Constraining {\it CP} violation in a softly broken $A_4$ symmetric Model

To understand the mass spectra of charged lepton and neutrino $A_4$ symmetry has been proposed in addition with the Standard $SU(2)_L\times U(1)_Y$ model. We break $A_4$ symmetry softly and the deviation from the tri-bimaximal mixing arises due to Zee mechanism. In the present work, we express two mixing angles $\theta_{13}$ and $\theta_{23}$ in terms of a single model parameter and experimental observables, such as, mixing angle $\theta_{12}$, mass squared differences $\Delta m^2_{21}$ and $\Delta m^2_{32}$. Using the experimental values of $\theta_{23}$, $\theta_{12}$, $\Delta m^2_{21}$ and $\Delta m^2_{32}$ we restrict the model parameter and we predict $\theta_{13}$. This model gives rise to $\theta_{13}\simeq 11^\circ$ if we allow $1\sigma$ deviation of $\theta_{23}$ and $2^\circ$ deviation of $\theta_{12}$ from their best fit values. Utilizing all those constraints, we explore the extent of CP violation parameter $J_{\rm CP}$ in the present model and found a value of $J_{\rm CP}\approx 2.65\times 10^{-3}$ (for $1\sigma$ deviation of $\theta_{23}$ and $2^\circ$ deviation of $\theta_{12}$) consistent with the other neutrino experimental results. We have studied the mass pattern of neutrino and neutrinoless double beta decay $(\beta\beta_{0\nu})$ parameter $|(M_\nu)_{ee}|$ in this model.