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

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 ${\delta }_{\mathrm{PDG}}+{\delta }_{\mathrm{KM}}=\pi -\alpha +\gamma$.

arXiv comment6 pages, published version in Chin. Phys. C