CP violation in non-leptonic $B_c$ decays to excited final states

We study the CP violation in two-body nonleptonic decays of $B_c$ meson. We concentrate on the decay channels which contain at least one excited heavy meson in the final states. Specifically, the following channels are considered: $B_c\to c\bar{c}(2S,2P)+\bar{c}q(1S,1P)$, $B_c\to c\bar{c}(1S)+\bar{c}q(2S,2P)$, $B_c\to c\bar{c}(1P)+\bar{c}q(2S)$, $B_c\to c\bar{c}(1D)+\bar{c}q(1S,1P)$, and $B_c\to c\bar{c}(3S)+\bar{c}q(1S)$. The improved Bethe-Salpeter method is applied to calculate the hadronic transition matrix element. Our results show that some decay modes have large branching ratios, which is of the order of ${10}^{-3}$. The CP violation effect in $B_c\to {\eta }_c(1S)+D(2S)$, $B_c \rightarrow \eta_c(1S)+D_0^{}(2P)$,and$B_c \rightarrow J/\psi+D^{}(2S)$ are most likely to be found. If the detection precision of the CP asymmetry in such channels can reach the $3\sigma$ level, at least ${10}^7$ $B_c$ events are needed.