Spontaneous CP violation in A₄ flavor symmetry and leptogenesis

We propose a simple renormalizable model for the spontaneous CP violation based on SU(2)_L×U(1)_Y×A₄ 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 the breaking of flavor symmetry, by a complex vacuum expectation value of the A₄ triplet and gauge-singlet scalar field. We show that the spontaneously generated CP phase could become a natural source of leptogenesis, and also investigate CP violation at low energies in the lepton sector and show how the CP phases in the Pontecorvo-Maki-Nakagawa-Sakata formalism could arise through a spontaneous symmetry-breaking mechanism. As a numerical study, interestingly, we show that the normal mass hierarchy favors relatively large values of θ₁₃, large deviations from maximality of θ₂₃<π/4, and the Dirac-CP phase 0 ≤δ _CP≤50 and 300 ≤δ_CP≤360. For the inverted hierarchy case, the experimentally measured values of θ₁₃ favors θ₂₃>π/4 and discrete values of δ_CP around 100, 135, 255, and 300. Finally, with a successful leptogenesis our numerical results give more predictive values on the Dirac CP phase: for the normal mass hierarchy 1 δ_CP 10 and for inverted one δ_CP∼100, 135, 300.