Phase Direct CP Violations and General Mixing Matrices

Abstract

I formulate expressions for amplitudes suitable for quantifying both modulus and phase direct CP violations. They result in Möbius transformation (MT) relations, which provide encouraging information for the search of direct CP violations in general. I apply the formulation to calculate the measurements of phase direct CP violations and strong amplitudes in $B^{\mp}\to K^{\mp}π^{\pm}π^{\mp}$ by the Belle collaboration. For the formulation, I show a versatile construction procedure for $N\times N$ Cabibbo-Kobayashi-Maskawa (CKM) matrices, Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrices, and general unitary matrices. It clarifies the $3\times3$ cases and is useful for the beyond.