Analysis of the right-handed Majorana neutrino mass in an SU(4) x SU(2)L x SU(2)R Pati-Salam model with democratic texture

In this paper, we attempt to build a unified model with the democratic texture, that has some unification between up-type Yukawa interactions Y-v and Y-u. Since the S-3L x S-3R flavor symmetry is chiral, the unified gauge group is assumed to be Pati-Salam type SU(4)(c) x SU(2)(L) x SU(2)(R). The breaking scheme of the flavor symmetry is considered to be S-3L x S-3R -> S-2L x S-2R -> 0. In this picture, the four-zero texture is desirable for realistic masses and mixings. This texture is realized by a specific representation for the second breaking of the S-3L x S-3R flavor symmetry. Assuming only renormalizable Yukawa interactions, type-I seesaw mechanism, and neglecting CP phases for simplicity, the right-handed neutrino mass matrix MR can be reconstructed from low energy input values. Numerical analysis shows that the texture of M-R basically behaves like the "waterfall texture." Since MR tends to be the "cascade texture" in the democratic texture approach, a model with type-I seesaw and up-type Yukawa unification Y-v similar or equal to Y-u basically requires fine-tunings between parameters. Therefore, it seems to be more realistic to consider universal waterfall textures for both Y-f and M-R, e. g., by the radiative mass generation or the Froggatt-Nielsen mechanism. Moreover, analysis of eigenvalues shows that the lightest mass eigenvalue M-R1 is too light to achieve successful thermal leptogenesis. Although the resonant leptogenesis might be possible, it also requires finetunings of parameters.