Discrete Flavor Symmetry, Dynamical Mass Textures, and Grand Unification
  Discrete flavor symmetry is explored for an intrinsic property of mass matrix forms of quarks and leptons. In this paper we investigate the S3 permutation symmetry and derive the general forms of mass matrices in various types of S3 theories. We also exhibit particular realizations of previous ansatze of mass matrices, which have often been applied in the literature to the standard model Yukawa sector. Discrete flavor symmetry is also advantageous for vanishing matrix elements being dynamically generated in the vacuum of scalar potential. This is due to the fact that group operations are discrete. While zero elements themselves do not explain mass hierarchies, we introduce an abelian flavor symmetry. A non-trivial issue is whether successful quantum numbers can be assigned so that they are compatible with other (non-abelian) flavor symmetries. We show typical examples of charge assignments which not only produce hierarchical orders of mass eigenvalues but also prohibit non-renormalizable operators which disturb the hierarchies in first-order estimation. As an explicit application, a flavor model is constructed in grand unification scheme with S3 and U(1) (or Z_N) flavor symmetries. 
Naoyuki Haba and Koichi Yoshioka