Quasi-two-body decays B(s )→P ρ'(1450 ),P ρ''(1700 )→P π π in the perturbative QCD approach

In this work, we calculate the C P -averaged branching ratios and direct C P -violating asymmetries of the quasi-two-body decays B_{(s )}→P ρ^'(1450 ),P ρ^''(1700 )→P π π by employing the perturbative QCD (PQCD) factorization approach, where P is a light pseudoscalar meson K , π , η , and η^'. The considered decay modes are studied in the quasi-two-body framework by parametrizing the two-pion distribution amplitude Φ_π^{π P}. The P -wave timelike form factor F_π in the resonant regions associated with the ρ^'(1450 ) and ρ^''(1700 ) is estimated based on available experimental data. The PQCD predictions for the C P -averaged branching ratios of the decays B_{(s )}→P ρ^'(1450 ),P ρ^''(1700 )→P π π are in the order of 10^-7-10^-9. The branching ratios of the two-body decays B_{(s )}→P ρ^'(1450 ),P ρ^''(1700 ) are extracted from the corresponding quasi-two-body decay modes. The whole pattern of the squared pion form factor |F_π|² measured by BABAR Collaboration could also be understood based on our studies. The PQCD predictions in this work will be tested by the precise data from the LHCb and the future Belle II experiments.