In this paper, we attempt to build an unified model with democratic texture, that have some unification between $Y_{ \nu}$ and $Y_{ u}$. Since the $S_{ 3L} \times S_{ 3R}$ flavor symmetry is chiral, the unified gauge group is assumed to be Pati-Salam type $SU(4)_{ c} \times SU(2)_{ L} \times SU(2)_{ R}$. The flavor symmetry breaking scheme is considered to be $S_{ 3L} \times S_{ 3R} \to S_{ 2L} \times S_{ 2R} \to 0$. In this picture, the four-zero texture is desirable for realistic mass and mixings. This texture is realized by a specific representation for the second breaking of the $S_{ 3L} \times S_{ 3R}$ flavor symmetry. Assuming only renormalizable Yukawa interactions, type-I seesaw mechanism, and neglecting $CP$ phases for simplicity, the right-handed neutrino mass matrix $M_{ R}$ can be reconstructed from low energy input values. Numerical analysis shows that the texture of $M_{ R}$ basically behaves like the "waterfall texture". Since $M_{ R}$ tends to be the "cascade texture" in the democratic texture approach, a model with type-I seesaw and up-type Yukawa unification $Y_{ \nu} \simeq Y_{ u}$ 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 fine-tunings of parameters.
Categories: hep-ph