# Non$-$zero $\theta_{13}$ and $\delta_{CP}$ phase with $A_{4}$ Flavor
Symmetry and Deviations to Tri$-$Bi$-$Maximal mixing via $Z_{2}\times Z_{2}$
invariant perturbations in the Neutrino sector.

Gayatri Ghosh<EMAIL_ADDRESS>Department of Physics, Gauhati
University, Jalukbari, Assam-781015, India  Department of Physics, Pandit
Deendayal Upadhayay Mahavidyalaya, Karimganj, Assam-788720, India

###### Abstract

In this work, a flavour theory of a neutrino mass model based on $A_{4}$
symmetry is considered to explain the phenomenology of neutrino mixing. The
spontaneous symmetry breaking of $A_{4}$ symmetry in this model leads to
tribimaximal mixing in the neutrino sector at a leading order. We consider the
effect of $Z_{2}\times Z_{2}$ invariant perturbations in neutrino sector and
find the allowed region of correction terms in the perturbation matrix that is
consistent with 3$\sigma$ ranges of the experimental values of the mixing
angles. We study the entanglement of this formalism on the other
phenomenological observables, such as $\delta_{CP}$ phase, the neutrino
oscillation probability $P(\nu_{\mu}\rightarrow\nu_{e})$, the effective
Majorana mass $|m_{ee}|$ and $|m^{eff}_{\nu e}|$. A $Z_{2}\times Z_{2}$
invariant perturbations in this model is introduced in the neutrino sector
which leads to testable predictions of $\theta_{13}$ and CP violation. By
changing the magnitudes of perturbations in neutrino sector, one can generate
viable values of $\delta_{CP}$ and neutrino oscillation parameters. Next we
investigate the feasibility of charged lepton flavour violation in type-I
seesaw models with leptonic flavour symmetries at high energy that leads to
tribimaximal neutrino mixing. We consider an effective theory with an
$A_{4}\times Z_{2}\times Z_{2}$ symmetry, which after spontaneous symmetry
breaking at high scale which is much higher than the electroweak scale leads
to charged lepton flavour violation processes once the heavy Majorana neutrino
mass degeneracy is lifted either by renormalization group effects or by a soft
breaking of the $A_{4}$ symmetry. In this context the implications for charged
lepton flavour violation processes like $\mu\rightarrow e\gamma$,
$\tau\rightarrow e\gamma$, $\tau\rightarrow\mu\gamma$ are discussed.

##  I Introduction

Ever since the discovery of neutrino oscillations, the aspects of lepton
masses, mixings and flavour violation GG  have been an active topic of
research and there have been a lot of updates on the results from a long
ongoing series of global fits to neutrino oscillation data a ; b ; c ; d .
Neutrino flavor conversion was first detected in solar 1  and atmospheric
neutrinos 2 . This discovery led to the Nobel prize in Physics in 2015 3 ; 4
and was confirmed by subsequent results from the KamLAND reactor experiment 5
as well as long baseline accelerator experiments.

The neutrinos change their flavour as they propagate in space and this
phenomenon is known as neutrino oscillation which occurs since the flavour
gauge eigenstates of neutrinos are mixture of mass eigenstates. The mixing is
described by PMNS matrix which can be parameterized in terms of three neutrino
mixing angles and CP violating phases. The experimental disovery of neutrino
oscillations constitutes not only neutrino mass squared differences, but the
probability of nearly degenerate neutrino spectrum is also contemplated.
Further neutrino oscillation has triggered the experimental and theoretical
endeavour to understand the aspects of lepton masses, mixings and flavour
violation in SUSY GUTs theories. The massive neutrinos are produced in their
gauge eigenstates $\left(\nu_{\alpha}\right)$ which is related to their mass
eigenstate $\left(\nu_{i}\right)$, where the gauge eigenstates take part in
gauge interactions.

$\mid\nu_{\alpha}>=\sum U_{\alpha_{i}}\mid\nu_{i}>$ (1)

where, $\alpha=e,\mu,\tau$ , $\nu_{i}$ is the neutrino of distinct mass
$m_{i}$.

In the physics of the dynamics of neutrino mass generation in the leptonic
sector, the flavour problem of particle physics, is one of the open
challeneges that the field of high energy physics faces today.

Since the flavour mixing happens due to the mixing between mass and flavour
eigenstates, neutrinos have nondegenerate mass. To put into effect this idea
into a renormalisable field theory what so ever symmetry used in generating
neutrino mass degeneracy must be broken. In this work $A_{4}$ symmetry 39d ;
39 ; 39a ; 39b ; 39c  which is the group of the even permutation of four
objects or equivalently that of a tetrahedron used to maintain this degeneracy
is broken spontaneously to produce the spectrum of different charged lepton
masses.

Many inferences have been intended to guess the actual pattern of lepton
mixings. Some of the phenomenological pattern of neutrino mixings incorporate
for example, Tri-bimaximal (TBM) 39a ; 39b ; 39c , Trimaximal (TM1/TM2) and
bi-large mixing patterns.

Over the past two decades, a lot of theoretical and experimental works have
been going on, which aimed at grasping the structure of lepton mixing matrix e
. Solar and atmospheric angle as conferred by accelerator and reactor data
indicated that the mixing in the lepton sector is very different from quark
mixings, given the large values of $\theta_{12}$ and $\theta_{23}$. These
observations were soon encrypted in the tribimaximal (TBM) mixing ansatz
presented by Harrison, Perkins, and Scott 6  and also dhrm  described by.

$U_{PMNS}\simeq\begin{bmatrix}\frac{2}{\sqrt{6}}&\frac{1}{\sqrt{3}}&0\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{2}}\\\
-\frac{1}{\sqrt{6}}&-\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\\\
\end{bmatrix}=U_{TBM}$ (2)

where, $Sin\hskip 2.84544pt\theta_{13}=0$. In this educated guess, mixing
angles have $Sin\hskip 2.84544pt\theta_{12}=\frac{1}{3}$,
$\theta_{23}=\frac{\pi}{4}$ and $Sin^{2}\hskip
2.84544pt\theta_{23}=\frac{1}{2}$ whose perspective is good bearing in mind
the latest neutrino oscillation global fit. Since the TBM ansatz was first
proposed so it became a touchstone convention for inspiring the pattern of
lepton masses and mixings. Unfortunately, it envisages $Sin\hskip
2.84544pt\theta_{13}=0$ and hence zero leptonic CP violation phase in neutrino
oscillation. Infact, data from reactors have stipulated that such sterling TBM
ansatz can not be the correct description of nature, since the reactor mixing
angle $\theta_{13}$ has been confirmed to be non-zero to a very high
significant content 7 ; 8 . Further, till now it is becoming increasingly
apparent that there has been compelling evidence for CP violation in neutrino
oscillations, allocating further hint that alteration or change of TBM mixing
ansatz is vital.

Neutrino oscillation experiments are a probe to measure neutrino mixing and
mass spectrum since the oscillation probability
$P(\nu_{\mu}\rightarrow\nu_{e})$ depends on mixing angles, Dirac CP Violation
phase and the mass square differences $m^{2}_{21},m^{2}_{23}$. Results from
earlier experiments stipulate that $\theta_{13}$ is very small, almost zero
and the lepton mixing matrix follows the TBM $($tri-bimaximal mixing$)$
ansatz. This ansatz tells $Sin\hskip 2.84544pt\theta_{13}=0$,
$Sin^{2}\theta_{23}=\frac{1}{2}$, $Tan^{2}\theta_{12}=\frac{1}{2}$. One can
conclude the neutrino mixing matrix as TBM type, with small deviations or
corrections to it due to perturbation in the charged-lepton or neutrino
sector. Current experimental observations of fairly large $\theta_{13}$ a ; b
; c ; d , deviated neutrino mixing a little away from TBM ansatz, but close to
the predictions of non-zero $\theta_{13}$ and $\delta_{CP}=\pm\frac{\pi}{2}$.
One can correlate the CP violation in neutrino oscillation with the octant of
the atmospheric mixing angle $\theta_{23}$. In this paper, we would like to
address a model based on $A_{4}$ symmetry which gives non-zero $\theta_{13}$,
$\delta_{CP}=\pm\frac{\pi}{2}$ and $Sin^{2}\theta_{23}=0.57$ via perturbations
in the form of $Z_{2}\times Z_{2}$ invariant symmetry in the neutrino sector
at leading order. In order to take into account the deviations in mixing
angles at a leading order consistent with the experimental results, we add a
perturbation in neutrino sector in the form of $Z_{2}\times Z_{2}$ invariant
symmetry including second order corrections in the PMNS matrix.

The predictions of vanishing $\theta_{13}$ by TBM is owing to its invariance
under $\mu-\tau$ exchange symmetry GG . Small explicit breaking of $\mu-\tau$
symmetry can generate large Dirac CP violating phase in neutrino oscillations
GC . Also some studies in the context of corrections to TBM mixing in $A_{4}$
symmetry is presented in BK . All CP violations (both Dirac and Majorana
types) emerge from a common origin in neutrino seesaw. $\mu-\tau$ symmetry
breaking shares the common origin with all CP violations, since in the limit
of $\mu-\tau$ symmetry mixing angle $\theta_{13}$ becomes non$-$zero and thus
CP conservation takes place Pcheng1 . Studies on common origin of soft
$\mu-\tau$ symmetry and CP breaking in neutrino seesaw, the origin of matter,
baryon asymmetry, hidden flavor symmetry are vividly illustrated in Pcheng1 ;
Pcheng2 . $\mu-\tau$ symmetry and its breaking together with CP violation,
correlation between $\theta_{13}$, $\theta_{23}$ and $\theta_{12}$, in
connection to hidden flavor symmetry $($including $Z_{2}\times Z_{2}$ $)$ are
extensively studied in Pcheng1 ; Pcheng2 ; Pcheng3 ; Pcheng4 . Octahedral
symmetric group $O_{h}$ is described as the flavor symmetry of neutrino-lepton
sector. Here the residual symmetries are $Z_{2}^{\mu-\tau}\bigotimes
Z_{2}^{s}$ and $Z_{l}^{4}$ and it prescribe the neutrinos and charged leptons,
respectively Pcheng3 . Studies on the notion of constrained maximal CP
violation (CMCPV) which predicts the features
$\delta_{CP}=\frac{-\pi}{2}\hskip 2.84544pt\text{and}\hskip
2.84544pt\theta_{23}=\frac{\pi}{4}$ and their origin in the context of flavor
symmetry is presented in Pcheng4 . With the discovery of non$-$zero value of
the reactor mixing angle $\theta_{13}$ by reactor experiments RENO 9  and Daya
Bay 10  the texture of tri$-$bimaximal mixings can be generalised.

A minimally asymmetric Yukawa texture based on the Frobenius group $T_{13}$
and SU$(5)$ GUT are presented in pa ; ka ; si ; ha , where the neutrino masses
and mixing angles are predicted from TBM seesaw mixing, (up to a sign), the CP
violating phase (1.32$\pi)$ and Jarlskog invariant are determined in agreement
with current global fits with definite prediction for neutrinoless double beta
experiments, and baryon asymmetry is explained from flavored leptogenesis.

The essence of knowing the exact symmetries behind the observed pattern of
neutrino oscillations is one of the challenging tasks in particle physics.

In this work we propose a $A_{4}$ family symmetry 11  $-$ the symmetry group
of even permutations of 4 objects or equivalently that of a tetrahedron, which
is used here to obtain neutrino mixing predictions within fundamental theories
of neutrino mass. This $A_{4}$ family symmetry was first introduced as a
possible family symmetry for the quark sector 12  and is now mostly used for
the lepton sector 13 ; 14 ; 15 ; 16 ; 17 . During last two decades many
neutrino oscillation experiments like KamLAND 18 , LBL+ATM+REAC 19 , SOL,
LBL+ATM, REAC, LBL, (LBL+REAC) and ATM 20  are being performed and the
oscillation parameters are being measured to a very good precision. On the
light of discovery of non zero $\theta_{13}$, the neutrino mass model
dictating TBM mixing pattern needs necessary modifications.

The discrete family symmetry groups necessitates the need of special vacuum
alignment condition to implement tribimaximal mixing pattern ansatz. Also one
can generate deviations from TBM mixing pattern by adding symmetry breaking
terms in the interactive Lagrangian of the specific discrete family symmetry
group. This results in partial and complete symmetry breaking. Residual
symmetries exist in neutrino and charged lepton sectors after such
perturbations.

In this work we also carry out studies on lepton flavour violation decay
$\mu\rightarrow e\gamma$, $\tau\rightarrow\mu\gamma$ and $\tau\rightarrow
e\gamma$ in $G_{SM}\times A_{4}\times U(1)_{X}$ incorporating $Z_{2}\times
Z_{2}$ invariant perturbation in both charged lepton sector and neutrino
sector as discussed below, and hence one can guess the sensitivity to test the
observation of sleptons and sparticles at future run of LHC. These charged
lepton flavour violation rates depend on the form of Dirac neutrino yukawa
couplings as fixed by most favourable predicted value of Dirac CPV phase of
this work and on the details of soft SUSY breaking parameters and Tan$\beta$.
We have used the Higgs mass as measured at LHC, non zero reactor mixing angle
$\theta_{13}$ for neutrinos, and latest present and future constraints on
BR($\mu\rightarrow e\gamma$) 14nov .

Persuaded by the prerequisite for departing from the simplest first$-$order
form for the TBM ansatz, Eq. (2), here we propose a generalized version of the
TBM ansatz in which the new ansatz is realised in a model based on $A_{4}$
group as suggested in 39c  by breaking $A_{4}$ symmetry spontaneously to
$Z_{2}$ in the neutrino sector, which correctly accounts for the non-zero
value of $\theta_{13}$ and introduces CP violation. We then incorporate a real
$Z_{2}\times Z_{2}$ perturbations in the neutrino sector leading to feasible
values of $\theta_{13}$ and $\delta_{CP}$. This results in predictions of
neutrino oscillation parameters and leptonic CPV phase that will be tested at
upcoming neutrino experiments. Appendix A summarizes the $A_{4}$ algebra.

##  II The $A_{4}$ model

We take a type I SeeSaw model based on $A_{4}$ symmetry 39c . Let us limit
ourselves to only leptonic sector. The field consists of three left handed
$SU(2)_{L}$ gauge doublets, three right handed charged gauge singlets, three
right handed neutrino gauge singlets. In addition there exists also four Higgs
doublets $\phi_{i}$ $\left(i=1,2,3\right)$ and $\phi_{0}$ and three scalar
singlets. The above fields can be represented under various irreducible
representations as:

Fields | $SU(2)_{L}$ | $U(1)_{Y}$ | $A_{4}$ | Representation
---|---|---|---|---
Left Handed Doublets | $\frac{1}{2}$ | $Y=-1$ | $\underline{3}$ | $Y_{iL}$
Right Handed Charged Lepton Singlets | 0 | $Y=-2$ | $\underline{1}\oplus\underline{1^{{}^{\prime}}}\oplus 1^{{}^{\prime\prime}}$ | $l_{iR}$
Right Handed Neutrino Singlets | 0 | $Y=0$ | $\underline{3}$ | $\nu_{iR}$
Higgs Doublet | $\frac{1}{2}$ | $Y=1$ | $\underline{3}$ | $\phi_{i}$
Higgs Doublet | $\frac{1}{2}$ | $Y=1$ | $\underline{1}$ | $\phi_{0}$
Real Gauge Singlet | $0$ | $Y=0$ | $\underline{3}$ | $F_{i}$
Table 1: Allocations under various irreducible representations of $SU(2)_{L}$,
$U(1)_{Y}$ and $A_{4}$.

The Yukawa Lagrangian of the leptonic fields of the model $G_{SM}\times
A_{4}\times U(1)_{X}$ 46  is

$\textit{L}=\textit{L}_{Charged\hskip 2.84544ptleptons\hskip
2.84544ptDirac}+\textit{L}_{Neutrino\hskip
2.84544ptDirac}+\textit{L}_{Neutrino\hskip 2.84544ptMajorana}$ (3)

where, $G_{SM}$ is the standard model gauge symmetry, $G_{SM}=U(1)_{Y}\times
SU(2)_{L}\times SU(3)_{C}$. Now,

$\begin{split}\textit{L}_{Charged\hskip 2.84544ptleptons\hskip
2.84544ptDirac}=&-[h_{1}\left(\bar{Y_{1L}}\phi_{1}\right)l_{1R}+h_{1}\left(\bar{Y_{2L}}\phi_{2}\right)l_{1R}+h_{1}\left(\bar{Y_{3L}}\phi_{3}\right)l_{1R}+h_{2}\left(\bar{Y_{1L}}\phi_{1}\right)l_{2R}\\\
&+\omega^{2}\left\\{h_{2}\left(\bar{Y_{2L}}\phi_{2}\right)l_{2R}\right\\}\allowbreak+\omega\left\\{h_{2}\left(\bar{Y_{3L}}\phi_{3}\right)l_{2R}\right\\}+h_{3}\left(\bar{Y_{1L}}\phi_{1}\right)l_{3R}\\\
&+\omega\left\\{h_{3}\left(\bar{Y_{2L}}\phi_{2}\right)l_{3R}\right\\}+\omega^{2}\left\\{h_{3}\left(\bar{Y_{3L}}\phi_{3}\right)l_{3R}\right\\}]+hc\end{split}$
(4)

where,

$\omega=exp(\frac{2\pi i}{3})=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$

$\textit{L}_{Neutrino\hskip
2.84544ptDirac}=-h_{0}\left(\bar{Y_{1L}}\nu_{1R}\right)\bar{\phi_{0}}-h_{0}\left(\bar{Y_{2L}}\nu_{2R}\right)\bar{\phi_{0}}-h_{0}\left(\bar{Y_{3L}}\nu_{2R}\right)\bar{\phi_{0}}+h.c$
(5)

and

$\begin{split}\textit{L}_{Neutrino\hskip
2.84544ptMajorana}=&-\frac{1}{2}[\left\\{M\nu_{1R}^{T}C^{-1}\nu_{1R}+M\nu_{2R}^{T}C^{-1}\nu_{2R}+M\nu_{3R}^{T}C^{-1}\nu_{3R}\right\\}+h.c+h_{F}F_{1}\nu_{2R}^{T}C^{-1}\nu_{3R}\\\
&+h_{F}F_{1}\nu_{3R}^{T}C^{-1}\nu_{2R}+h_{F}F_{2}\nu_{3R}^{T}C^{-1}\nu_{1R}\\\
&+h_{F}F_{2}\nu_{1R}^{T}C^{-1}\nu_{3R}+h_{F}F_{3}\nu_{1R}^{T}C^{-1}\nu_{2R}+h_{F}F_{3}\nu_{2R}^{T}C^{-1}\nu_{1R}]\end{split}$
(6)

where $C$ is the charge conjugation matrix. $\textit{L}_{Charged\hskip
2.84544ptleptons\hskip 2.84544ptDirac}$ is the Dirac mass matrix in the
charged leptonic fields, $\textit{L}_{Neutrino\hskip 2.84544ptDirac}$ is the
Dirac mass matrix in the neutrino sector, $\textit{L}_{Neutrino\hskip
2.84544ptDirac}$ is the Dirac mass matrix in neutrino sector.

The model here is accompanied by an additional $U(1)_{X}$ symmetry which
prevents the existence of Yukawa interactions of the form
$\bar{Y_{iL}}\nu_{iR}\tilde{\phi_{i}}$ and
$\bar{Y_{iL}}\nu_{iR}\tilde{\phi_{0}}$ as $Y_{iL},l_{iR},\tilde{\phi_{0}}$
have quantum numbers $X=1$ and all other fields have quantum numbers $X=0$.
This phenomenology disfavours Nambu Goldstone boson to arise in this case as
$U(1)_{X}$ symmetry does not break spontaneously but explicitly. Thus the
Yukawa Lagrangian for the leptonic sector are of the form as described by  Eq.
(3), (4), (5), (6) under the symmetry $G_{SM}\times A_{4}\times U(1)_{X}$.

Some studies on Cosmological Domain Wall Problem are done in newc  , where it
is shown that if a discrete symmetry is embedded with a continuous gauge or
global gauge group, $($in this case only $U(1)_{X}$ $)$, then on account of
the phenomenon Lazarides-Shafi mechanism the electroweak phase transition of
the apparent discrete symmetry $A_{4}$ $($which is a subgroup of the centre of
the continuous lie group$)$, results in only a network of domain walls bounded
by strings to form and then quickly collapse.

Here a symmetry of the form $U(1)_{X}$ exists. Under this symmetry
$Y_{iL},l_{iR}$ and $\phi_{i}$ have quantum numbers $X=1$ and all other fields
have X = 0. This symmetry does not permit the terms like,
$\bar{Y_{L}}\nu_{R}\tilde{\phi_{i}}$ which are invariant under $G_{SM}\times
A_{4}$ and contributes to Dirac mass matrix for neutrinos. Spontaneous
symmetry breaking leads to the following Vacuum expectation values for
scalars, $\upsilon_{1},\upsilon_{2},\upsilon_{3}$ for $\phi_{i}^{\prime s}$,
$u_{i}$ for $F_{i}^{\prime s}$, $\upsilon_{0}$ for $\phi_{0}$. Let,
$\upsilon_{1},\upsilon_{2},\upsilon_{3}=\upsilon$. Here
$Y_{iL}=(\nu_{i},l_{i})\sim\underline{3}$,
$l_{iR}\sim\underline{1},\underline{1^{{}^{\prime}}},\underline{1^{{}^{\prime\prime}}}$,
$\phi_{i}=(\phi_{i}^{0},\phi_{i}^{-})\sim\underline{3},(i=1,2,3)$. Along with
these vacuum expectation values, the superpotential for different mass terms
are:

$-\bar{l_{L}}M^{0}_{l}l_{R}-\bar{\nu_{L}}M_{D}\nu_{R}+\frac{1}{2}\nu_{R}^{T}C^{-1}M_{R}\nu_{R}+h.c$

where,

$M^{0}_{l}=\begin{bmatrix}h_{1}\upsilon_{1}&h_{2}\upsilon_{1}&h_{3}\upsilon_{1}\\\
h_{1}\upsilon_{2}&h_{2}\omega^{2}\upsilon_{2}&h_{3}\omega\upsilon_{2}\\\
h_{1}\upsilon_{3}&h_{2}\omega\upsilon_{3}&h_{3}\omega^{2}\upsilon_{3}\\\
\end{bmatrix},M_{R}=\begin{bmatrix}M&h_{s}u_{3}&h_{s}u_{2}\\\
h_{s}u_{3}&M&h_{s}u_{1}\\\ h_{s}u_{2}&h_{s}u_{1}&M\\\ \end{bmatrix}$ (7)

and $M_{D}=h_{0}\upsilon_{0}I$. A special vacuum alignment is needed in
tribimaximal mixing, which is given by

$\upsilon_{1}=\upsilon_{2}=\upsilon_{3}=\upsilon,\hskip
2.84544ptu_{1}=u_{3}=0,\hskip 2.84544pt\text{and}\hskip
2.84544pth_{F}u_{2}=M^{{}^{\prime}}.$ (8)

The charged lepton mass matrix $M^{0}_{l}$ is now put in a diagonal form by
the transformation,

$M_{l}^{0d}=U_{\omega}M_{l}^{0}I$ (9)

where $M_{l}^{0d}$ is the diagonal form of $M_{l}^{0}$. Here

$U^{C\textit{W}}_{\omega}=\frac{1}{\sqrt{3}}\begin{bmatrix}1&1&1\\\
1&\omega&\omega^{2}\\\ 1&\omega^{2}&\omega\\\ \end{bmatrix}\text{and}\hskip
2.84544pt\omega=exp(\frac{i2\pi}{3})=-\frac{1}{2}+i\frac{\sqrt{3}}{2}.$ (10)

Now,

$M_{l}^{0d}=\frac{1}{\sqrt{3}}\begin{bmatrix}1&1&1\\\ 1&\omega&\omega^{2}\\\
1&\omega^{2}&\omega\\\
\end{bmatrix}\begin{bmatrix}h_{1}\upsilon&h_{2}\upsilon&h_{3}\upsilon\\\
h_{1}\upsilon&h_{2}\omega^{2}\upsilon&h_{3}\omega\upsilon\\\
h_{1}\upsilon&h_{2}\omega\upsilon&h_{3}\omega^{2}\upsilon\\\
\end{bmatrix}=\frac{1}{\sqrt{3}}\begin{bmatrix}3h_{1}\upsilon&0&0\\\
0&3h_{2}\upsilon&0\\\ 0&0&3h_{3}\upsilon\\\
\end{bmatrix}=\begin{bmatrix}\sqrt{3}h_{1}\upsilon&0&0\\\
0&\sqrt{3}h_{2}\upsilon&0\\\ 0&0&\sqrt{3}h_{3}\upsilon\\\ \end{bmatrix}$ (11)

Or,

$M_{l}^{0d}=\begin{bmatrix}m_{e}&0&0\\\ 0&m_{\mu}&0\\\ 0&0&m_{\tau}\\\
\end{bmatrix}.$ (12)

$M_{R}$ is diagonalised by the orthogonal transformation,

$U_{\nu}M_{R}U_{\nu}^{\dagger}=\begin{bmatrix}\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}\\\
0&1&0\\\ \frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}\\\
\end{bmatrix}\begin{bmatrix}M&0&M^{{}^{\prime}}\\\ 0&M&0\\\
M^{{}^{\prime}}&0&M\\\
\end{bmatrix}\begin{bmatrix}\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}\\\
0&1&0\\\ -\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}\\\
\end{bmatrix}=\begin{bmatrix}M-M^{{}^{\prime}}&0&0\\\ 0&M&0\\\
0&0&M+M^{{}^{\prime}}\\\ \end{bmatrix}.$ (13)

The choice of the vacuum alignment for scalar fields is to break $A_{4}$
spontaneously along two incompatible directions. $(111)$ with residual
symmetry $Z_{3}$ and $(100)$ with residual symmetry $Z_{2}$. The vacuum
alignment breaks $A_{4}$ in charged lepton sector coupling only with
$\phi_{i}$ to $Z_{3}$ group. Also the vacuum alignment breaks $A_{4}$ in
neutrino sector coupling only with $\phi_{0}$ and $F$, the residual symmetry
is $Z_{2}$ group. The PMNS matrix, tribimaximal with phases is

$\frac{1}{\sqrt{3}}\begin{bmatrix}1&0&0\\\ 0&\omega&0\\\ 0&0&\omega\\\
\end{bmatrix}\begin{bmatrix}\sqrt{\frac{2}{3}}&\frac{1}{\sqrt{3}}&0\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{2}}\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\\\
\end{bmatrix}\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&-i\\\ \end{bmatrix}.$ (14)

##  III Perturbations in Neutrino Sector

In this section, we consider the effect of perturbations to mass matrices due
to higher order corrections in the form of $Z_{2}\times Z_{2}$ invariant
perturbations. In the model discussed till now, the PMNS matrix has the
tribimaximal form with zero $\theta_{13}$ and zero $\delta_{CP}$ phase. To
generate non$-$zero values for these, small perturbation in the form of
$Z_{2}\times Z_{2}$ symmetry is added to our model. We first instigate a
symmetry breaking term in the charged lepton sector which is invariant under
the symmetry $Z_{2}\times Z_{2}$, which is a normal subgroup of $A_{4}$ with
four elements. The three non trivial singlet representation of $Z_{2}\times
Z_{2}$ are $\underline{\widehat{1}}^{{}^{\prime\prime\prime}}(1,1,-1,-1)$,
$\underline{\widehat{1}}^{{}^{\prime\prime}}(1,-1,1,-1)$,
$\underline{\widehat{1}}^{{}^{\prime}}(1,-1,-1,1)$, whereas the one trivial
singlet representation has the form $\underline{\widehat{1}}(1,1,1,1)$. The
breaking of $A_{4}$ triplet into $Z_{2}\times Z_{2}$ irreducible
representations is given as

$(\underline{3})\hskip 2.84544ptof\hskip
2.84544ptA_{4}\longrightarrow(\underline{\widehat{1}}^{{}^{\prime}}\oplus\underline{\widehat{1}}^{{}^{\prime\prime}}\oplus\underline{\widehat{1}}^{{}^{\prime\prime\prime}})\hskip
2.84544ptof\hskip 2.84544ptZ_{2}\times Z_{2},$ (15)
$(\underline{1}^{{}^{\prime\prime}},\underline{1}^{{}^{\prime}},\underline{1})\hskip
2.84544ptof\hskip 2.84544ptA_{4}\longrightarrow(\underline{\widehat{1}})\hskip
2.84544ptof\hskip 2.84544ptZ_{2}\times Z_{2}.$ (16)

If we want to break $A_{4}$ into $Z_{2}\times Z_{2}$ irreducible
representations, it could be breaking of $A_{4}$ triplet of right handed
neutrino singlets into $\underline{\widehat{1}}(1,1,1,1)$ trivial
representations of $Z_{2}\times Z_{2}$.

The general $Z_{2}\times Z_{2}$ invariant perturbations are of the form

$h_{1}\bar{Y_{L}}M_{1}\phi l_{1R}+h_{2}\bar{Y_{L}}M_{2}\phi
l_{2R}+h_{3}\bar{Y_{L}}M_{3}\phi l_{3R}$ (17)

where, $\bar{Y_{L}}$, $\phi$ are the three dimensional reducible
representations of $Z_{2}\times Z_{2}$ and $l_{R}^{{}^{\prime}}$s are the
trivial singlets. Since we have considered here $Z_{2}\times Z_{2}$ invariant
perturbations, then the matrices $M_{1}$, $M_{2}$, $M_{3}$ must commute with
the matrices in Eq. (A11) in Appendix A. The $U_{e3}$ element of the PMNS
matrix in its TBM form is zero because the $11$ and $13$ elements of
$U_{\omega}$ are same. The perturbation terms in Eq. (17) can disturb the
balance between the $11$ and $13$ elements of $U_{\omega}$ and this
phenomenology leads to non$-$zero $\theta_{13}$. The value of $\theta_{13}$
relates to the elements of the mass matrices, $M_{1}$, $M_{2}$, $M_{3}$. We
prefer the form of $M_{i}$’s as $M_{i}=diag(\bar{z},0,\omega^{i-1}z)$ to
generate simple form of perturbed charged lepton mass matrices $M_{l}$. $z$ is
a complex number and $|z|<1$. After spontaneous symmetry breaking the
resulting $M_{l}=M_{l}^{0}+\delta M_{l}$ where, $M_{l}^{0}$ has the form in
Eq. (7), and $\delta M_{l}$ has the form

$\delta
M_{l}=\begin{bmatrix}h_{1}\upsilon\bar{z}&h_{2}\upsilon\bar{z}&h_{3}\upsilon\bar{z}\\\
0&0&0\\\ h_{1}\upsilon z&h_{2}\upsilon z\omega&h_{3}\upsilon z\omega^{2}\\\
\end{bmatrix}.$ (18)

$\delta M_{l}$ arises from the higher order effects of the theory. We
parameterize all the higher order perturbations in the terms of the complex
number $z$ 39c . There is no residual symmetry remaining in the charged lepton
sector after the spontaneous symmetry breaking. Constraining $U_{\omega}$ to
be unitary, we limit $z$ as

$z=-1\pm\sqrt{1-S^{2}}+iS.$ (19)

For $z<1$ one gets perturbation to be of the order $S$. Using the
parametrization $S=Sin\hskip 2.84544pt\alpha$, $U_{\omega}$ to

$U_{\omega}=\frac{1}{\sqrt{3}}\begin{bmatrix}e^{i\alpha}&1&e^{-i\alpha}\\\
e^{i\alpha}&\omega&\omega^{2}e^{-i\alpha}\\\ e^{i\alpha}&\omega^{2}&\omega
e^{-i\alpha}\\\ \end{bmatrix},$ (20)

we introduce a $Z_{2}\times Z_{2}$ invariant perturbations in the neutrino
sector, and study its influence on $\theta_{13}$ and $\delta_{CP}$. The
perturbing matrix is diagonal since it should satisfy $Z_{2}\times Z_{2}$
symmetry. We chose the perturbation 50  to be as follows:

$M\nu_{R}^{T}C^{-1}\begin{bmatrix}\frac{1}{\rho}e^{-}{i\varphi}&0&0\\\
0&0&0\\\ 0&0&\frac{1}{\rho}e^{-}{i\varphi}\\\ \end{bmatrix}\nu_{R}$

where, $\frac{1}{\rho}e^{-}{i\varphi}$ characterises the soft brreaking of
$A_{4}$. M is $A_{4}$ invariant soft term in the Lagrangian. The perturbing
term is $A_{4}$ breaking but $Z_{2}\times Z_{2}$ invariant soft term in the
Lagrangian. The perturbed matrix is now

$\begin{bmatrix}M+\frac{1}{\rho}e^{-}{i\varphi}M&0&M^{{}^{\prime}}\\\ 0&M&0\\\
M^{{}^{\prime}}&0&M-\frac{1}{\rho}e^{-}{i\varphi}M\\\ \end{bmatrix}.$ (21)

We can diagonalise it by rotation angle $x$, where,

$Tan\hskip
2.84544pt2x=\frac{M^{{}^{\prime}}}{\frac{1}{\rho}e^{-}{i\varphi}M}.$ (22)

Thus, we see that the ratio, $\frac{M^{{}^{\prime}}}{M}$ is a physical
observable in the rotation angle, $Tan\hskip 2.84544pt2x$ which helps us to
diagonalise the perturbing matrix Eq. (21). The introduction of the perturbing
terms like $\frac{1}{\rho}e^{-}{i\varphi}$ and the ratio,
$\frac{M^{{}^{\prime}}}{M}$ has helped us to derive non zero $\theta_{13}$ and
other neutrino oscillation parameters in terms of $x,\alpha,\rho$, Eq.
(24)-(37). The input range of Majorana phases used in our calculation is from
0 to 2$\pi$. The heavy right handed Majorana neutrino used here for the
computation of various LFV decay rates like $\mu\rightarrow e\gamma$,
$\tau\rightarrow\mu\gamma$, $\tau\rightarrow e\gamma$ is $10^{15}$ GeV.

The most interesting feature of our work is that, from Eq. (24)-(37) we can
extract meaningful extract of current pattern of neutrino flavour mixing in
the sense that the favoured value of $\delta_{CP}$ phase from our results
attached with non$-$zero $\theta_{13}$ can induce signatures of various decay
rates of charged lepton flavour violation processes like $\mu\rightarrow
e\gamma$, $\tau\rightarrow\mu\gamma$, $\tau\rightarrow e\gamma$ after
spontaneous symmetry breaking of our model $G_{SM}\times A_{4}\times U(1)_{X}$
incorporating $Z_{2}\times Z_{2}$ invariant perturbations into account. The
sleptons and gauginos so constrained are shown in Figs. 11-14. The prospect to
test these sparticles at future run of LHC will favour or rule out our model.

After introducing $Z_{2}\times Z_{2}$ perturbations in both charged leptonic
sector and neutrino sector 50 ; ra ; pat , and setting $U_{\omega}$ as defined
by Eq. (20), PMNS matrix after perturbation becomes

$U_{PMNS}=\frac{1}{\sqrt{3}}\begin{bmatrix}e^{i\alpha}&1&e^{-i\alpha}\\\
e^{i\alpha}&\omega&\omega^{2}e^{-i\alpha}\\\ e^{i\alpha}&\omega^{2}&\omega
e^{-i\alpha}\\\ \end{bmatrix}\begin{bmatrix}Cosx&0&-Sinx\\\ 0&1&0\\\
Sinx&0&Cosx\\\ \end{bmatrix}.$ (23)

From above it is seen that, after computation matching with the actual PMNS
matix one gets,

$\text{Sin}\theta_{13}e^{-i\delta_{CP}}=\frac{1}{\sqrt{3}}\left(e^{-i\alpha}\text{Cosx}-e^{i\alpha}\text{Sinx}\right)=\frac{1}{\sqrt{3}}\left(\text{Cos}\alpha\text{Cosx}-\text{Cos}\alpha\text{Sinx}-i\text{Sin}\alpha\text{Cosx}-i\text{Sin}\alpha\text{Sinx}\right).$
(24)

Therefore,

$\text{Sin}\theta_{13}Cos\delta_{CP}=\frac{1}{\sqrt{3}}\left(\text{Cos}\alpha\text{Cosx}-\text{Cos}\alpha\text{Sinx}\right),$
(25)

and

$\text{Sin}\theta_{13}Sin\delta_{CP}=\frac{1}{\sqrt{3}}\left(\text{Sin}\alpha\text{Cosx}+\text{Sin}\alpha\text{Sinx}\right).$
(26)

Squaring and adding Eq. (25) and Eq. (26) we get

$\begin{split}Sin^{2}\theta_{13}(Cos^{2}\delta_{CP}+Sin^{2}\delta_{CP})=\frac{1}{(\sqrt{3})^{2}}\left\\{\left(Cos\hskip
2.84544pt\alpha\hskip 2.84544ptCos\hskip 2.84544ptx-Cos\hskip
2.84544pt\alpha\hskip 2.84544ptSin\hskip 2.84544ptx\right)^{2}+\left(Sin\hskip
2.84544pt\alpha\hskip 2.84544ptCos\hskip 2.84544ptx+Sin\hskip
2.84544pt\alpha\hskip 2.84544ptCos\hskip 2.84544ptx\right)^{2}\right\\}\\\
=\frac{1}{3}\left\\{Cos\hskip 2.84544pt^{2}\alpha\hskip 2.84544ptCos\hskip
2.84544pt^{2}x+Cos\hskip 2.84544pt^{2}\alpha\hskip 2.84544ptSin\hskip
2.84544pt^{2}x-2Cos\hskip 2.84544pt^{2}\alpha Cos\hskip 2.84544ptx\hskip
2.84544ptSin\hskip 2.84544ptx+Sin\hskip 2.84544pt^{2}\alpha\hskip
2.84544ptCos\hskip 2.84544pt^{2}x+Sin\hskip 2.84544pt^{2}\alpha\hskip
2.84544ptSin\hskip 2.84544pt^{2}x+2Sin^{2}\hskip 2.84544pt\alpha Cos\hskip
2.84544ptx\hskip 2.84544ptSin\hskip 2.84544ptx\right\\}\\\
=\frac{1}{3}\left\\{1-Cos\hskip 2.84544pt2\alpha\hskip 2.84544ptSin\hskip
2.84544pt2x\right\\}=\frac{1}{3}\left\\{1-Sin\hskip 2.84544pt2x(1-2Sin\hskip
2.84544pt^{2}\alpha)\right\\}=\frac{1}{3}\left\\{1-Sin\hskip
2.84544pt2x(1-2S^{2})\right\\}\\\
=\frac{1}{3}\left\\{1-\frac{1}{\sqrt{1+Cot\hskip
2.84544pt^{2}2x}}(1-2S^{2})\right\\}=\frac{1}{3}\left\\{1-\frac{1}{\sqrt{1+\kappa^{2}}}(1-2S^{2})\right\\}=\frac{1}{3}\left\\{1-(1+\frac{-1}{2}\kappa^{2})(1-2S^{2})\right\\}=\frac{\kappa^{2}}{6}+\frac{2}{3}S^{2}-\frac{\kappa^{2}S^{2}}{3}\\\
\end{split},$ (27)

or

$Sin^{2}\theta_{13}=\frac{1}{3}\left(1-Cos\hskip 2.84544pt2\alpha\hskip
2.84544ptSin\hskip
2.84544pt2x\right)=\frac{\kappa^{2}}{6}+\frac{2}{3}S^{2}-\frac{\kappa^{2}S^{2}}{3},$
(28)

where perturbations in neutrino sector is defined by

$\kappa=\frac{1}{\rho}e^{-}{i\varphi}\frac{M}{M^{{}^{\prime}}}=Cot2x.$ (29)

and $S=Sin\hskip 2.84544pt\alpha$. Therefore, in terms of soft breaking
parameters, one gets

$Sin^{2}\theta_{13}=\frac{1}{6\rho^{2}}e^{-}{2i\varphi}\frac{M^{2}}{M^{{}^{\prime}2}}+\frac{2}{3}Sin^{2}\hskip
2.84544pt\alpha-\frac{1}{3\rho^{2}}e^{-}{2i\varphi}{\frac{M^{2}}{M^{{}^{\prime}2}}}^{2}Sin^{2}\hskip
2.84544pt\alpha.$ (30)

Similarly one has,

$Sin^{2}\hskip 2.84544pt\theta_{12}Cos^{2}\hskip
2.84544pt\theta_{13}=\frac{1}{3},$ (31)

and also

$Cos^{2}\hskip 2.84544pt\theta_{13}=\frac{1}{3}(2+Sin\hskip
2.84544pt2xCos\hskip 2.84544pt2\alpha).$ (32)

Thus,

$Sin^{2}\hskip 2.84544pt\theta_{12}=\frac{1}{2+Sin\hskip 2.84544pt2xCos\hskip
2.84544pt2\alpha}.$ (33)

Inflating the above expression for $Sin^{2}\hskip 2.84544pt\theta_{12}$ upto
the order $\kappa^{2}$ and $S^{2}$, one gets

$Sin^{2}\hskip
2.84544pt\theta_{12}=\frac{1}{3}+\frac{2}{9}S^{2}+\frac{\kappa^{2}}{18}-\frac{\kappa^{2}S^{2}}{27}=\frac{1}{3}+\frac{2}{9}Sin^{2}\alpha+\frac{1}{18\rho^{2}}e^{-}{2i\varphi\frac{M^{2}}{M^{{}^{\prime}2}}}-\frac{1}{27\rho^{2}}e^{-}{2i\varphi\frac{M^{2}}{M^{{}^{\prime}2}}Sin^{2}\alpha}.$
(34)

Similarly we get in terms of soft breaking parameters, after computation
matching with the actual PMNS matix,

$Sin^{2}\hskip
2.84544pt\theta_{23}=\frac{\sqrt{3}Sin2xSin2\alpha+2+Sin2xCos2\alpha}{4+Sin2xCos2\alpha}=0.5+\frac{Sin\hskip
2.84544pt\alpha}{\sqrt{3}}-\frac{1}{3\sqrt{3}\rho^{2}}e^{-}{2i\varphi\frac{M^{2}}{M^{{}^{\prime}2}}}Sin\hskip
2.84544pt\alpha.$ (35)

Finally, we have

$Cos\hskip 2.84544pt\delta_{CP}=\sqrt{1-\frac{Cos^{2}\hskip
2.84544pt2x\left(2+Cos\hskip 2.84544pt2\alpha\hskip 2.84544ptSin\hskip
2.84544pt2x\right)^{2}}{\left(1-Cos^{2}\hskip 2.84544pt2\alpha\hskip
2.84544ptSin^{2}\hskip 2.84544pt2x\right)\left[4+4Cos\hskip
2.84544pt2\alpha\hskip 2.84544ptSin\hskip
2.84544pt2x+(-1+2Cos4\alpha)Sin^{2}2x\right]}}$ (36)

or

$Cos\hskip
2.84544pt\delta_{CP}=\sqrt{1-\frac{(-\kappa)^{2}}{4Sin^{2}\alpha+\kappa^{2}-16\frac{\kappa^{2}Sin^{2}\alpha}{3}}}=\sqrt{1-\frac{\frac{1}{\rho^{2}}e^{-}{2i\varphi}\frac{M^{2}}{M^{{}^{\prime}2}}^{2}}{4Sin^{2}\alpha+\frac{1}{\rho^{2}}e^{-}{2i\varphi}\frac{M^{2}}{M^{{}^{\prime}2}}^{2}-16\frac{\frac{1}{\rho^{2}}e^{-}{2i\varphi}\frac{M^{2}}{M^{{}^{\prime}2}}^{2}Sin^{2}\alpha}{3}}},$
(37)

keeping the leading powers in numerators and denominator. For no perturbation
in neutrino sector, the value of $\delta_{CP}$ becomes zero as $\kappa$ tends
to zero. Owing to perturbation inhibition only in neutrino sector, one sets
$S=0$ or $\delta_{CP}=\pm\frac{\pi}{2}$. In conformity with T2k data from the
$\nu_{e}$ appearance data, the value of $\delta_{CP}$ is favoured to be in the
lower half plane. In the wake of perturbations only in the neutrino sector,
one gets

$Sin\hskip 2.84544pt\delta_{CP}=-Sin\frac{\pi}{2}.$ (38)

This implies $S\longrightarrow 0$ or $\kappa$ is positive and perturbations in
charged leptonic sector is imperceptible.

As we work in the type I seesaw framework, the heavy Majorana neutrino mass
scale M is much higher than the electroweak scale and the light neutrinos are
simply given by the well-known effective mass matrix

$m_{\nu}=-M_{D}M_{R}^{-1}m_{D}^{T},$ (39)

where $M_{D}$ is the Dirac $-$ type neutrino mass matrix in the weak basis.
The charged lepton masses are real and diagonal.

$M_{D}=\upsilon_{0}U_{\omega}^{\dagger}Y_{\nu},$ (40)

where $\upsilon_{0}$ denotes the vacuum expectation value of the usual SM
Higgs doublet, $<\phi_{0}>=\upsilon_{0}$. We thus compute the form of
$Y_{\nu}$, the Dirac neutrino yukawa couplings (DNY) from
$M_{D}=h_{0}\upsilon_{0}I$ and Eq. (40), after spontaneous breaking of $A_{4}$
symmetry incorporating $Z_{2}\times Z_{2}$ invariant peturbations in the
neutrino sector and thus generating non zero $\theta_{13}$ and $\delta_{CP}$.

$Y_{\nu}=h_{0}|U_{\omega}^{\dagger}|^{-1}.$ (41)

Here $h_{0}$ is the arbitrary coupling constant giving the lepton masses.
Taking $h_{0}$ of the order of 0.08 we can construct a user defined neutrino
Yukawa coupling $Y_{\nu}$ corresponding to favoured value of $\alpha$ from our
results.

Charged lepton flavour violation (CLFV) and hence neutrino oscillations and
mixings are real phenomenon. In terms of low energy observables, the lepton
flavour violating entries in the SO(10) SUSY GUT framework can be expressed as

$\left(m^{2}_{\tilde{L}}\right)_{i\neq
j}=\frac{-3m_{o}^{2}+A_{o}^{2}}{8\pi^{2}}\sum_{k}\left(Y_{\nu}^{\star}\right)_{ik}\left(Y_{\nu}\right)_{jk}\\\
log\left(\frac{M_{X}}{M_{R_{k}}}\right),$ (42)

here $M_{X}$ is the GUT scale, $M_{R_{k}}$ is the scale of the $k^{th}$ heavy
RH majorana neutrino, $m_{0}$ and $A_{0}$ are universal soft mass and
trilinear terms at the high scale. $Y_{\nu}$ are the Dirac neutrino Yukawa
couplings. The flavor violating off-diagonal entries at the weak scale are
found by using $Y_{\nu}$. The branching ratio of a charged lepton flavour
violating decay GG  $l_{i}$ $\rightarrow$ $l_{j}$ is

$\text{BR}\left(l_{i}\rightarrow
l_{j}+\gamma\right)\approx\alpha^{3}\frac{|\delta^{LL}_{ij}|^{2}}{G_{F}^{2}M^{4}_{SUSY}}\hskip
2.84544ptTan^{2}\hskip 0.28436pt\beta\linebreak\hskip
2.84544pt\text{BR}\left(l_{i}\rightarrow l_{j}\nu_{i}\tilde{\nu_{j}}\right).$
(43)

The most interesting feature of this work is that we predicted form of Dirac
neutrino Yukawa coupling $Y_{\nu}$ corresponding to favoured value of
$\alpha\sim 60^{0}$ which could realise signatures of rare CLFV decays like
$\mu\rightarrow e\gamma$, $\tau\rightarrow\mu\gamma$, $\tau\rightarrow
e\gamma$ after spontaneous symmetry breaking of our model $G_{SM}\times
A_{4}\times U(1)_{X}$ incorporating $Z_{2}\times Z_{2}$ invariant
perturbations into account. In this context we used the value of Higgs mass as
measured at LHC, latest global data on the reactor mixing angle $\theta_{13}$
for neutrinos, and latest constraints on BR($\mu\rightarrow e\gamma$) as
projected by MEG at PSI and MEG II PSI 14nov  planning to achieve sensitivity
to BR$(\mu\rightarrow e\gamma)\sim 10^{-14}$. Beyond Standard Model Physics
that could yield CLFV embraces SUSY sparticles, $Z^{{}^{\prime}}$ vector
bosons with flavour non diagonal couplings which is an indication of lepton
flavour violation.

Signatures of CLFV could be tested by the late 2020s at next run of High $-$
Luminosity LHC, if SUSY sparticles are observed within few TeV range, as
discussed in detail below. It is worth mentioning here that, during last run
of LHC, no SUSY partner of SM has been observed, and this could point to a
high scale SUSY theory. Split SUSY offers a dark matter candidate and unifies
the fundamental forces at high energies, it doesn’t address the stability of
Higgs boson .

Figure 1: In Fig. 1(a) the points in $\rho-\alpha$ space which satisfy the
3$\sigma$ constraints on $Sin^{2}\theta_{13}$ for
$\frac{M^{{}^{\prime}}}{M}=10^{-2}$ are presented. Fig. 1(b) shows the points
in $Sin^{2}\theta_{13}-\rho-\alpha$ space corresponding to the 3$\sigma$
bounds on $Sin^{2}\theta_{13}$ for $\frac{M^{{}^{\prime}}}{M}=10^{-3}$.

To show that the model predicts the neutrino mixing angles compatible with the
observed data, we obtain the allowed parameter space for the correction terms
in the perturbation matrix in the form of $Z_{2}\times Z_{2}$ invariant
symmetry compatible with the 3$\sigma$ range of the observed $\nu$ oscillation
data by varying the parameters. Considering, definite values for
$\frac{M^{{}^{\prime}}}{M}=10^{-2},10^{-3}$ and setting forth the values for
soft breaking phase as $\varphi=0,\pm\frac{\pi}{2},\pi$, we generate the soft
breaking parameter space for $\rho$ and $\alpha$ from the $3\sigma$
constraints on mixing angles, $Sin^{2}\theta_{12}$, $Sin^{2}\theta_{23}$,
$Sin^{2}\theta_{13}$, and CP violating Phase, $\delta_{CP}$. As can be seen
from Eq. (30) that to generate non-zero values of $\theta_{13}$, soft breaking
phase $\varphi$ should take the values as $0,\pm\frac{\pi}{2},\pi$ as
$\varphi$ is present in the form of $\frac{1}{6\rho^{2}}e^{-}{2i\varphi}$ in
the expression of $\theta_{13}$. $e^{-}{2i\varphi}$ will take real values only
for $\varphi=0,\pm\frac{\pi}{2},\pi$, since, other values of $\varphi$, such
as $\pm\frac{\pi}{4}$ will generate imaginary values of $\theta_{13}$ in the
form of $iSin2\varphi$ which is absurd. Similarly Eq. (34), Eq. (35) and Eq.
(37) will take real favoured values of mixing angles $\theta_{12}$,
$\theta_{23}$ and CP violating phase $\delta_{CP}$ respectively for
$\varphi=0,\pm\frac{\pi}{2},\pi$.

Owing to the Eq. (37) one finds that the value of $\delta_{CP}$ depends on the
comparative supremacy between the parameters $Sin\hskip 2.84544pt\alpha$ and
$\kappa$ or $\rho$. This obsession between the parameters $Sin\hskip
2.84544pt\alpha$ and $\kappa$ or $\rho$ in procuring the mixing angles and CP
violation phase, $\delta_{CP}$ in terms of Eqs. (28), (29), (30), (34), (35),
(37) by virtue of perturbations in neutrino sector and charged leptonic sector
has been plotted in Figs. 1-10.

Figure 2: In Fig. 2(a) the allowed range of $\rho-\alpha$ space which satisfy
the 3$\sigma$ constraints on $Sin^{2}\theta_{12}$ for
$\frac{M^{{}^{\prime}}}{M}=10^{-2}$ are shown. Fig. 2(a) shows the correlation
between $\rho$ and $\alpha$ space corresponding to the 3$\sigma$ bounds on
$Sin^{2}\theta_{12}$ for $\frac{M^{{}^{\prime}}}{M}=10^{-3}$.

.

In Fig. 1 the predicted dependence of $Sin^{2}\theta_{13}$ on the soft
breaking parameter, $\rho$ and $\alpha$ is shown. Owing to the constrained
nature of the mixing angle, $Sin^{2}\theta_{13}$ varying within its 3$\sigma$
range indicated by current neutrino oscillation global fit pc , one finds the
correlation between mixing angle, $Sin^{2}\theta_{13}$, and $\rho$, $\alpha$
breaking parameter space corresponding to $\frac{M^{{}^{\prime}}}{M}=10^{-2}$
and $\frac{M^{{}^{\prime}}}{M}=10^{-3}$ in the left and right panel
respectively. Owing to $Sin\hskip 2.84544pt\alpha$ and $Z_{2}\times Z_{2}$
perturbations in the neutrino sector the allowed range of $\rho$ space lies in
the range $[0,50]$ and $[0,560]$ for $\frac{M^{{}^{\prime}}}{M}=10^{-2}$ and
$\frac{M^{{}^{\prime}}}{M}=10^{-3}$ respectively. Fig. 2, shows the variation
of $Sin^{2}\theta_{12}$ within its 3 $\sigma$ range with respect to the soft
breaking parameter, $\rho$ and $\alpha$. The curtailment of the mixing angle,
$Sin^{2}\theta_{12}$ differing within its 3$\sigma$ range allowed by the
current neutrino oscillation global fit, one finds the whole experimentally
allowed range of $\rho$, $\alpha$ breaking parameter space corresponding to
$\frac{M^{{}^{\prime}}}{M}=10^{-2}$ and $\frac{M^{{}^{\prime}}}{M}=10^{-3}$ in
the left and right panel respectively. On account of $Sin\hskip
2.84544pt\alpha$ and $Z_{2}\times Z_{2}$ perturbations in the neutrino sector
the predicted range of $\rho$ space lies in the range $[0,600]$ and
$[1500,4500]$ for $\frac{M^{{}^{\prime}}}{M}=10^{-2}$ and
$\frac{M^{{}^{\prime}}}{M}=10^{-3}$ respectively. From Fig. 1, one finds that
the value of $Sin^{2}\theta_{13}=0.0216$ is near $\rho=560$. For this value of
$\rho$, the change in $Sin^{2}\theta_{12}$ is comparatively small $(\sim
6)\%$. Thus, one finds that the value of $\rho$ is comparatively large but the
parameter $\rho e^{-}{i\varphi}=\kappa\frac{M^{{}^{\prime}}}{M}$ appraising
the perturbation in the neutrino sector is utterly small because
$\frac{M^{{}^{\prime}}}{M}<<1$ and also owing to the contributions from
$\frac{1}{\rho}$ factor. In the limiting case, $Sin\hskip
2.84544pt\alpha\rightarrow 0$, the value of $Sin^{2}\theta_{23}$ becomes
$\frac{1}{2}$.

Figure 3: In Fig. 3(a), the values of $\delta_{CP}$ within its $3\sigma$
bounds phase for different regions in $\alpha$ space for
$\frac{M^{{}^{\prime}}}{M}=10^{-2}$ are shown. Fig. 3(b) shows the values of
$\delta_{CP}$ within its $3\sigma$ bounds as indicated by current neutrino
oscillation global fit pc  for different regions in $\alpha$ space for
$\frac{M^{{}^{\prime}}}{M}=10^{-3}$.

.

Figure 4: In Fig. 4 the allowed range of electron neutrino appearance
probability at T2K which covers a more restricted region is depicted. The blue
region is our model prediction.

Figure 5: Fig. 5(a) presents the values of $\delta_{CP}$ within its $3\sigma$
bounds phase for different regions in $\rho$ space for
$\frac{M^{{}^{\prime}}}{M}=10^{-2}$. In Fig. 5(b) the values of $\delta_{CP}$
within its $3\sigma$ bounds as indicated by current neutrino oscillation
global fit pc , for different regions in $\rho$ space for
$\frac{M^{{}^{\prime}}}{M}=10^{-3}$ are illustrated.

.

The value of $\delta_{CP}$ depends on the relative predominance between the
parameters $Sin\hskip 2.84544pt\alpha$ and $\kappa$ or $\rho$. This dependance
is plotted in Figs. 3, 5. From the right panel in  Fig. 3 the results of our
present analysis suggests $\delta_{CP}$ violation phase to be around
$144^{{\circ}}$, corresponding to $\frac{M^{{}^{\prime}}}{M}=10^{-3}$ with
$\alpha\sim 60^{\circ}$. The analysis of No$\nu$A results shows a preferance
for $\delta_{CP}\sim 0.8\pi$ suggests our present analysis of $\delta_{CP}$
phase $\sim 144^{\circ}$ exactly coincides with the preferred value. The
separate analysis of neutrino and antineutrino channels can not provide, at
present, a sensitive measurement of $\delta_{CP}$ phase. The CPV phase can
therefore be measured by the long-baseline accelerator experiments T2K and
NO$\nu$A, and also by Super-Kamiokande atmospheric neutrino data. Similarly,
for $\frac{M^{{}^{\prime}}}{M}=10^{-2}$, we obtain the best fit value of
$\delta_{CP}\sim 0.8\pi$ in our present analysis corresponding to $\alpha\sim
310^{\circ}$. The predictions made in this analysis can also be tested in
currently running and upcoming neutrino oscillation experiments. The
predictions made by our model to electron neutrino appearance probability
oscillation experiments are displayed in  Fig. 4. This conjecture is for the
T2K setup, neglecting matter effects, as an approximation. Clearly, the
allowed range of electron neutrino appearance probability at T2K is
significantly constrained with respect to the generic expectation. One finds
from the right panel in  Fig. 5, that NO$\nu$A preferance of $\delta_{CP}\sim
0.8\pi$ propounds the parameter of perturbation in neutrino sector
$\frac{1}{\rho}$ to be around 5$\times 10^{-4}$ and 2$\times 10^{-3}$. From
the left panel it is found that preferance of $\delta_{CP}\sim 0.8\pi$
constrains the parameter of perturbation in neutrino sector $\frac{1}{\rho}$
to limit itself around 4$\times 10^{-43}$. The inclusion of reactor data can
help to improve the determination of $\delta_{CP}$ phase, owing to the
existing correlation between the CP phase and $\theta_{13}$. From the results
presented in this work, we obtain the best fit value for the CP phase at
$\delta_{CP}\sim 0.8\pi$ for NO. The CP conserving value $\delta$ = 0 is
favored with $\frac{1}{\rho}$ in the region, $1\times 10^{-3}$ to $5\times
10^{-4}$ in NO as can be seen from the left panel in  Fig. 6.

Figure 6: The plot of sine squared values of mixing angles for maximal
$\delta_{CP}$ through a $Z_{2}\times Z_{2}$invariant perturbations in neutrino
sector are presented in Fig. 6(a) and Fig. 6(b) which corresponds to
$\frac{M^{{}^{\prime}}}{M}=10^{-2}$ and $\frac{M^{{}^{\prime}}}{M}=10^{-3}$
respectively.

Figure 7: In Fig. 7 the plot of sine squared values of mixing angles for
maximal $\delta_{CP}$ through a $Z_{2}\times Z_{2}$invariant perturbations in
neutrino sector is presented.

Figure 8:  Fig. 8(a) demonstrates the region in $\rho-\alpha$ space which is
consistent with the 3$\sigma$ constraints on $Sin^{2}\theta_{23}$ for
$\frac{M^{{}^{\prime}}}{M}=10^{-2}$. Fig. 8(b) illustrates the region in
$Sin^{2}\theta_{23}-\rho$ space corresponding to the 3$\sigma$ bounds on
$Sin^{2}\theta_{23}$ for $\frac{M^{{}^{\prime}}}{M}=10^{-3}$.

.

Figure 9: $|m_{ee}|$ prediction for lightest neutrino mass $m_{1}$ (eV) and
$\alpha$ space in our model. Figure 10: $|m_{\nu_{e}}^{eff}|$ prediction for
lightest neutrino mass $m_{1}$ (eV) and $\alpha$ space in our model.

Next, we discuss the perturbations in determining the atmospheric mixing
angle, $Sin^{2}\theta_{23}$. Accelerator and atmospheric oscillation
experiments estimate the disappearance of muon (anti)neutrinos and are mostly
sensitive to $Sin^{2}2\theta_{23}$. Thus, one cannot resolve the octant GGG ;
GG1 ; GG2 ; GG3  of the angle. In other words, one cannot decide whether
$Sin^{2}2\theta_{23}>0.5$ or $Sin^{2}2\theta_{23}<0.5$. Nonetheless, on
account of matter effects in the neutrino trajectories inside the Earth, this
degeneracy is slightly broken for atmospheric neutrino oscillation experiment.
The quantity $Sin^{2}\theta_{23}$ finds itself in the expressions for
appearance channels of these probability experiments. Examining the data from
long-baseline accelerators, one finds two essentially degenerate solutions for
$Sin^{2}2\theta_{23}$ for both mass orderings, The best fit is obtained for
$Sin^{2}\theta_{23}=0.46$, and a local minimum appears at
$Sin^{2}2\theta_{23}$ = 0.57 with $\Delta\chi^{2}=0.3(0.7)$ for normal mass
(inverted mass) ordering. In the present analysis the best fit experimental
value of $Sin^{2}\theta_{23}=0.57$ is $8.59\%$ larger than the TBM value of
0.5. To procure this deviation, one needs $\frac{1}{\rho}$ to be 0.0111 and
$\frac{1}{\rho}$ to be $1.54\times 10^{-3}$ for
$\frac{M^{{}^{\prime}}}{M}=10^{-2}$ and $\frac{M^{{}^{\prime}}}{M}=10^{-3}$
respectively as seen from Fig. 8. The above values of $\frac{1}{\rho}$ lead to
small values of $\kappa$ and hence of $\delta_{CP}$ phase. Thus there is a
strain in obtaining large values of $\delta_{CP}$ phase and best fit
experimental value of $Sin^{2}\theta_{23}$. For $\kappa>2Sin\hskip
2.84544pt\alpha$ we have large CPV phase, and that keeps the value of
$Sin^{2}\theta_{23}$ close to the TBM value of 0.5 as is evident from Eq.
(35).

Owing to the constrained limited nature of the mixing angles and perturbing
factors like $\frac{1}{\rho}$ and $\alpha$ in our model, one also gets
predictions for $|m_{ee}|$ and $|m_{\nu_{e}}^{eff}|$, as shown in Fig. 9 and
Fig. 10 respectively.

Figure 11: In Fig. 11a, 11b, different horizontal lines black and violet
represent the present MEG bound at PSI and future MEG II PSI bounds for
BR($\mu$ $\rightarrow$ e+ $\gamma$) respectively.

.

To test various CLFV processes, like $\mu\rightarrow e\gamma$,
$\tau\rightarrow e\gamma$, $\tau\rightarrow\mu\gamma$, the parameters we use
in our model are scalar masses $m_{0}$, triliniear coupling, $A_{0}$ and
unified gaugino mass $M_{1/2}$. There is also the Higgs potential parameter
$\mu$ and the undetermined ratio of the Higgs VEVs, Tan$\beta$. The present
MEG bound at PSI i.e, $<4.2\times 10^{-13}$ together with a non zero values of
$\theta_{13}$ pc  puts notable constraints on SUSY parameter space. As can be
seen from Fig. 11a, only small part of the heavy $m_{0}$ space around 15000
GeV survives for Tan $\beta$ = 10 in our model restricted by future MEG II
bound at PSI for BR($\mu\rightarrow e\gamma$) $\sim 10^{-14}$. Fig. 11b
reveals that the parameter space $M_{1/2}\geq$ 1 TeV is allowed by present MEG
PSI bounds on BR($\mu\rightarrow e\gamma$), while future MEG limit
BR($\mu\rightarrow e\gamma$) $\sim 10^{-14}$ excludes almost whole of
$M_{1/2}$ space. Very few points around $\sim$ 1 $-$ 2 TeV are favourable. In
Fig. 12a, 12b, we show the correlation among the branching ratios of
BR($\tau\rightarrow\mu+\gamma$) versus BR($\mu$ $\rightarrow$ e+ $\gamma$) and
BR($\tau\rightarrow e+\gamma$) versus BR($\mu$ $\rightarrow$ e+ $\gamma$)
respectively.

Figure 12: In Fig. 12a, 12b correlations between different LFV decays,
BR($\tau\rightarrow\mu+\gamma$) versus BR($\mu$ $\rightarrow$ e+ $\gamma$) and
BR($\tau\rightarrow e+\gamma$) versus BR($\mu$ $\rightarrow$ e+ $\gamma$) are
shown respectively.

.

Figure 13: In Figs. 13a, 13b we have shown feasible SUSY parameters space
allowed by present bound at MEG at PSI. Different horizontal lines denote the
range of Higgs mass as given by the data at LHC, i.e 122.5 $\text{GeV}\leq
m_{h}\leq$ 129.5 GeV GG .

.

The current upper limit on BR($\mu$ $\rightarrow$ e+ $\gamma$) implies an
upper limit BR($\tau$ $\rightarrow$ $\mu$\+ $\gamma$) $\sim 10^{-13}$ which is
notably smaller than the sensitivity of current generation experiments. Thus,
any signatures of CLFV decay $\tau$ $\rightarrow$ $\mu$\+ $\gamma$ may rule
out the present favoured value of $\alpha\sim 60^{0}$. In Fig. 13a, 13b we
plot the lightest Higgs mass $m_{h}$ as a function of $m_{0}/M_{1/2}$,
$A_{0}/M_{1/2}$ respectively. For the allowed range of Higgs mass as given by
the data at LHC, i.e 122.5 $\text{GeV}\leq m_{h}\leq$ 129.5 GeV,
$m_{0}/M_{1/2}$ should be around 5 as allowed by present MEG PSI bounds on
BR($\mu\rightarrow e\gamma$).

Figure 14: In Figs. 14a, 14b we have shown feasible SUSY parameters space
allowed by present bound at MEG at PSI. Different horizontal lines denote the
range of Higgs mass as given by the data at LHC, i.e 122.5 $\text{GeV}\leq
m_{h}\leq$ 129.5 GeV GG .

.

Figure 15: In Fig. 15a, different horizontal lines represent the present and
future bounds for hadronic flavour violation BR(b $\rightarrow$ s + $\gamma$).
In Fig. 15b we have shown SUSY parameters space allowed by present bounds on
BR(b $\rightarrow$ s + $\gamma$).

.

In Fig. 14a, for the allowed range of Higgs mass, i.e 122.5 $\text{GeV}\leq
m_{h}\leq$ 129.5 GeV, $A_{0}/m_{0}$ should be around -1 to +1. Asymmetry in
the value of $A_{0}$, can be seen in Fig. 14a. The space $M_{1/2}\geq$ 4 TeV
is allowed as can be seen from Fig. 14b. In Fig. 15a we have presented results
for the decay $b\rightarrow s\gamma$. In Fig. 15b we have shown SUSY
parameters space allowed by present bounds on BR(b $\rightarrow$ s +
$\gamma$).

##  IV Conclusion

We consider a model based on $A_{4}$ symmetry which gives the corrections to
the TBM form for the leading order neutrino mixing matrix. We present here the
phenomenology of a model with $A_{4}$symmetry which envisage the tribimaximal
form for the PMNS matrix. In this model, we have instigated a $Z_{2}\times
Z_{2}$ invariant perturbations in both the charged lepton in the form of
$Sin\hskip 2.84544pt\alpha$ and the neutrino sectors in the form of $\kappa$.
We perceive that perturbations in the neutrino sector leads to allowable
values of non zero $\theta_{13}$ varying within its 3$\sigma$ range as
indicated by current neutrino oscillation global fit pc  and maximal CP
violation for $Sin\hskip 2.84544pt\alpha=0$. The desired value of the CP
violating phase $\delta_{CP}$ lying within its 3$\sigma$ range can be procured
by choosing the fitting and pertinent values for $Sin\hskip 2.84544pt\alpha$
term and for the perturbations in neutrino sectors. However, there is a strain
in obtaining large values of $\delta_{CP}$ phase and best fit experimental
value of $Sin^{2}\theta_{23}$. For $\kappa>2Sin\hskip 2.84544pt\alpha$ we have
large CPV phase, and that keeps the value of $Sin^{2}\theta_{23}$ close to the
TBM value of 0.5. Also, our analysis of $\delta_{CP}$ phase $\sim 144^{\circ}$
in this model exactly coincides with the preferred value of $\delta_{CP}\sim
0.8\pi$ by the analysis of No$\nu$A results 20 .

We have considered leading order corrections in the form of $Z_{2}\times
Z_{2}$ invariant perturbations in neutrino sector after spontaneous breaking
of $A_{4}$ symmetry. The neutrino mixing angles, thus obtained are found to be
within the $3\sigma$ ranges of their experimental values. The CP violating
phase $\delta_{CP}$ is around $\sim 144^{\circ}$ in this model. We also
studied the variation of the the neutrino oscillation probability
$P(\nu_{\mu}\rightarrow\nu_{e})$, the effective Majorana mass $|m_{ee}|$ and
$|m^{eff}_{\nu e}|$ with the lightest neutrino mass $m_{1}$ in the case of
normal hierarchy and found its value to be lower than the experimental upper
limit for all allowed values of $m_{1}\in[0\hskip 2.84544pteV,10^{-5}\hskip
2.84544pteV]$.

We show that our predicted value of $\delta_{CP}\sim 144^{0}$ corresponding to
$\frac{M^{{}^{\prime}}}{M}=10^{-3}$ and $\alpha=60^{0}$ indicates signatures
of various charged LFV channels in a class of $G_{SM}\times A_{4}\times
U(1)_{X}$ model incorporating $Z_{2}\times Z_{2}$ invariant perturbations in
charged lepton and neutrino sector, which is the most interesting feature of
our work. These ratios depend on the form of Dirac neutrino yukawa couplings
as fixed by Dirac CP phase and on the details of soft SUSY breaking parameters
and Tan$\beta$. We have used the Higgs mass as measured at LHC, non zero
reactor mixing angle $\theta_{13}$ for neutrinos, and latest present and
future constraints on BR($\mu\rightarrow e\gamma$). We find that very heavy
$m_{0}$ region is allowed by future MEG bound of BR($\mu\rightarrow e\gamma$)
$\sim 10^{-14}$ and $M_{1/2}$ values greater than 1 TeV is allowed. We have
also shown the predictions for neutrinoless double beta decay in terms of
lightest neutrino mass for a mass range of $m_{1}\in[0\hskip
2.84544pteV,10^{-5}\hskip 2.84544pteV]$, which we epitomize in Fig. 9, 10.
Ultimately we have estimated the parameters of our model where the branching
ratios of $\mu\rightarrow e\gamma$ of the order of $\sim 10^{-14}$ are
calculated, which is well within the latest experimental constraints and is
summarised in Fig. 11.

To conclude, we have proposed a pragmatic generalization of the TBM ansatz,
which in addition to explaining nonzero $\theta_{13}$, also makes exact and
certain testable predictions for the other parameters of the lepton mixing
matrix, including CP violating and CP conserving phases. The $Z_{2}\times
Z_{2}$ invariant perturbations in neutrino sector are characterized in terms
of three independent parameters, $\frac{1}{\rho}$, $\alpha$ and $\varphi$
which determine all three mixing angles and CP violating phase, leading to
several testable predictions. A more comprehensive version of the generalized
CP methodology and its potential to produce other hypothetically and realistic
ansatz forms for the lepton mixing matrix will be presented in our future
work.

##  V Acknowledgement

GG would like to thank would like to thank University Grants Commission RUSA,
MHRD, Government of India for financial support. GG would also like to thank
Prof. Probir Roy for useful discussion on this topic.

##  Appendix A Basics of $A_{4}$ group

$A_{4}$ is the smallest non Abelian group with an irreducible triplet
representation. Alternating group $A_{4}$ is a group of even permutations of
four objects, with a three one dimensional irreducible representation which
makes it one of the most favoured group in neutrino mass models.

$A_{4}$ group is a non $-$ Abelian, and it is not a direct product of cyclic
groups. Group $A_{4}$ has twelve elements and it is isomorphic to tetrahedral
$T_{d}$.

$A_{4}$ group consists of twelve elements which are written in terms of
generators of the group S and $T$, the generators satisfy the relation,

$\textbf{{S}}^{2}=\left(\textbf{{S}}T\right)^{3}=T^{3}=1.$ (44)

There are three one dimensional irreducible representations of the $A_{4}$
group defined as

$1,\hskip 14.22636pt\textbf{{S}}=1,\hskip 14.22636ptT=1$
$1^{{}^{\prime}},\hskip 14.22636pt\textbf{{S}}=1,\hskip
14.22636ptT=\omega^{2}$ $1^{{}^{\prime\prime}},\hskip
14.22636pt\textbf{{S}}=1,\hskip 14.22636ptT=\omega.$

The three dimensionals unitary representations of T and S are,

$T=\begin{bmatrix}1&0&0\\\ 0&\omega^{2}&0\\\ 0&0&\omega\\\ \end{bmatrix},$
(45)

and

$\textbf{{S}}=\frac{1}{3}\begin{bmatrix}-1&2&2\\\ 2&-1&2\\\ 2&2&-1\\\
\end{bmatrix}.$ (46)

The multiplication rules for the singlet and triplet representations of two
generators S and T of $A_{4}$ are,

$1\otimes 1=1,\hskip 8.5359pt1^{{}^{\prime\prime}}\otimes
1^{{}^{\prime\prime}}=1^{{}^{\prime}}$ $1^{{}^{\prime}}\otimes
1^{{}^{\prime\prime}}=1,\hskip 8.5359pt1^{{}^{\prime}}\otimes
1^{{}^{\prime}}=1^{{}^{\prime\prime}}$ $3\otimes
3=3_{1}+3_{2}+1+1^{{}^{\prime}}+1^{{}^{\prime\prime}}.$

$A_{4}$ is a symmetry group of tetrahedron. There are twelve independent
transformations of the tetrahedron and hence there are twelve group elements
as follows:
a. four rotations by $120^{0}$ clockwise $\left(\text{as seen from the
vertex}\right)$ which are T-type,
b. four rotations by $120^{0}$ anticlockwise $\left(\text{as seen from the
vertex}\right)$,
c. three rotations by $180^{0}$ $-$ S type,
d. 1 unit operator 1.

$A_{4}$ has four irreducible representations which are three singlets
$-1,1^{{}^{\prime}},1^{{}^{\prime\prime}}$ and one triplet. The products of
singlets are:

$1\otimes 1=1,\hskip 8.5359pt1^{{}^{\prime\prime}}\otimes
1^{{}^{\prime\prime}}=1^{{}^{\prime}}$ $1^{{}^{\prime}}\otimes
1^{{}^{\prime\prime}}=1,\hskip 8.5359pt1^{{}^{\prime}}\otimes
1^{{}^{\prime}}=1^{{}^{\prime\prime}}.$

If we consider two triplets $a=\left(a_{1},a_{2},a_{3}\right)$,
$b=\left(b_{1},b_{2},b_{3}\right)$ then one can write,

$\displaystyle\left(ab\right)_{1}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3},$ (47)
$\displaystyle\left(ab\right)_{1^{{}^{\prime}}}=a_{1}b_{1}+\omega^{2}a_{2}b_{2}+\omega
a_{3}b_{3},$ (48)
$\displaystyle\left(ab\right)_{1^{{}^{\prime\prime}}}=a_{1}b_{1}+\omega
a_{2}b_{2}+\omega^{2}a_{3}b_{3},$ (49)
$\displaystyle\left(ab\right)_{3_{1}}=a_{2}b_{3}+a_{3}b_{1}+a_{1}b_{2},$ (50)
$\displaystyle\left(ab\right)_{3_{2}}=a_{3}b_{2}+a_{1}b_{3}+a_{2}b_{1},$ (51)
$\displaystyle\omega^{3}=1.$ (52)

In the basis of triplet representations

$\textbf{{S}}=\begin{bmatrix}1&0&0\\\ 0&-1&0\\\ 0&0&-1\\\ \end{bmatrix},$ (53)
$T=\begin{bmatrix}0&1&0\\\ 0&0&1\\\ 1&0&0\\\ \end{bmatrix}.$ (54)

One generates 12 real $3\times 3$ matrix group elements, after multiplication
of the matrices together in all possible ways, like

$\textbf{{S}}^{2}=\frac{1}{9}\begin{bmatrix}-1&2&2\\\ 2&-1&2\\\ 2&2&-1\\\
\end{bmatrix}\begin{bmatrix}-1&2&2\\\ 2&-1&2\\\ 2&2&-1\\\
\end{bmatrix}=\frac{1}{9}\begin{bmatrix}9&0&0\\\ 0&9&0\\\ 0&0&9\\\
\end{bmatrix}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\ \end{bmatrix}.$ (55)

$A_{4}$ has four classes 39  denoted by,

$a)\hskip 5.69046ptC_{1}:\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\
\end{bmatrix},$ (56)

which is the $3\times 3$ matrix representation of $A_{4}$ elements in $C_{1}$.
Likewise we have,

$b)\hskip 5.69046ptC_{2}:\begin{bmatrix}1&0&0\\\ 0&-1&0\\\ 0&0&-1\\\
\end{bmatrix},\begin{bmatrix}-1&0&0\\\ 0&1&0\\\ 0&0&-1\\\
\end{bmatrix},\begin{bmatrix}-1&0&0\\\ 0&-1&0\\\ 0&0&1\\\ \end{bmatrix}.$ (57)
$c)\hskip 5.69046ptC_{2}:\begin{bmatrix}0&1&0\\\ 0&0&1\\\ 1&0&0\\\
\end{bmatrix},\begin{bmatrix}0&-1&0\\\ 0&0&-1\\\ 1&0&0\\\
\end{bmatrix},\begin{bmatrix}0&-1&0\\\ 0&0&1\\\ -1&0&0\\\
\end{bmatrix},\begin{bmatrix}0&1&0\\\ 0&0&-1\\\ -1&0&0\\\ \end{bmatrix}.$ (58)
$d)\hskip 5.69046ptC_{2}:\begin{bmatrix}0&0&1\\\ 1&0&0\\\ 0&1&0\\\
\end{bmatrix},\begin{bmatrix}0&0&-1\\\ 1&0&0\\\ 0&-1&0\\\
\end{bmatrix},\begin{bmatrix}0&0&1\\\ -1&0&0\\\ 0&-1&0\\\
\end{bmatrix},\begin{bmatrix}0&0&-1\\\ -1&0&0\\\ 0&1&0\\\ \end{bmatrix},$ (59)

where $Z_{3}$ elements are

$\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\
\end{bmatrix},\begin{bmatrix}0&1&0\\\ 0&0&1\\\ 1&0&0\\\
\end{bmatrix},\begin{bmatrix}0&0&1\\\ 1&0&0\\\ 0&1&0\\\ \end{bmatrix}$

and $Z_{2}\times Z_{2}$ elements in this classification are;

$\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\
\end{bmatrix},\begin{bmatrix}1&0&0\\\ 0&-1&0\\\ 0&0&-1\\\
\end{bmatrix},\begin{bmatrix}-1&0&0\\\ 0&1&0\\\ 0&0&-1\\\
\end{bmatrix},\begin{bmatrix}-1&0&0\\\ 0&-1&0\\\ 0&0&1\\\ \end{bmatrix}.$ (60)

## References

  * (1) Kalpana Bora, Gayatri Ghosh, Charged Lepton Flavor Violation $\mu\rightarrow e\gamma$ in $\mu$-$\tau$ Symmetric SUSY SO(10) mSUGRA, NUHM, NUGM and NUSM Theories and LHC, Eur. Phys. J. C75, 9, 428, (2015), arXiv:1410.1265 [hep-ph].
  * (2) P.F. de Salas, D.V. Forero, C.A. Ternes, M. Tortola and J.W.F. Valle, Status of neutrino oscillations 2018: 3$\sigma$ hint for normal mass ordering and improved CP sensitivity, Phys. Lett. B 782 (2018) 633 [arXiv:1708.01186] [ IN SPIRE ].
  * (3) D.V. Forero, M. Tortola and J.W.F. Valle, Neutrino oscillations refitted, Phys. Rev. D 90 (2014) 093006 [arXiv:1405.7540] [ IN SPIRE ].
  * (4) D.V. Forero, M. Tortola and J.W.F. Valle, Global status of neutrino oscillation parameters after Neutrino-2012, Phys. Rev. D 86 (2012) 073012 [arXiv:1205.4018] [ IN SPIRE ].
  * (5) T. Schwetz, M. Tortola and J.W.F. Valle, Where we are on $\theta_{13}$: addendum to ‘Global neutrino data and recent reactor fluxes: status of three-flavour oscillation parameters’, New J. Phys. 13 (2011) 109401 [arXiv:1108.1376] [ IN SPIRE ].
  * (6) SNO collaboration, Direct evidence for neutrino flavor transformation from neutral current interactions in the Sudbury Neutrino Observatory, Phys. Rev. Lett. 89 (2002) 011301 [nucl-ex/0204008] [ IN SPIRE ].
  * (7) Super-Kamiokande collaboration, Evidence for oscillation of atmospheric neutrinos, Phys. Rev. Lett. 81 (1998) 1562 [hep-ex/9807003] [ IN SPIRE ].
  * (8) A.B. McDonald, Nobel Lecture: The Sudbury Neutrino Observatory: Observation of flavor change for solar neutrinos, Rev. Mod. Phys. 88 (2016) 030502.
  * (9) T. Kajita, Nobel Lecture: Discovery of atmospheric neutrino oscillations, Rev. Mod. Phys. 88 (2016) 030501.
  * (10) KamLAND collaboration, First results from KamLAND: Evidence for reactor anti-neutrino disappearance, Phys. Rev. Lett. 90 (2003) 021802 [hep-ex/0212021] [ IN SPIRE ].
  * (11) K. Babu, E. Ma, and J. Valle, Underlying A(4) symmetry for the neutrino mass matrix and the quark mixing matrix,” Phys.Lett. B552, 207213, 2003.
  * (12) E. Ma and G. Rajasekaran, ”Softly broken A(4) symmetry for nearly degenerate neutrino masses,” Phys.Rev. , D64, p. 113012, 2001.
  * (13) G. Altarelli and F. Feruglio, Tri-bimaximal neutrino mixing from discrete symmetry in extra dimensions,” Nucl.Phys., B720, 6488, 2005.
  * (14) G. Altarelli and F. Feruglio, Tri-bimaximal neutrino mixing, A(4) and the modular symmetry,” Nucl.Phys., B741, 215235, 2006;
  * (15) X.-G. He, Y.-Y. Keum, and R. R. Volkas, A(4) avor symmetry breaking scheme for understanding quark and neutrino mixing angles,” JHEP, 0604, 039, 2006.
  * (16) J. Schechter and J. W. F. Valle,  Phys. Rev. D 22, 2227 (1980).
  * (17) P. F. Harrison, D. H. Perkins, and W. G. Scott, Phys. Lett. B 530, 167 (2002),[arXiv: hep-ph/0202074].
  * (18) I. Stancu, D. V. Ahluwalia, L/E-Flatness of the Electron-Like Event Ratio in Super-Kamiokande and a Degeneracy in Neutrino Masses,  Phys.Lett.B460, 1999, arXiv:hep-ph/9903408.
  * (19) F. P. An et al., Phys. Rev. D 95, 072006 (2017).
  * (20) M. Y. Pac (RENO Collaboration), Proc. Sci., NuFact2017, 38 (2018), https://pos.sissa.it/295/038/pdf; Y. Abe et al. (Double Chooz Collaboration), J. High Energy Phys. 10 (2014) 086; 02 (2015) 074.
  * (21) Gayatri Ghosh, Significance of Broken $\mu-\tau$ Symmetry in correlating $\delta_{CP},\theta_{13}$, Lightest Neutrino Mass and Neutrinoless Double Beta Decay, ,  Adv.High Energy Phys. 2021 (2021) 9563917.
  * (22) Biswajit Karmakar, Arunansu Sil, Spontaneous CP Violation in Lepton-sector a common origin for $\theta_{13}$, Dirac CP phase and leptogenesis,  Phys. Rev. D 93, 013006 (2016), arXiv:1509.07090 [hep-ph].
  * (23) Hong-Jian He and Fu-Rong Yin, Common Origin of $\mu-\tau$ and CP Breaking in Neutrino Seesaw, Baryon Asymmetry, and Hidden Flavor Symmetry,  Phys. Rev. D84 (2011) 033009 [arXiv:1104.2654].
  * (24) S.F. Ge, H.J. He, and F.R. Yin, Common Origin of Soft $\mu-\tau$ and CP Breaking in Neutrino Seesaw and the Origin of Matter, JCAP 1005 (2010) 017 [arXiv:1001.0940]
  * (25) Hong-Jian He and Xunjie Xu, Octahedral Symmetry with Geometrical Breaking: New Prediction for Neutrino Mixing Angle $\theta_{13}$ and CP Violation,  Phys. Rev. D86 (2012) 111301 (R) [arXiv:1203.2908].
  * (26) Hong-Jian He, Werner Rodejohann, Xun-Jie Xu, Origin of Constrained Maximal CP Violation in Flavor Symmetry,  Phys. Lett. B751 (2015) 586 [arXiv:1507.03541].
  * (27) RENO collaboration, Measurement of Reactor Antineutrino Oscillation Amplitude and Frequency at RENO, Phys. Rev. Lett. 121 (2018) 201801 [arXiv:1806.00248] [ INSPIRE ].
  * (28) Daya Bay collaboration, Measurement of the Electron Antineutrino Oscillation with 1958 Days of Operation at Daya Bay, Phys. Rev. Lett. 121 (2018) 241805 [arXiv:1809.02261] [ IN SPIRE ].
  * (29) Moinul Hossain Rahat, Pierre Ramond, and Bin Xu, Asymmetric tribimaximal texture Phys. Rev. D 98, 055030, arXiv:1805.10684 [hep-ph] [ INSPIRE ].
  * (30) M. Jay Pérez, Moinul Hossain Rahat, Pierre Ramond, Alexander J. Stuart, and Bin Xu,  Phys. Rev. D 100, 075008, arXiv:1907.10698 [hep-ph] [ INSPIRE ].
  * (31) Moinul Hossain Rahat, Leptogenesis from the asymmetric texture, Phys. Rev. D 103, 035011, arXiv:2008.04204[hep-ph].
  * (32) M.Jay Perez, Moinul Hossain Rahat, Pierre Ramond, Alexander J.Stuart and Bin Xu, Tribimaximal mixing in the $SU(5)\times\tau_{13}$ texture,  Phys. Rev.D101, 075018; Chee Sheng Fong, Moinul Hossain Rahat, Shaikh Saad, Low-scale Resonant Leptogenesis in SU(5) GUT with $\tau_{13}$ Family Symmetry, arXiv:2103.14691 [hep-ph].
  * (33) G.C. Branco, R. Gonzalez Felipe, M.N. Rebelo, H. Serodio  Phys.Rev. D79 (2009) 093008.
  * (34) D. Wyler, Phys. Rev. D 19, 3369 (1979); G. C. Branco, H. P. Nilles and V. Rittenberg, Phys. Rev. D 21, 3417 (1980).
  * (35) E. Ma and G. Rajasekaran, Phys. Rev. D 64, 113012 (2001) [arXiv:hep-ph/0106291]; K. S. Babu, E. Ma and J. W. F. Valle, Phys. Lett. B 552, 207 (2003) [arXiv:hep-ph/0206292]; E. Ma, Phys. Rev. D 70, 031901 (2004) [arXiv:hep-ph/0404199].
  * (36) G. Altarelli and F. Feruglio, Nucl. Phys. B 741, 215 (2006) [arXiv:hep-ph/0512103].
  * (37) X. G. He, Y. Y. Keum and R. R. Volkas, JHEP 0604, 039 (2006) [arXiv:hep-ph/0601001].
  * (38) I. de Medeiros Varzielas, S. F. King and G. G. Ross,  Phys. Lett. B 644, 153 (2007) [arXiv:hep-ph/0512313].
  * (39) M. Hirsch, S. Morisi and J. W. F. Valle, Phys. Rev. D 78, 093007 (2008) [arXiv:0804.1521 [hep-ph]].
  * (40) KamLAND collaboration, Precision Measurement of Neutrino Oscillation Parameters with KamLAND, Phys. Rev. Lett. 100 (2008) 221803 [arXiv:0801.4589] [ IN SPIRE ]; KamLAND collaboration, Constraints on $\theta_{13}$ from A Three-Flavor Oscillation Analysis of Reactor Antineutrinos at KamLAND, Phys. Rev. D 83 (2011) 052002 [arXiv:1009.4771] [ IN SPIRE ]; KamLAND collaboration, Reactor On-Off Antineutrino Measurement with KamLAND, Phys. Rev. D 88 (2013) 033001 [arXiv:1303.4667] [ IN SPIRE ].
  * (41) M. Dentler, A. Hernández-Cabezudo, J. Kopp, M. Maltoni and T. Schwetz, Sterile neutrinos or flux uncertainties — Status of the reactor anti-neutrino anomaly, JHEP 11 (2017) 099 [arXiv:1709.04294] [ IN SPIRE ]; S. Gariazzo, C. Giunti, M. Laveder and Y.F. Li, Model-independent $\bar{\nu_{e}}$ short-baseline oscillations from reactor spectral ratios, Phys. Lett. B 782 (2018) 13 [arXiv:1801.06467] [ IN SPIRE ].
  * (42) P.F. de Salas, D.V. Forero, S. Gariazzo, P. Martínez-Miravé, O. Mena, C.A. Ternes, M. Tórtola, J.W.F. Valle, global reassessment of the neutrino oscillation picture; JHEP 2102 (2021) 071, arXiv:2006.11237 [hep-ph].
  * (43) MEG Collaboration, Manuel Meucci, Nuovo Cim.C 43(2020) 48, arXiv:1912.08656.
  * (44) W. Grimus, Theory of Neutrino Masses and Mixing, Phys.Part.Nucl.,42, pp. 566 576, 2011.
  * (45) S.A. Abel, Subir Sarkar, P.L.White, On the cosmological domain wall problem for the minimally extended supersymmetric standard model, Nucl.Phys.B 454 (1995), hep$-$ph$/$9506359[hep-ph].
  * (46) F. Feruglio, C. Hagedorn, and R. Ziegler, Lepton Mixing Parameters from Discrete and CP Symmetries, JHEP , 1307, 027, 2013.
  * (47) M.$-$C. Chen, J. Huang, K. Mahanthappa, and A. M. Wijangco, JHEP 1310, 112, 2013.
  * (48) Y. BenTov, X.-G. He, and A. Zee, ”An $A_{4}\times Z_{4}$ model for neutrino mixing,” JHEP 1212, 093, 2012.
  * (49) M.C. Gonzalez-Garcia, M. Yokoyama, neutrino Masses, Mixings and oscillations, in review of PDG-2019.
  * (50) Kalpana Bora, Gayatri Ghosh, Debajyoti Dutta, Octant Degeneracy and Quadrant of Leptonic CPV Phase at Long Baseline $\nu$ Experiments and Baryogenesis, Adv.High Energy Phys. 2016 (2016) 9496758.
  * (51) Gayatri Ghosh, Kalpana Bora, Resolving Entanglement of CPV Phase with Octant of $\theta_{23}$, and Leptogenesis, Springer Proc.Phys. 174 (2016) 287-29.
  * (52) Naoyuki Haba, Yukihiro Mimura, Toshifumi Yamada , $\theta_{23}$ octant measurement in 3+1 neutrino oscillations in T2HKK, Phys.Rev. D101 (2020) 7, 075034.
  * (53) T2K Collaboration (K. Abe et al.), Measurement of neutrino and antineutrino oscillations by the T2K experiment including a new additional sample of $\nu_{e}$interactions at the far detector  Phys.Rev. D96 (2017) no.9, 092006, Erratum: Phys.Rev. D98 (2018) no.1, 019902, arXiv:1707.01048 [hep-ex].