Nonabelian Discrete Family Symmetry to Soften the SUSY Flavor Problem and to Suppress Proton Decay

Family symmetry could explain large mixing of the atmospheric neutrinos. The same symmetry could explain why the flavor changing current processes in supersymmetric standard models can be so suppressed. It also may be able to explain why the proton is so stable. We investigate these questions in a supersymmetric, renormalizable extension of the standard model, which possess a family symmetry based on a binary dihedral group Q_6. We find that the amplitude for \mu \to e+\gamma enjoys a suppression factor proportional to |(V_{MNS})_{e3}| ~ m_e/(\sqrt{2}m_\mu) ~ 3.4\times 10^{-3}, and that B(p \to K^0 \mu^+)/B(p \to K^0 e^+) ~ |(V_{MNS})_{e3}|^2 ~ 10^{-5}, where V_{MNS} is the neutrino mixing matrix.