Quasi-two-body decays $B \to η_c {(1S ,2S)}\;[ρ(770),ρ(1450),ρ(1700) \to ]\; ππ$ in the perturbative QCD approach

Abstract

In this paper, we calculated the branching ratios of the quasi-two-body decays $B \to η_c (1S ,2S)$ $[ρ(770), ρ(1450),ρ(1700)\to ] ππ$ by employing the perturbative QCD (PQCD) approach. The contributions from the $P$-wave resonances $ρ(770)$, $ρ(1450)$ and $ρ(1700)$ were taken into account. The two-pion distribution amplitude $Φ_{ππ}^{\rm P}$ is parameterized by the vector current time-like form factor $F_π$ 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 ${\cal B}(B \to η_c{(1S,2S)} (ρ(1450),ρ(1700)))$ are extracted from those for the corresponding quasi-two-body decays; (b) the whole pattern of the pion form factor-squared $|F_π|^2$ measured by the BABAR Collaboration could be understood based on our theoretical results; (c) the general expectation based on the similarity between $B \to η_c ππ$ and $B \to J/ψππ$ decays are confirmed: $R_2(η_c)\approx 0.45$ is consistent with the measured $R_2(J/ψ)\approx 0.56\pm 0.09$ within errors; and (d) new ratios $R_3(η_c(1S))$ and $R_4(η_c(2S))$ among the branching ratios of the considered decay modes are defined and could be tested by future experiments.

10 pages, 3 figures