An $SU(2)_{L}\times U(1)_{Y} \times S_{3} \times D$ model for atmospheric and solar neutrino deficits

Motivated by the recent Super-Kamiokande experiment on atmospheric and solar neutrino oscillation we propose a see-saw model of three generations of neutrinos based on the gauge group $SU(2)_{L}\times U(1)_{Y}$ with discrete symmetries $(S_{3} \times D)$ and three right handed singlet neutrinos so that this model can accommodate the recent Super-Kamiokande data on atmospheric and solar neutrino oscillations. The model predicts maximal mixing between $\nu_{\mu}$ and $\nu_{\tau}$ with $sin^{2}2{\theta_{\mu\tau}}$ = 1 as required by the atmospheric neutrino data and small mixing between $\nu_{e}$ and $\nu_{\mu}$ with $sin^{2}2{\theta_{e\mu}} \sim (10^{-2}-10^{-3})$ as a possible explanation of the solar neutrino deficit through the MSW mechanism. The model admits two mass scales of which one breaks the electroweak symmetry and the other is responsible for the breaking of the lepton number symmetry at GUT scale leading to small Majorana mass of the left handed doublet neutrinos.