Breaking of the flavor permutational symmetry: Mass textures and the CKM matrix

Different Ansätze for the breaking of flavor permutational symmetry according to $S_L\mathrm{}\left(3\right)\mathrm{}\otimes S_R\mathrm{}\left(3\right)\mathrm{}\supset S_L\mathrm{}\left(2\right)\mathrm{}\otimes S_R\mathrm{}\left(2\right)\mathrm{}$ give different Hermitian mass matrices of the same modified Fritzsch type, which differ in the symmetry breaking pattern. In this work we obtain a clear and precise indication on the preferred symmetry breaking scheme from a fit of the predicted $|{\mathbf{V}}^{\mathrm{th}}|$ to the experimentally determined absolute values of the elements of the CKM matrix. The preferred scheme leads to simple mass textures and allows us to compute the CKM mixing matrix, the Jarlskog invariant $J$, and the three inner angles of the unitarity triangle in terms of four quark mass ratios and only one free parameter: the $\mathrm{CP}$ violating phase $\Phi$. Excellent agreement with the experimentally determined absolute values of the entries in the CKM matrix is obtained for $\Phi =90\mathrm{}°$. The corresponding computed values of the Jarlskog invariant and the inner angles are $J=3.00\mathrm{}\times {10}^{-5},$ $\alpha =84\mathrm{}°,$ $\beta =24\mathrm{}°$, and $\gamma =72\mathrm{}°$ in very good agreement with current data on $\mathrm{CP}$ violation in the neutral kaon-antikaon system and oscillations in the $B_s^0-B_s^0$ system.