Flavor Unification and Discrete Nonabelian Symmetries

Grand unified theories with fermions transforming as irreducible representations of a discrete nonabelian flavor symmetry can lead to realistic fermion masses, without requiring very small fundamental parameters. We construct a specific example of a supersymmetric GUT based on the flavor symmetry $\Delta(75)$ --- a subgroup of $SU(3)$ --- which can explain the observed quark and lepton masses and mixing angles. The model predicts $\tan\beta \simeq 2-5$ and gives a $\tau$ neutrino mass $m_\nu\simeq M_p/G_F M_{GUT}^2 = 10$ eV, with other neutrino masses much lighter. Combined constraints of light quark masses and perturbative unification place flavor symmetry breaking near the GUT scale; it may be possible to probe these extremely high energies by continuing the search for flavor changing neutral currents.