Rephasing invariant formulae for CP phases in general parameterizations of flavor mixing matrix and exact sum rules with unitarity triangles

TL;DR

This paper derives rephasing invariant formulas for CP phases in various parameterizations of flavor mixing matrices and establishes exact sum rules linking these phases with unitarity triangle angles, enhancing understanding of CP violation.

Contribution

It introduces new rephasing invariant expressions for CP phases and exact sum rules connecting these phases with unitarity triangle angles in general parameterizations.

Findings

Derived invariant formulas for CP phases in nine parameterizations.

Established sum rules relating CP phases and unitarity triangle angles.

Generalized previous CP phase relations to broader parameterizations.

Abstract

In this letter, we present rephasing invariant formulae $\delta^{(\alpha i)} = \arg [ { V_{\alpha 1} V_{\alpha 2} V_{\alpha 3} V_{1i} V_{2i} V_{3i} / V_{\alpha i }^{3} \det V } ] $ for CP phases $\delta^{(\alpha i)}$ associated with nine Euler-angle-like parameterizations of a flavor mixing matrix. Here, $\alpha$ and $i$ denote the row and column carrying the trivial phases in a given parameterization. Furthermore, we show that the phases $\delta^{(\alpha i)}$ and the nine angles $\Phi_{\alpha i}$ of unitarity triangles satisfy compact sum rules $ \delta^{(\alpha, i+2)} - \delta^{(\alpha, i+1)} = \Phi_{\alpha-2, i} - \Phi_{\alpha-1, i}$ and $ \delta^{(\alpha-2, i)} - \delta^{(\alpha-1, i)} = \Phi_{\alpha, i+2} - \Phi_{\alpha, i+1}$ where all indices are taken cyclically modulo three. These twelve relations are natural generalizations of the previous result $\delta_{\mathrm{PDG}}+\delta_{\mathrm{KM}}=\pi-\alpha+\gamma$.