Spontaneous breaking of $SO(3)$ to finite family symmetries with supersymmetry - an $A_4$ model

We discuss the breaking of $SO(3)$ down to finite family symmetries such as $A_4$, $S_4$ and $A_5$ using supersymmetric potentials for the first time. We analyse in detail the case of supersymmetric $A_4$ and its finite subgroups $Z_3$ and $Z_2$. We then propose a supersymmetric $A_4$ model of leptons along these lines, originating from $SO(3)\times U(1)$, which leads to a phenomenologically acceptable pattern of lepton mixing and masses once subleading corrections are taken into account. We also discuss the phenomenological consequences of having a gauged $SO(3)$, leading to massive gauge bosons, and show that all domain wall problems are resolved in this model.