Study of $B_c \to \chi_{cJ}\ (P, V)$ decays in the improved perturbative QCD formalism

Motivated by the recent LHCb measurements of the ratios between the branching ratios (BRs), $R_{\chi_{c2}/J/\psi} \equiv {\rm BR}(B_c^+ \to \chi_{c2} \pi^+)/{\rm BR}(B_c^+ \to J/\psi \pi^+)$ and $R_{\chi_{c1}/\chi_{c2}} \equiv {\rm BR}(B_c^+ \to \chi_{c1} \pi^+)/{\rm BR}(B_c^+ \to \chi_{c2} \pi^+)$, we analyze the decays $B_c^+ \to \chi_{cJ} (P, V)^+$ in the improved perturbative QCD (iPQCD) formalism, where $\chi_{cJ}$ denotes the $p$-wave charmonia with $J=0, 1, 2$, and $P (V)$ denotes the pseudoscalars $\pi$ and $K$ (the vectors $\rho$ and $K^*$). Our results $R_{\chi_{c2}/J/\psi} = 0.32 \pm 0.05$ and $R_{\chi_{c1}/\chi_{c2}}= 0.11 \pm 0.01$ at leading order in the strong coupling $\alpha_s$ agree well with the data within uncertainties. The result $R_{\chi_{c0}/J/\psi} \equiv {\rm BR}(B_c^+ \to \chi_{c0} \pi^+)/{\rm BR}(B_c^+ \to J/\psi \pi^+) = 0.17 \pm 0.02$ is lower than $1.41^{+0.50}_{-0.45}$ inferred by the $B_c^+ \to \chi_{c0} \pi^+$ measurement. The large longitudinal polarization fractions imply that these components dominate the $B_c^+ \to \chi_{c1,c2} \rho^+$ BRs. A peculiar constructive (destructive) interference between the twist-2 and twist-3 contributions is identified in the $B_c^+\to \chi_{c1} \rho^+$ ($B_c^+\to \chi_{c1} \pi^+$) mode, but not in the corresponding ones with $\chi_{c0,c2}$ mesons. Inputting the BRs of the strong decays $\chi_{c0,c2} \to \pi^+ \pi^- / K^+ K^-$ and $\chi_{cJ} \to \pi^+\pi^-(\pi^+ \pi^-/K^+K^-)$ reported by the BESIII Collaboration, we obtain those of $B_c^+ \to \chi_{cJ} (\rho, \pi)^+$ followed by secondary decay chains under the narrow-width approximation. All the above iPQCD predictions can be confronted by more precise experiments in the future to deepen our understanding of QCD dynamics in $B_c\to\chi_{cJ}$ transitions.