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Spontaneous CP Violation in Lepton-sector: a common origin for $θ_{13}$, Dirac CP phase and leptogenesis

A possible interplay between the two terms of the general type-II seesaw formula is exercised which leads to the generation of nonzero $θ_{13}$. The specific flavor structure of the model, guided by the $A_4 \times Z_4 \times Z_3$ symmetry and accompanied with the Standard Model singlet flavons, yields the conventional seesaw contribution to produce the tribimaximal lepton mixing which is further corrected by the presence of the $SU(2)_L$ triplet contribution to accommodate $θ_{13}$. We consider the CP symmetry to be spontaneously broken by the complex vacuum expectation value (vev) of a singlet field $S$. While the magnitude of its complex vev is responsible for generating $θ_{13}$, its phase part induces the low energy CP violating phase ($δ$) and the CP violation required for leptogenesis. Hence the triplet contribution, although sub-dominant, plays crucial role in providing a common source for non-zero $θ_{13}$, $δ$ and CP-violation required for leptogenesis. We find that the recent hint for $δ$ close to $3π/2$ is somewhat favored in this set-up though it excludes the exact equality with $3π/2$. We also discuss the generation of lepton asymmetry in this scenario.