$CP$~Violation in $B \to J/\psi X$ Modes

In the $B^0$~system, $CP$~violation occurs dominantly via the CKM mechanism, which produces little violation of $CP$ in the $B^0_s$ system. We report on the effort to measure the parameters: $\sin 2\beta$, which applies in the $B^0$~system (and which has already been tightly constrained at the $B$~factories), and $\sin 2\beta_s$ which applies in the $B^0_s$~system (and which is expected to be small in the absence of additional, non-CKM sources of $CP$~violation).