Horizontal symmetries of leptons with a massless neutrino

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) $\mathrm{\Sigma }(3N^3)$, (b) $\mathrm{\Sigma }(2N^2)$ and © $S_4(2)\equiv A_4⋊Z_4$. $\mathrm{\Sigma }(3N^3)$ groups are able to reproduce either the second or the third column of $U_{PMNS}$ correctly. $\mathrm{\Sigma }(2N^2)$ groups lead to prediction ${\theta }_{13}=0$, ${\theta }_{23}=\frac{\pi }{4}$ for the reactor and atmospheric mixing angles respectively if neutrino mass hierarchy is inverted. Solar angle remains undetermined in this case. This also gets determined when $G=S_4(2)$ which can give bi-maximal mixing for inverted hierarchy. Examples (b) and © provide a good zeroth order approximation to realistic leptonic mixing with a massless neutrino. We also present an example of the specific model based on $S_4(2)$ symmetry in which a massless neutrino and viable leptonic mixing angles are obtained.