Revisiting nonfactorizable contributions to factorization-forbidden decays of $B$ mesons to charmonium

Motivated by the large rates of $B\rightarrow (\chi_{c0}, \chi_{c2}, h_c)K$ decays observed by the $BABAR$ and Belle collaborations, we investigate the nonfactorizable contributions to these factorization-forbidden decays, which can occur through a gluon exchange between the $c\bar c$ system and the spectator quark. Our numerical results demonstrate that the spectator contributions are capable of producing a large branching ratio consistent with the experiments. As a by-product, we also study the Cabibbo-suppressed decays, such as $B\rightarrow (\chi_{c0}, \chi_{c2}, h_c)\pi$ and the U-spin-related $B_s$ decay, which have so far received less theoretical and experimental attention. The calculated branching ratios reach the order of $10^{-6}$, which in within the scope of the Belle-II and LHCb experiments. Further, the $CP$-asymmetry parameters are also calculated for these decays. The obtained results are compared with the available experimental data and numbers from other predictions. We also investigate the sources of theoretical uncertainties in our calculation.