Quasi-two-body decays B → Pf 0 (500)→ Pπ+π- in the perturbative QCD approach

In this paper, we study the quasi-two-body decays B -> Pf(0) (500) -> P pi(+)pi(-) [with P = (pi, K, eta, eta ')] within framework of perturbative QCD (PQCD) factorization approach. With the help of pi - pi distribution amplitude and scalar form factor F-pi pi(omega(2)), we calculate the CP averaged branching fraction and the CP asymmetry for the quasi-two-body decays B -> Pf(0) (500) -> P pi(+)pi(-). Taking the quasi-two-body decay B -> pi(+) f(0)(500) -> pi(+)pi(+)pi(-) as an explicit example, we present the behavior of differential branching fraction and direct CP violation versus the pi-pi invariant mass. The total branching fraction and direct CP violation are B(B+ -> pi(+) [sigma ->]pi(+)pi(-)) = (1.78 +/- 0.41 +/- 0.51) x 10(-6) and A(CP)(B+ -> pi(+) [sigma ->]pi(+)pi(-)) = 29.8 +/- 11.1 +/- 13.0)% respectively. Our results could be tested by further experiments.