Flavor in SU(5)$SU(5)$ Finite Grand Unified Models.

Four SU(5)N=1$SU(5) \nobreakspace N=1$ supersymmetric models which exhibit S3$S_3$ and/or ZN$Z_N$ symmetries are studied, that are finite to two or all loops, and their corresponding mass matrices. The first is an all‐loop finite model based on an S3×Z3×Z2$S_3\times Z_3\times Z_2$ flavor symmetry, which leads to phenomenologically nonviable mass matrices. The remaining models, based on cyclic symmetries, show various mass textures, some of which are phenomenologically promising. For the two‐loop finite models, the parametric solutions to the finiteness conditions determine completely some of the Yukawa couplings, and lead to a restricted range of values for other ones at the GUT scale, with a considerable reduction in the number of free parameters. One particular solution of the two‐loop models shows an enhanced symmetry, leading to an all‐loop finite model, which has a significant parameter reduction and could in principle reproduce the observed quark masses and mixing pattern. In this case the finiteness conditions determine the absolute value of all the Yukawa couplings at the unification scale. Finally, the minimum number of phases in the mass matrices and their position are determined, a task not previously done in Finite Unified Theories, which contributes towards the reduction of parameters and a better understanding of the Yukawa couplings. [ABSTRACT FROM AUTHOR]