Quasi-two-body decays $B_c^+ \to \chi_{c0,c1} [\rho(K^*) \to] \pi\pi(K\pi)$ in the PQCD approach

Abstract

We study the quasi-two-body decays $B_c^+ \to \chi_{c0,c1} [\rho(K^*) \to] \pi^+\pi^0(K^0\pi^+)$ in the perturbative QCD framework at leading order. The $\pi\pi$ and $K\pi$ pairs are produced via the P-wave resonances $\rho(770)$, $\rho(1450)$, $\rho(1700)$, and $K^*(892)$, parametrized by the Gounaris-Sakurai and relativistic Breit-Wigner models. The $\rho$ channels, dominated by the $\rho(770)$ resonance, yield $\mathcal{B}(B_c^+ \to \chi_{c0} \pi^+\pi^0) = 7.33 \times 10^{-4}$ and $\mathcal{B}(B_c^+ \to \chi_{c1} \pi^+\pi^0) = 1.02 \times 10^{-3}$, with constructive $\rho$ interference making the coherent total $\sim 26\%$--$30\%$ larger than the $\rho(770)$ contribution alone. The Cabibbo-suppressed $K\pi$ channels yield branching ratios roughly a factor of five below those of the $\pi\pi$ modes, $\mathcal{B}(B_c^+ \to \chi_{c0} K^0\pi^+) = 1.49 \times 10^{-4}$ and $\mathcal{B}(B_c^+ \to \chi_{c1} K^0\pi^+) = 1.94 \times 10^{-4}$, the Cabibbo suppression being only partially compensated by the narrow $K^*(892)$ resonance and the normalization factor $N_{K^*}=1.48$. For $\chi_{c1}$ modes, $f_L \approx 93\%$ dominates and decreases with increasing resonance mass. The ratio $R_{\chi_{c1}/\chi_{c0}}^{\pi\pi} \approx 1.38$ reflects the common resonant $\rho$ production mechanism shared by $\chi_{c0}$ and $\chi_{c1}$, with the charmonium decay-constant hierarchy and phase space determining the relative yield of P-wave charmonia. The analogous ratio in the Cabibbo-suppressed $K\pi$ channel, $R_{\chi_{c1}/\chi_{c0}}^{K\pi} \approx 1.30$, is consistent with this value, confirming that the spin ratio is governed by charmonium-side dynamics.