The U(1) symmetry of the non-tribimaximal pattern in the degenerate mass spectrum case of the neutrino mass matrix

On account of the new neutrino oscillation data signalling a non-zero value for the smallest mixing angle ($θ_z$), we present an explicit realization of the underlying U(1) symmetry characterizing the maximal atmospheric mixing angle ($θ_y = π/ 4$) pattern with two degenerate masses but now with generic values of $θ_z$. We study the effects of the form invariance with respect to U(1), and/or $Z_3$, $Z_2$ subgroups, on the Yukawa couplings and the mass terms. Later on, we specify $θ_z$ to its experimental best fit value ($ \sim 8^o$), and impose the symmetry in an entire model which includes charged leptons, and many Higgs doublets or standard model singlet heavy scalars, to show that it can make room for the charged lepton mass hierarchies. In addition, we show for the non-tribimaximal value of $θ_z \neq 0$ within type-I seesaw mechanism enhanced with flavor symmetry that neutrino mass hierarchies can be generated. Furthermore, lepton/baryogenesis can be interpreted via type-II seesaw mechanism within a setup meeting the flavor U(1)-symmetry.