Geometric representation of CP phases $δ_{\rm PDG}, δ_{\rm KM}$ in flavor mixing matrix and its sum rule by alternative unitarity triangle and quadrangle (2509.11596v1)

Abstract: In this letter, we present a geometric representation of the CP phases $\delta_{\rm PDG}$ and $\delta_{\rm KM}$ in the PDG and Kobayashi--Maskawa parameterizations of the mixing matrix as angles on the complex plane. The sum rule with the unitarity triangle $\delta_{\rm PDG} + \delta_{\rm KM} = \pi - \alpha + \gamma$ 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 $U^{\dagger} = U^{-1}$. Through these unitarity triangle and quadrangle, the CP phases are no longer abstract entities but are identified with specific geometric angles.