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

Abstract

We study the quasi-two-body decays $B_c^{+}\to {\chi }_{c0,c1}[\rho (K^{\ast })\to ]{\pi }^{+}{\pi }^0(K^0{\pi }^{+})$ 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 $K^{\ast }(892)$, 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 $K^{\ast }(892)$ resonance and the normalization factor $N_{K^{\ast }}=1.48$. 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.

Comments: 21 pages, 5 figures