Wolfenstein Parametrization Re-examined

Abstract

The Wolfenstein parametrization of the $3\times 3$ Kobayashi-Maskawa (KM) matrix $V$ is modified by keeping its unitarity up to the accuracy of $O(\lambda^{6})$. This modification can self-consistently lead to the off-diagonal asymmetry of $V$: $|V_{ij}|^{2}-|V_{ji}|^{2}$ = $Z\displaystyle\sum_{k}\epsilon^{~}_{ijk}$ with $Z=\approx A^{2}\lambda^{6} (1-2\rho)$, which is comparable in magnitude with the Jarlskog parameter of $CP$ violation $J\approx A^{2}\lambda^{6}\eta$. We constrain the ranges of $J$ and $Z$ by using the current experimental data, and point out that the possibility of a symmetric KM matrix has almost been ruled out.