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

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 $\barρ + {\rm i} \barη = -\left(V^{*}_{ub} V^{}_{ud}\right)/\left(V^{*}_{cb} V^{}_{cd}\right)$ and $\widetildeρ + {\rm i} \widetildeη = -\left(V^{*}_{ub} V^{}_{tb}\right)/\left(V^{*}_{us} V^{}_{ts}\right)$, are located on a circular arc whose center and radius are given by $O = \left(0.5, 0.5 \cotα\right)$ and $R = 0.5 \cscα$ with $α$ being their common inner angle. The small difference between $(\barρ, \barη)$ and $(\widetildeρ, \widetildeη)$ is characterized by $\widetildeρ - \barρ \sim \widetildeη - \barη \sim {\cal O}(λ^2)$ with $λ\simeq 0.22$ being the Wolfenstein expansion parameter, and these two vertices are insensitive to the two-loop renormalization-group running effects up to the accuracy of ${\cal O}(λ^4)$. Some comments are also made on similar features of three pairs of the rescaled unitarity triangles of lepton flavor mixing and CP violation.

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