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

In this letter, we present rephasing invariant formulae ${\delta }^{(\alpha i)}=\arg [V_{\alpha 1}{V}_{\alpha 2}{V}_{\alpha 3}{V}_{1i}{V}_{2i}{V}_{3i}\mathrm{/}{V}_{\alpha i}^3\det V]$ 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 ${\mathrm{\Phi }}_{\alpha i}$ of unitarity triangles satisfy compact sum rules ${\delta }^{(\alpha ,i+2)}-{\delta }^{(\alpha ,i+1)}={\mathrm{\Phi }}_{\alpha -2,i}-{\mathrm{\Phi }}_{\alpha -1,i}$ and ${\delta }^{(\alpha -2,i)}-{\delta }^{(\alpha -1,i)}={\mathrm{\Phi }}_{\alpha ,i+2}-{\mathrm{\Phi }}_{\alpha ,i+1}$ where all indices are taken cyclically modulo three. These twelve relations are natural generalizations of the previous result ${\delta }_{\mathrm{P}\mathrm{D}\mathrm{G}}+{\delta }_{\mathrm{K}\mathrm{M}}=\pi -\alpha +\gamma$.