Unified explanation of the Solar and Atmospheric neutrino Puzzles in a minimal supersymmetric SO(10) model

We generalize the idea that the minimal supersymmetric SO(10) model defined by Higgs multiplets in a single ${\bf 10}$, ${\bf 126 + \overline{126}}$ and ${\bf 45+54}$ dimensional representations enable a complete prediction of the neutrino masses and mixings without requiring any extra symmetries or fields by including the additional left-handed triplet contribution to the light neutrino mass matrix. The neutrino mass matrix then has only two free parameters, $v_R$, the scale of $B-L$ symmetry breaking and $v_L$, the induced triplet Higgs vev. We find that there exist $SO(10)$ scenarios capable of accommodating both the small angle MSW solution to the solar neutrino problem as well as the $\nu_{\mu}-\nu_{\tau}$ oscillation solution to the atmospheric neutrino puzzle. The model cannot reproduce the mass and mixing parameters required to explain the LSND observations neither does the model have have a neutrino hot dark matter.