Hybrid Seesaw Neutrino Masses with A_4 Family Symmetry

We consider the scenario in which neutrino data are explained by the interplay of type I and II seesaw terms in the Majorana neutrino mass matrix ${\mathcal{M}}_{\nu }={\mathcal{M}}_L-{\mathcal{M}}_D{\mathcal{M}}_R^{-1}{\mathcal{M}}_D^T$. We construct a predictive model with ${\mathcal{M}}_L$ proportional to the unit matrix, 3 diagonal texture zeros in ${\mathcal{M}}_R$, and ${\mathcal{M}}_D$ diagonal. We show how this pattern can be maintained by the non-Abelian discrete symmetry $A_4$, and discuss its phenomenological consequences. It turns out that the two types of seesaw give contributions of the same order to ${\mathcal{M}}_{\nu }$. In the CP conserving case, we find $\sin\theta_{13}\approx 2/(\tan 2\theta_{23} \tan 2\theta_{12})$ and we predict inverted (normal) ordering of the mass spectrum for ${\tan }^2{\theta }_{12}<0.5(>0.5)$.