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Spontaneous CP Violation in $A_4$ Flavor Symmetry and Leptogenesis

We propose a simple renormalizable model for the spontaneous CP violation based on $SU(2)_{L}\times U(1)_{Y}\times A_{4}$ symmetry in a radiative seesaw mechanism, which can be guaranteed by an extra $Z_{2}$ symmetry. In our model CP is spontaneously broken at high energies, after breaking of flavor symmetry, by a complex vacuum expectation value of $A_{4}$-triplet and gauge singlet scalar field. We show that the spontaneously generated CP phase could become a natural source of leptogenesis, %CP violation for neutrino oscillation and leptogenesis, respectively. Then, and also investigate CP violation at low energies in the lepton sector and show how the CP phases in PMNS could be arisen through spontaneous symmetry breaking mechanism. As a numerical study, interestingly, we show that the normal mass hierarchy favors relatively large values of $θ_{13}$, large deviations from maximality of $θ_{23}<π/4$ and Dirac-CP phase $0^{\circ}\leqδ_{CP}\leq50^{\circ}$ and $300^{\circ}\leqδ_{CP}\leq360^{\circ}$. For the inverted hierarchy case, the experimentally measured values of $θ_{13}$ favors $θ_{23}>π/4$ and discrete values of $δ_{CP}$ around $100^{\circ},135^{\circ},255^{\circ}$ and $300^{\circ}$. Finally, with a successful leptogenesis our numerical results give more predictive values on the Dirac CP phase: for the normal mass hierarchy $1^{\circ}\lesssimδ_{CP}\lesssim10^{\circ}$ and for inverted one $δ_{CP}\sim100^{\circ},135^{\circ},300^{\circ}$.