We extend our recent works on the P-wave twopion resonant contributions to the kaon-pion cases in the hadronic charmless B meson decays by employing the perturbative QCD approach. The concerned decay modes are analysed in the quasi-two-body framework by parameterizing the kaon-pion distribution amplitude P K π , which contains the final state interactions between the kaon and pion in the resonant region. The relativistic Breit-Wigner formula for the Pwave resonant state K * (892) is adopted to parameterize the time-like form factor F K π. We calculate the C P-averaged branching ratios and direct C P-violating asymmetries of the quasi-two-body decays B (s) → K * (892)h → K π h, with h = (π, K), in this work. It is shown that the agreement of theoretical results with the experimental data can be achieved, through which Gegenbauer moments of the Pwave kaon-pion distribution amplitudes are determined. The predictions in this work will be tested by the precise data from the LHCb and the future Belle II experiments.

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We present the differential branching fractions for the B → ψππ decays with the charmonia ψ = J/ψ, ψ(2S) in the invariant mass of the P -wave pion pairs in the perturbative QCD approach. The two-pion distribution amplitudes (DAs) corresponding to both longitudinal and transverse polarizations are constructed to capture important final state interactions in the processes. The timelike form factors, normalizing the two-pion DAs, contain contributions from the ρ resonance and radial excitations fitted to the BABAR e + e − annihilation data. Given the hadronic parameters for the two-pion DAs associated with the longitudinal polarization which were determined in our previous study, and tuning those associated with the transverse polarization, we accommodate well the observed branching ratios and polarization fractions of the B → J/ψππ decays. Our predictions for the B → ψ(2S)ππ modes from the same set of parameters can be tested in future LHCb and BelleII experiments. We also investigate the sources of theoretical uncertainties in our calculation.

We study the quasi-two-body decays B → DK * (892) → DK π by employing the perturbative QCD approach. The two-meson distribution amplitudes P-wave K π are adopted to describe the final state interactions of the kaon-pion pair in the resonance region. The resonance line shape for the P-wave K π component K * (892) in the timelike form factor F K π (s) is parameterized by the relativistic Breit-Wigner function. For most considered decay modes, the theoretical predictions for their branching ratios are consistent with currently available experimental measurements within errors. We also disscuss some ratios of the branching fractions of the concerned decay processes. More precise data from LHCb and Belle-II are expected to test our predictions.