Remarks on the Qin-Ma Parametrization of Quark Mixing Matrix

TL;DR

This paper analyzes the Qin-Ma parametrization of the quark mixing matrix, showing its relation to Wolfenstein and CKM parametrizations, and discusses phase and high-order discrepancies in quark mixing descriptions.

Contribution

It demonstrates how the Qin-Ma parametrization can be derived from Wolfenstein and CKM forms, clarifies phase relations, and explores high-order differences in quark mixing parametrizations.

Findings

Qin-Ma parametrization can be obtained from Wolfenstein after phase redefinition.

The phase in QM and CKM is related to unitarity angles as δ=β+γ or π−α.

Discrepancies between Wolfenstein and QM appear at high order in λ.

Abstract

Recently, Qin and Ma (QM) have advocated a new Wolfenstein-like parametrization of the quark mixing matrix based on the triminimal expansion of the Cabibbo-Kobayashi-Maskawa (CKM) parametrization. The CP-odd phase in the QM parametrization is around $90^\circ$ just as that in the CKM parametrization. We point out that the QM parametrization can be readily obtained from the Wolfenstein parametrization after appropriate phase redefinition for quark fields and that the phase $\delta$ in both QM and CKM parametrizations is related to the unitarity angles $\alpha$, $\beta$ and $\gamma$, namely, $\delta= \beta+\gamma$ or $\pi-\alpha$. We show that both QM and Wolfenstein parametrizations can be deduced from the CKM and Chau-Keung-Maiani ones. By deriving the QM parametrization from the Fritzsch-Xing (FX) parametrization of the quark mixing matrix, we find that the phase of the FX form is in the vicinity of $-270^\circ$ and hence $\sin\delta\approx 1$. We discuss the seeming discrepancy between the Wolfenstein and QM parametrizations at the high order of $\lambda\approx |V_{us}|$.