $M_{12}=0$ for X2 CP symmetry in IH

(b) $M_{13}=0$ for X2 CP symmetry in IH

(c) $M_{22}=0$ for X2 CP symmetry in IH

(d) $M_{23}=0$ for X2 CP symmetry in IH

(e) $M_{33}=0$ for X2 CP symmetry in IH

Figure 9: Predicted values of the mixing angles ($\theta_{12}$, $\theta_{23}$,
and $\theta_{13}$) and CP-violating phase $\delta$ for one texture zeros with
X2 CP symmetry in IH. All angles and phases are in degrees.

(a) $M_{11}=0$ for X1 CP symmetry in NH

(b) $M_{12}=0$ for X1 CP symmetry in NH

(c) $M_{13}=0$ for X1 CP symmetry in NH

(d) $M_{22}=0$ for X1 CP symmetry in NH

(e) $M_{23}=0$ for X1 CP symmetry in NH

(f) $M_{33}=0$ for X1 CP symmetry in NH

Figure 10: Predicted values of the CP-violating phases ($\phi_{12}$,
$\phi_{13}$, and $\delta$) for one texture zeros with X1 CP symmetry in NH.
All phases are in degrees.

(a) $M_{12}=0$ for X1 CP symmetry in IH

(b) $M_{13}=0$ for X1 CP symmetry in IH

(c) $M_{22}=0$ for X1 CP symmetry in IH

(d) $M_{23}=0$ for X1 CP symmetry in IH

(e) $M_{33}=0$ for X1 CP symmetry in IH

Figure 11: Predicted values of the CP-violating phases ($\phi_{12}$,
$\phi_{13}$, and $\delta$) for one texture zeros with X1 CP symmetry in IH.
All phases are in degrees.

(a) $M_{11}=0$ for X2 CP symmetry in NH

(b) $M_{12}=0$ for X2 CP symmetry in NH

(c) $M_{13}=0$ for X2 CP symmetry in NH

(d) $M_{22}=0$ for X2 CP symmetry in NH

(e) $M_{23}=0$ for X2 CP symmetry in NH

(f) $M_{33}=0$ for X2 CP symmetry in NH

Figure 12: Predicted values of the CP-violating phases ($\phi_{12}$,
$\phi_{13}$, and $\delta$) for one texture zeros with X2 CP symmetry in NH.
All phases are in degrees.

(a) $M_{12}=0$ for X2 CP symmetry in IH

(b) $M_{13}=0$ for X2 CP symmetry in IH

(c) $M_{22}=0$ for X2 CP symmetry in IH

(d) $M_{23}=0$ for X2 CP symmetry in IH

(e) $M_{33}=0$ for X2 CP symmetry in IH

Figure 13: Predicted values of the CP-violating phases ($\phi_{12}$,
$\phi_{13}$, and $\delta$) for one texture zero with X2 CP symmetry in IH. All
phases are in degrees.