$B_c$ meson decays into $S$-wave charmonium plus light meson pairs in the perturbative QCD approach

In this work, we explore the $P$-wave resonance contributions to the three-body charmonium decays of $B_c\to \Psi (V\to) P_1P_2$ using the perturbative QCD formalism at leading order, where $\Psi$ denotes a $S$-wave charmonium state, such as $\eta_c(1S,2S),J/\psi$, and $\psi(2S)$. Here, $P_1P_2$ represents a collinear $\pi\pi$ ($K\pi$) pair in the final state, which was primarily produced through the vector resonance $\rho(770)$ ($K^*(892)$ ). With the improved two-meson distribution amplitudes determined from our previous works, we examined the $CP$-averaged branching ratios and polarization fractions of the considered three-body decays. The longitudinal polarization fractions of the $B_c\to [J/\psi, \psi(2S)] (V\to) P_1P_2$ decays are found to be as large as $\sim 90\%$, since the transverse amplitudes from the dominant factorizable emission diagrams are always power suppressed with respect to the longitudinal ones. The direct $CP$ violations in $B_c\to \Psi (V\to) P_1P_2$ decays are predicted naturally to be zero as they solely receive contributions from tree diagrams. Several interesting relative ratios among the branching fractions of the concerned processes are investigated. In particular, the obtained ratio $R^{\rm PQCD}_{2\pi/\pi}\equiv \mathcal{B}(B^+_c \to J/\psi(\rho\to)\pi^+\pi^0)/{\mathcal{B}(B^+_c \to J/\psi\pi^+)}=2.67^{+0.21}_{-0.14}$ is consistent well with the LHCb measurement $R^{\rm exp}_{2\pi/\pi}=2.80\pm0.25$. Other similar ratios proposed in this work can be tested by LHCb experiments in the near future.