Horizontal symmetries of leptons with a massless neutrino

Understand

Residual symmetry $G_\nu$ of neutrino mass matrix with a massless neutrino and embedding of $G_\nu$ and the residual symmetry $G_l$ of the charged lepton mass matrix into finite discrete groups $G$ is discussed.

• Massless neutrino results if $G_\nu$ and hence $G$ are subgroups of $U(3)$ rather than of $SU(3)$.
• Structure of the resulting leptonic mixing matrix $U_{PMNS}$ is discussed in three specific examples based on groups (a) $\Sigma(3N^3)$, (b) $\Sigma(2N^2)$ and (c) $S_4(2) \equiv A_4\rtimes Z_4$.
• $\Sigma(3N^3)$ groups are able to reproduce either the second or the third column of $U_{PMNS}$ correctly.