Motivated by the recent LHCb measurements of the ratios between the branching ratios (BRs), $R_{{\chi }_{c2}/J/\psi }\equiv \mathrm{BR}(B_c^{+}\to {\chi }_{c2}{\pi }^{+})/\mathrm{BR}(B_c^{+}\to J/\psi {\pi }^{+})$ and $R_{{\chi }_{c1}/{\chi }_{c2}}\equiv \mathrm{BR}(B_c^{+}\to {\chi }_{c1}{\pi }^{+})/\mathrm{BR}(B_c^{+}\to {\chi }_{c2}{\pi }^{+})$, 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 $R_{{\chi }_{c0}/J/\psi }\equiv \mathrm{BR}(B_c^{+}\to {\chi }_{c0}{\pi }^{+})/\mathrm{BR}(B_c^{+}\to J/\psi {\pi }^{+})=0.17\pm 0.02$ is lower than ${1.41}_{-0.45}^{+0.50}$ 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.

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• Branching fraction
• Particle decays
• Particle phenomena