Geometric representation of CP phases δPDG, δKM in flavor mixing matrix and its sum rule by alternative unitarity triangle and quadrangle
Masaki J. S. Yang · Physics Letters B · 2025
In this letter, we present a geometric representation of the CP phases δ PDG and δ KM in the PDG and Kobayashi–Maskawa parameterizations of the flavor mixing matrix in the complex plane. The sum rule with the unitarity triangle δ P D G + δ K M = π − α + γ is expressed as a quadrangle, which is a combination of a unitarity triangle and an alternative triangle. Through the unitarity quadrangle, the CP phases are also identified with specific geometric angles. Furthermore, a new set of inverse unitarity triangles is defined from the inversion formula of a unitary matrix U † = U − 1 . These novel triangles contain standard angles of the form arg [ U α i U β j U α j * U β i * ] and new angles arg [ U α i U β j U γ k / det U ] , which directly determine nontrivial arguments of the mixing matrix elements.