Flavor Symmetry and Neutrino Oscillations

We show how the nearly bi-maximal mixing scenario comes out naturally from gauged $SO(3)_{F}$ flavor symmetry via spontaneous symmetry breaking. An interesting relation between the neutrino mass-squared differences and the mixing angle, i.e., $\Delta m_{\mu e}^2 / \Delta m_{\tau\mu}^2 \simeq 2|U_{e 3}|^2$ is obtained. The smallness of the ratio (or $|U_{e 3}|$) can also naturally be understood from an approximate permutation symmetry. Once the mixing element $|U_{e 3}|$ is determined, such a relation will tell us which solution will be favored within this model. The model can also lead to interesting phenomena on lepton-flavor violations.