Quasi-two-body decays $B\to {\eta }_c(1S,2S)[\rho (770),\rho (1450),\rho (1700)\to ]\pi \pi$ in the perturbative QCD approach

Abstract

In this paper, we calculated the branching ratios of the quasi-two-body decays $B\to {\eta }_c(1S,2S)$ $[\rho (770),\rho (1450),\rho (1700)\to ]\pi \pi$ by employing the perturbative QCD (PQCD) approach. The contributions from the $P$-wave resonances $\rho (770)$, $\rho (1450)$ and $\rho (1700)$ were taken into account. The two-pion distribution amplitude ${\Phi }_{\pi \pi }^{\mathrm{P}}$ is parameterized by the vector current time-like form factor $F_{\pi }$ to study the considered decay modes. We found that (a) the PQCD predictions for the branching ratios of the considered quasi-two-body decays are in the order of $10^{-7}\sim 10^{-6}$, while the two-body decay rates $\mathcal{B}(B\to {\eta }_c(1S,2S)(\rho (1450),\rho (1700)))$ are extracted from those for the corresponding quasi-two-body decays; (b) the whole pattern of the pion form factor-squared $|F_{\pi }{|}^2$ measured by the BABAR Collaboration could be understood based on our theoretical results; (c) the general expectation based on the similarity between $B\to {\eta }_c\pi \pi$ and $B\to J/\psi \pi \pi$ decays are confirmed: $R_2({\eta }_c)\approx 0.45$ is consistent with the measured $R_2(J/\psi )\approx 0.56\pm 0.09$ within errors; and (d) new ratios $R_3({\eta }_c(1S))$ and $R_4({\eta }_c(2S))$ among the branching ratios of the considered decay modes are defined and could be tested by future experiments.

Comments: 10 pages, 3 figures