Speaker
Description
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 apexes of the rescaled versions of these two triangles, described respectively by $\overline{\rho} + {\rm i} \overline{\eta} = -\left(V^{*}_{ub} V^{}_{ud}\right)/\left(V^{*}_{cb} V^{}_{cd}\right)$ 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 $(\overline{\rho}, \overline{\eta})$ and $(\widetilde{\rho}, \widetilde{\eta})$ is characterized by $\widetilde{\rho} - \overline{\rho} \sim \widetilde{\eta} - \overline{\eta} \sim {\cal O}(\lambda^2)$ with $\lambda \simeq 0.22$ being the Wolfenstein expansion parameter, and these two apexes are insensitive to the two-loop renormalization-group running effects up to the accuracy of ${\cal O}(\lambda^4)$. We suggest that the second $b$-flavored unitarity triangle can be established with the observables from $B^{\pm}_u$ and $B^0_s$-$\bar{B}^0_s$ systems based on the high-precision measurements to be done at the upgraded LHCb and Belle II experiments, and a comparison between the twin $b$-flavor unitarity triangles will provide a consistency check of the CKM picture for CP violation and probe possible new physics in this connection.
Based on arXiv:1911.03292 accepted for publication in Phys. Lett. B.
