Quasi-two-body decays B → ηc(1S,2S) [ρ(770),ρ(1450),ρ(1700)→] ππ in the perturbative QCD approach

In this paper, we calculated the branching ratios of the quasi-two-body decays B→ηc(1S,2S)[ρ(770),ρ(1450),ρ(1700)→]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 ΦππP is parameterized by the vector current time-like form factor Fπ 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∼10−6, while the two-body decay rates B(B→ηc(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 |Fπ|2 measured by the BABAR Collaboration could be understood based on our theoretical results; (c) the general expectation based on the similarity between B→ηcππ and B→J/ψππ decays are confirmed: R2(ηc)≈0.45 is consistent with the measured R2(J/ψ)≈0.56±0.09 within errors; and (d) new ratios R3(ηc(1S)) and R4(ηc(2S)) among the branching ratios of the considered decay modes are defined and could be tested by future experiments.