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Abstract Based on the fitting results of the LHCb collaboration on the contributions of various intermediate resonances to the B^+ ^+ ^+ ^- B + → π + π + π - decay, we calculate the branching fractions and localized CP asymmetries of the quasi-two-body B^+ ^+ ( (770), (782), (1450), f₂ (1270) ) ^+ ^- B + → π + ρ (770), ω (782), ρ (1450), f 2 1270 → π + π - decays within the perturbative QCD (PQCD) approach systematically. Our theoretical predictions for the branching fractions agree well with the data within the errors. In order to further test the framework of the three-body B meson decays in PQCD and the wave functions of π π pair, we also calculate the branching fractions of the corresponding two-body B^+ ^+ (770) B + → π + ρ (770), B^+ ^+ (782) B + → π + ω (782) and B^+ ^+f₂ (1270) B + → π + f 2 1270 decays under the narrow-width-approximation, which are in consistence with the experimental data. The direct CP asymmetries of B^+ ^+ ( (770/1450) ) ^+ ^- B + → π + (ρ (770 / 1450) →) π + π - decays are found to be very small because these decay modes are tree-dominated. However, due to the large penguin contributions from the chiral enhanced annihilation diagrams, B^+ ^+ (f₂ (1270) ) ^+ ^- B + → π + (f 2 1270 →) π + π - decay has a large direct CP asymmetry, which is also discovered by LHCb collaboration. A relatively large CP asymmetry is also expected in the B^+ ^+ () ^+ ^- B + → π + (ω →) π + π - decay occurring via - ρ - ω mixing, which would be measured LHCb and Belle-II experiments in future.
Chang et al. (Sat,) studied this question.