The $\beta$ angle as the CP violating phase in the CKM matrix

TL;DR

This paper proposes using the experimentally well-measured CP violating angle β as the phase parameter in the CKM matrix, providing a new explicit parameterization and an approximate Wolfenstein-like expression.

Contribution

It introduces a novel CKM matrix parameterization based on the measurable angle β, addressing the parametrization dependence issue of the CP phase.

Findings

Explicit β-based CKM parameterization derived.

Approximate Wolfenstein-like expression obtained.

Addresses the measurement and parametrization dependence of CP phase.

Abstract

The CKM matrix describing quark mixing with three generations can be parameterized by three Euler 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 the most accurately measured CP violating angle $\beta$ in the unitarity triangleas the phase in the CKM matrix, and construct an explicit $\beta$ parameterization. We also derive an approximate Wolfenstein-like expression for this parameterization.