text:
\section{Introduction}\label{sec1}

The left-right symmetric model (LRSM) which was initially proposed as
the most economical approach to restore parity came a long way in explaining neutrino mass, lepton number violation,
dark matter, baryon asymmetry of the universe thereby gaining popularity~\cite{Mohapatra:1974gc, Pati:1974yy, Senjanovic:1975rk, Senjanovic:1978ev,
Mohapatra:1980yp,Akhmedov:1995vm,Heeck:2015qra,Garcia-Cely:2015quu,Patra:2015vmp,Patra:2015qny}. All accolades to its gauge group,
i.e. $SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}$ which naturally gives room to a right handed neutrino
and obeys a complete symmetry between left and right chirality untill spontaneous symmetry breaking
occurs. The model predicts
$W_R^{\pm}$ and $Z^\prime$ gauge bosons which couple to Standard Model fields and heavy Majorana neutrino $N$
and these exotic states at low scale offer rich collider phenomenology. Neutrino mass
can be explained within LRSM via various seesaw mechanisms like the canonical seesaw \cite{Minkowski:1977sc,Mohapatra:1979ia,Dev:2013oxa,Agostinho:2017wfs,Magg:1980ut,Schechter:1980gr,Cheng:1980qt,Lazarides:1980nt} and its lowscale variants like inverse seesaw~\cite{Dev:2009aw}, linear seesaw,
extended seesaw etc~\cite{Mohapatra:2005wg}. The issue with canonical seesaw is that it links the smallness of neutrino mass to a very heavy right-handed scale which can't be
verified by the current or planned experiments. Whereas in case of low scale seesaw heavy neutrino mass can lie in the TeV range which is
experiment friendly and moreover it offers rich phenomenology like lepton flavor violation~\cite{Cirigliano:2004mv,Cirigliano:2004tc,Riazuddin:1981hz,Pal:1983bf,Mohapatra:1992uu} and new physics contributions to lepton number violating decays like
neutrinoless double beta decay~\cite{Deppisch:2017vne, Hati:2018tge, Pritimita:2016fgr,Ge:2015bfa, Deppisch:2014zta, Tello:2010am,Deppisch:2014qpa,Patra:2014goa,Borah:2013lva,Awasthi:2013ff,Patra:2012ur,Chakrabortty:2012mh,Dev:2013vxa,Barry:2013xxa,Nemevsek:2011hz,Dev:2014iva,Keung:1983uu,Das:2012ii,Bertolini:2014sua,Beall:1981ze,Ge:2015yqa,Deppisch:2015cua,Hirsch:1996qw,Dev:2013oxa,Dev:2015pga,Bajc:2009ft}.
Inverse seesaw and linear seesaw can be realized with the introduction of extra singlet fermions per generation
to the LRSM where the mass matrix in the basis $(\nu_L, ~\nu^c_R, ~S_L)$ can be written as,
\begin{equation}
	\mathbb{M} =
	\begin{pmatrix}
		0 & m_\text{D} & m_\text{LS} \\
		m_\text{D}^\text{T} & 0 & m_{RS} \\
		m_\text{LS}^\text{T} & m^\text{T}_{RS} & \mu
	\end{pmatrix}.
\end{equation}
Thus the light neutrino mass formula becomes,
$m^{\rm inv}_\nu = \left(\frac{m_D}{m_{RS}} \right) \mu \left(\frac{m_D}{m_{RS}} \right)^T$ in the former case and
$m^{\rm lin}_\nu = m_\text{D} m_{RS}^{-1} m_\text{LS}^\text{T} \mbox{+ transpose}$ in the later case. It can be
interpreted from the formula that it allows
order one magnitude of Dirac Yukawa coupling, large light-heavy neutrino mixing and heavy neutrinos of mass few TeV.
In \cite{Pritimita:2016fgr}, LRSM has been extended to study natural
type-II seesaw dominance which allows large light-heavy neutrino mixing thus leading to many new physics contributions
to neutrinoless double beta decay along with constraints on light neutrino mass. Another interesting variant is
universal seesaw  which allows all the quarks and leptons to get mass from a common seesaw due to the addition of
vector-like fermions in LRSM \cite{Deppisch:2017vne}.

However we aim to study here the $A_4$ extension of LRSM which offers the advantage of studying neutrino mass, non-unitarity effect in linear seesaw,
lepton flavour violation and CP violation within one framework. $A_4$, the discrete group of even permutations of four objects is the smallest non-Abelian group
containing triplet irreducible representations. While it was first proposed in \cite{Ma:2001dn} to study lepton masses and mixing,
several other $A_4$-based flavour models have been suggested after that mostly to shed light on the
flavour problem \cite{Babu:2002dz,Altarelli:2005yp,Holthausen:2012wz,King:2013hj}. In one such recent work \cite{Karmakar:2016cvb} the authors
have elegantly explained the origin of non-zero $\theta_{13}$
and leptogenesis via inverse seesaw with $A_4$ extension of SM. However in the model the light-heavy neutrino
mixing is proportional to identity and thus the branching ratios of
LFV decays become vanishingly small. Similarly, another paper which considers realization of linear seesaw with $A_4$
symmetry gives suppressed contributions to lepton flavour violating (LFV) decays due to the chosen model parameters \cite{Sruthilaya:2017mzt}. This can be ameliorated by
considering $A_4$ realization of LRSM where large non-unitarity effect can be achieved and thus it can lead to
dominant contributions to LFV decays, which is the primary motive of this work.
The embedding of $A_4$ group into left-right flavour symmetry has been attempted previously in order to explain charged fermion mass hierarchies
and quark and lepton mixing angles \cite{Morisi:2007ft,Ma:2006wm}.

In this work, we have considered $A_4$ realization of left-right symmetric linear seesaw mechanism. The fermion
content of the model includes usual quarks, leptons along with additional sterile fermion
per generation while the scalar content includes Higgs doublets with $B-L=1$ and scalar bidoublet
with $B-L=0$. Within this scenario all the fermion masses get simpler mass structure for neutrino phenomenology.
While the right-handed Higgs doublet $H_R$ plays the crucial role of left-right symmetry breaking, its left handed partner $H_L$ has merely any role.
Thus the non-zero VEV of $H_L$ can be taken too small. As usual the scalar bidoublet $\Phi$ plays the role of electroweak
symmetry breaking. The introduction of $A_4$ symmetry helps to avoid the $\mu$ term and hence, inverse seesaw term
for light neutrino masses is absent. The other good points of the model are large light-heavy neutrino mixing, prominent non-unitarity effects,
dominant lepton flavour violating effects and CP-violating effects. The work also contains a number of plots showing
correlation among model parameters and the experimentally observed parameters.

The paper is organised as follows. In Sec \ref{sec:LRSM} we briefly note the features of manifest left-right symmetric model and then move to explain
the realization of linear seesaw structure with $A_4$ extension of the left-right model. In Sec \ref{sec:mass-mixing} we discuss neutrino masses and mixing. We do so by
setting up the flavour structure of neutrino mass matrices and establish analytically the correlation among model parameters. In Sec \ref{sec:numerical result} we estimate
numerically the correlation among model parameters by using the values of experimentally measued neutrino parameters. Sec \ref{sec:non unitarity} explains non-unitarity effects
in linear seesaw and Sec \ref{sec:LFV} has a discussion on various low energy lepton flavour violating processes that the model facilitates. In Sec \ref{sec:cp} we study
leptonic CP violation for active neutrinos using Jarlskog invariants and in Sec \ref{sec:conclusion} we summerize the work.

\section{Left-right symmetric model with linear sessaw}
\label{sec:LRSM}
The gauge group of left-right symmetric model (LRSM) is given by
\begin{equation}
 SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}
\end{equation}
where $B-L$ stands for the difference between baryon number and lepton number. The standard lepton and quark content of the model
is given by

\begin{eqnarray}
 \ell_L=\left(
\begin{array}{c}
\nu_{{L}}\\
e_{L}
\end{array}
\right)\sim (1,2,1,-1), \hspace{1cm}
\ell_R=\left(
\begin{array}{c}
\nu_{{R}}\\
e_{R}
\end{array}
\right)\sim (1,1,2,-1)\\
 q_L=\left(
\begin{array}{c}
u_{{L}}\\
d_{L}
\end{array}
\right)\sim (3,2,1,\frac{1}{3}), \hspace{1cm}
q_R=\left(
\begin{array}{c}
u_{{R}}\\
d_{R}
\end{array}
\right)\sim (3,1,2,\frac{1}{3}).
\end{eqnarray}

The scalar sector of a general LRSM contains two Higgs doublets and a bidoublet
\begin{eqnarray}
 H_L=\left(
\begin{array}{c}
h^0_{L}\\
h^-_{L}
\end{array}
\right)\sim (1,2,1,-1), \hspace{1cm}
H_R=\left(
\begin{array}{c}
h^0_{R}\\
h^-_{R}
\end{array}
\right)\sim (1,1,2,-1),
\end{eqnarray}

\begin{eqnarray}
 \Phi=\left(
 \begin{array}{cc}
 \phi^0_1 & \phi^+_2\\
 \phi^{-}_1 & \phi^{0}_2
 \end{array}
\right)\sim (1,2,2,0).
\end{eqnarray}

In order to generate neutrino mass through linear seesaw mechanism within LRSM, we will have to add one left-right gauge singlet
neutral fermion $S_L$ to the model. Now the complete Lagrangian for the leptonic sector becomes,

\begin{eqnarray}
 -\mathcal{L}_{\rm lepton}&=&
\overline{\ell_L}\left(Y \Phi+\tilde{Y} \tilde{\Phi}\right)\ell_R
                        \nonumber\\
                         & &+Y_L \overline{\ell_L} H_L S_L + Y_R \overline{\ell_R} H_R S_L
                         +h.c.
\end{eqnarray}
Once the scalars $H_L, H_R$ and $\Phi$ obtain VEV as,
\begin{eqnarray}
\langle H_L \rangle=\left(
\begin{array}{c}
0\\
v_L
\end{array}
\right), \hspace{.5cm}
\langle H_R \rangle=\left(
\begin{array}{c}
0\\
v_R
\end{array}
\right), \hspace{0.5cm}
\langle \Phi \rangle=\left(
 \begin{array}{cc}
 v_1 & 0\\
 0 & v_2
 \end{array}
\right),
\end{eqnarray}
the above Lagrangian can be written as,
\begin{eqnarray}
 -\mathcal{L}_{\rm lepton}&=& m_{LR} \overline{\nu_L}\nu_R
                              + m_{RS} \overline{\nu_R^c}S_L+ m_{LS} \overline{\nu_L} S_L + h.c.
\end{eqnarray}
Hence in the the basis  $(\nu_L, \nu_R^c, S_L)$, the effective $9\times 9$ mass
matrix
can be written as
\begin{eqnarray}
M_{\nu} = \left(
\begin{array}{ccc}
         0     & m_{LR}     &m_{LS}\\
         m^T_{LR} & 0       &m_{RS}\\
         m^T_{LS}     & m^T_{RS}     & 0
\end{array}
\right),
\end{eqnarray}
where $m_{LR}=Y_1 v_1 + Y_2 v_2^*$
and $m_{RS}=y_1 v_R$.
In the above matrix, one may wonder why the $\mu$ term is absent, which will be clarified in the next section once we introduce $A_4$ symmetry.
Now, with $m_{LS} << m_{LR} < m_{RS}$, the light neutrino masses are obtained from the formula,
\begin{eqnarray}\label{mnu1}
 m_{\nu}&=&m_{LR} m^{-1}_{RS} m^T_{LS} + \mbox{transpose}
\end{eqnarray}

\subsection{An $A_4$ realization of left-right symmetric linear seesaw mechanism}

\begin{table}
\begin{center}
\begin{tabular} {|c|c|c|c|c|c|c|c|c|}\hline
& Fields & $SU(3)_c$ & $SU(2)_L$ & $SU(2)_R$ & $B-L$ & $A_4$ & $Z_4$ & $Z_3$\\
\hline
& $\ell_{L_{1,2,3}}$ & {\bf 1} & {\bf 2} & {\bf 1} & $-1$ & $1,1'',1'$ & $-1$ &1\\
& $\ell_{R}$ & {\bf 1} & {\bf 1} & {\bf 2} & $-1$ & $3$ & $i$
&1\\
& $S_L$ & {\bf 1} & {\bf 1} & {\bf 1} & 0 & 3 & $i$ &$\omega^2$ \\
\hline
& $\Phi$ & {\bf 1} & {\bf 2} & {\bf 2} & 0 & 1  &1 &1\\
& $H_L$ & {\bf 1} & {\bf 2} & {\bf 1} & $-1$ & 1 & 1 & $\omega$\\
& $H_R$ & {\bf 1} & {\bf 1} & {\bf 2} & $-1$ & 1 & 1 &$\omega^2$\\
\hline
& $\phi_{S}$ & {\bf 1} & {\bf 1} & {\bf 1} & $0$ & 3 & $-1$ &1\\
& $\phi_{T}$ & {\bf 1} & {\bf 1} & {\bf 1} &  $0$ & 3 & $i$ &1\\
& $\xi$ & {\bf 1} & {\bf 1} & {\bf 1} & $0$ & 1 & $-1$ &1\\
& $\xi'$ & {\bf 1} & {\bf 1} & {\bf 1} &  $0$ & $1'$  & $-1$ &1\\
\hline
\end{tabular}
\end{center}
\caption{Particle content and transformation properties under
$A_4$ based flavour left-right symmetric model. }
\label{tab:1}
\end{table}

$A_4$ symmetry group has four irreducible representations; three singlets, namely $1, 1^\prime, 1^{\prime\prime}$ and a triplet $3$.
The multiplication rules for the irreducible representations can be written as;
$1\otimes1=1$, $1^\prime\otimes1^\prime=1^{\prime\prime}$, $1^{\prime\prime}\otimes1^{\prime\prime}=1^\prime$, $1^\prime\otimes1^{\prime\prime}=1$.
The product of two $A_4$ triplets; ($a_1,a_2,a_3$) and ($b_1,b_2,b_3$)
can be written as,
\begin{eqnarray}
3\times 3&=&1+1^{\prime}+1^{\prime\prime}+3_A+3_S
\end{eqnarray}
\begin{eqnarray}
1\sim a_1b_1+a_2b_3+a_3b_2 \\ \nonumber
1^{\prime}\sim a_3b_3+a_1b_2+a_2b_2 \\ \nonumber
1^{\prime\prime}\sim a_2b_2+a_3b_1+a_1b_3 \\ \nonumber
3_S \sim \frac{1}{3}\left[\begin{array}{c}
2a_1b_1-a_2b_3-a_3b_2\\
2a_3b_3-a_1b_2-a_2b_1\\
2a_2b_2-a_1b_3-a_3b_1
\end{array} \right]\\ \nonumber
3_A\sim \frac{1}{2}\left[\begin{array}{c}
a_2b_3-a_3b_2\\
a_1b_2-a_2b_1\\
a_3b_1-a_1b_3
\end{array} \right]\\ \nonumber
\end{eqnarray}
With the particle content and symmetries mentioned in Table \ref{tab:1}, the
Lagrangian involved in generation of the mass matrices in a left-right $A_4$ flavor symmetric framework can be written as,
\begin{align}\label{a4lag}
 -\mathcal{L}_{\rm lepton}=&\mathcal{L}_{\nu_L \nu_R}+\mathcal{L}_{\nu_R S_L}+\mathcal{L}_{\nu_L S_L}\,,
\end{align}
where
\begin{align}
 \mathcal{L}_{\nu_L \nu_R}= \label{lag:diracmass}
&\frac{1}{\Lambda}(\overline{\ell_{L_1}})_{1} \left(Y \Phi+\widetilde{Y} \tilde{\Phi}\right) \left(\ell_R \phi_T\right)_{1} \nonumber \\
&+\frac{1}{\Lambda} (\overline{\ell_{L_2}})_{1^{\prime \prime}} \left(Y \Phi+\widetilde{Y} \tilde{\Phi}\right) \left(\ell_R \phi_T\right)_{1^\prime} \nonumber \\
&+\frac{1}{\Lambda} (\overline{\ell_{L_3}})_{1^\prime} \left(Y \Phi+\widetilde{Y} \tilde{\Phi}\right) \left(\ell_R \phi_T\right)_{1^{\prime \prime}}
\end{align}

With the vevs for the scalar and flavon fields as,
$\langle \phi_S \rangle= v_S(1,1,1), \langle \phi_T \rangle= v_T(1,0,0), \langle\xi \rangle=v_{\xi}, \langle\xi^{\prime}\rangle
=v_{\xi^{\prime}}$,
we obtain  the flavor structures of the involved mass matrices.

\begin{eqnarray}\label{mCL}
M_{\ell} =\frac{v_T}{\Lambda} \left(
\begin{array}{ccc}
         Y_{11} v_2+\widetilde{Y}_{11} v_1^* & 0                 &  0\\
         0                 & Y_{22} v_2+\widetilde{Y}_{22} v_1^* &  0\\
         0                 & 0                 & Y_{33} v_2+\widetilde{Y}_{33} v_1^*
\end{array}
\right).
\end{eqnarray}
In analogy to the charged lepton mass matrix, the Dirac mass for light neutrinos can be written as,
\begin{eqnarray}
m_{LR} &=&\frac{v_T}{\Lambda} \left(
\begin{array}{ccc}
         Y_{11} v_1+\widetilde{Y}_{11} v_2^* & 0                 &  0\\
         0                 & Y_{22} v_1+\widetilde{Y}_{2} v_2^* &  0\\
         0                 & 0                 &  Y_{33} v_1+\widetilde{Y}_{33} v_2^*
\end{array}
\right)\\
     &=&\frac{v_T}{\Lambda} Y_{D}v \label{md} \left(
\begin{array}{ccc}
         1 & 0  &  0\\
         0 & 1  &  0\\
         0 & 0  &  1
\end{array}
\right),
\end{eqnarray}
where we have considered, $Y_D v \equiv Y v_1+\widetilde{Y} v_2^*=Y_{11,22,33} v_1+\widetilde{Y}_{11,22,33} v_2^*$.

It is seen from Table-1 that under $A_4$, $\ell_L$ transforms as $1, 1^{\prime}, 1^{\prime \prime}$ for 1st, 2nd and 3rd generation of left-handed doublet
respectively, $S_{L}$ transforms as triplet and $H_L$, $H_R$ as singlets. Thus the $\nu_L-S_L$ mixing term; $\overline{\ell^c_L} H_L S_L$  and
the $\nu_R-S_L$ mixing term $\overline{\ell_R} H_R S_L$ are not allowed at tree level and are generated at dimension five level as follows,
\begin{align}
 \mathcal{L}_{\nu_L S_L} &= \frac{Y^{11}_L}{\Lambda} (\overline{\ell_{L_1}})_{1}
   H_L (S_L \phi_T)_{1}  \nonumber \\
&+\frac{Y^{22}_L}{\Lambda} (\overline{\ell_{L_2}})_{1^{\prime \prime}}
   H_L (S_L \phi_T)_{1^{\prime}} \nonumber \\
&+\frac{Y^{33}_L}{\Lambda} (\overline{\ell_{L_3}})_{1^{ \prime}}
   H_L (S_L \phi_T)_{1^{\prime \prime}}
\end{align}
Once $\langle H_L \rangle$ and $\langle \phi_T \rangle$ get vev, the $\nu_L-S_L$ mixing matrix becomes,
\begin{eqnarray}
m_{LS} &=&\frac{v_T}{\Lambda} \left(
\begin{array}{ccc}
         Y^{11}_L v_L & 0                 &  0\\
         0                 & Y^{22}_L v_L &  0\\
         0                 & 0                 &  Y^{33}_L v_L
\end{array}
\right)\\
     &=&\frac{v_T}{\Lambda} Y_{L} v_L \label{mLS} \left(
\begin{array}{ccc}
         1 & 0  &  0\\
         0 & 1  &  0\\
         0 & 0  &  1
\end{array}
\right),
\end{eqnarray}
where we have considered, $Y_L v_L =Y^{11,22,33}_L v_L$. As we have mentioned earlier, the scalar $H_R$ plays the crucial role of
left-right symmetry breaking, and $H_L$ is required only for left-right invariance. Thus $H_L$ gets a small induced vev which is much smaller than $H_R$.
This clarifies why $m_{LS}$ term is much smaller than the $m_{RS}$ term.

Similarly, the $\nu_R-S_L$ mixing term is generated at dimension five level as follows

\begin{align}
\mathcal{L}_{\nu_R S_L} & =  \frac{1}{\lambda}  \left(\lambda^{\phi_s} \phi_s
                            +\lambda^{\xi} \xi+
                          \lambda^{\xi^{\prime}} \xi^{\prime} \right) \overline{\ell_R} H_R S_L\;.
\end{align}
The advantage of forbidding $\overline{\nu_L}-S_L$ and $\overline{\nu_R}-S_L$ terms at tree level and generating them by dimension five operator is that
it helps in achieving large light-heavy neutrino mixing which gives large non-unitarity effects and lepton flavour violation.
It should be noted that the terms $\mathcal{L}_{\nu_L \nu_R}$, $\mathcal{L}_{\nu_L S_L}$ and $\mathcal{L}_{\nu_R S_L}$ represent the contributions
for Dirac neutrino mass connecting $\nu_L-\nu_R, \nu_L-S_L$, $\nu_R-S_L$ mixing, respectively.
If one looks at the mass formula for light neutrinos governed by linear seesaw mechanism given in Eq.\ref{mnu1},
one can use the mass hierarchy $m_{RS} \gg m_{LR}, m_{LS}$.

Using the   following vevs for the scalar and flavon fields
\begin{eqnarray}\label{e4a}
\langle \phi_S \rangle= v_S(1,1,1), \langle \phi_T \rangle= v_T(1,0,0), \langle\xi \rangle=v_{\xi},~ \langle\xi^{\prime}\rangle
=v_{\xi^{\prime}},~~
\end{eqnarray}
the various mass matrices are found to be,
\begin{eqnarray}
&&m_\text{LR}= \frac{Y_D v v_T}{\Lambda}\left(
\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 &0 \\
0 & 0 &1
\end{array}
\right)
,~~~~m_\text{LS}=\frac{Y_L v_L v_T}{\Lambda}
\left(
\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 &0 \\
0 & 0 &1
\end{array}
\right), \label{e4-b} \\ \label{e4-a}
&&m_{RS}=\frac{a}{3}\left(
\begin{array}{ccc}
2 & -1 & -1\\
-1 & 2 &-1 \\
-1 & -1 &2
\end{array}
\right)+
b\left(
\begin{array}{ccc}
1 & 0& 0\\
0 & 0 &1 \\
0& 1 &0
\end{array}
\right)+
d\left(
\begin{array}{ccc}
0 & 0& 1\\
0 & 1 &0 \\
1& 0 &0
\end{array}
\right),
\end{eqnarray}
where $a=\lambda^{\phi}v_S v_R/\Lambda$, $b=\lambda^{\xi}v_{\xi} v_R/\Lambda$ and $d=\lambda^{\xi^{\prime}} v_{\xi^{\prime}} v_R /\Lambda$.

\section{Neutrino Masses and Mixing}
\label{sec:mass-mixing}
In this section we focus on studying the correlation between different model parameters and their dependence on experimentally determined neutrino
parameters. We start by rewriting the mass matrix $m_\text{RS}$ (\ref{e4-a}) for calculational convenience as,
\begin{equation}
m_{RS}=\left(
\begin{array}{ccc}
2a/3+b & -a/3& -a/3\\
-a/3 & 2a/3 &-a/3+b \\
-a/3& -a/3+b &2a/3
\end{array}
\right)+
\left(
\begin{array}{ccc}
0 & 0& d\\
0 & d &0 \\
d& 0 &0
\end{array}
\right).
\end{equation}
The importance of this matrix is that it dictates the flavour structure of the light neutrino mass matrix $m_\nu$ in the linear seesaw scenario.
Moreover, the structure of this matrix is such that, it leads to lepton mixing consistent with
neutrino oscillation data ~\cite{Capozzi:2017ipn,Aker:2019qfn}.

Using Eqns.(\ref{e4a}),(\ref{e4-b}) and (\ref{e4-a}),
 one can write the light neutrino mass matrix as,
\begin{eqnarray}
m_{\nu}&=&m_{LR} m_{RS}^{-1} m_{LS}^T+\text{transpose} \nonumber\\
&=&k_1k_2\left(\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 &0 \\
0 & 0 &1
\end{array}\right) m_{RS}^{-1}\left(\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 &0 \\
0 & 0 & 1
\end{array}\right),
\end{eqnarray}
where the parameters $k_1$ and $k_2$  are related to the vevs through
\begin{eqnarray}
k_1=\sqrt{2} Y_D v \frac{v_T}{\Lambda}\;, ~~~~k_2=\sqrt{2} Y_L v_L \frac{v_T}{\Lambda}.\nonumber
\end{eqnarray}
 The inverse of light neutrino mass matrix becomes,
\begin{eqnarray}
m_{\nu}^{-1}
&=&\frac{1}{k_1k_2}\left(
\begin{array}{ccc}
2a/3+b & -a/3& -a/3\\
-a/3 & 2a/3 &-a/3+b \\
-a/3& -a/3+b &2a/3
\end{array}
\right)+\frac{1}{k_1k_2}\left(
\begin{array}{ccc}
0 & d& 0\\
d & 0 &0 \\
0& 0 &d
\end{array}
\right),
\end{eqnarray}
which in TBM~\cite{Harrison:2002er,Harrison:1999cf,Rashed:2011zs} basis  will have the form, i.e., $m_{\nu}^{-1^{\prime}}=U_\text{TBM}^T m_{\nu}^{-1}U_\text{TBM}$,
\begin{equation}
m_{\nu}^{-1^{\prime}}=\left(\begin{array}{ccc}
a+b-d/2 & 0 & -\frac{\sqrt{3}}{2}d\\
0 &~~ b+d~~ &0 \\
-\frac{\sqrt{3}}{2}d & 0 &a-b+d/2
\end{array}\right)\;.
\end{equation}
The above matrix $m_{\nu}^{-1^{\prime}}$ can be  diagonalized by $U_{13}^*$. Which means $m_\nu^{-1}$
can be diagonalized by $U_{TBM}\cdot U_{13}^*$ and $m_{\nu}$ by $U_{TBM}\cdot U_{13}$, while $m_{RS}$  by $U_{TBM}\cdot U_{13}^T$.
The matrix $U_{13}$ and $U_{TBM}$ are of the form
\begin{equation}
 U_{TBM}=\left(
 \begin{array}{ccc}
 \sqrt{\frac{2}{3}} & \sqrt{\frac{1}{3}} & 0 \\
 -\sqrt{\frac{1}{6}} & \sqrt{\frac{1}{3}} & \sqrt{\frac{1}{2}}\\
 -\sqrt{\frac{1}{6}}& \sqrt{\frac{1}{3}} & -\sqrt{\frac{1}{2}}
 \end{array}
 \right),\label{eqnTBM}
 \end{equation}
\begin{equation}
 U_{13}=\left(
 \begin{array}{ccc}
 \cos\theta & 0 & \sin\theta e^{-i \delta}\\
 0 & 1 & 0 \\
 -\sin\theta e^{i\delta}& 0 & \cos\theta
 \end{array}
 \right),\label{eqn14}
 \end{equation}
 where internal mixing angle $\theta$ and phase $\delta$ are expressed in terms of the
 mass matrix parameters  $d/b=\lambda_1e^{\phi_{db}}$, $a/b=\lambda_2e^{\phi_{ab}}$ as
 \begin{eqnarray}
 \tan 2\theta=-\frac{\sqrt{3}\lambda_1\cos\phi_{db}}{(\lambda_1 \cos\phi_{db}-2)\cos\delta +(2\lambda_2\sin\phi_{ab})\sin\delta}\;,\label{tant}
 \end{eqnarray}
 and
 \begin{eqnarray}
 \tan\delta=\frac{\sin\phi_{db}}{\lambda_2\cos(\phi_{ab}-\phi_{db})}\;.\label{tanpsi}
 \end{eqnarray}
 The purpose of rotating the $m_\nu$ matrix by $U_{TBM}$ followed by $U_{13}$ is to achieve non-zero reactor mixing angle $\theta_{13}$ and see the possible
 correlations between various parameters.

 Again, from Eq.3.1 and 3.2 it is found that eigenvalues of $m_{\nu}$ and $m_{RS}$ are related to each other as
 \begin{equation}
 m_i=\frac{k_1k_2}{M_i}\;.\label{e16}
 \end{equation}
  where $m_i$ and $M_i$ are $i^\text{th}$ eigenvalues of $m_{\nu}$ and $m_{RS}$ respectively. The eigenvalues of $m_{RS}$  can be expressed using
  defined parameters in terms of $\lambda_1$ and $\lambda_2$ as,
  \begin{eqnarray}
M_1&=&b\left[\lambda_2 e^{i\phi_{ab}}-\sqrt{1+\lambda_1^2e^{2i\phi_{db}}-\lambda_1 e^{i\phi_{db}}}\right], \nonumber\\
M_2&=&b\left[1+\lambda_1 e^{i\phi_{db}}\right], \nonumber\\
M_3&=&b\left[\lambda_2 e^{i\phi_{ab}}+\sqrt{1+\lambda_1^2e^{2i\phi_{db}}-\lambda_1 e^{i\phi_{db}}}\right],\label{eq17a}
 \end{eqnarray}

After some simple calculations, one can  write the heavy neutrino masses as
 \begin{eqnarray}
 M_1&=& |b|\left[(\lambda_2\cos\phi_{ab}-C)^2+(\lambda_2\sin\phi_{ab}-D)^2\right]^{1/2} ,\nonumber\\
 M_2&=& |b|\left[1+\lambda_1^2+2\lambda_1\cos\phi_{db}\right]^{1/2}, \nonumber\\
 M_3&=& |b|\left[(\lambda_2\cos\phi_{ab}+C)^2+(\lambda_2\sin\phi_{ab}+D)^2\right]^{1/2}\;,\label{eq17}
 \end{eqnarray}
 where
 \begin{eqnarray}
 C&=&\left[ \frac{A+\sqrt{A^2+B^2}}{2}\right]^{1/2}\;,~~~~~~~
 D=\left[ \frac{-A+\sqrt{A^2+B^2}}{2}\right]^{1/2}\;,\nonumber\\
 A&=&1+\lambda_1^2\cos 2\phi_{db}-\lambda_1\cos\phi_{db}\;,~~~~
 B=\lambda_1^2\sin 2\phi_{db}-\lambda\sin\phi_{db}\;.\label{abcd}
 \end{eqnarray}
 and the phases ($\phi_i$'s) of $M_i$, i.e., $M_i=|M_i|e^{i \phi_i}$  as
 \begin{eqnarray}
 \phi_1&=&\tan^{-1}\left[\frac{\lambda_2\sin\phi_{ab}-D}{\lambda_2\cos\phi_{ab}-C}\right]\;,\nonumber \\
 \phi_2&=&\tan^{-1}\left[\frac{\lambda_1\sin\phi_{db}}{1+\lambda_1\cos\phi_{db}}\right]\;, \nonumber\\
  \phi_3&=&\tan^{-1}\left[\frac{\lambda_2\sin\phi_{ab}+D}{\lambda_2\cos\phi_{ab}+C}\right].
 \end{eqnarray}

 The matrix which diagonalizes active neutrino mass matrix, $U_{\nu}$ is given by
 \begin{eqnarray}

 U_{\nu}&=&U_{TBM}\cdot U_{13}\cdot P\;,
\end{eqnarray}
with $P=\text{diag}(e^{-i\phi_1/2},e^{-i\phi_2/2},e^{-i\phi_3/2})$.
\\
 and the lepton mixing matrix, known as PMNS matrix is given by \cite{Pontecorvo:1957qd,Maki:1962mu}
\begin{equation}
U_{\rm PMNS}=U_\ell^{\dagger}\cdot U_{\nu}\;,
\end{equation}
Here $U_\ell=\mathbb{I}$, which implies,
\begin{equation}
U_{\rm PMNS}=U_{\rm TBM}\cdot U_{13}\cdot P, \label{eqn15}
\end{equation}
and this looks to be in good agreement with the experimental observations \cite{Chao:2011sp,M.:2014kca}.
The PMNS matrix can be parametrized in terms of three mixing angles ($\theta_{13}$, $\theta_{23}$ and $\theta_{12}$)
and three phases (one Dirac phase $\delta_{CP}$, and two Majorana phases $\rho$ and $\sigma$) as
\begin{equation}
U_\text{PMNS}=\left( \begin{array}{ccc} c^{}_{12} c^{}_{13} & s^{}_{12}
c^{}_{13} & s^{}_{13} e^{-i\delta_{CP}} \\ -s^{}_{12} c^{}_{23} -
c^{}_{12} s^{}_{13} s^{}_{23} e^{i\delta_{CP}} & c^{}_{12} c^{}_{23} -
s^{}_{12} s^{}_{13} s^{}_{23} e^{i\delta_{CP}} & c^{}_{13} s^{}_{23} \\
s^{}_{12} s^{}_{23} - c^{}_{12} s^{}_{13} c^{}_{23} e^{i\delta_{CP}} &
-c^{}_{12} s^{}_{23} - s^{}_{12} s^{}_{13} c^{}_{23} e^{i\delta_{CP}} &
c^{}_{13} c^{}_{23} \end{array} \right) P^{}_\nu \;,\label{pmns}
\end{equation}
where $c_{ij}=\cos\theta_{ij}$ and $s_{ij}=\sin\theta_{ij}$ and $P_{\nu}=\text{diag}(1,e^{i\rho/2},e^{i\sigma/2})$.
From Eqns. (\ref{eqn15}) and (\ref{pmns}), one can find
\begin{eqnarray}
&&\sin\theta =\sqrt{\frac{3}{2}}\sin\theta_{13}~, \nonumber \\
&&\sin\delta_\text{CP}=-\frac{\sin\delta}{\displaystyle{\sqrt{1-\frac{3(2-3\sin^2\theta_{13})}{(1-\sin^2\theta_{13})^2}\sin^2\theta_{13}\cos^2\delta}}}\approx -\sin\delta\;.
\end{eqnarray}
The advantage of expressing $\theta$ and $\delta$ in this manner is that they become related to the mixing observables $\sin^2\theta_{13}$ and $\delta_{CP}$
respectively. $\sin^2\theta_{13}$ is known more precisely than $\delta_\text{CP}$, and thus in our calculation we fix $\theta$ by taking the best fit value
of $\sin^2\theta_{13}$ and consider all possible values of $\delta$ for which $\delta_\text{CP}$ falls within its $3\sigma$ experimental range.
 Even though the solar mixing angle lies slightly on the higher side of the observed central value in this case,
 i.e., $\displaystyle{\sin^2 \theta_{12} = 1/\left (3-2\sin^2 \theta \right )}$, it is still within the $3 \sigma$ range of the  observed data. \\

\section{Numerical results}
\label{sec:numerical result}
 In the previous section we set up the flavour structure of neutrino mass matrices and analytically established correlation among model parameters by
 fixing $\delta$, $\theta$ and other parameters like $\phi_{ab}$, $\phi_{db}$. In this section, we intend to estimate numerically the inter-relation among
 the model parameters by using the measured values of the ratio of two mass squared differences, r and the different mixing angles,
 $\theta_{12}$, $\theta_{13}$, $\theta_{23}$. From Eqns. (\ref{e16}) the light neutrino masses are found to be
 \begin{equation}
  m_i=\frac{|k_1k_2|}{M_i}\;.\label{lmass}
\end{equation}
 Since the solar mass squared difference,$\Delta m^2_{21}$ and atmospheric mass squared difference, $|\Delta m^2_{32}|$
 are measured in neutrino oscillation experiments, we calculate the mass squared differences from
 Eqn. (\ref{lmass}) as,
\begin{eqnarray}
\Delta m^2_{21}&=&\left|\frac{k_1k_2}{b}\right|^2\left(\frac{1}{{M_2}^2}-\frac{1}{{M_1}^2}\right)\;, \nonumber \\
 \left|\Delta m^2_{31}\right|&=&\left|\frac{k_1k_2}{b}\right|^2\left|\left(\frac{1}{{M_3}^2}-\frac{1}{{M_1}^2}\right)\right|\;. \label{mdiff}
\end{eqnarray}
In order to find the ratio of the two mass squared differences,(r), one may substituting the set of Eqns. (\ref{eq17}) in the above equations, so that
 \begin{eqnarray}
 r&=&\frac{\Delta m_{21}^2}{|\Delta m_{31}^2|}=\left[\frac{(\lambda_2\cos\phi_{ab}+C)^2+(\lambda_2\sin\phi_{ab}+D)^2}{1+\lambda_1^2+2\lambda_1\cos\phi_{db}}\right] \nonumber\\
 &\times &\left[\frac{(\lambda_2\cos\phi_{ab}-C)^2+(\lambda_2\sin\phi_{ab}-D)^2-\left(1+\lambda_1^2+2\lambda_1\cos\phi_{db}\right)}{4\lambda_2|C \cos\phi_{ab}+D\sin\phi_{ab}|}\right]\;.\label{ra}
 \end{eqnarray}
Now using Eqs. (\ref{tant}), (\ref{tanpsi}), (\ref{eq17}), (\ref{abcd}) and (\ref{ra}), and by fixing the
parameters $\phi_{db}$, $\delta$ and $\theta$, one can find numerical values of $M_i$. Once $M_i$ are
known $\left|\displaystyle{\frac{k_1k_2}{b}}\right|$ can be calculated from (\ref{mdiff}) as
\begin{equation}
\left|\frac{k_1k_2}{b}\right|=\sqrt{\frac{\Delta m^2_{21}}{\displaystyle{ \left (\frac{1}{M_2^{2}}-\frac{1}{M_1^{ 2}}\right )}}}
 =\sqrt{\left|\frac{\Delta m^2_{31}}{\left (\frac{1}{M_3^{2}}-\frac{1}{M_1^{ 2}}\right )}\right|}\;,
\end{equation}
which will also give the absolute value of light neutrino masses as all the quantities on the right hand side of (\ref{lmass}) are now known.

 We now rewrite the expression  $\tan \delta$ Eq(\ref{tanpsi})  in terms of $\phi_{db}$ as
\begin{equation}
\phi_{db}=0,\pi,~~~\text{for}~ \tan\delta=0\;,
\end{equation}
and
\begin{equation}
\phi_{ab}=\phi_{db}+ \cos^{-1}\left(\frac{\sin\phi_{db}}{\lambda_2\tan\delta}\right),~~~\text{for}~~ \tan\delta\neq 0,
\end{equation}
 and consider the following cases to see the implications.\\

\subsection{Correlation between model parameters with $\tan\delta=0$}
The input model parameters for neutrino mass analysis are,
$$\lambda_1, \lambda_2, \phi_{db}, \phi_{ab}, \delta$$

For simplification, we chose $\tan\delta = 0$ and $\phi_{db}$ will be taken either $0$
or $\pi$. \\
{\underline{\bf {Case-I- $\tan\delta = 0$,\, $\phi_{db}=0$}}};\\

From eq. (\ref{tant}), the expressions that relates $\lambda_1$ with internal mixing angle $\theta$ is,
 \begin{equation}
 \lambda_1=\frac{2\tan2\theta}{\sqrt{3}+\tan2\theta}\;,
 \label{eq:l1}
 \end{equation}
 The ratio of the mass square differences $r$ (\ref{ra}), satisfies the relation
 \begin{eqnarray}
 &&r=\left[\frac{\lambda_2^2+2\lambda_2C\cos\phi_{ab}+C^2}{(1+\lambda_1)^2}\right]\left[\frac{\lambda_2^2-2\lambda_2C\cos\phi_{ab}+C^2-(1+\lambda_1)^2}{4\lambda_2|C\cos\phi_{ab}|}\right], \label{rb} \nonumber\\
 &\mbox{where,}& C=\sqrt{\frac{1-\lambda_1+\lambda_1^2}{2}} \, .
 \end{eqnarray}
 \begin{figure}[t!]
	\centering
	\includegraphics[width=0.48\textwidth]{Plot_lambda1_theta_phidb0.pdf}
		\includegraphics[width=0.48\textwidth]{Plot_lambda1_theta_phidbpi.pdf}
	\caption{Plots for variation of $\lambda_1$ and internal mixing angle $\theta$ with $\tan{\delta}=0$
	and $\phi_{db}=0$ (left panel), $\phi_{db}=\pi$ (right panel).
	}
	\label{fig:l1-theta}
\end{figure}

The physical mass eigenvalues of $m_{RS}$ in this case become
 \begin{eqnarray}
 M_1&=&|b|\sqrt{\lambda_2^2-2\lambda_2C\cos\phi_{ab}+C^2}\;, \nonumber\\
 M_2&=&|b|(1+\lambda_1)\;,\nonumber\\
 M_3&=&|b|\sqrt{\lambda_2^2+2\lambda_2C\cos\phi_{ab}+C^2}\;.
 \end{eqnarray}
Fig.\ref{fig:l1-theta} displays variation of input model parameter $\lambda_1$ with change in internal mixing angle $\theta$ in the range 0 to 180 degree.
It is seen from the figures that those values of $\theta$ which are multiples of $\frac{\pi}{4}$ are divergent or not allowed. The figures
are plotted using equations (\ref{eq:l1}) and (\ref{eq:l11}) and it comes out that the figures are mirror images of each other due to the $`+ve'$ and $`-ve'$ signs of
$\tan\,2\theta$ in the denomenators of the respective equations.
In Fig.\ref{fig:Ratio-phiab2} using eq.(\ref{rb}) it is shown that input model parameters with $\phi_{db}=0, \tan\delta=0$ and
variation of phase angle $\phi_{ab}$ from $0-2\pi$ are consistent with experiment measured value of $r=0.03$~\cite{Capozzi:2017ipn}.
In right-panel of Fig.\ref{fig:Ratio-phiab2}, it is shown that the ratio is divergent for $\phi_{ab}$ around 90 degree, which means $\phi_{ab}$ around 90 degree is not allowed.
If we examine $\phi_{ab}$ from two different ranges, $0-\pi$ to $0-2\pi$ it is observed that for both case I and II the values
of $\phi_{ab}$ around $(2n+1) \frac{\pi}{2} , n=1,2,3...$ are not allowed since at these values r diverges. So the constraints obtained on $\phi_{ab}$ is that $\phi_{ab} \neq (2n+1) \frac{\pi}{2} $.

{\underline{\bf{Case-II-$\tan\delta = 0$,\, $\phi_{db}=\pi$ }}} \\
For this case the relation between $\lambda_1$ and $\theta$ becomes,
\begin{equation}
 \lambda_1=\frac{2\tan2\theta}{\sqrt{3}-\tan2\theta}\;,
 \label{eq:l11}
 \end{equation}
 and $r$ obeys the relation
 \begin{equation}
 r=\left[\frac{\lambda_2^2+2\lambda_2C\cos\phi_{ab}+C^2}{(1-\lambda_1)^2}\right]\times\left[\frac{\lambda_2^2-2\lambda_2C\cos\phi_{ab}+C^2
 -(1-\lambda_1)^2}{4\lambda_2|C\cos\phi_{ab}|}\right], \label{r}
 \end{equation}
  with $C=\sqrt{\frac{1+\lambda_1+\lambda_1^2}{2}}$.

The eigenvalues of $m_{RS}$ can be written as
 \begin{eqnarray}
 M_1&=&|b|\sqrt{\lambda_2^2-2\lambda_2C\cos\phi_{ab}+C^2}\;, \nonumber\\
 M_2&=&|b|(1-\lambda_1)\;,\nonumber\\
 M_3&=&|b|\sqrt{\lambda_2^2+2\lambda_2C\cos\phi_{ab}+C^2}\;.
 \end{eqnarray}
For the above two cases we have shown the correlation plots in Fig.\ref{fig:l1-theta}. It should be noted from (\ref{rb}) that $r$ will be divergent near  $\phi_{ab}=\pi/2$ and thus,
the values of $\phi_{ab}$ around $\pi/2$ are not allowed.
 \begin{figure}[t!]
	\centering
	\includegraphics[width=0.48\textwidth]{Plot_Ratio_phiab_theta120_phidb0_aa.pdf}
		\includegraphics[width=0.48\textwidth]{Plot_Ratio_phiab_theta160_phidbpi_bb.pdf}
	\caption{Contour plots for ratio of mass squared difference, r and $\phi_{ab}$ from $0$ to $\pi$ with $\phi_{db}=0$ (left panel) and $\phi_{db}=\pi$ (right panel)
	In these plots $\phi_{ab}$ is taken from $0$ to $2\pi$}
	\label{fig:Ratio-phiab2}
\end{figure}

Similarly we can find Correlation between model parameters with $\tan\delta \neq 0$ . In this case The analytic expression for $\lambda_1$ is given by
 \begin{equation}
 \lambda_1=\frac{2\lambda_2\tan 2\theta\cos\phi_{ab}\sin\phi}{\sin\phi_{ab}\left[\sqrt{3}+\tan 2\theta\cos\phi\right]}\;.
 \end{equation}

\subsection{Analysis on Neutrino mixing angles}

In the present left-right symmetric model with linear seesaw mechanism, the light neutrino masses are diagonalized by $\mathbb{U}_{\rm TBM}$, $\mathbb{U}_{13}$ containing the
 mixing angle $\theta$  and phases. The form of the mixing matrix  is expressed in terms of  $\theta$, $\delta$  and other phases in the
 following way \cite{Sruthilaya:2017mzt, Karmakar:2016cvb,Mishra:2019oqq,Sethi:2019bfu},
 \begin{eqnarray}
 \mathbb{U}
&&=\begin{pmatrix}
 \frac{2}{\sqrt{6}}  \cos\theta & \frac{1}{\sqrt{3}} & \frac{2}{\sqrt{6}} \sin\theta e^{-i \delta} \nonumber \\
 -\frac{1}{\sqrt{6}} \cos\theta+\frac{1}{\sqrt{2}}  \sin\theta e^{i\delta} & \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{6}} \sin\theta e^{-i\delta}-\frac{1}{\sqrt{2}}  \cos\theta \\
 -\frac{1}{\sqrt{6}}  \cos\theta-\frac{1}{\sqrt{2}}  \sin\theta e^{i\delta} & \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{6}}  \sin\theta e^{-i\delta}+\frac{1}{\sqrt{2}}  \cos\theta
  \end{pmatrix}
  \cdot \begin{pmatrix}
 1 & 0 & 0\\
 0 & e^{\frac{i\alpha}{2}} & 0\\
 0 & 0 & e^{\frac{i\beta}{2}}
 \end{pmatrix}.
 \end{eqnarray}
 where $\alpha$ and $\beta$ are the two Majorana phases.

\begin{figure}[t!]
	\centering
	\includegraphics[width=0.65\textwidth]{mixingangle_theta_degree.pdf}
	\caption{Variation of measured neutrino oscillation parameters like solar mixing angle ($\theta_{12}$), reactor mixing angle ($\theta_{13}$), and atmospheric mixing angle ($\theta_{23}$) with the change of internal mixing angle $\theta$ by fixing phase $\delta$.}
	\label{fig:mixingangles-theta}
\end{figure}

The neutrino mixing angles like solar mixing angle $\theta_{12}$, atmospheric mixing angle $\theta_{23}$, reactor mixing angle $\theta_{13}$
and Dirac CP-phase are related to the elements of the $U_ {\rm PMNS}$ through the following set of equations
$\sin^2 \theta_{13} = \mid \mathbb{U}_{e3} \mid^2$ , $\sin^2\theta_{12} =  \frac{\mid \mathbb{U}_{e2} \mid^2}{1 - \mid \mathbb{U}_{e3} \mid^2}$,
$\tan^2 {\theta_{23}} = \frac{\mid \mathbb{U}_{\mu 3} \mid^2}{1 - \mid \mathbb{U}_{e3} \mid^2}$. The final expressions for these mixing angles can also be expressed in
terms of the  model parameters like internal mixing angle $\theta$ and phase $\delta$
as,
\begin{eqnarray}
&& \sin^2 \theta_{13}  =\frac{2}{3}\sin^2\theta , \sin^2\theta_{12}  =\frac{1}{2+\cos2\theta}, \sin^2 {\theta_{23}} =\frac{1}{2}(1+\frac{\sqrt{3} \sin 2\theta \cos \delta}{2+\cos 2\theta}). \nonumber
 \end{eqnarray}
The other known quantiny in neutrino sector is Jarlskog rephrasing invariant~\cite{Jarlskog:1985cw} which can be expressed in terms $\theta$ and $\delta$ as,
 \begin{equation}
 J_{\rm CP}=\mbox{Im} \Big[\mathbb{U}_{e1} \mathbb{U}_{\mu 2} \mathbb{U}^*_{e2} \mathbb{U}^*_{\mu 1} \Big] =\frac{\sin\theta_{13}}{3\sqrt{2}}\sin\delta \sqrt{1-\frac{3}{2}\sin^2\theta_{13}}\, ,
 \end{equation}
 %
 Using $\sin\theta_{13}\simeq 0.16$ and $|\sin \delta| > \frac{1}{2}$, the allowed range $0.026 < |J_{\rm CP} | < 0.036$ is obtained.

Fig.\ref{fig:mixingangles-theta}, shows the variation of neutrino parameters like solar mixing angle ($\theta_{12}$), reactor mixing angle ($\theta_{13}$),
and atmospheric mixing angle ($\theta_{23}$) along with the Jarlskog rephrasing invariant $J_{\rm CP}$ with the change of internal mixing angle $\theta$
. The solid green line represents $\theta_{12}$, solid blue line is for $\theta_{23}$ and
solid magenta line is for $\theta_{13}$ while dashed lines show different predictions for rephrasing invariant $J_{\rm CP}$ by
fixing phase $\delta=0,60,100$ for yellow,magenta and green respectively. The experimentally measured $\sigma$ allowed region for $\theta_{23}, \theta_{12}$ and $\theta_{13}$~\cite{Esteban:2016qun}
are displayed in blue, green and magenta bands respectively. We have done random scan of internal mixing angle $\theta$ and the phase angle $\delta$ in the
range $0-2\pi$ by fixing $M_1$ at 25~GeV, $M_2$ at 800~GeV and $M_3$ at 2~TeV,  and considered only those values which fall within the experimental range and shown the correlation between predicted neutrino oscillation
parameters in Fig.\ref{fig:mixing1}.

\begin{figure}[t!]
	\centering
	\includegraphics[width=0.45\textwidth]{sin13_deltaCP.pdf}
	\includegraphics[width=0.45\textwidth]{sin13_JCP.pdf}
	\includegraphics[width=0.48\textwidth]{sinsq23_deltaCP.pdf}
	\includegraphics[width=0.48\textwidth]{sinsq23_JCP.pdf}
	\includegraphics[width=0.48\textwidth]{sinsq12_deltaCP.pdf}
	\includegraphics[width=0.48\textwidth]{sinsq12_JCP.pdf}
	\caption{The plots show the inter-relation between the Dirac CP violating phase $\delta_{CP}$ and rephrasing invariant $J_{\rm CP}$  with other mixing angles $\theta_{12}$, $\theta_{23}$, $\theta_{13}$.}
	\label{fig:mixing1}
\end{figure}

\section{Non-unitarity effects in linear seesaw}
\label{sec:non unitarity}
The linear seesaw mechanism allows large mixing between light and heavy neutrinos, which gives dominant contributions to lepton flavor violating decays and Jarlskog invariants ${\bf J_{\rm CP}}$. These processes are related to non-unitarity effects in neutrino masses and mixing.\cite{Antusch:2006vwa,Antusch:2008tz,Antusch:2009pm,Forero:2011pc,FernandezMartinez:2007ms,Kanaya:1980cw,Kersten:2007vk,Malinsky:2009gw,Altarelli:2008yr,delAguila:2008hw,delAguila:2009bb,Cabrera:2019xkf}
The complete neutral fermion spectrum with flavor and mass eigenstates are related in the following way \begin{eqnarray}
 \mid \Psi \rangle_f = \mathbb{V}^\dagger \mid \Psi \rangle_m\, \quad,  \mid \Psi \rangle_f =  \begin{pmatrix}
               \nu_L \\ \nu^c_R  \\ S_L
              \end{pmatrix}
\quad, \mid \Psi \rangle_m  =
              \begin{pmatrix}
               \nu_i \\ N_k
              \end{pmatrix}
\end{eqnarray}
Here we assume $\nu_L$ with $L=e, \mu, \tau$ for flavour eigenstates, $\nu_i$ with $i=1,2,3$ for mass eigenstates,
$\nu^c_R$ and $S_L$ as flavour eigenstates, $N_{k}$ with $k=1,2,...,6$ for mass eigenstates.
After complete diagonalization process the physical neutral fermions are comprised of three Majorana neutrinos and three Dirac neutrinos which come up after
six heavy neutrinos pair up. The mass formula for light as well as heavy neutrinos are given by
\begin{eqnarray}\label{mnu}
&&m_{\nu}\simeq m_{LR} m^{-1}_{RS} m^{T}_{LS} + \mbox{transpose}\nonumber \\
&&M \simeq m_{RS} + \cdots
\end{eqnarray}

The complete $9 \times 9$ mixing matrix is of the following form\cite{Agostinho:2017wfs},
\begin{eqnarray}
 \mathbb{V} = \mathbb{W} \cdot \mathbb{U}
    = \begin{pmatrix} \sqrt{1+X X^\dagger} U_\nu & X\,U_N \nonumber \\ X^\dagger\,U_\nu & \sqrt{1+X^\dagger X} U_N
      \end{pmatrix} \cdot \begin{pmatrix} U_\nu & 0 \\ 0 & U_N
\end{pmatrix}
\end{eqnarray}

\begin{figure}[t]
  \begin{minipage}[b]{0.5\linewidth}
    \includegraphics[width=.80\linewidth]{Contour_M1_M2M3_eta12.pdf}
  \end{minipage}
  \begin{minipage}[b]{0.5\linewidth}
    \includegraphics[width=.80\linewidth]{Contour_M1_M2M3_eta13.pdf}
  \end{minipage}
   \begin{minipage}[b]{0.5\linewidth}
    \includegraphics[width=.80\linewidth]{Contour_M1_M2M3_eta23.pdf}
  \end{minipage}
  \begin{minipage}[b]{0.5\linewidth}
    \includegraphics[width=.80\linewidth]{Contour_M1_M2M3_eta33.pdf}
  \end{minipage}
\caption{Contours plots in the plane of $M_2$ and $M_3$ for different fixed values of $M_1$ by saturating the experimental values of
unitarity violating parameter $\eta$ in $e\mu, e\tau, \mu\tau, \tau \tau$ sectors.}
\label{fig:eta-contour}
\end{figure}

The unitary mixing matrices $U_\nu$ and $U_N$ are required to diagonalise the light neutrino mass matrix $m_\nu$ and heavy neutrino
mass matrix $M$. In usual case, the light active Majorana neutrino mass matrix is diagonalized by the PMNS mixing
matrix $U_{\rm PMNS}$ as $U_{\rm PMNS}^{\dagger}\, m_{\nu}\, U^*_{\rm PMNS} = \text{diag}
\left(m_{1},m_{2},m_{3}\right)$ where $m_1,m_2,m_3$ are mass eigenvalues for light neutrinos.
However, due to the presence of extra heavy neutrinos, the diagonalizing mixing matrix in case of linear
seesaw mechanism ~\cite{Hettmansperger:2011bt,FernandezMartinez:2007ms,Antusch:2009pm,Forero:2011pc}(where the neutral
lepton sector comprises of light active Majorana neutrinos plus two right-handed sterile neutrinos) is no longer unitary
and is given by,
\begin{equation}
\mathbb{N} = \left(1-\eta \right) U_{\nu} \equiv  \left(1-\eta \right) U_{\rm PMNS}\, ,
\end{equation}
Here,$\eta$ is a measure of deviation from unitarity in the PMNS mixing matrix in the light neutrino sector. The non-unitarity effect can be recast in terms of  $m_{LR}$ and $m_{RS}$ as~\cite{Forero:2011pc},
\begin{equation}
\eta=\frac{1}{2}m_{LR}^*{m_{RS}^{\dagger~ -1}} m_{RS}^{-1}m_{LR}^T\;.\label{et}
\end{equation}

In linear seesaw scheme invoked with $A_4$ flavor symmetry, the structure of $m_{LR}$ and $m_{LS}$ are proportional to the identity matrix
and the other matrix $m_{RS}$ is diagonalized in the following way
\begin{equation}
m^d_{RS} = \Big(U_{\rm TBM} U_{13} \Big)^T m_{RS} \Big(U_{\rm TBM} U_{13} \Big)
\end{equation}
where,
\begin{eqnarray}
&&U_{\rm TBM} U_{13} = \begin{pmatrix}
 \frac{2}{\sqrt{6}}  \cos\theta & \frac{1}{\sqrt{3}} & \frac{2}{\sqrt{6}} \sin\theta e^{-i \delta} \nonumber \\
 -\frac{1}{\sqrt{6}} \cos\theta+\frac{1}{\sqrt{2}}  \sin\theta e^{i \delta} & \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{6}} \sin\theta e^{-i \delta}-\frac{1}{\sqrt{2}}  \cos\theta \\
 -\frac{1}{\sqrt{6}}  \cos\theta-\frac{1}{\sqrt{2}}  \sin\theta e^{i \delta} & \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{6}}  \sin\theta e^{-i\delta }+\frac{1}{\sqrt{2}}  \cos\theta
  \end{pmatrix}
\nonumber \\
&&
m^d_{RS} = \begin{pmatrix}
M_1 & 0 & 0   \\
0      & M_2  & 0 \\
0 & 0 & M_3
\end{pmatrix}
\end{eqnarray}

 \begin{figure}[t]
  \begin{minipage}[b]{0.5\linewidth}
    \includegraphics[width=.80\linewidth]{Contour_M1_thetaSigma_eta12.pdf}
  \end{minipage}
  \begin{minipage}[b]{0.5\linewidth}
    \includegraphics[width=.80\linewidth]{Contour_M1_thetaSigma_eta13.pdf}
  \end{minipage}
   \begin{minipage}[b]{0.5\linewidth}
    \includegraphics[width=.80\linewidth]{Contour_M1_thetaSigma_eta23.pdf}
  \end{minipage}
  \begin{minipage}[b]{0.5\linewidth}
    \includegraphics[width=.80\linewidth]{Contour_M1_thetaSigma_eta33.pdf}
  \end{minipage}
\caption{Correlation plot between the internal mixing angle $\theta$ and phase $\delta$ for observable unitarity effects
at current and planned experiments  in $e\mu, e\tau, \mu\tau, \tau \tau$ sectors.}
\label{fig:delta-contour}
\end{figure}

 The linear seesaw formula provides sub-eV scale (of order 0.1~eV) masses for light active neutrinos with values of
 model parameters, $m_{LR} \sim 0.5$~GeV, $m_{RS} \sim 10^3$ GeV and $m_{LS} \sim 100$~eV.
 The unitarity violation in $e\mu$,$e\tau$,$\mu \tau$, $\tau\tau$ sector can be expressed by saturating the experimental
bound which are $|\eta_{e \mu}| < 3.5\times 10^{-5}$, $|\eta_{e\tau}| < 8.0\times 10^{-4}$, $|\eta_{\mu \tau}| < 5.1\times 10^{-3}$
and $|\eta_{\tau\tau}| < 2.7\times 10^{-3}$ \cite{Awasthi:2013ff,LalAwasthi:2011aa,Miranda:2019ynh} as
\begin{eqnarray}
\pmb{\eta_{e\mu}}&=& m^2_d \times \Bigg[
\frac{1}{M^2_1} \Bigg(\sqrt{\frac{2}{3}} \cos \theta \bigg(
\frac{-\cos \theta}{\sqrt{6}} + \frac{\cos\delta \sin\theta}{\sqrt{2}}
+ \frac{i \sin\delta \sin\theta}{\sqrt{2}}
\bigg)
  \Bigg)
    + \frac{1}{3 M^2_2} \nonumber \\
&&\hspace*{-0.5cm}+\frac{1}{M^2_3} \Big(\sqrt{\frac{2}{3}} \cos \delta \sin\theta -
i \sqrt{\frac{2}{3}} \sin\delta \sin\theta \Big) \Big(
\frac{-\cos \theta}{\sqrt{2}} - \frac{\cos\delta \sin\theta}{\sqrt{6}}
+ \frac{i \sin\delta \sin\theta}{\sqrt{6}}
\Big)
\Bigg]
\label{eq:eta_emu}
\end{eqnarray}
\noindent
\begin{eqnarray}
\pmb{\eta_{e\tau}}&=& m^2_d \times \Bigg[
\frac{1}{M^2_1} \Bigg(\sqrt{\frac{2}{3}} \cos \theta \bigg(
\frac{-\cos \theta}{\sqrt{6}} - \frac{\cos\delta \sin\theta}{\sqrt{2}}
+ \frac{i \sin\delta \sin\theta}{\sqrt{2}}
\bigg)
  \Bigg)
    + \frac{1}{3 M^2_2} \nonumber \\
&&\hspace*{-0.5cm}+\frac{1}{M^2_3} \Big(\sqrt{\frac{2}{3}} \cos \delta \sin\theta -
i \sqrt{\frac{2}{3}} \sin\delta \sin\theta \Big) \Big(
\frac{\cos \theta}{\sqrt{2}} - \frac{\cos\delta \sin\theta}{\sqrt{6}}
- \frac{i \sin\delta \sin\theta}{\sqrt{6}}
\Big)
\Bigg]
\label{eq:eta_etau}
\end{eqnarray}
\begin{eqnarray}
\pmb{\eta_{\mu\tau}}&=& m^2_d \times \Bigg[
\frac{1}{M^2_1} \Bigg(\bigg(
\frac{-\cos \theta}{\sqrt{6}} - \frac{\cos\delta \sin\theta}{\sqrt{2}}
+ \frac{i \sin\delta \sin\theta}{\sqrt{2}}
\bigg)\bigg(
\frac{-\cos \theta}{\sqrt{6}} + \frac{\cos\delta \sin\theta}{\sqrt{2}}
+ \frac{i \sin\delta \sin\theta}{\sqrt{2}}
\bigg)
  \Bigg)
    + \frac{1}{3 M^2_2} \nonumber \\
&&\hspace*{-0.5cm}+\frac{1}{M^2_3}  \Big(
\frac{\cos \theta}{\sqrt{2}} - \frac{\cos\delta \sin\theta}{\sqrt{6}}
- \frac{i \sin\delta \sin\theta}{\sqrt{6}}
\Big)
\Big(
\frac{-\cos \theta}{\sqrt{2}} - \frac{\cos\delta \sin\theta}{\sqrt{6}}
+\frac{i \sin\delta \sin\theta}{\sqrt{6}}\Big)
\Bigg]
\label{eq:eta_mutau}
\end{eqnarray}
\begin{eqnarray}
\pmb{\eta_{\tau\tau}}&=& m^2_d \times \Bigg[
\frac{1}{M^2_1} \Bigg(\bigg(
\frac{-\cos \theta}{\sqrt{6}} - \frac{\cos\delta \sin\theta}{\sqrt{2}}
- \frac{i \sin\delta \sin\theta}{\sqrt{2}}
\bigg)\bigg(
\frac{-\cos \theta}{\sqrt{6}} - \frac{\cos\delta \sin\theta}{\sqrt{2}}
+ \frac{i \sin\delta \sin\theta}{\sqrt{2}}
\bigg)
  \Bigg)
    + \frac{1}{3 M^2_2} \nonumber \\
&&\hspace*{-0.5cm}+\frac{1}{M^2_3}  \Big(
\frac{\cos \theta}{\sqrt{2}} - \frac{\cos\delta \sin\theta}{\sqrt{6}}
- \frac{i \sin\delta \sin\theta}{\sqrt{6}}
\Big)
\Big(
\frac{\cos \theta}{\sqrt{2}} - \frac{\cos\delta \sin\theta}{\sqrt{6}}
+\frac{i \sin\delta \sin\theta}{\sqrt{6}}\Big)
\Bigg]
\label{eq:eta_tautau}
\end{eqnarray}

In Fig.\ref{fig:eta-contour} we have used experimental values of $\eta$ in the
$e\mu$,$e\tau$,$\mu\tau$,$e\tau\tau$ sectors \cite{Awasthi:2013ff,LalAwasthi:2011aa,Miranda:2019ynh} and plotted $M_3$
versus $M_2$  where the curves show the allowed values of $M_1$. We have used the equations \ref{eq:eta_emu}, \ref{eq:eta_etau}, \ref{eq:eta_mutau},
\ref{eq:eta_tautau} for the four plots and set $\theta=120$ degree, $\delta=60$ degree. Whereas in Fig.\ref{fig:delta-contour} we have
fixed $M_2$ at 100 GeV, $M_3$ at 2 TeV and plotted $\delta$ versus $\theta$ for observable $\eta$ at experiments.

\section{Low Energy Lepton Flavour Violating Processes}
\label{sec:LFV}

The observation of neutrino oscillations strongly hints that lepton flavor violation might be occuring in other processes as well.
In our model, the mechanism of Majorana neutrino mass generation is associated with the occurrence of
charged lepton flavor violation (LFV). LFV is highly suppressed by GIM mechanism, that is, $(\Delta m^2_\nu/m_W^2) \approx 10^{-50}$ and is
well below any experimental sensitivity in case only light neutrinos contribute to them. However, in the considered left-right symmetric framework due to the
contribution from heavy right-handed neutrinos sizable charged lepton flavor violation occurs. For a discussion we focus here on low energy LFV processes
$\mu\to e\gamma$, $\mu\to eee$ and $\mu\to e$~conversion in nuclei because of their sensitivity and omit LFV $\tau$ decays.For a review of LFV and new physics scenarios,
one may refer \cite{Deppisch:2012vj}.

 \begin{figure}[t]
 \centering
  \begin{minipage}[b]{0.89\linewidth}
    \includegraphics[width=.80\linewidth]{Contour_BR_thetaSigma_eta12.pdf}
  \end{minipage}
\caption{Correlation plot between the internal mixing angle $\theta$ and phase $\delta$ for branching ratios for the
lepton flavour violating processes, ${\rm BR}(\mu \to e \gamma)$.}
\label{fig:branchingR1}
\end{figure}
As discussed in the previous section, unitarity violation has implications on prediction for lepton flavor violation. Since the measure of unitarity violation is of the order
of $M_{LR}^2/M^2$, $\mu \to e\gamma$ term plays a vital role  in deriving constraints on input parameters like internal mixing angle $\theta$ and phases $\delta$.
The branching ratio for this particular process $\mu \to e \gamma$ is given by \cite{Ibarra:2011xn}
\begin{align}
	{\rm BR}(\mu \to e \gamma) =
		\frac{3\alpha}{32\pi} \sum_{i=1}^3 f\left(\frac{M_i}{M_W}\right)
		\left| \Theta_{\mu i}^\ast \, \Theta _{e i}\right|^2 \,,
\end{align}
Here, $M_i$ denotes for physical masses for pseudo-Dirac neutrinos, the other loop factor $f(M^2_i/M^2_W)$ is the order of one and this results,
\begin{align}
	{\rm BR}(\mu \to e \gamma) \simeq
		8.4 \times 10^{-14} \cdot
		\left(\frac{|(\Theta\Theta^\dagger)_{e\mu}|}{10^{-5}}\right)^2\,.
\end{align}
We examined how the input model parameters are correlated by saturating the experimental bounds on these LFV processes.
The term $\Theta_{\alpha i}\Theta_{\beta i}^\dagger \simeq \pmb{\eta_{\alpha\beta}}$ in the above equation represents deviation of unitarity in the
lepton sector which has been discussed in previous section . It has also been demonstrated in contour plots in the plane of internal mixing angle $\theta$ and phase $\delta$ in Fig.\ref{fig:branchingR1}.

 \begin{figure}[t]
  \begin{minipage}[b]{0.5\linewidth}
    \includegraphics[width=.80\linewidth]{Contour_BRtau-mu_thetaSigma0_180_eta32}
  \end{minipage}
  \begin{minipage}[b]{0.5\linewidth}
    \includegraphics[width=.80\linewidth]{Contour_BRtau-e_thetaSigma_eta31.pdf}
  \end{minipage}
\caption{Correlation plot between the internal mixing angle $\theta$ and phase $\delta$ for branching ratios for the lepton flavour
violating processes like ${\rm BR}(\tau \to e \gamma)$ and ${\rm BR}(\tau \to \mu \gamma)$.}
\label{fig:branchingR2}
\end{figure}

Left-right symmetric model with linear seesaw mechanism can mediate other LFV processes like $Br(\mu\to eee)$ and conversion rate process $R^N(\mu\to e)$ in a
nucleus which is discussed in reference~\cite{Cirigliano:2004mv}. The experimental bounds on these LFV processes are as follows~\cite{TheMEG:2016wtm,Nomura:2019xsb,Aubert:2009ag,Adam:2013mnn},
\begin{align}
\label{eq:Bexpllgamma}
	Br_{\rm exp}(\mu\to e\gamma) &< 5.7 \cdot 10^{-13}, \nonumber\\
	R^{Au}_{\rm exp}(\mu\to e)   &< 8.0 \cdot 10^{-13}, \\
	Br_{\rm exp}(\mu\to eee)     &< 1.0 \cdot 10^{-12}. \nonumber
\end{align}
At present the process $Br(\mu\to eee)$ gives the most restrictive bound while the currently running MEG experiment~\cite{TheMEG:2016wtm,Adam:2011ch} may provide
a better sensitivity with
\begin{equation}
\label{eq:BexpllgammaMEG}
	Br_{\rm MEG} (\mu\to e\gamma) \approx 10^{-13},
\end{equation}
Other planned experiments like COMET and Mu2e aim to reach~\cite{Kutschke:2011ux, Kurup:2011zza}
\begin{equation}
\label{eq:RmueCOMET}
	R^{Al}_{\rm COMET} (\mu\to e) \approx 10^{-16}.
\end{equation}
In Fig.\ref{fig:branchingR1} we have presented a correlation plot between the internal mixing angle $\theta$ and phase $\delta$ by fixing $M_1$,$M_2$ and $M_3$
at 35~GeV,100~GeV and 2~TeV respectively for branching ratios
for the LFV process $(\mu \to e \gamma)$. The curves in the plot represent different allowed values of branching ratios for the process for different
values of $\delta$ and $\theta$. It is seen that the values are sensitive to the current experimental bound on branching ratio as mentioned
in eq.\ref{eq:Bexpllgamma}. Similarly in Fig.\ref{fig:branchingR2} we have shown correlation plots between $\theta$ and $\delta$ for branching ratios of the processes
$(\tau \to e \gamma)$ and $(\tau \to \mu \gamma)$ by fixing $M_1$,$M_2$ and $M_3$ at 2.5~GeV,100~GeV and 2~TeV respectively .

\section{CP-violation for active neutrinos via Jarlskog invariants}
\label{sec:cp}
The CP-violating effects in neutrino oscillation are studied mostly in various long-baseline experiments with neutrinos
$\nu_{\mu}$ and anti-neutrinos $\overline{\nu}_{\mu}$. This effect is characterized by the $\mbox{PMNS}$ mixing
matrix $\mathbb{N}$ containing non-unitarity information rather than the $U_{\rm PMNS}$ matrix through Jarlskog invariant~\cite{Jarlskog:1985cw,PhysRevLett.55.1039}.
They measure the strength of leptonic CP-violation in neutrino oscillations.
Using usual PMNS mixing matrix $U_{\rm PMNS} \equiv U$, the standard contribution
to these CP-violating effects is determined by the rephasing invariant
$J_{\rm CP}$ associated with the Dirac phase $\delta_{CP}$ and matrix elements of the PMNS matrix
$$J_{\rm CP}\equiv \text{Im}\left(U_{\alpha\, i} U_{\beta\, j} U^*_{\alpha\,j} U^*_{\beta\, j}\right)
= \cos \theta_{12}\, \cos^2 \theta_{13}\, \cos \theta_{23}\,
                      \sin \theta_{12}\, \sin \theta_{13}\, \sin \theta_{23}\, \sin \delta_{\rm CP}.$$

\begin{figure}[t!]
	\centering
	\includegraphics[width=0.45\textwidth]{Contourplot_JCP_theta_delta.pdf}
	\caption{Plot showing relation between internal mixing angle $\theta$ and phase angle $\delta$ while fixing the rephasing invariant
$J_{\rm CP}$ within observable range.}
\label{fig:jcp1}
\end{figure}

However, in extended seesaw schemes like linear seesaw mechanism which we follow, the leptonic CP-violation can be written
in terms of $\mathbb{N}$ as,
\begin{eqnarray}
\mathcal{J}^{ij}_{\alpha \beta} =  \text{Im}\left(\mathbb{N}_{\alpha\, i}
\mathbb{N}_{\beta\, j} \mathbb{N}^*_{\alpha\,j} \mathbb{N}^*_{\beta\, j}\right)
\simeq J_{\rm CP} + \Delta J^{ij}_{\alpha \beta}\,
\end{eqnarray}
Here the indices $\alpha \neq \beta$ run over $e, \mu, \tau$ while indices $i,j$ run over $1,2,3$.
Assuming $\sin\theta_{13}$ small and non-unitarity parameter $\eta$ (up to second order), the derived
expression for $\Delta J^{ij}_{\alpha \beta}$ is given by~\cite{Awasthi:2013ff}
\begin{eqnarray}
\Delta J^{ij}_{\alpha \beta} = - \sum_{\gamma=e, \mu, \tau} & &\text{Im} \bigg[
                      \eta_{\alpha \gamma}\, U_{\gamma i}\, U_{\beta j}\, U^*_{\alpha j}\, U^*_{\beta i}
                    + \eta_{\beta \gamma}\, U_{\alpha i}\, U_{\gamma j}\, U^*_{\alpha j}\, U^*_{\beta i}
\nonumber \\  &&     + \eta^*_{\alpha \gamma}\, U_{\alpha i}\, U_{\beta j}\, U^*_{\gamma j}\, U^*_{\beta j}
                    + \eta^*_{\beta \gamma}\, U_{\alpha i}\, U_{\beta j}\, U^*_{\alpha j}\, U^*_{\gamma j} \bigg]\, .
\end{eqnarray}

Table.\ref{tab:JCP} shows the extra contributions to $\Delta J^{ij}_{\alpha \beta}$ due to unitary violation in the neutrino sector.
This has been worked out by choosing different values of $M_1, M_2, M_3$ while fixing $\theta$ and $\delta$ and following the mass hierarchy
for linear seesaw mechanism i.e, $m_{RS} >> m_{LR} >>  m_{LS}$.

\begin{table}[htb]
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline\hline
M&$\Delta{J}^{12}_{e\mu}$& $\Delta{J}^{12}_{\mu \tau}$
&$\Delta{J}^{13}_{e\tau}$&$\Delta{J}^{23}_{e\tau}$ &$\Delta{J}^{13}_{\mu \tau }$\\ \hline
(a) &$2.77\times 10^{-6}$ &$7.2\times 10^{-5}$ &$-1.62\times 10^{-5}$ &$-2.4\times 10^{-5}$&$-7.2\times 10^{-5}$ \\
(b) &$2.8\times 10^{-6}$ &$5.54\times 10^{-6}$&$-5.86\times 10^{-7}$ &$-1.01\times 10^{-6}$&$-5.54\times 10^{-4}$  \\
(c) &$1.38\times 10^{-6}$ &$1.38\times 10^{-6}$ &$1.38\times 10^{-6}$&$-1.38\times 10^{-6}$&$-1.38\times 10^{-6}$\\ \hline
\hline
\end{tabular}
\caption{Estimated CP-violating effects for three cases, (a) $M=(10,50,1500)$~GeV, (b) partially degenerate masses
$M=(50, 50, 1000)$~GeV and (c) fully degenerate masses
$M=(500, 500, 500)$~GeV. We have fixed the internal mixing angle and phase $\delta$ as $120$~degree and $60$~degree, respectively. }
\label{tab:JCP}
\end{table}

\begin{figure}[t!]
	\centering
    \includegraphics[width=0.45\textwidth]{Contour_JCP1212_10m6_theta_delta_Deg100.pdf}
	\includegraphics[width=0.45\textwidth]{Contour_JCP1213_10m6_theta_delta_Deg100.pdf}
	\caption{The contour plot for the rephasing inavariant $\Delta J^{12}_{e \mu}$ due to unitary violation in the neutrino sector in the plane of
    internal mixing angle $\theta$ and internal phase angle $\delta$. The value of $\Delta J^{12}_{e \mu}$ are considered here around $10^{-6}$ by taking degenerate values of $M_{1}, M_2, M_3$.}
	\label{fig:jcp2}
\end{figure}

The numerical results for relation between the internal mixing angle $\theta$ and the phase $\delta$ are displayed in Fig.\ref{fig:jcp1} and \ref{fig:jcp2}.
In Fig.\ref{fig:jcp1} the range for the value of $J_{\rm CP}$ comes out to be 0.02 to 0.04 which matches with the experimental observable range.
Since $J_{\rm CP}$ has a modulus, both negative and positive values are shown in the figure. In Fig.\ref{fig:jcp2}
the value of $\Delta J^{12}_{e \mu}$ lies around $10^{-6}$ for degenerate values of $M_1, M_2, M_3$.
The allowed range of rephasing invariant $J_{\rm CP}$ is $0.026 < |J_{\rm CP} | < 0.036$ and that of $\Delta J_{\rm CP}$ is  $10^{-5} < |\Delta J_{\rm CP} | < 10^{-7}$ .

\section{Conclusion}
\label{sec:conclusion}
We have studied a left-right symmetric model with discrete $A_4$-flavour symmetry where neutrino masses and mixing are explained via linear
seesaw mechanism. Even though $A_4$ flavour based models have been studied before, we have shown here that the $A_4$ extension of LRSM simpler
analytical expressions for large non-unitarity effect which can lead to dominant contributions to LFV decays.
We have shown correlation among model parameters like internal mixing angle $\theta$, internal phase $\delta$ and their dependence
on experimentally determined parameters like mixing angles $\theta_{12}$, $\theta_{23}$, $\theta_{13}$ and sum of neutrino masses both
analytically as well as numerically.

The model facilitates sizable charged lepton flavour violation due to contributions from heavy right handed neutrinos.
We have studied non-unitarity effects in linear seesaw which has implications on prediction for LFV decays
like $\mu\to e\gamma$, $\mu\to eee$ and $\mu\to e$ and by saturating the experimental bounds on these decays
we have derived constraints on input model parameters. Finally we have studied CP-violation for active neutrinos via Jarlskog invariants
and shown extra contributions to CP violating effects that the model generates due to unitarity violation in the neutrino sector.
Again by saturating the experimental values of unitarity violating parameter $\eta$ in $e\mu, e\tau, \mu\tau, \tau\tau$ sectors
we have generated contour plots to show constraints on model parameters. Interestingly, the range for the value of $J_{\rm CP}$ comes out
to be 0.02 to 0.04 which matches with the experimental observable range.

\section{Acknowledgement}
Purushottam Sahu would like to acknowledge Ministry of Human Resource Development (MHRD), Govt of India for financial support.

\bibliographystyle{utcaps_mod}

redpajama_set_name: RedPajamaArXiv


text:
Q: How to set default file permissions? I have ran these codes:
sudo chown -R /
sudo chmod 755 /

And now my server is not working. I need to get backup copy of MySQL database
How can I fix it?

A: That was a big failure to do so. There is a tool for macos to restore permissions for the filesystem, but as long as you don't have a backup of your server you are lost.

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The gate valve has holes for Dritz Quilting Pins (yellow heads) which are about 0.024″ dia. These valves need a handwheel 27-30″ dia. An O scale railroad brake wheel is approx. 22″ dia.
This was an assemblage of the HO 24-inch proof-printing. The rough texture is from the primer I used .. which went into the trash afterwards. The parts are smooth.


This domain is free from any Chinese Communist slave labor's products trading or support.
EVERY MEMBER OF BIZNETMALL CAN MARKET YOUR PRODUCTS AT THEIR WEB SITES - IT GIVES YOU MULTIPLICATION OF YOUR MARKETING - THIS IS VERY POWERFUL FEATURE - WE ARE THE LARGEST COOP ON THE INTERNET - PLEASE JOIN US FOR FREE!
You can become distributor for every line of products sold here - right now - just apply here. You can sell your products or service here by hundreds of our distributors - just place our shopping page at your web site and contact us with our prices and link to biznetmall shopping page on your web site!


Davi'Ana Kirkland – For Kids Foundation – A Charitable Organization that provides financial assistance to the children of Northern Nevada for medial services, educational needs, mental health and more.
I just want to thank the Kids Foundation for paying my whole balance at Absolute Dental. The payments were extremely high and my grandmother and I could not afford them. If it wasn't for The Kids Foundation I'm sure I would still have braces and try to scrape money together every month just to be able to attend my visits. The Kids Foundation made my 2 year experience with braces worth it!
When Sparks High School senior's grandmother needed help to finish paying for orthodontia that had been initiated but which had grown to be too much of a financial burden for her, she reached out to For Kids Foundation. For Kids Foundation approved the application . Now, just look at her beautiful smile as she heads off to Atlanta for college!


text:
16 October 2020 in Decision making, Policy making, Risk Management0by Garry Honey
Compromise risk: 'cake-and- eat-it' syndrome
Viral epidemics have been part of human civilisation for centuries. Black Death in the fourteenth and Plague in the seventeenth, are only the most famous but there were many more…
30 September 2020 in Decision making, Risk Management, Strategy1by Garry Honey
Political risk & the right strategy
Selecting the right risk mitigation strategy is always a matter of choice, the future is uncertain and the default human response tends to be risk aversion. Some uncertainties can be…
5 February 2020 in Decision making, Governance, Risk Management, Strategy0by Garry Honey
Do we really understand risk?
A recent report from the National Audit Office revealed the government failed to understand the risk to the Treasury for the HS2 rail project from the outset in 2010. Shortly…
13 January 2021 by Huw Williams
In this third blogpost on the future of cities, we look at city economics. Although tied up with political decisions, notably on devolution, it is worth considering the current dynamics of city financing. Cities contribute massively to the economy. London…
18 December 2020 by Garry Honey
Regular readers of our blogs will know that single point forecasts are not scenarios – and so pretty much useless for planning for the future. That said, in this blog, Garry Honey, a SAMI associate and founder of Chiron Risk,…
16 December 2020 by Huw Williams
In this second blogpost on the future of cities, we look at the impacts of new technology. The application of technology to city development often goes by the name of "smart cities". But it's important to look at the unintended…
Reasonable Worst-Case Scenarios
9 December 2020 by Huw Williams and Jonathan Blanchard-Smith
We're hearing a lot from Government about "reasonable worst-case scenarios" at the moment. At first sight it seems sensible. We need to think beyond the "base case" or "central case", but no-one wants to bother with the most extreme case.…
2 December 2020 by Huw Williams
A key "megatrend" identified by SAMI and many others is increasing urbanisation, globally. This interacts with a great many other trends and forces in complex ways. In this series of posts, I'll look at the factors affecting the future of…
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SAMI Associate Keith Leslie's new book "A Question of Leadership " is to be launched on the 18th February. He provides a wide range of illustrative case studies derived from both research and his first-hand experience in the public and private sectors as a former partner at both Deloitte and McKinsey.
We are now running our public Futures Thinking course online. The next course will run from 22-26 March 2021 in five sessions, starting at 10:00am (UK time) each day and lasting one and a half to two hours each. The course fee is £490 + VAT, and discounts may be available for self-funded individuals. For more information, or if you wish to make a booking, please email us at training@samiconsulting.co.uk
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Fireman. Firestarter. When the two come together, their world ignites!
Pete Santori is a New York City fireman committed to his job who doesn't take time for personal relationships. That all changes one night when he saves a man from a fire the victim appears to have started. But holding the man in his arms, a strange connection passes between them.
Alan Delaine is a firestarter, who manifests powerful energy when he experiences strong emotion. He's been held captive in a facility, where he was tested physically and psychologically. When he meets Pete, he's living on the run after escaping the lab.
As the two men come together, physical attraction grows into something deeper. Can love strike a balance between passion and incineration?
Note: Previously included in the Hearts Afire anthology.
Alan triggers in Pete the ability to deal with the heat he generates, otherwise this story would give a whole new bent to the idea of a "combustible" love scene. The burst of action at the end of this story is very entertaining and makes the story well worth reading.
A must read, you certainly won't regret IGNITE.
If you enjoy gay firefighters, who aren't scared of love and commitments, then you won't want to miss reading this story.
Pete felt like all the air had been sucked from his body. Was this a confession of arson from the man whose life he'd just saved? He glanced at the organized chaos of the scene: the pump crew still extinguishing the last of the flames while other firefighters continued to search the damaged building for survivors, the paramedics, the cops, the media; actually, only a couple of reporters. Since the blaze had taken place in an abandoned building, it was hardly newsworthy. Red lights flashed and emergency beams illuminated the dark night, reflecting off the pall of smoke shrouding the area.
"What is it like?" Irrationally, Pete wanted to smooth the furrow between the man's slanting brows, but settled for squeezing his shoulder, offering comfort while coaxing a confession.
"I can't help it. It's out of my control." The dark gaze fixed on his so intently he couldn't have moved if he tried.
"Starting fires?" Pete clarified. It was worse than he'd thought. The guy was a firebug, a psycho who thrived on the sight of flames and the destruction they caused.
Silence surrounded them. It was as if the noise and bustle of the world had faded away. Pete could only hear his own breathing and that of the man who lay on the muddy ground.
The man's eyes closed, cutting off the strange connection between them once more, then there was no time to talk as the emergency team took over, surrounding the stranger and blocking him from Pete's sight.
He returned to his duties, helping his crew douse the smoking building to prevent another flare-up. As he tramped through the jumble of charred beams and collapsed walls, checking to make sure the fire was completely extinguished, he thought of what the man had said. Firestarter. Spontaneous combustion. Crazy talk.
He should report this to the fire marshal who'd be investigating the scene right away. Knowing what combustible substance the man had used and where and how he'd started the blaze would save a lot of time. Yet Pete kept the man's insane tale to himself. There was something about the guy that was so melancholy, so vulnerable Pete didn't want to cause him further pain. In fact, he found himself wanting to protect the man he'd carried through blinding black smoke from the burning building.


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Issie Veisfeld
Issie Veisfeld describes what happened during the Iasi pogrom in 1941.
Source: Montreal Holocaust Museum, 1998; editing: Montreal Holocaust Museum, 2015
Issie Veisfeld was born in 1928 in Iasi, Romania. As a child, he encountered a lot of antisemitism, including physical assault of Jews by the Romanians.
Issie and his father managed to hide to avoid being killed or rounded up during the Iasi Pogrom in June 1941. The Romanian authorities rounded up Jews, killed many on the spot and deported the survivors to Transnistria. Since Issie's father worked as an upholsterer for the Romanian Army, he was warned by an officer about the impending pogrom, and the family was able to hide until it was over. It was also due to his father's connections that the family was not deported from Iasi during the war.
So, therefore, they had a lot of help, and nobody couldn't escape.
Issie, his mother and sister survived, but his father was killed during an Allied bombing in 1944, just weeks before the Red Army liberated Iasi. Issie decided to leave Romania when he saw that former collaborationists and fascists started gaining influence in the Communist Party.
He illegally crossed the border to Hungary and Austria with his mother and sister in 1947, and arrived in Canada the following year. He settled in Montreal and was one of the founders of the Montreal Holocaust Memorial Centre. He was president of the Association of Survivors of Nazi Oppression, as well as the Association of Romanian Jewish Survivors of the Holocaust.
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Where There's Always Something New in Bluegrass!
We're Gonna Rise
Crosspoint On The Rise With New Single "Going Home"
published by BMNN on Fri, 03/28/2014 - 00:31
Nashville, TN -- Rural Rhythm Records is proud to announce the release of the single "Going Home" from Bluegrass Gospel band Crosspoint. Hailing from Southeast Kentucky / East Tennesse and formed in 2008 Crosspoint will make their album debut on Rural Rhythm Christian with the release of their album We're Gonna Rise releasing on May 13th. Produced by acclaimed bluegrass artist Steve Gulley and recorded at the Curve Studio in Cumberland Gap Tennessee, We're Gonna Rise is the real deal as it goes for authentic, inspirational and fresh bluegrass gospel music.
Read more about Crosspoint On The Rise With New Single "Going Home"


The Jaguar F-Type V6 S AWD Coupe is sure to steal hearts with its gorgeous looks and good all-round performance. But it comes at a premium over faster rivals.
Stunning looks, guaranteed to turn heads regardless of what else is around.
The supercharged V6 sounds great, has plenty of power and the eight-speed automatic gearbox is smooth.
Adding all-wheel drive does little to dampen the F-Type's agile handling, but adds security on slippery roads.
Steering and brakes are excellent, the Jaguar F-Type doesn't just look like a sports car, it feels like one to drive as well.
Costs quite a lot, particularly as a lot of desirable equipment is optional.
Practicality. Forgo a spare wheel and there's reasonable space under the rear hatch, but you're still going to want to pack light for a weekend away.
Not as outright fast as rivals like the BMW M4 and Mercedes-AMG C63.
All-wheel drive adds almost $17K and is unlikely to be necessary for most drivers.
Click here to find out more about the Jaguar F-Type V6 S AWD Coupe.


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By Land, Air, and Steed
For almost seven decades, dedicated volunteers have traversed the mountains and canyons of the Coachella Valley to rescue the injured or lost.
Miranda Caudell December 16, 2018 Current Digital, Hiking
PHOTOGRAPH COURTESY PALM SPRINGS MOUNTED POLICE SEARCH AND RESCUE
Greater Palm Springs is known as a hiker's paradise, with more than 300 days of sunshine a year and countless trails winding across the beautiful desert landscape and even up into the pine forests of the San Jacinto Mountains. Though the panoramic views only get better the higher you climb, so too, does the potential for danger. Every once in a while, the faint whirring of a helicopter's blades passes overhead as the Coachella Valley's go-to team of search and rescue operators saves the day.
Officially known as Palm Springs Mounted Police Search and Rescue (PSMP-SAR), the volunteer unit has performed more than 1,000 technical rescues in Palm Springs since 1949. What began as a few men riding on horseback up into the mountains to help stranded hikers in their spare time has grown into a 13-member team. (To learn more about one of the founders, Jim Maynard, see p. 80.) PSMP-SAR once served as the honor guard for President Dwight D. Eisenhower and even appeared in a 1995 episode of the national television show Rescue 911.
The unit depends mostly on private donations and fundraising (all members are volunteers and purchase their own gear), and with a $2,000 average cost per rescue, the community's support is crucial to their continued service. The Agua Caliente Band of Cahuilla Indians is, in the words of Doug Stevens, a 37-year search and rescue volunteer, their "biggest donor" and has contributed to the organization for at least 20 years. "They donated to us before they ever had the casinos," Stevens says. "They are amazing — we wouldn't have half of what we have if it weren't for them."
PHOTOGRAPH COURTESY PALM SPRINGS HISTORICAL SOCIETY
Palm Springs Mounted Police Search and Rescue, circa 1960s. The volunteer unit was founded in 1949, when a small group of men began helping stranded hikers in their spare time.
Often, Agua Caliente Tribal Rangers and the Tribal Emergency Services team are on-site to coordinate with the volunteer rescue officers in locating a missing or stranded hiker, as several popular trails are located on the Tribe's Reservation and in the Indian Canyons.
"The rangers are a great asset — they really know the terrain," Stevens says, recalling a recent three-day search for a hiker who had set off from Highway 74 and ended up getting lost in the canyons. The Tribe helped provide food and water for the countless officers and volunteers stationed on the reservation. "That was a team effort," he adds. "They were such a help during that search."
Though horses and motorcycles were used throughout that three-day stretch, Stevens notes the use of a drone as a key part of the successful outcome. When he began volunteering in 1982, the rescue unit primarily worked on horseback and on foot, but as technology has advanced, helicopters and other airborne crafts have come to play a crucial role in their missions. For instance, thanks to an 8-pound-capacity drone, Stevens can deliver small bottles of water to stranded hikers to help them stay hydrated while they wait for help.
"They just don't understand the extremity of the heat," he says. All it takes is a few minutes of walking in 80-plus degrees and your body already begins to lose moisture, and when hiking up the side of a mountain with practically no shade, even more so. "We've started carrying umbrellas in the summertime," he adds with a light laugh, "because it's the only shade we have."
"Once Stevens and his team have located missing or injured hikers, their primary concern is to get them to safety and to paramedics as soon as possible, though all volunteers are trained in basic first aid and law enforcement if the need arises. (Many are even trained in rappelling and rock climbing, which alone can take up to 18 months of instruction.) So why do these men and women do it?
"I had an interest in law enforcement early on," Stevens says, "and one of my friends was a search and rescue volunteer, so he got me [into it]. Back then you had to be sponsored. But it's really to give back to the community. That's how it all got started."
The Palm Springs Mounted Police Search and Rescue volunteers use various means, from helicopters to rock climbing and rappelling to walking on foot, to rescue stranded or injured hikers.
Before hitting the trails, brush up on a few hiking tips from Palm Springs Mounted Police Search and Rescue. For more information, call 760-778-8470.
Always hike in an approved, open area with well-marked trails.
Be prepared for the trip by packing lots of water, even when it's cool out. Once half of your water supply is gone, it's time to turn back.
Always hike with another person.
Wear sturdy shoes; hiking in flip-flops, sandals, or other slippery-soled shoes has spelled disaster for many of the victims rescued by PSMP-SAR.
Tell someone where you are going, when you expect to get back, how long you will be gone, what you are wearing, and who is going with you.
If you are visiting from another state or country, turn your cellphone off and back on when you arrive to your hiking area to reset the calibration; if you find yourself in need of help, call 911 immediately.
This story originally appeared in MeYah Whae, The Magazine of the Agua Caliente Band of Cahuilla Indians, Fall/Winter 2018-19. To read the current digital edition, click HERE.
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