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 $V_{ub}^{\ast }{V}_{ud}^{}+V_{cb}^{\ast }{V}_{cd}^{}+V_{tb}^{\ast }{V}_{td}^{}=0$ and $V_{ud}^{\ast }{V}_{td}^{}+V_{us}^{\ast }{V}_{ts}^{}+V_{ub}^{\ast }{V}_{tb}^{}=0$ in the complex plane. We find that vertices of the rescaled versions of these two triangles, described respectively by $\bar{\rho }+\mathrm{i}\bar{\eta }=-\left(V_{ub}^{\ast }{V}_{ud}^{}\right)/\left(V_{cb}^{\ast }{V}_{cd}^{}\right)$ and $\tilde{\rho }+\mathrm{i}\tilde{\eta }=-\left(V_{ub}^{\ast }{V}_{tb}^{}\right)/\left(V_{us}^{\ast }{V}_{ts}^{}\right)$, 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 $(\tilde{\rho },\tilde{\eta })$ is characterized by $\tilde{\rho }-\bar{\rho }\sim \tilde{\eta }-\bar{\eta }\sim \mathcal{O}({\lambda }^2)$ 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 $\mathcal{O}({\lambda }^4)$. Some comments are also made on similar features of three pairs of the rescaled unitarity triangles of lepton flavor mixing and CP violation.

Comments: 15 pages, 6 figures, 1 table, accepted for publication in PLB