A massless neutrino and lepton mixing patterns from finite discrete subgroups of U(3)

Finite discrete subgroups of $U(3)$ as possible flavour symmetries $G_f$ for a massless neutrinos with predictive mixing angles are studied. This is done by assuming that a residual symmetry $S_\nu$ appropriate for describing a massless neutrino is contained in $G_f$. It is shown that all the groups $G_f$ admitting three dimensional faithful irreducible representation and generated from a specific set of $3\times 3$ matrices imply only one of the three flavour compositions for the massless state namely, unmixed, maximally mixed with equal probabilities and bimaximally mixed with probabilities (0,1/2,1/2) and their permutations. This result holds irrespective of the order of $G_f$ and the choice of $S_{\nu}$ within it. All of these lead to unfavourable leading order prediction for the solar mixing angle. Neutrino mixing pattern is then numerically investigated in case of subgroups of $U(3)$ with order less than 512 and it is found that only one of these can lead to a massless neutrino and leading order predictions for all the mixing angles close to their experimental values. Ways to correct for the solar angle prediction are proposed and two concrete examples giving the observed mixing pattern are discussed.