Horizontal symmetries of leptons with a massless neutrino

Residual symmetry $G_ν$ of neutrino mass matrix with a massless neutrino and embedding of $G_ν$ and the residual symmetry $G_l$ of the charged lepton mass matrix into finite discrete groups $G$ is discussed. Massless neutrino results if $G_ν$ 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) $Σ(3N^3)$, (b) $Σ(2N^2)$ and (c) $S_4(2) \equiv A_4\rtimes Z_4$. $Σ(3N^3)$ groups are able to reproduce either the second or the third column of $U_{PMNS}$ correctly. $Σ(2N^2)$ groups lead to prediction $θ_{13}=0$, $θ_{23}=\fracπ{4}$ 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 (c) 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.

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