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

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 $P({\nu }_{\mu }\to {\nu }_e)$, the effective Majorana mass $\mathrm{\mid }{m}_{ee}\mathrm{\mid }$ and $\mathrm{\mid }{m}_{\nu e}^{eff}\mathrm{\mid }$. 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 $\mu \to e\gamma$, $\tau \to e\gamma$, $\tau \to \mu \gamma$ are discussed.