Probing the $P$-wave charmonium decays of $B_c$ meson

Motivated by the large number of $B_c$ meson decay modes observed recently by several detectors at the LHC, we present a detailed analysis of the $B_c$ meson decaying to the $P$-wave charmonium states and a light pseudoscalar($P$) or vector ($V$) meson within the framework of perturbative QCD factorization. The $P$-wave charmonium distribution amplitudes are extracted from the $n=2,l=1$ Schr$\ddot{o}$dinger states for a Coulomb potential,which can be taken as the universal nonperturbative objects to analyze the hard exclusive processes with $P$-wave charmonium production. It is found that these decays have large branching ratios of the order of $10^{-5}\sim 10^{-2}$, which seem to be in the reach of future experiments. We also provide predictions for the polarization fractions and relative phases of $B_c\rightarrow (\chi_{c1},\chi_{c2},h_c)V$ decays. It is expected that the longitudinal polarization amplitudes dominate the branching ratios according to the quark helicity analysis, and the magnitudes and phases of parallel polarization amplitude are approximately equal to the perpendicular ones. The obtained results are compared with available experimental data, our previous studies, and numbers from other approaches.