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A4-based see-saw model for realistic neutrino masses and mixing

We present an $A4$-based model where neutrino masses arise from a combination of see-saw mechanisms. The model is motivated by several small mixing and mass parameters indicated by the data. These are $θ_{13}$, the solar mass splitting, and the small deviation of $θ_{23}$ from maximal mixing (= $π/4$). We take the above as indications that at some level the small quantities are well-approximated by zero. In particular the mixing angles, to a zero order, should be either 0 or $π/4$. Accordingly, in this model the Type-II see-saw dominates and generates the larger atmospheric mass splitting and sets $θ_{23} = π/4$. The other mixing angles are vanishing as is the solar splitting. We show how the $A4$ assignment for the lepton doublets leads to this form. We also specify the $A4$ properties of the right-handed neutrinos which result in a smaller Type-I see-saw contribution that acts as a perturbation and shifts the angles $θ_{12}$ and $θ_{13}$ into the correct range and the desired value of $Δm^2_{solar}$ is produced. The $A4$ symmetry results in relationships between these quantities as well as with a small deviation of $θ_{23}$ from $π/4$. If the right-handed neutrino mass matrix, $M_R$, is chosen real then there is no leptonic CP-violation and only Normal Ordering is admissible. If $M_R$ is complex then Inverted Ordering is also allowed with the proviso that the CP-phase, $δ$, is large, i.e., $\sim π/2$ or $-π/2$. The preliminary results from NO$ν$A favouring Normal Ordering and $δ$ near $-π/2$ imply quasi-degenerate neutrino masses in this model.