The $\alpha$, $\beta$ and $\gamma$ parameterizations of CP violating CKM phase

TL;DR

This paper introduces new parameterizations of the CKM matrix using directly measurable CP violating quantities, namely alpha, beta, and gamma, instead of the traditional phase, providing clearer experimental relevance.

Contribution

It proposes explicit alpha, beta, and gamma parameterizations of the CKM matrix based on measurable CP violation parameters, with approximate Wolfenstein-like forms.

Findings

New parameterizations using alpha, beta, gamma

Explicit forms for CKM matrix with these parameters

Approximate Wolfenstein-like expressions provided

Abstract

The CKM matrix describing quark mixing with three generations can be parameterized by three mixing angles and one CP violating phase. In most of the parameterizations, the CP violating phase chosen is not a directly measurable quantity and is parametrization dependent. In this work, we propose to use experimentally measurable CP violating quantities, $\alpha$, $\beta$ or $\gamma$ in the unitarity triangle as the phase in the CKM matrix, and construct explicit $\alpha$, $\beta$ and $\gamma$ parameterizations. Approximate Wolfenstein-like expressions are also suggested.