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 m₁=0, and sum rules for the mixing angles and phase which are characterised by the solar angle being on the low side θ₁₂∼31^∘ and the Dirac (oscillation) CP phase δ being either about ±45^∘ or ±π.