# Parallel Lepton Mass Matrices with Texture/Cofactor Zeros

Weijian Wang  Department of Physics, North China Electric Power University,
Baoding 071003, P. R. China<EMAIL_ADDRESS>
###### Abstract

In this paper we investigate the parallel texture structures containing
texture zeros in charged lepton mass matrix $M_{l}$ and cofactor zeros in
neutrino mass matrix $M_{\nu}$. These textures are interesting since they are
related to the $Z_{n}$ flavor symmetries. Using the weak basis permutation
transformation, the 15 parallel textures are grouped as 4 classes (class
I,II,III and IV) with the matrices in each class sharing the same physical
implications. Under the current experimental data, the class I, III with
inverted mass hierarchy and class II with normal mass hierarchy are
phenomenologically acceptable. The correlations between some important
physical variables are presented, which are essential for the model selection
and can be text by future experiments. The model realization is illustrated by
means of $Z_{4}\times Z_{2}$ flavor symmetry.

PACS: 14.60.Pq, 12.15.Ff, 11.30.Hv

##  I Introduction

The discovery of neutrino oscillations have provided us with convincing
evidences for massive neutrinos and leptonic flavor mixing with high degree of
accuracyneu1 (1, 2, 3). The understanding of the leptonic flavor structure is
one of the major open questions in particle physics. Several attempts have
been proposed to explain the origin of neutrino mass and the observed pattern
of leptonic mixing by introducing the flavor symmetries within the framework
of seesaw modelsseesaw (4). The flavor symmetry often reduces the number of
free parameters and leads to the specific structures of fermion mass matrices
including texture zeroszero (5, 6, 7, 8, 9), hybrid textureshybrid (10, 11),
zero tracesum (12), zero determinantdet (13), vanishing minorsminor1 (14, 15,
16), two traceless submatricestra (17), equal elements or cofactorsco (18),
inverse hybrid textureshyco (19). Among these models, the matrices with
texture or cofactor zeros are particularly interesting due to their
connections to the flavor symmetries. The phenomenological examination of
texture zeros or cofactor zeros in flavor basis have been widely studies in
Ref.zero (5, 6, 14, 15, 16) where the charged lepton mass matrices $M_{l}$ are
diagonal. However, no universal principle is required that the flavor basis is
necessary and the more general cases should be considered in no diagonal
$M_{l}$ basis. In this scenario, the lepton mass matrices with texture zeros
in both lepton mass $M_{l}$ and neutrino mass matrix $M_{\nu}$ have been
systematically investigated by many authorsGCB (7, 8)(for a review, see z1
(9)).

In this paper, we propose the new possible texture structures where there are
two texture zeros in $M_{l}$ and two cofactor zeros in $M_{\nu}$ (We denote
them the matrices with texture/cofactor zeros). It seems that such mass
matrices are rather unusual because one instinctively expects the type of
texture structures to be the same for both $M_{l}$ and $M_{\nu}$. However, one
reminds the type-I seesaw model as $M_{\nu}=-M_{D}M_{R}^{-1}M_{D}^{T}$. Then
the texture or cofactor zeros of $M_{\nu}$ can be attributed to the texture
zeros in $M_{D}$ and $M_{R}$. Generally, this can be realized by $Z_{n}$
flavor symmetryzn (20, 14). Therefore from the point of flavor symmetry, both
texture zeros and cofactor zeros structures manifest the same flavor symmetry
in different ways. It is our main motivation to carry out this work and a
concrete model will be constructed in the following section. Furthermore, we
take the so-called the parallel $Ans\ddot{a}tze$ that the positions of texture
zeros in $M_{l}$ are chosen to be the same as the cofactor zeros in $M_{\nu}$.
Although there is no priori reason requiring the parallel structures, they are
usually regarded in many literatures as an esthetical appeal and the precursor
of the more general cases. The lepton mass matrices with parallel texture zero
structures have been systematically investigated in Ref.GCB (7). Subsequently,
the idea is generalized to more complicated situations such as parallel hybrid
textureshyp (21), parallel cofactor zero texturesmyco (22). In our case, there
exists $C^{2}_{6}=15$ logically possible patterns for two texture/cofactor
zeros in mass matrices. It is indicated that the 15 textures can be grouped
into 4 classes with the matrices in each class connected by $S_{3}$
permutation transformation and sharing the same physical implications. Among
the 4 classes, one of them is not viable phenomenologically. Therefore we
focus on the other three nontrivial classes.

The paper is organized as follow. In Sec. II, we present the classification of
mass matrices and relate them to the current experimental results. In Sec.
III, we diagonalize the mass matrices, confront the numerical results with the
experimental data and discuss their predictions. In Sec. IV, the model
realization is given under the $Z_{4}\times Z_{2}$ flavor symmetry. We
summarize the results in Sec. V.

##  II Formalism

###  II.1 Weak basis equivalent classes

As shown in Ref.GCB (7), there exists the general weak basis (WB)
transformations leaving gauge currents invariant i.e

$M_{l}\rightarrow M_{l}^{\prime}=W^{\dagger}M_{l}W_{R}\quad\quad\quad
M_{\nu}\rightarrow M_{\nu}^{\prime}=W^{T}M_{\nu}W$ (1)

where the neutrinos are assumed to be Majorana fermions and $W$, $W_{R}$ are
$3\times 3$ unitary matrices. Two matrices related by WB transformations have
the same physical implications. Therefore the parallel matrices with
texture/cofactor zeros located at different positions can be connected by
$S_{3}$ permutation matrix $P$ as a specific WB transformation

$M_{l}^{\prime}=P^{T}M_{l}P\quad\quad\quad M_{\nu}^{\prime}=P^{T}M_{\nu}P$ (2)

It is noted that $P$ changes the positions of cofactor zero elements but still
preserves the parallel structures for both charged lepton and neutrino mass
textures. Then the texture/cofactor zeros matrices are classified into 4
classes:

Class I:

$\begin{split}\left(\begin{array}[]{ccc}0/\bigtriangleup&\times&0/\bigtriangleup\\\
\times&\times&\times\\\
0/\bigtriangleup&\times&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}0/\bigtriangleup&0/\bigtriangleup&\times\\\
0/\bigtriangleup&\times&\times\\\
\times&\times&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&0/\bigtriangleup&\times\\\
0/\bigtriangleup&0/\bigtriangleup&\times\\\
\times&\times&\times\end{array}\right)\\\
\left(\begin{array}[]{ccc}\times&\times&\times\\\
\times&0/\bigtriangleup&0/\bigtriangleup\\\
\times&0/\bigtriangleup&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&\times&0/\bigtriangleup\\\
\times&\times&\times\\\
0/\bigtriangleup&\times&0/\bigtriangleup\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&\times&\times\\\
\times&\times&0/\bigtriangleup\\\
\times&0/\bigtriangleup&0/\bigtriangleup\end{array}\right)\end{split}$ (3)

Class II:

$\begin{split}\left(\begin{array}[]{ccc}0/\bigtriangleup&\times&\times\\\
\times&\times&0/\bigtriangleup\\\
\times&0/\bigtriangleup&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&\times&0/\bigtriangleup\\\
\times&0/\bigtriangleup&\times\\\
0/\bigtriangleup&\times&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&0/\bigtriangleup&\times\\\
0/\bigtriangleup&\times&\times\\\
\times&\times&0/\bigtriangleup\end{array}\right)\end{split}$ (4)

Class III:

$\begin{split}\left(\begin{array}[]{ccc}0/\bigtriangleup&\times&\times\\\
\times&0/\bigtriangleup&\times\\\
\times&\times&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}0/\bigtriangleup&\times&\times\\\
\times&\times&\times\\\
\times&\times&0/\bigtriangleup\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&\times&\times\\\
\times&0/\bigtriangleup&\times\\\
\times&\times&0/\bigtriangleup\end{array}\right)\end{split}$ (5)

Class IV:

$\begin{split}\left(\begin{array}[]{ccc}\times&0/\bigtriangleup&0/\bigtriangleup\\\
0/\bigtriangleup&\times&\times\\\
0/\bigtriangleup&\times&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&0/\bigtriangleup&\times\\\
0/\bigtriangleup&\times&0/\bigtriangleup\\\
\times&0/\bigtriangleup&\times\end{array}\right)\quad\quad\left(\begin{array}[]{ccc}\times&\times&0/\bigtriangleup\\\
\times&\times&0/\bigtriangleup\\\
0/\bigtriangleup&0/\bigtriangleup&\times\end{array}\right)\end{split}$ (6)

where ”$0/\bigtriangleup$” at $(i,j)$ position represents the texture zero
condition $M_{ij}=0$ and the cofactor zero condition $C_{ij}=0$; The
”$\times$” denotes arbitrary element. One can check that the matrices with
cofactor zeros in class I are equivalent to the texture zero ones. Choosing
the first matrix of class I as an example, we have

$\begin{split}M_{\nu}=\left(\begin{array}[]{ccc}\bigtriangleup&\times&\bigtriangleup\\\
\times&\times&\times\\\
\bigtriangleup&\times&\times\end{array}\right)\Rightarrow
M_{\nu}^{-1}=\left(\begin{array}[]{ccc}0&\times&0\\\ \times&\times&\times\\\
0&\times&\times\end{array}\right)\Rightarrow
M_{\nu}=\left(\begin{array}[]{ccc}\times&\times&\times\\\ \times&0&0\\\
\times&0&\times\end{array}\right)\end{split}$ (7)

Thus the parallel texture structures of class I are equivalent to the no-
parallel structures with two texture zeros. Although the parallel texture zero
structures has been explored extensivelyGCB (7, 8, 9), the analysis of the no-
parallel two texture zero structure has not yet been reported. On the other
hand, as having been pointed out in Ref.GCB (7, 22), the class IV leads to the
decoupling of a generation of lepton from mixing and thus not experimentally
viable.

###  II.2 Useful notations

As we have mentioned, among the 4 classes only class I, II and III are
nontrivial. We represent them as

$\begin{split}M_{l/\nu}^{I}=\left(\begin{array}[]{ccc}0/\bigtriangleup&\times&0\bigtriangleup\\\
\times&\times&\times\\\
0\bigtriangleup&\times&\times\end{array}\right)\quad\quad
M_{l/\nu}^{II}=\left(\begin{array}[]{ccc}0/\bigtriangleup&\times&\times\\\
\times&\times&0/\bigtriangleup\\\
\times&0/\bigtriangleup&\times\end{array}\right)\quad\quad
M_{l/\nu}^{III}=\left(\begin{array}[]{ccc}0/\bigtriangleup&\times&\times\\\
\times&0/\bigtriangleup&\times\\\
\times&\times&\times\end{array}\right)\end{split}$ (8)

In the analysis, we consider $M_{l}$ is to be Hermitian and the Majorana
neutrino mass texture $M_{\nu}$ is complex and symmetric. The $M_{l}$ and
$M_{\nu}$ are diagonalized by unitary matrix $V_{l}$ and $V_{\nu}$

$M_{l}=V_{l}M_{l}^{D}V_{l}^{\dagger}\quad\quad
M_{\nu}=V_{\nu}M_{\nu}^{D}V_{\nu}^{T}$ (9)

where $M_{l}^{D}=Diag(m_{e},m_{\mu},m_{\tau})$,
$M_{\nu}^{D}=Diag(m_{1},m_{2},m_{3})$. The Pontecorvo-Maki-Nakagawa-Sakata
matrixPMNS (23) $U_{PMNS}$ is given by

$U_{PMNS}=V_{l}^{\dagger}V_{\nu}$ (10)

and parameterized as

$U_{PMNS}=UP_{\nu}=\left(\begin{array}[]{ccc}c_{12}c_{13}&c_{13}s_{12}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{13}s_{23}e^{i\delta}&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}&c_{13}s_{23}\\\
s_{23}s_{12}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-c_{23}s_{12}s_{13}e^{i\delta}&c_{13}c_{23}\end{array}\right)\left(\begin{array}[]{ccc}1&0&0\\\
0&e^{i\alpha}&0\\\ 0&0&e^{i(\beta+\delta)}\end{array}\right)$ (11)

where we use the abbreviation $s_{ij}=\sin\theta_{ij}$ and
$c_{ij}=\cos\theta_{ij}$. The ($\alpha$,$\beta$) in $P_{\nu}$ represents the
two Majorana CP-violating phases and $\delta$ denotes the Dirac CP-violating
phase. In order to facilitate our calculation, we treat the Hermitian matrix
$M_{l}$ factorisable. i.e

$M_{l}=K_{l}M_{l}^{r}K_{l}^{\dagger}$ (12)

where $K_{l}$ is the unitary phase matrix parameterized as
$K_{l}=diag(1,e^{i\phi_{1}},e^{i\phi_{2}})$. The $M_{l}^{r}$ becomes a real
symmetric matrix which can be diagonalized by real orthogonal matrix $O_{l}$.
Then we have

$V_{l}=K_{l}O_{l}$ (13)

and

$U_{PMNS}=O_{l}^{T}K_{l}^{\dagger}V_{\nu}$ (14)

From (9), (10) and (14), the neutrino mass matrix $M_{\nu}$ is given by

$M_{\nu}=K_{l}VP_{\nu}M_{\nu}^{D}P_{\nu}V^{T}K_{l}^{\dagger}$ (15)

where $V\equiv O_{l}U$. From (15) and solving the cofactor zero conditions of
$M_{\nu}$

$M_{\nu(pq)}M_{\nu(rs)}-M_{\nu(tu)}M_{\nu(vw)}=0\quad\quad
M_{\nu(p^{\prime}q^{\prime})}M_{\nu(r^{\prime}s^{\prime})}-M_{\nu(t^{\prime}u^{\prime})}M_{\nu(v^{\prime}w^{\prime})}=0$
(16)

we get

$\frac{m_{1}}{m_{2}}e^{-2i\alpha}=\frac{K_{3}L_{1}-K_{1}L_{3}}{K_{2}L_{3}-K_{3}L_{2}}$
(17)
$\frac{m_{1}}{m_{3}}e^{-2i\beta}=\frac{K_{2}L_{1}-K_{1}L_{2}}{K_{3}L_{2}-K_{2}L_{3}}e^{2i\delta}$
(18)

where

$K_{i}=(V_{pj}V_{qj}V_{rk}V_{sk}-V_{tj}V_{uj}V_{vk}V_{wk})+(j\leftrightarrow
k)$ (19)
$L_{i}=(V_{p^{\prime}j}V_{q^{\prime}j}V_{r^{\prime}k}V_{s^{\prime}k}-V_{t^{\prime}j}V_{u^{\prime}j}V_{v^{\prime}k}V_{w^{\prime}k})+(j\leftrightarrow
k)$ (20)

with $(i,j,k)$ a cyclic permutation of (1,2,3). With the help of Eq.(17) and
(18), the magnitudes of neutrino mass radios are given by

$\rho=\Big{|}\frac{m_{1}}{m_{3}}e^{-2i\beta}\Big{|}$ (21)
$\sigma=\Big{|}\frac{m_{1}}{m_{2}}e^{-2i\alpha}\Big{|}$ (22)

with the two Majorana CP-violating phases

$\alpha=-\frac{1}{2}arg\Big{(}\frac{K_{3}L_{1}-K_{1}L_{3}}{K_{2}L_{3}-K_{3}L_{2}}\Big{)}$
(23)
$\beta=-\frac{1}{2}arg\Big{(}\frac{K_{2}L_{1}-K_{1}L_{2}}{K_{3}L_{3}-K_{2}L_{3}}e^{2i\delta}\Big{)}$
(24)

The results of Eq. (21),(22), (23) and (24) imply that the two mass ratio
($\rho$ and $\sigma$) and two Majorana CP-violating phases ($\alpha$ and
$\beta$) are fully determined in terms of the real orthogonal matrix $O_{l}$,
$U$($\theta_{12},\theta_{23},\theta_{13}$ and $\delta$). The neutrino mass
ratios $\rho$ and $\sigma$ are related to the ratio of two neutrino mass-
squared differences defined as

$R_{\nu}\equiv\frac{\delta m^{2}}{\Delta
m^{2}}=\frac{2\rho^{2}(1-\sigma^{2})}{|2\sigma^{2}-\rho^{2}-\rho^{2}\sigma^{2}|}$
(25)

where $\delta m^{2}\equiv m_{2}^{2}-m_{1}^{2}$ and $\Delta m^{2}\equiv\mid
m_{3}^{2}-\frac{1}{2}(m_{1}^{2}+m_{2}^{2})\mid$. The three neutrino mass
eigenvalues $m_{1},m_{2}$ and $m_{3}$ are given by

$m_{2}=\sqrt{\frac{\delta m^{2}}{1-\sigma^{2}}}\quad\quad m_{1}=\sigma
m_{2}\quad\quad m_{3}=\frac{m_{1}}{\rho}$ (26)

In the following numerical analysis, we utilize the recent 3$\sigma$
confidential level global-fit data from the neutrino oscillation
experimentsdata (25).i.e

$\begin{split}\sin^{2}\theta_{12}/10^{-1}=3.08^{+0.51}_{-0.49}\quad\sin^{2}\theta_{23}/10^{-1}=4.25^{+2.16}_{-0.68}\quad\sin^{2}\theta_{13}/10^{-2}=2.34^{+0.63}_{-0.57}\\\
\delta m^{2}/10^{-5}=7.54^{+0.64}_{-0.55}eV^{2}\quad\quad\bigtriangleup
m^{2}/10^{-3}=2.44^{+0.22}_{-0.22}eV^{2}\end{split}$ (27)

for normal hierarchy (NH) and

$\begin{split}\sin^{2}\theta_{12}/10^{-1}=3.08^{+0.51}_{-0.49}\quad\sin^{2}\theta_{23}/10^{-1}=4.25^{+2.22}_{-0.74}\quad\sin^{2}\theta_{13}/10^{-2}=2.34^{+0.61}_{-0.61}\\\
\delta m^{2}/10^{-5}=7.54^{+0.64}_{-0.55}eV^{2}\quad\quad\bigtriangleup
m^{2}/10^{-3}=2.40^{+0.21}_{-0.23}eV^{2}\end{split}$ (28)

for inverted hierarchy(IH). By this time, no constraint is added on the Dirac
CP-violating phase $\delta$ at $3\sigma$ level, however the recent numerical
analysisdata (25) tends to give the best-fit value $\delta\approx 1.40\pi$. In
neutrino oscillation experiments, the CP violation effect is usually reflected
by the Jarlskog rephasing invariant quantityJas (26) defined as

$J_{CP}=s_{12}s_{23}s_{13}c_{12}c_{23}c_{13}^{2}\sin\delta$ (29)

The Majorana nature of neutrino can be determined if any signal of
neutrinoless double decay($0\nu\beta\beta$) is observed, implying the
violation of leptonic number violation. The decay ratio is related to the
effective Majorana neutrino mass $m_{ee}$, which is written as

$m_{ee}=|m_{1}c_{12}^{2}c_{13}^{2}+m_{2}s_{12}^{2}c_{13}^{2}e^{2i\alpha}+m_{3}s_{13}^{2}e^{2i\beta}|$
(30)

Although a $3\sigma$ result of $m_{ee}=(0.11-0.56)$ eV is reported by the
Heidelberg-Moscow CollaborationHM (27), this result is criticizedNND2 (28) and
shall be checked by the forthcoming experiment. It is believed that that the
next generation $0\nu\beta\beta$ experiments, with the sensitivity of $m_{ee}$
being up to 0.01 eVNDD (29), will open the window to not only the absolute
neutrino mass scale but also the Majorana-type CP violation. Besides the
$0\nu\beta\beta$ experiments, a more severe constraint was set from the recent
cosmology observation. Recently, an upper bound on the sum of neutrino mass
$\sum m_{i}<0.23$ eV is reported by Plank CollaborationPlanck (30) combined
with the WMAP, high-resolution CMB and BAO experiments.

##  III Numerical analysis

We have proposed a detailed numerical analysis for class I, II and III. In
this section we presented the main predictions of all the classes.

###  III.1 Class I

Let’s start from the factorisable formation of charged lepton matrix
$M_{l}^{r}$

$\begin{split}(M_{l}^{r})^{I}=\left(\begin{array}[]{ccc}0&a&0\\\ a&b&c\\\
0&c&d\end{array}\right)\end{split}$ (31)

As proposed in Ref.GCB (7, 22), the coefficients $a,b$ and $c$ are assumed to
be real and positive without losing generality. The real coefficient $d$ is
treated as a free parameter. Then the matrix (31) can be diagonalized by an
orthogonal matrix $O_{l}$

$O_{l}^{T}(M_{l}^{r})^{I}O_{l}=diag(m_{e},-m_{\mu},m_{\tau})$ (32)

where the minus sign in (32) is introduced to facilitate the analytical
calculation and has no physical meaning since it originates from the phase
transformation of Dirac fermions. Following the same strategy of Ref.GCB (7)
and using the invariant Tr$(M_{l}^{r})$, Det$(M_{l}^{r})$ and
Tr$(M_{l}^{r})^{2}$, the nozero elements of $M_{l}^{r}$ can be expressed in
terms of three mass eigenvalues $m_{e},m_{\mu}$, $m_{\tau}$ and $d$

$a=\sqrt{\frac{m_{e}m_{\mu}m_{\tau}}{d}}$ (33) $b=m_{e}-m_{\mu}+m_{\tau}-d$
(34) $c=\sqrt{-\frac{(d-m_{e})(d+m_{\mu})(d-m_{\tau})}{d}}$ (35)

Using the expression (33), (34) and (35), $O_{l}$ can be constructed. Here we
adopt the result of GCB (7) i.e

$\begin{split}O_{l}=\left(\begin{array}[]{ccc}\sqrt{\frac{m_{\mu}m_{\tau}(d-m_{e})}{d(m_{\mu}+m_{e})(m_{\tau}-m_{e})}}&\sqrt{\frac{m_{e}m_{\tau}(m_{\mu}+d)}{d(m_{\mu}+m_{e})(m_{\tau}+m_{\mu})}}&\sqrt{-\frac{m_{e}m_{\mu}(d-m_{\tau})}{d(m_{\tau}-m_{e})(m_{\tau}+m_{\mu})}}\\\
\sqrt{-\frac{m_{e}(m_{e}-d)}{(m_{\mu}+m_{e})(m_{\tau}-m_{e})}}&-\sqrt{-\frac{m_{\mu}(d+m_{\mu})}{(m_{\mu}+m_{e})(m_{\tau}+m_{\mu})}}&\sqrt{\frac{m_{\tau}(m_{\tau}-d)}{(m_{\tau}-m_{e})(m_{\tau}+m_{\mu})}}\\\
-\sqrt{-\frac{m_{e}(d+m_{\mu})(d-m_{\tau})}{d(m_{\mu}+m_{e})(m_{\tau}-m_{e})}}&\sqrt{\frac{m_{\mu}(d-m_{e})(m_{\tau}-d)}{d(m_{\mu}+m_{e})(m_{\tau}-m_{e})}}&\sqrt{\frac{m_{\tau}(d-m_{e})(d+m_{\mu})}{d(m_{\tau}-m_{e})(m_{\tau}+m_{\mu})}}\end{array}\right)\end{split}$
(36)

Replacing the (21), (22), (23), (24) and (25) with the $O_{l}$ obtained in
(43), we can see that the ratios of mass ($\rho,\sigma$), two Majorana CP-
violating phases $(\alpha,\beta)$ and the ratio of mass squared difference
$R_{\nu}$ can be expressed via eight parameters: three mixing angle
$\theta_{12},\theta_{23},\theta_{13}$, one Dirac CP violating phase $\delta$,
three charged lepton mass $(m_{e},m_{\mu},m_{\tau})$ and the parameter $d$.
Here we choose the three charged lepton mass at the electroweak
scale($\mu\simeq M_{Z}$) i.ezzx2 (31)

$m_{e}=0.486570154MeV\quad\quad m_{\mu}=102.7181377MeV\quad\quad
m_{\tau}=1746.17MeV$ (37)

Figure 1: The correlation plots for class I(IH). The blue horizontal bands
represent the 1$\sigma$ uncertainty in determination of
$\theta_{12},\theta_{23}$ and $\theta_{13}$ while they plus the green
horizontal bands correspond to the 2$\sigma$ uncertainty.

In the numerical analysis, a set of random numbers are generated for the three
mixing angles $(\theta_{12},\theta_{23},\theta_{13})$ and mass square
differences ($\delta m^{2},\Delta m^{2}$) in their $3\sigma$ range. We also
randomly vary the parameter $d$ in its appropriate range. Since at 3 $\sigma$
level the Dirac CP-violating phase $\delta$ is unconstrained in neutrino
oscillation experiments, we vary it randomly in the range of $[0,2\pi)$. With
the random number and using Eq. (21), (22) and (25), neutrino mass ratios
$(\rho,\sigma)$ and the mass-squared difference ratio $R_{\nu}$ are
determined. Then the input parameters is empirically acceptable when the
$R_{\nu}$ falls inside the the $3\sigma$ range of experimental data, otherwise
they are ruled out. Finally, we get the value of neutrino mass and Majorana
CP-violating $\alpha$ and $\beta$ though Eq.(23), (24) and (26). Once the the
absolute neutrino mass $m_{1,2,3}$ are obtained , the further constraint from
cosmology should be considered. In this paper, the upper bound on the sum of
neutrino mass $\Sigma m_{i}$ is set to be less than 0.23 eV. It turns out that
class I are phenomenologically acceptable only for inverted mass hierarchy.

Figure 2: The correlation plots $(\theta_{23},\theta_{12})$ and
$(\theta_{23},\theta_{13})$ for class I(NH). The horizontal and vertical lines
respectively denote the 3$\sigma$ upper and lower bound of $\theta_{12}$ and
$\theta_{23}$

The predictions of class I with inverted mass hierarchy are presented in
Fig.1. From the diagrams, one can see that the three neutrino mixing angles
$\theta_{12}$, $\theta_{23}$ and $\theta_{13}$ fully cover their $3\sigma$
experimental data. Although there is no bound on the Dirac CP-violating phase
$\delta$, a numerical preference appears at around $0^{\circ}\sim
50^{\circ}(360^{\circ}\sim 310^{\circ})$. The unrestricted $\delta$ leads to
the $J_{CP}$ varying in the range of $0\sim 0.04$. There also exists a strong
correlation between $\delta$ and the lightest neutrino mass $m_{3}$.
Especially, the range $0.002$eV$<m_{3}<0.02$eV is derived for $\delta$ lying
around $0^{\circ}(360^{\circ})$, indicating that both strong and mild mass
hierarchy are allowed. On the other hand, the mild mass hierarchy is much more
appealing for $100^{\circ}<\delta<260^{\circ}$. Although both Majorana CP-
violating phase $\alpha$ and $\beta$ is allowed in the range of
$-90^{\circ}\sim 90^{\circ}$, there shows a preferable distribution for
$\alpha$ in $\pm 90^{\circ}\sim\pm 50^{\circ}$ and a strong correlation
between $\delta$ and $\beta$. There exists an upper bound of $0.05$eV on the
effective Majorana neutrino mass $m_{ee}$, leaving the possible space for
detecting in future neutrinoless double beta decay $(0\nu\beta\beta)$
experiments.

The class I with inverted hierarchy is ruled out by $3\sigma$ data. To see
this, we show the correlated plots $(\theta_{23},\theta_{12})$ and
$(\theta_{23},\theta_{13})$ in Fig.2. From the diagrams, one can see that even
though $\theta_{13}$ fully covers its $3\sigma$ range, the common parameter
spaces $(\theta_{23},\theta_{12})$ fails to provide a allowed region to
saturate the experimental constraint. Moveover, one always obtains
$\theta_{23}>40^{\circ}$, which means a rather large correction of
$\theta_{12}$ is needed to reconcile the observed PMNS matrix.

###  III.2 Class II

The factorisable formation of charged lepton matrix of class I is given by
expression:

$\begin{split}(M_{l}^{r})^{II}=\left(\begin{array}[]{ccc}0&a&c\\\ a&b&0\\\
c&0&d\end{array}\right)\end{split}$ (38)

It can be diagonalized by an orthogonal matrix $O_{l}$

$O_{l}^{T}(M_{l}^{r})^{II}O_{l}=diag(m_{e},-m_{\mu},m_{\tau})$ (39)

Without losing generality, the coefficients $a,c,d$ are set to be real and
positive. Using the invariant Tr$(M_{l}^{r})$, Det$(M_{l}^{r})$ and
Tr$(M_{l}^{r})^{2}$, the nozero elements of $M_{l}^{r}$ are expressed as

$a=\sqrt{-\frac{(m_{e}-m_{\mu}-d)(m_{e}+m_{\tau}-d)(-m_{\mu}+m_{\tau}-d)}{m_{e}-m_{\mu}+m_{\tau}-2d}}$
(40) $b=m_{e}-m_{\mu}+m_{\tau}-d$ (41)
$c=\sqrt{\frac{(d-m_{e})(d+m_{\mu})(d-m_{\tau})}{m_{e}-m_{\mu}+m_{\tau}-2d}}$
(42)

where the parameter $d$ is allowed in the range of $m_{e}-m_{\mu}<d<m_{e}$ and
$m_{\tau}-m_{\mu}<d<m_{\tau}$. Then the $O_{l}$ can be easily constructed as

$\begin{split}O_{l}=\left(\begin{array}[]{ccc}\frac{(b-m_{e})(d-m_{e})}{N_{1}}&\frac{(b+m_{\mu})(d+m_{\mu})}{N_{2}}&\frac{(b-m_{\tau})(d-m_{\tau})}{N_{3}}\\\
-\frac{a(d-m_{e})}{N_{1}}&-\frac{a(d+m_{\mu})}{N_{2}}&-\frac{a(d-m_{\tau})}{N_{3}}\\\
-\frac{c(b-m_{e})}{N_{3}}&-\frac{c(b+m_{\mu})}{N_{3}}&-\frac{c(b-m_{\tau})}{N_{3}}\end{array}\right)\end{split}$
(43)

where $N_{1}$, $N_{2}$ and $N_{3}$ are the normalized coefficients given by

$N_{1}^{2}=(b-m_{e})^{2}(d-m_{e})^{2}+a^{2}(d-m_{e})^{2}+c^{2}(b-m_{\tau})^{2}$
(44)
$N_{2}^{2}=(b+m_{\mu})^{2}(d+m_{\mu})^{2}+a^{2}(d+m_{\mu})^{2}+c^{2}(b+m_{\mu})^{2}$
(45)
$N_{3}^{2}=(b-m_{\tau})^{2}(d-m_{\tau})^{2}+a^{2}(d-m_{\tau})^{2}+c^{2}(b-m_{\tau})^{2}$
(46)

Figure 3: The correlation plots for class II(NH). The blue horizontal bands
represent the 1$\sigma$ uncertainty in determination of
$\theta_{12},\theta_{23}$ and $\theta_{13}$ while they plus the green
horizontal bands correspond to the 2$\sigma$ uncertainty.

The numerical results of class II for normal hierarchy are presented in Fig.3.
We can see from the figures that the three neutrino mixing angle
$\theta_{12}$, $\theta_{23}$ $\theta_{13}$ and Dirac CP-violating phase
$\delta$ vary arbitrarily in its $3\sigma$ range. There exhibits a strong
correlation between $\delta$ and $\theta_{23}$. Only when $\delta$ is located
in the range of $100^{\circ}\sim 260^{\circ}$, the $\theta_{23}$ has the
possibility to be less then $45^{\circ}$. This is particularly interesting
since the recent global fit trends to give the $\theta_{23}<45^{\circ}$ at
2$\sigma$ level. The strong $\delta-\theta_{23}$ correlation is essential for
the model selection and will be confirmed or ruled out by future long-baseline
neutrino oscillation experiments. The similar correlations also holds for
$\delta$, $m_{ee}$ and the lightest neutrino mass $m_{1}$. Moveover, there
exists a constrained range of $0$eV$<m_{1}<$$0.06$eV, indicating that both
strong and mild neutrino mass hierarchy are possible. There are strong
correlations between $\alpha$, $\beta$ and $\delta$. Especially, the Majorana
CP-violating phase $\alpha$ is restricted in the range of
$-5^{\circ}\sim+5^{\circ}$ and $\pm 90^{\circ}\sim\pm 50^{\circ}$. The
effective Majorana neutrino mass $m_{ee}$ is highly constrained in the two
ranges of $0$eV$\sim 0.008$eV and $0.01$eV$\sim 0.025$eV. The later reaches
the accuracy of the future neutrinoless double beta decay $(0\nu\beta\beta)$
experiments. We also observed that the allowed range of Jarlskog rephasing
invariant $|J_{CP}|$ is $0\sim 0.04$, which is potentially detected by future
long-baseline neutrino oscillation experiments.

The IH case, as we can see from Fig.4, is phenomenologically ruled by
$3\sigma$ experimental data. As class I, the theoretical prediction of
$(\theta_{23},\theta_{12})$ common space fails to be located in its
experimental region. Moreover, the possibility distribution of $\theta_{23}$
shows a strong preference of $\theta_{23}<33^{\circ}$ or
$\theta_{23}>50^{\circ}$, which means a large correction of $\theta_{23}$
angle is needed to produce the $2\sigma$ global-fit value.

Figure 4: The correlation plots $(\theta_{23},\theta_{12})$ and
$(\theta_{23},\theta_{13})$ for class II(IH). The horizontal and vertical
lines respectively denote the 3$\sigma$ upper and lower bound of $\theta_{12}$
and $\theta_{23}$ Figure 5: The correlation plots for class III(IH). The blue
horizontal bands represent the 1$\sigma$ uncertainty in determination of
$\theta_{12},\theta_{23}$ and $\theta_{13}$ while they plus the green
horizontal bands correspond to the 2$\sigma$ uncertainty. Figure 6: The
correlation plots $(\theta_{23},\theta_{12})$ and $(\theta_{23},\theta_{13})$
for class III(NH). The horizontal and vertical lines respectively denote the
3$\sigma$ upper and lower bound of $\theta_{12}$ and $\theta_{23}$

###  III.3 Class III

In the case of class III, the factorisable charged lepton matrix is written by

$\begin{split}(M_{l}^{r})^{III}=\left(\begin{array}[]{ccc}0&a&b\\\ a&0&c\\\
b&c&d\end{array}\right)\end{split}$ (47)

where $a,b,c$ and $d$ are real number and $b,c$ are set to be positive. The
matrix $(M_{l}^{r})^{III}$ is diagonalized by the orthogonal matrix $O_{l}$

$O_{l}^{T}(M_{l}^{r})^{III}O_{l}=diag(m_{e},-m_{\mu},m_{\tau})$ (48)

Here we choose $a$ as the free parameter because $d$ has been fixed by
Tr$(M_{l}^{r})$. i.e

$d=m_{e}-m_{\mu}+m_{\tau}$ (49)

With the help of other two invariant quantity Det$(M_{l}^{r})$ and
Tr$(M_{l}^{r})^{2}$, $b,c$ are determined by three charged leptonic mass
eigenvalues$(m_{e},m_{\mu},m_{\tau})$ and $a$

$(b\pm
c)^{2}=-(-m_{e}m_{\mu}+m_{e}m_{\tau}-m_{\mu}m_{\tau})-a^{2}\pm\frac{a^{2}(m_{e}-m_{\mu}+m_{\tau})-m_{e}m_{\mu}m_{\tau}}{a}$
(50)

Then diagonalization matrix can be constructed as

$\begin{split}(M_{l}^{r})^{III}=\left(\begin{array}[]{ccc}\frac{O(11)}{N_{1}}&\frac{O(12)}{N_{2}}&\frac{O(13)}{N_{3}}\\\
\frac{O(21)}{N_{1}}&\frac{O(22)}{N_{2}}&\frac{O(23)}{N_{3}}\\\
\frac{O(31)}{N_{1}}&\frac{O(32)}{N_{2}}&\frac{O(33)}{N_{3}}\end{array}\right)\end{split}$
(51)

The matrix elements are given by

$\begin{split}O(11)=&am_{e}^{-1}(bm_{e}^{-1}+ca^{-1})+bm_{e}^{-1}(m_{e}a^{-1}-m_{e}^{-1}a)\\\
O(12)=&-am_{\mu}^{-1}(-bm_{\mu}^{-1}+ca^{-1})-bm_{\mu}^{-1}(-m_{\mu}a^{-1}+m_{\mu}^{-1}a)\\\
O(13)=&am_{\tau}^{-1}(bm_{\tau}^{-1}+ca^{-1})+bm_{\tau}^{-1}(m_{\tau}a^{-1}-m_{\tau}^{-1}a)\\\
&O(21)=bm_{e}^{-1}+ca^{-1}\\\ &O(22)=-bm_{\mu}^{-1}+ca^{-1}\\\
&O(23)=bm_{\tau}^{-1}+ca^{-1}\\\ &O(31)=m_{e}a^{-1}-m_{e}^{-1}a\\\
&O(32)=-m_{\mu}a^{-1}+m_{\mu}^{-1}a\\\ &O(33)=m_{\tau}a^{-1}-m_{\tau}^{-1}a\\\
\end{split}$ (52)

with the normalized coefficients

$\begin{split}N_{1}^{2}=O(11)^{2}+O(21)^{2}+O(31)^{2}\\\
N_{2}^{2}=O(12)^{2}+O(22)^{2}+O(32)^{2}\\\
N_{3}^{2}=O(13)^{2}+O(23)^{2}+O(33)^{2}\end{split}$ (53)

Repeating the previous analysis, the class III with inverted hierarchy are now
found to be acceptable by current experimental data while the NH case are
excluded. In Fig.5, we show the the main predictions for IH case. One can
observe that no bounds are founded on three mixing angles and Dirac CP-
violating phase $\delta$, leading to the Jarlskog rephasing invariant
$0<|J_{CP}|<0.04$. One the other hand, there is a correlation between $\delta$
and the lightest neutrino mass $m_{3}$. One obtains $0$eV$<m_{3}<$$0.05$eV for
$0^{\circ}<\delta<100^{\circ}(260^{\circ}<\delta<360^{\circ})$ while
$0$eV$<m_{3}<$$0.02$eV for $100^{\circ}<\delta<260^{\circ}$, implying that
both strong and mild mass hierarchy are allowed. Interestingly, although the
correlations of $(\delta,\alpha)$ and $(\delta,\beta)$ are complicated, there
exists a lower bound of $0.01$eV on the effective Majorana neutrino mass
$m_{ee}$ which is achievable in future $0\nu\beta\beta$ experiments.

In Fig.6, we present the common space of $(\theta_{23},\theta_{12})$ and
$(\theta_{23},\theta_{13})$ for NH case. One easily observes that parameter
space of $(\theta_{23},\theta_{12})$ is outside the $3\sigma$ allowed region
and a large corrections of $\theta_{23}$ or $\theta_{12}$ is needed.

##  IV The $Z_{4}\times Z_{2}$ flavor symmetry realization

In general, all phenomenologically viable lepton mass matrices with with
parallel texture/cofactor zeros can be realized in seesaw models with Abelian
flavor symmetry. The lepton mass matrices of class I are equivalent to the
ones with no-parallel texture zeros. The symmetry realization of such texture
structures has been performed in Ref.zn (20). Thus we only consider class II
and III. In this section, we take the first matrix pattern of class II as a
illustration. It is shown that the lepton mass matrix can be realized based on
the type-I seesaw models with the $Z_{4}\times Z_{2}$ flavor symmetry. We take
the same strategy of Ref.minor (15, 14, 16). In flavor basis, $M_{\nu}$
belonging to class II is realized under $Z_{8}$ symmetryminor1 (14). Different
from Ref.minor1 (14), we build the model under the basis where $M_{l}$ is
nodiagonal. Under the $Z_{4}\times Z_{2}$ symmetry, the three charged lepton
doublets $D_{iL}=(\nu_{iL},l_{iL})$, three right-handed charged lepton
singlets $l_{iR}$ and three right-handed neutrinos $\nu_{iR}$ (where
$i=e,\mu,\tau$) transform as

$\begin{split}\nu_{eR}\sim(\omega,1),\quad\quad\nu_{\mu
R}\sim(1,1),\quad\quad\nu_{\tau R}\sim(\omega^{2},1)\\\
D_{eL}\sim(\omega,-1),\quad\quad D_{\mu L}\sim(1,-1),\quad\quad D_{\tau
L}\sim(\omega^{2},-1)\\\ l_{eR}\sim(\omega^{3},-1),\quad\quad l_{\mu
R}\sim(1,-1),\quad\quad l_{\tau R}\sim(\omega^{2},-1)\end{split}$ (54)

where $\omega=e^{i\pi/2}$. Then, under $Z_{4}$ symmetry, the bilinears of
$\overline{D}_{iL}l_{jR}$, $\overline{D}_{iL}\nu_{jR}$, and
$\nu_{iR}^{T}\nu_{jR}$, transform respectively as

$\left(\begin{array}[]{ccc}-1&-i&i\\\ i&1&-1\\\
-i&-1&1\end{array}\right)\quad\quad\quad\left(\begin{array}[]{ccc}1&-i&i\\\
i&1&-1\\\
-i&-1&1\end{array}\right)\quad\quad\quad\left(\begin{array}[]{ccc}-1&i&-i\\\
i&1&-1\\\ -i&-1&1\end{array}\right)$ (55)

To generate the fermion mass, we need introduce the three Higgs doublets
$\Phi_{12},\Phi_{23},\Phi$ for charged lepton matrix $M_{l}$, one the Higgs
doublet $\Phi^{\prime}$ for Dirac neutrino mass matrix $M_{D}$ and a scalar
singlet $\chi$ for the Majorana neutrino mass matrix $M_{R}$, which transform
under $Z_{4}\times Z_{2}$ symmetry as

$\begin{split}\Phi_{12}\sim(\omega,1),&\quad\quad\Phi_{13}\sim(\omega^{3},1),\quad\quad\Phi\sim(1,1)\\\
&\Phi^{\prime}\sim(1,-1),\quad\quad\quad\quad\chi\sim(\omega,1)\end{split}$
(56)

To maintain the invariant Yukawa Lagrange under the flavor symmetry , the
$\Phi_{12}$ and $\Phi_{13}$ couple to the bilnears $\overline{D}_{eL}l_{\mu
R}$ and $\overline{D}_{eL}l_{\tau R}$ to produce the (1,2) and (1,3) nozero
matrix elements in $M_{l}$ while $\Phi$ couples to $\overline{D}_{\mu L}l_{\mu
R}$ $\overline{D}_{\tau L}l_{\tau R}$ to produce the (2,2) and (3,3) no zero
matrix elements. The zero matrix elements in $M_{l}$ is obtained because there
are no appropriate scalars to generate them. For the Dirac neutrino mass
sector, there exists only one scalar doublet $\Phi^{\prime}$ transforming
invariantly under $Z_{4}$. Therefore the $\Phi^{\prime}$ will contribute only
to the (1,1), (2,2), (3,3) no zero elements leading to a diagonal $M_{D}$.
Here the $Z_{2}$ symmetry is used to distinguish the set of scalar doublets
$(\Phi_{12},\Phi_{13},\phi)$ from $\Phi^{\prime}$ so that they are
respectively in charge of the mass generation of $M_{l}$ and $M_{D}$ without
any crossing. In order to produce the Majorana neutrino mass term, we
introduce a complex scalar singlet $\chi$. The $\chi$ couples to
$\nu_{eR}^{T}\nu_{\tau R}$ while $\chi^{\ast}$ couples to
$\nu_{eR}^{T}\nu_{\mu R}$, leading to the (1,2) and (1,3) no zero elements in
$M_{R}$. From (55), the $\nu_{\mu R}^{T}\nu_{\mu R}$ and $\nu_{\tau
R}^{T}\nu_{\tau R}$ is invariant under $Z_{4}$, thus we can directly write
them in the Lagrange without needing the singlets. The zero elements in
$M_{R}$ are obtained by not introducing other scalar singlets. Therefore the
mass matrices $M_{l}$, $M_{D}$ and $M_{R}$ is given by

$M_{l}\sim\left(\begin{array}[]{ccc}0&\times&\times\\\ \times&\times&0\\\
\times&0&\times\end{array}\right)\quad\quad\quad
M_{D}\sim\left(\begin{array}[]{ccc}\times&0&0\\\ 0&\times&0\\\
0&0&\times\end{array}\right)\quad\quad\quad
M_{R}\sim\left(\begin{array}[]{ccc}0&\times&\times\\\ \times&\times&0\\\
\times&0&\times\end{array}\right)$ (57)

Using the neutrino mass formula of type-I seesaw mechanism
$M_{\nu}=-M_{D}M_{R}M_{D}^{T}$, we obtain

$M_{\nu}\sim\left(\begin{array}[]{ccc}\Delta&\times&\times\\\
\times&\times&\Delta\\\ \times&\Delta&\times\end{array}\right)$ (58)

Together with the $M_{l}$ in (57), we have realized the leptonic mass matrices
of class II with parallel texture/cofactor zeros under $Z_{4}\times Z_{2}$
flavor symmetry. The symmetry realization of class III can be similarly
performed.

##  V Conclusion and discussion

We have investigated the parallel texture structures with two texture zeros in
lepton mass matrix $M_{l}$ and two cofactor zeros in neutrino mass matrix
$M_{\nu}$. The 15 possible textures are grouped into class I, II, III, and IV,
where the matrices in each class are related by means of permutation
transformation and share the same physical implications. We found only class
I, II, III are notrivial. Using the recent results of the neutrino oscillation
and cosmology experiments, a phenomenological analysis are systematically
proposed for each class and mass hierarchy. We demonstrate the correlation
plots between Dirac CP-violating phase $\delta$, three mixing angles
$\theta_{12},\theta_{23}$ and $\theta_{13}$, the effective Majorana neutrino
mass $m_{ee}$, the lightest neutrino mass, Majorana CP-violating phase
$\alpha,\beta$ and the neutrino mass ratio, leading to the predictions to be
confirmed by future experiments. A realization of the model base on
$Z_{4}\times Z_{2}$ flavor symmetry is illustrated.

Finally we would like to mention that in the spirit of Ref. GCB (7, 21), the
parallel texture structures are treated as a natural precursor of more general
cases. A systematic analysis of all possible combinations deserves further
study and will be published in pre (32).

###### Acknowledgements.

This work is supported by the Fundamental Research Funds for the Central
Universities The author would like to thank Shu-Yuan Guo for the helpful
discussion.

## References

  * (1) Q.R. Ahmad et al.(SNO Collaboration), Phys. Rev. Lett 89, 011301(2002); K. Eguchi et al. (KamLAND Collaboration), Phys. Rev. Lett 90, 021802(2003); M.H. Ahn et al. (K2K Collaboration), Phys. Rev. Lett 90, 041801(2003).
  * (2) F.P. An et al. (DAYA-BAY Xollaboration), Phys. Rev. Lett. 108, 171803(2012).
  * (3) J.K. Ahn et al. (RENO Collaboration), Phys. Rev. Lett.108, 191802(2012).
  * (4) H. Fritzsch, M. Gell-Mann, and P. Minkowski, Phys. Lett. B59, 256(1975); P. Minkowski, Phys. Lett. B67, 421(1977); T. Yanagida, in Proceedings of Workshop on Unified Theory and the Baryon Number of the Universe, edited by O. Sawada and A. Sugamoto(KEK, Tsukuba, 1979), p. 95; M. Gell-Mann, P. Ramond, and Slansky, in Supergravity, edited by P. van. Nieuwenhuizen and D.Z. Freeman (North-Holland, Amsterdam,1979), p. 315; R.N. Mohapatra and G. Senjanovic, Phys, Rev. Lett. 44, 912(1980); J. Schechter and J. W. F. Valle, Phys. Rev. D22, 2227(1980); J. Schechter and J. W. F. Valle, Phys. Rev. D25, 774(1982).
  * (5) P.H. Frampton, S. L. Glashow, and D. Marfatia, Phys. Lett. B536, 79(2002); H. Fritzsch, Z.-z. Xing, and S. Zhou, J. High Energy Phys. 09 (2011)083.
  * (6) Z,-z. Xing, Phys. Lett. B530, 159(2002); A. Merle, and W. Rodejohann, Phys. Rev. D73, 073012(2006); S. Dev, S. Kumar, S. Verma, and S. Gupta, Phys. Rev. D76, 013002(2007); S. Dev, S. Kumar, S. Verma, and S. Gupta, Nucl. Phys. B784, 103(2007);S. Dev, S. Kumar, Mod. Phys, Lett. A22, 1401(2007); S. Kumar, Phys. Rev. D84, 077301(2011); P.O. Ludl, S. Morisi, and E. Peinado, Nucl. Phys. B857, 411(2012); W. Grimus. and P.O. Ludl, arXiv:1208.4515; D. Meloni, and G. Blankenburg, Nucl. Phys. B867, 749(2013);
  * (7) G. C. Branco, D. Emmannuel-Costa, R. Gonz$\acute{a}$lez Felipe, and H. Ser$\hat{o}$dio, Phys. Lett. B670, 340(2009);
  * (8) M. Randhawa, G. Ahuja, and M. Gupta, Phys. Lett. B643, 175(2006); G. Ahuja, S. Kumar, M. Randhawa, M. Gupta, and S. Dev, Phys. Rev. D76, 013006(2007); G. Ahuja, M. Gupta, M. Randhawa, and R. Verma, Phys. Rev. D79, 093006(2009); J. A. Acosta, Alfredo Aranda, M. A. Buen-Abad, A. D. Rojas, Phys. Lett. B18, 1413(2013) S. Sharma, P. Fakay, G. Ahuja, M. Gupta, arXiv: 1402.0628; S. Sharma, P. Fakay, G. Ahuja, M. Gupta, arXiv: 1402.1598.
  * (9) M. Gupta, G. Ahuja; Int. Jour. Mod. Phys. A23, 1270033(2012).
  * (10) S. Kaneko, H. Sawanaka, and M. Tanimoto, J. High Energy Phys. 08 (2005)073; S. Dev, S. Verma, and S. Gupta, Phys. Lett. B687, 53(2010); S. Goswami, S. Khan, and A. Watanable, Phys. Lett. B687, 53(2010), W. Grimus, and P. O. Ludl, arXiv: 1208.4515.
  * (11) J.-Y. Liu and S. Zhou, Phys. Rev. D87, 093010(2013).
  * (12) X.-G. He and A. Zee, Phys. Rev. D68, 037302(2003).
  * (13) G.C. Branco, R. Gonzalez Felipe, F.R. Joaquim, and T. Yanagida, Phys. Lett. B562, 265(2003); B.C. Chauhan, J. Pulido, and M. Picariello, Phys. Rev. D73, 053003(2006).
  * (14) L. Lavoura, Phys. Lett. B609, 317(2005); E.I. Lashin and N. Chamoun, Phys. Rev. D78, 073002(2008); E.I. Lashin and N. Chamoun, Phys. Rev. D80, 093004(2009);
  * (15) S. Dev, S. Gupta, and R.R. Gautam, Mod. Phys, Lett. A26, 501(2011); S. Dev, S. Gupta, R.R. Gautam, and L. Singh, Phys. Lett. B706, 168(2011); T. Araki, J. Heeck, and J. Kubo, J. High Energy Phys. 07 (2012)083; S. Verma, Nucl. Phys. B854, 340(2012); S. Dev, R.R. Gautam, and L. Singh, arXiv: 1309.4219;
  * (16) S. Dev, S. Verma, S. Gupta, and R.R. Gautam, Phys. Rev. D81, 053010(2010); J. Liao,D. Marfatia, K. Whisnant, arXiv: 1311.2639.
  * (17) H.A. Alhendi, E.I. Lashin, and A.A. Mudlej, Phys. Rev. D77, 013009(2008).
  * (18) S. Dev, R.R. Gautam, and L. Singh, Phys. Rev. D87, 073011(2013).
  * (19) S. Dev, R.R. Gautam, and L. Singh, Phys. Rev. D88, 033008(2013); W. Wang, Eur. Phys. J. C73, 2551(2013).
  * (20) W. Grimus, A.S. Joshipura, L. Lavoura, M. Tanimoto, Eur. Phys. J. C36, 227(2004).
  * (21) S. Dev, S. Gupta, and R.R. Gautam, Phys. Rev. D82, 073015(2010);
  * (22) W. Wang, Phys. Lett. B733, 320(2014);
  * (23) B. Pontecorvo, Zh. Eksp. Teor. Fiz. 33, 549(1957); Z. Maki, M. Nakagawa, and N. Sakata, Prog. Theor. Phys. 28, 870(1962).
  * (24) Z. Z. Xing. Phys. Lett. B550, 178(2002); Z. Z. Xing, and S. Zhou Phys. Lett. B593, 156(2004); S. Zhou, and Z. Z. Xing, Eur. Phys. J. C38, 495(2005);
  * (25) F. Capazzi, G.L. Fogli, E. Lisi, A. Marrone, D. Montanino, and A. Palazzo, Phys. Rev. D89, 093018(2014).
  * (26) C. Jarlskog, Phys, Rev. Lett. 55, 1039(1985).
  * (27) H.V. Klapdor-Kleingrothaus, A. Dietz, H.L. Harney, and I.V. Krivosheina, Mod. Phys. Lett. A16, 2409(2001).
  * (28) C.E. Aalseth et al. Mod. Phys. Lett. A17, 1475(2002); F. Feruglio, A. Strumia, and F. Vissani, Nucl. Phys. B637, 345(2002).
  * (29) S.M. Bilenky and C. Giunti, Mod. Phys, Lett. A16, 1230015(2012).
  * (30) P.A.R. Ade et al. (Planck Collaboration), arXiv: 1303.5076.
  * (31) Z. Z. Xing, H. Zhang, and S. Zhou, Phys. Rev. D86, 013013(2012).
  * (32) in prepearation.