Predictive SUSY GUT Model for CP Violation, Fermion Masses and Mixings

CP violation, fermion masses and mixing angles including that of neutrinos are studied in an SUSY SO(10)$\times \Delta (48)\times$ U(1) model. The nonabelian SU(3) discrete family symmetry $\Delta(48)$ associated with a simple scheme of U(1) charge assignment on various fields concerned in superpotential leads to unique Yukawa coupling matrices with zero textures. Thirteen parameters involving masses and mixing angles in the quark and charged lepton sector are successfully predicted by only four parameters. The masses and mixing angles for the neutrino sector could also be predicted by constructing an appropriate heavy Majorana neutrino mass matrix without involving new parameters. It is found that the atmospheric neutrino deficit, the mass limit put by hot dark matter and the LSND $\bar{\nu}_{\mu} \to \bar{\nu}_{e}$ events may simultaneously be explained, but solar neutrino puzzle can be solved only by introducing a sterile neutrino. An additional parameter is added to obtain the mass and mixing of the sterile neutrino. The hadronic parameters $B_{K}$ and $f_{B}\sqrt{B}$ are extracted from the observed $K^{0}$-$\bar{K}^{0}$ and $B^{0}$-$\bar{B}^{0}$ mixings respectively. The direct CP violation ($\epsilon'/\epsilon$) in kaon decays and the three angles $\alpha$, $\beta$ and $\gamma$ of the unitarity triangle in the CKM matrix are also presented. More precise measurements of $\alpha_{s}(M_{Z})$, $|V_{cb}|$, $|V_{ub}/V_{cb}|$, $m_{t}$, as well as various CP violation and neutrino oscillation experiments will provide an important test for the present model and guide us to a more fundamental theory.