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

TL;DR

The paper tackles the lack of a direct geometric interpretation of CP phases in the flavor-mixing matrix by introducing a unitarity quadrangle that represents $\delta_{\rm PDG}$ and $\delta_{\rm KM}$ as angles in the complex plane. It demonstrates a sum rule $\delta_{\rm PDG} + \delta_{\rm KM} = \pi - \alpha + \gamma$ arising from combining a standard unitarity triangle with an alternative triangle, and shows how CP phases correspond to specific quadrangle angles. It further defines a set of inverse unitarity triangles from the inversion relation $U^{\dagger} = U^{-1}$, yielding nine triangles whose angles include $\arg [U_{\alpha i} U_{\beta j} U_{\gamma k} / \det U]$, providing direct access to nontrivial mixing-element arguments. Collectively, these constructions recast CP violation in a purely geometric framework and extend the unitarity-triangle picture to quadrangles with potential applicability to generalized sum rules and lepton mixing.

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.

Figures (1)

• Figure 1: Geometric understanding of the CP phases and the sum rule by a unitarity quadrangle and a new unitarity triangle. This figure is drawn without assuming any specific mixings of quarks or leptons, and the sum of vectors is defined in the counterclockwise direction.