Horizontal symmetries of leptons with a massless neutrino

Residual symmetry Gν of neutrino mass matrix with a massless neutrino and embedding of Gν and the residual symmetry Gl of the charged lepton mass matrix into finite discrete groups G is discussed. Massless neutrino results if Gν and hence G are subgroups of U(3) rather than of SU(3). Structure of the resulting leptonic mixing matrix UPMNS is discussed in three specific examples based on groups (a) Σ(3N3), (b) Σ(2N2) and (c) S4(2)≡A4⋊Z4. Σ(3N3) groups are able to reproduce either the second or the third column of UPMNS correctly. Σ(2N2) groups lead to prediction θ13=0, θ23=π4 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=S4(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 S4(2) symmetry in which a massless neutrino and viable leptonic mixing angles are obtained.