Distinguishing between the twin b-flavored unitarity triangles on a circular arc

Abstract

With the help of the generalized Wolfenstein parametrization of quark flavor mixing and CP violation, we calculate fine differences between the twin b-flavored unitarity triangles defined by Vub⁎Vud+Vcb⁎Vcd+Vtb⁎Vtd=0 and Vud⁎Vtd+Vus⁎Vts+Vub⁎Vtb=0 in the complex plane. We find that apexes of the rescaled versions of these two triangles, described respectively by ρ‾+iη‾=−(Vub⁎Vud)/(Vcb⁎Vcd) and ρ˜+iη˜=−(Vub⁎Vtb)/(Vus⁎Vts), are located on a circular arc whose center and radius are given by O=(0.5,0.5cot⁡α) and R=0.5csc⁡α with α being their common inner angle. The small difference between (ρ‾,η‾) and (ρ˜,η˜) is characterized by ρ˜−ρ‾∼η˜−η‾∼O(λ2) with λ≃0.22 being the Wolfenstein expansion parameter, and these two apexes are insensitive to the two-loop renormalization-group running effects up to the accuracy of O(λ4). Some comments are also made on similar features of three pairs of the rescaled unitarity triangles of lepton flavor mixing and CP violation.