A Symmetry Pattern of Maximal CP-Violation and a Determination of the Unitarity Triangle

Abstract

Within a specific texture of the quark mass matrix the notion of a maximal violation of the $CP$ symmetry can be defined. The experimental constraints from weak decays imply that in reality one is close to the case of maximal $CP$ violation, which vanishes as the mass of the $u$--quark approaches zero. The unitarity triangle of the Cabibbo--Kobayashi--Maskawa matrix elements is determined. It is related to the triangle relating the Cabibbo angle to the quark mass ratios in the complex plane. The angle $\alpha $ describing the $CP$ violation in the decay $B_d \rightarrow \pi^+ \pi^-$ is close to 90$^{\circ }$.