Spontaneous CP Violation in Lepton-sector: a common origin for $\theta_{13}$, 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 $\theta_{13}$. 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 $\theta_{13}$. 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 $\theta_{13}$, its phase part induces the low energy CP violating phase ($\delta$) 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 $\theta_{13}$, $\delta$ and CP-violation required for leptogenesis. We find that the recent hint for $\delta$ close to $3\pi/2$ is somewhat favored in this set-up though it excludes the exact equality with $3\pi/2$. We also discuss the generation of lepton asymmetry in this scenario.