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

TL;DR

This work addresses how CP-violating phases can be described consistently across nine Euler-angle-like parameterizations of the flavor-mixing matrix and how these phases relate to unitarity-triangle angles. It introduces rephasing-invariant CP phases $\delta^{(\alpha i)}$ given by $\delta^{(\alpha i)} = \arg \left[ { V_{\alpha 1} V_{\alpha 2} V_{\alpha 3} V_{1i} V_{2i} V_{3i} / ( V_{\alpha i}^{3} \det V ) } \right]$ for all nine parameterizations $V^{(\alpha i)}$, establishing them as irreducible fifth-order invariants. The paper then derives exact sum rules linking these phases to the nine unitarity-triangle angles $\Phi_{\alpha i}$ via a transfer-matrix formalism, such as $\delta^{(\alpha , i+2)} - \delta^{(\alpha, i+1)} = \Phi_{\alpha+1,i} - \Phi_{\alpha+2,i}$ and $\delta^{(\alpha+1,i)} - \delta^{(\alpha+2,i)} = \Phi_{\alpha,i+2} - \Phi_{\alpha,i+1}$, with the PDG–KM relation recovered as a special case. The results provide a general, order-preserving framework connecting phases and triangles across parameterizations, with numerical checks using CKM data supporting the consistency of the invariants and sum rules.

Abstract

In this letter, we present rephasing invariant formulae $δ^{(αi)} = \arg [ { V_{α1} V_{α2} V_{α3} V_{1i} V_{2i} V_{3i} / V_{αi }^{3} \det V } ] $ for CP phases $δ^{(αi)}$ associated with nine Euler-angle-like parameterizations of a flavor mixing matrix. Here, $α$ and $i$ denote the row and column carrying the trivial phases in a given parameterization. Furthermore, we show that the phases $δ^{(αi)}$ and the nine angles $Φ_{αi}$ of unitarity triangles satisfy compact sum rules $ δ^{(α, i+2)} - δ^{(α, i+1)} = Φ_{α-2, i} - Φ_{α-1, i}$ and $ δ^{(α-2, i)} - δ^{(α-1, i)} = Φ_{α, i+2} - Φ_{α, i+1}$ where all indices are taken cyclically modulo three. These twelve relations are natural generalizations of the previous result $δ_{\mathrm{PDG}}+δ_{\mathrm{KM}}=π-α+γ$.