Extracting $\gamma$ from $B_{s(d)}\to J/\psi K_S$ and $B_{d(s)}\to D^{+}_{d(s)} D^{-}_{d(s)}$

A completely general parametrization of the time-dependent decay rates of the modes $B_s\to J/\psi K_S$ and $B_d\to J/\psi K_S$ is given, which are related to each other through the $U$-spin flavour symmetry of strong interactions. Owing to the interference of current--current and penguin processes, the $B_s\to J/\psi K_S$ observables probe the angle $\gamma$ of the unitarity triangle. Using the $U$-spin symmetry, the overall normalization of the $B_s\to J/\psi K_S$ rate can be fixed with the help of the CP-averaged $B_d\to J/\psi K_{\rm S}$ rate, providing a new strategy to determine $\gamma$. This extraction of $\gamma$ is not affected by any final-state-interaction effects, and its theoretical accuracy is only limited by $U$-spin-breaking corrections. As a by-product, this strategy allows us to take into account also the penguin effects in the determination of $\beta$ from $B_d\to J/\psi K_S$, which are presumably very small, and to predict the direct CP asymmetry arising in this mode. An analogous strategy is provided by the time-dependent $B_d\to D^+ D^-$ rate, if its overall normalization is fixed through the CP-averaged $B_s\to D^+_s D^-_s$ rate.