Quasi-two-body decays $B\to P{f}_0(500)\to P{\pi }^{+}{\pi }^{-}$ in the perturbative QCD approach

In this paper, we study the quasi-two-body decays $B\to P{f}_0(500)\to P{\pi }^{+}{\pi }^{-}$ [with $P=(\pi ,K,\eta ,{\eta }^{\mathrm{\prime }})$] 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 ${\mathcal{A}}_{CP}(B^{+}\to {\pi }^{+}[\sigma \to ]{\pi }^{+}{\pi }^{-})=(29.8\pm 11.1\pm 13.0)\mathrm{\%}$ respectively. Our results could be tested by further experiments.