Quasi-two-body decays $B\to P f_0(500)\to Pπ^+π^-$ in the perturbative QCD approach

Abstract: In this paper, we study the quasi-two-body decays $B\to P f_0(500)\to P\pi^+\pi^-$ [with $P=(\pi, K, \eta, \eta^{\prime})$] 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 $B\to P f_0(500)\to P\pi^+\pi^-$. Taking the quasi-two-body decay $B^+ \to \pi^+ f_0(500) \to \pi^+ \pi^+ \pi^-$ 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 $\mathcal{B}(B^+\to \pi^+ [\sigma\to]\pi^+\pi^-) = (1.78 \pm 0.41\pm 0.51) \times 10^{-6}$ and $\mathcal{A}_{CP}(B^+\to \pi^+ [\sigma\to]\pi^+\pi^-) = (29.8\pm 11.1\pm 13.0)\%$ respectively. Our results could be tested by further experiments.