Occam's Razor in Quark Mass Matrices

Abstract

On the standpoint of Occam's Razor approach, we consider the minimum number of parameters in the quark mass matrices needed for the successful CKM mixing and CP violation. We impose three zeros in the down-quark mass matrix with taking the diagonal up-quark mass matrix to reduce the number of free parameters. The three zeros are maximal zeros in order to have a CP violating phase in the quark mass matrix. Then, there remain six real parameters and one CP violating phase, which is the minimal number to reproduce the observed data of the down-quark masses and the CKM parameters. The twenty textures with three zeros are examined. Among them, the thirteen textures are viable for the down-quark mass matrix. As a representative of these textures, we discuss a texture $M_d^{(1)}$ in details. By using the experimental data on $\sin 2\beta$, $\theta_{13}$ and $\theta_{23}$ together with the observed quark masses, the Cabibbo angle is predicted to be close to the experimental data. It is found that this surprising result remains unchanged in all other viable textures. We also investigate the correlations among $|V_{ub}/V_{cb}|$, $\sin 2\beta$ and $J_{CP}$. For all textures, the maximal value of the ratio $|V_{ub}/V_{cb}|$ is $0.09$, which is smaller than the upper-bound of the experimental data, $0.094$. We hope that this prediction will be tested in future experiments.