Geometric representation of CP phases δPDG, δKM in flavor mixing matrix and its sum rule by alternative unitarity triangle and quadrangle

In this letter, we present a geometric representation of the CP phases δPDG and δKM in the PDG and Kobayashi--Maskawa parameterizations of the mixing matrix as angles on the complex plane. The sum rule with the unitarity triangle δPDG + δKM = π- α+ γ is geometrically expressed as a quadrangle on the complex plane, combination of a unitarity triangle and an alternative triangle. Furthermore, a new set of inverse unitarity triangles is defined from the inversion formula of a unitary matrix U^ = U⁻¹. Through these unitarity triangle and quadrangle, the CP phases are no longer abstract entities but are identified with specific geometric angles.