CP-Violation and the Quark Mass Matrices

Abstract

We study a class of quark mass matrix models with the $\mathcal{C}\mathcal{P}$-violating phase determined through making the $\mathcal{C}\mathcal{P}$-violating parameter $J$ an extremum. These models assume that $m_u\ll m_c\ll m_t$ and that $m_d\ll m_s\ll m_b$. They have $\left|\frac{V_{ub}}{V_{cb}}\right|\approx \sqrt{\frac{m_u}{m_c}}$, $\left|\frac{V_{td}}{V_{ts}}\right|\approx \sqrt{\frac{m_d}{m_s}}$ and $|V_{us}|\approx |V_{cd}|\approx \sqrt{\frac{m_d}{m_s+\frac{m_u}{m_c}}}$. The Wolfenstein parameters $\rho$ and $\eta$ are found to be related by $\rho \approx {\eta }^2$. Finally, we examine a special class of such models where the masses are constrained to be roughly in geometric progression. Further application of the extremal condition to $J$ then leads to $\sqrt{\frac{m_d}{m_s}}\approx 3\sqrt{\frac{m_u}{m_c}}$ and hence $\eta \approx \frac{1}{3}$, for maximal $J$.

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