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Discrete flavour symmetries for degenerate solar neutrino pair and their predictions

Flavour symmetries appropriate for describing a neutrino spectrum with degenerate solar pair and a third massive or massless neutrino are discussed. We demand that the required residual symmetries of the leptonic mass matrices be subgroups of some discrete symmetry group $G_f$. $G_f$ can be a subgroup of SU(3) if the third neutrino is massive and we derive general results on the mixing angle predictions for various discrete subgroups of SU(3). The main results are: (a) All the SU(3) subgroups of type C fail in simultaneously giving correct $θ_{13}$ and $θ_{23}$. (b) All the groups of type D can predict a relation $\cos^2θ_{13} \sin^2θ_{23}=\frac{1}{3}$ among the mixing angles which appears to be a good zeroth order approximation. Among these, various $Δ(6n^2)$ groups with $n\geq 8$ can simultaneously lead also to $\sin^2 θ_{13}$ in agreement with global fit at 3$σ$. (c) The group $Σ(168)\cong PSL(2,7)$ predicts near to the best fit value for $θ_{13}$ and $θ_{23}$ within the 1$σ$ range. All discrete subgroups of U(3) with order $<512$ and having three dimensional irreducible representation are considered as possible $G_f$ when the third neutrino is massless. Only seven of them are shown to be viable and three of these can correctly predict $θ_{13}$ and/or $θ_{23}$. The solar angle remains undetermined at the leading order in all the cases due to degeneracy in the masses. A class of general perturbations which can correctly reproduce all the observables are discussed in the context of several groups which offer good leading order predictions.