Quasi-two-body decays $B_c^+ \to χ_{c0,c1} [ρ(K^*) \to] ππ(Kπ)$ in the PQCD approach

Abstract

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