Nearly right unitarity triangle and CP phase in quark and lepton flavor mixings

Abstract

The nearly right unitarity triangle can be simply obtained if a quark mass matrix is written as a linear combination of real rank-1 matrices and the coefficient which gives the mass of the second generation is pure imaginary. Supposing that the source of the CP violation is in the Yukawa coupling to an additional Higgs field which provides a factor for the strange quark and muon mass ratio, we obtain that the angle $\alpha= \phi_2$ of the unitarity triangle shifts from $90^{\rm o}$ by units of $V_{ub}/V_{us}$, and is postdicted as $\alpha \simeq 87^{\rm o}$ or $91^{\rm o}$. The Dirac CP phase $\delta$ which appears in the three-flavor neutrino oscillations is obtained to be $|\delta| \simeq 80^{\rm o}$ if the neutrino mass matrix gives tri-bimaximal-like mixing form. If the factor 3 for the muon and strange quark mass is considered in a simple manner in the quark-lepton unification, we obtain three distinctive prediction of $\delta$ for nearly right-angled phase as $|\delta| \simeq 70^{\rm o}$, $90^{\rm o}$, or $110^{\rm o}$ in an idealistic orthogonal structure of the neutrino mass matrix. The deviation from $90^{\rm o}$ is roughly given by $\arcsin(1/3)$ by inputting the experimental measurements.