Geometric representation of CP phases ${\delta }_{\mathrm{PDG}},{\delta }_{\mathrm{KM}}$ 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 ${\delta }_{\mathrm{PDG}}$ and ${\delta }_{\mathrm{KM}}$ in the PDG and Kobayashi--Maskawa parameterizations of the flavor mixing matrix in the complex plane. The sum rule with the unitarity triangle ${\delta }_{\mathrm{PDG}}+{\delta }_{\mathrm{KM}}=\pi -\alpha +\gamma$ 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 $\arg [U_{\alpha i}{U}_{\beta j}{U}_{\alpha j}^{\ast }{U}_{\beta i}^{\ast }]$ and new angles $\arg [U_{\alpha i}{U}_{\beta j}{U}_{\gamma k}/\det U]$, which directly determine nontrivial arguments of the mixing matrix elements.

Comments: 6 pages, 2 figures, published version in Phys. Lett. B