Quasi-two-body decays B→ P f₀(500)→ Pπ⁺π⁻ in the perturbative QCD approach

In this paper, we study the quasi-two-body decays B→ P f₀(500)→ Pπ⁺π⁻ (with P=(π, K, η, η^)) within framework of perturbative QCD (PQCD) factorization approach. With the help of π-π distribution amplitude and scalar form factor Fππ(ω²), we calculate the CP averaged branching fraction and the CP asymmetry for the quasi-two-body decays B→ P f₀(500)→ Pπ⁺π⁻. Taking the quasi-two-body decay B⁺ → π⁺ f₀(500) → π⁺ π⁺ π⁻ as an explicit example, we present the behaviour of differential branching fraction and direct CP violation versus the π-π invariant mass. The total branching fraction and direct CP violation are B(B⁺→ π⁺ [σ→]π⁺π⁻) = (1.78 ± 0.41± 0.51) × 10⁻⁶ and ACP(B⁺→ π⁺ [σ→]π⁺π⁻) = (29.8± 11.1± 13.0)% respectively. Our results could be tested by further experiments.