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Direct and Semi-Direct Approaches to Lepton Mixing with a Massless Neutrino

We discuss the possibility of enforcing a massless Majorana neutrino in the direct and semi-direct approaches to lepton mixing, in which the PMNS matrix is partly predicted by subgroups of a discrete family symmetry, extending previous group searches up to order 1535. We find a phenomenologically viable scheme for the semi-direct approach based on $Q(648)$ which contains $Δ(27)$ and the quaternion group as subgroups. This leads to novel predictions for the first column of the PMNS matrix corresponding to a normal neutrino mass hierarchy with $m_1=0$, and sum rules for the mixing angles and phase which are characterised by the solar angle being on the low side $θ_{12}\sim 31^{\circ}$ and the Dirac (oscillation) CP phase $δ$ being either about $\pm 45^\circ$ or $\pm π$.