An exact CKM matrix related to the approximate Wolfenstein form

Abstract

Noting the hierarchy between three mixing angles, $\theta_{2,3}={\cal O}(\theta_1^2)$, we present an exact form of the quark mixing matrix, replacing Wolfenstein's approximate form. In addition, we suggest to rotate the unitarity triangle, using the weak CP phase convention where the phase is located at the (31) element $\sin\theta_1\sin\theta_2 e^{i\delta}$ while the (13) element $\sin\theta_1\sin\theta_3$ is real. For the $(ab)$ unitarity triangle, the base line (x-axis) is defined from the product of the first row elements, $V_{1a}V^*_{1b}$, and the angle between two sides at the origin is defined to be the phase $\delta$. This is a useful definition since every Jarlskog triangle has the angle $\delta$ at the origin, defined directly from the unitarity condition. It is argued that $\delta$ represents the barometer of the weak CP violation, which can be used to relate it to possible Yukawa textures.