Discrete Flavor Symmetries and Mass Matrix Textures

There is a unique set of U(1) flavor symmetry quantum numbers that agree with the phenomenological requirements in the standard model quark sector. We show how introducing discrete Abelian flavor symmetries can produce texture zeros in the fermion mass matrices, while preserving the correct relationships with the low-energy data on quark and lepton masses. We categorize all phenomenologically viable solutions with at least four texture zeros in the quark sector. These texture zeros can explain the observed large mixing in atmospheric neutrino oscillations and offer the possibility of alignment of the quark and squark mass matrices, and thus giving a solution to the supersymmetric flavor problem. Requiring the flavor symmetry commute with the SU(5) grand unified group can be used to explain the lepton mass hierarchies as well as the neutrino parameters, including the large mixing observed in the atmospheric neutrino data. We present one such model that yields a large atmospheric neutrino mixing angle, as well as a solar neutrino mixing angle of order $\lambda \simeq 0.22$.