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¥₀₋₏₇₀ ₁ V¥₀₋₏₇₀ ₂ V¥₀₋₏₇₀ ₃ V₁₈ V₂₈ V₃₈ / V¥₀₋₏₇₀ ₈ ^{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¥₀₋₏₇₀ ₈ of unitarity triangles satisfy compact sum rules ¥delta^ (¥alpha, i+2) - ¥delta^ (¥alpha, i+1) = ¥Phi¥₀₋₏₇₀+₁, ₈ - ¥Phi¥₀₋₏₇₀+₂, ₈and ¥delta^ (¥alpha+1, i) - ¥delta^ (¥alpha+2, i) = ¥Phi¥₀₋₏₇₀, ₈+₂ - ¥Phi¥₀₋₏₇₀, ₈+₁where all indices are taken cyclically modulo three. These relations are natural generalizations of the previous result ¥delta¥₌₀ₓ₇ₑ₌₏₃₆+¥delta¥₌₀ₓ₇ₑ₌₊₌=¥pi-¥alpha+¥gamma.