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

Abstract

In this letter, we present a geometric representation of the CP phases $δ_{\rm PDG}$ and $δ_{\rm KM}$ in the PDG and Kobayashi--Maskawa parameterizations of the flavor mixing matrix in the complex plane. The sum rule with the unitarity triangle $δ_{\rm PDG} + δ_{\rm KM} = π- α+ γ$ 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 $U^{\dagger} = U^{-1}$. These novel triangles contain standard angles of the form $\arg [U_{αi } U_{βj} U_{αj}^{*} U_{βi}^{*}]$ and new angles $\arg [U_{αi } U_{βj} U_{γk} / \det U]$, which directly determine nontrivial arguments of the mixing matrix elements.