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 $\theta_{13}$ and $\theta_{23}$. (b) All the groups of type D can predict a relation $\cos^2\theta_{13} \sin^2\theta_{23}=\frac{1}{3}$ among the mixing angles which appears to be a good zeroth order approximation. Among these, various $\Delta(6n^2)$ groups with $n\geq 8$ can simultaneously lead also to $\sin^2 \theta_{13}$ in agreement with global fit at 3$\sigma$. (c) The group $\Sigma(168)\cong PSL(2,7)$ predicts near to the best fit value for $\theta_{13}$ and $\theta_{23}$ within the 1$\sigma$ 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 $\theta_{13}$ and/or $\theta_{23}$. 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. 
Anjan S. Joshipura, Ketan M. Patel
hep-ph