Tri-bimaximal neutrino mixing from A4 and θ13∼θC

It is a common believe that, if the tri-bimaximal mixing (TBM) pattern is explained by vacuum alignment in an $A_4$ model, only a very small reactor angle, say ${\theta }_{13}～O({\lambda }_C^2)$ being ${\lambda }_C\equiv {\theta }_C$ the Cabibbo angle, can be accommodated. This statement is based on the assumption that all the flavon fields acquire VEVs at a very similar scale and the departures from exact TBM arise at the same perturbation level. From the experimental point of view, however, a relatively large value ${\theta }_{13}～O({\lambda }_C)$ is not yet excluded by present data. In this paper, we propose a seesaw $A_4$ model in which the previous assumption can naturally be evaded. The aim is to describe a ${\theta }_{13}～O({\lambda }_C)$ without conflicting with the TBM prediction for ${\theta }_{12}$ which is rather close to the observed value (at ${\lambda }_C^2$ level). In our model the deviation of the atmospherical angle from maximal is subject to the sum-rule: ${\sin }^2{\theta }_{23}\approx 1/2+\sqrt{2}/2\cos \delta \sin {\theta }_{13}$ which is a next-to-leading order prediction of our model.