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

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 $δ_{\mathrm{PDG}}+δ_{\mathrm{KM}}=π-α+γ.$