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 $V^{*}_{ub} V^{}_{ud} + V^{*}_{cb} V^{}_{cd} + V^{*}_{tb} V^{}_{td} = 0$ and $V^{*}_{ud} V^{}_{td} + V^{*}_{us} V^{}_{ts} + V^{*}_{ub} V^{}_{tb} = 0$ in the complex plane. We find that vertices of the rescaled versions of these two triangles, described respectively by $\bar{\rho} + {\rm i} \bar{\eta} = -\left(V^{*}_{ub} V^{}_{ud}\right)/\left(V^{*}_{cb} V^{}_{cd}\right)$ and $\widetilde{\rho} + {\rm i} \widetilde{\eta} = -\left(V^{*}_{ub} V^{}_{tb}\right)/\left(V^{*}_{us} V^{}_{ts}\right)$, are located on a circular arc whose center and radius are given by $O = \left(0.5, 0.5 \cot\alpha\right)$ and $R = 0.5 \csc\alpha$ with $\alpha$ being their common inner angle. The small difference between $(\bar{\rho}, \bar{\eta})$ and $(\widetilde{\rho}, \widetilde{\eta})$ is characterized by $\widetilde{\rho} - \bar{\rho} \sim \widetilde{\eta} - \bar{\eta} \sim {\cal O}(\lambda^2)$ with $\lambda \simeq 0.22$ being the Wolfenstein expansion parameter, and these two vertices are insensitive to the two-loop renormalization-group running effects up to the accuracy of ${\cal O}(\lambda^4)$. Some comments are also made on similar features of three pairs of the rescaled unitarity triangles of lepton flavor mixing and CP violation.