Study of CP violations in \(B^-\rightarrow K^- \pi ^+\pi ^-\) and \(B^-\rightarrow K^- f_0(500)\) decays in the QCD factorization approach

Within the QCD factorization approach, we study the CP violations in \(B^-\rightarrow K^-\pi ^+\pi ^-\) and \(B^-\rightarrow K^- f_0(500)\) decays. We find the experimental data of the localized CP asymmetry in \(B^-\rightarrow K^-\pi ^+\pi ^-\) decays in the region \(m_{K^-\pi ^+}^2<15\) \(\mathrm {GeV}^2\) and \(0.08pi ^+\pi ^-}^2<0.66\) \(\mathrm {GeV}^2\) can be explained by the interference of two intermediate resonances, \(\rho ^0(770)\) and \(f_0(500)\) when the parameters in our interference model are in the allowed ranges, i.e. the relative strong phase \(\delta \in [2.124, 5.976]\) and the end-point divergence parameters \(\rho _S\in [5.692, 8]\) and \(\phi _S \in [0, 2\pi ]\). With the obtained allowed ranges for \(\rho _S\) and \(\phi _S\), we obtain the predictions for the CP asymmetry parameter \(A_{CP} \in [-0.115, -0.151]\) and the branching fraction \({\mathcal {B}} \in [3.763, 20.014]\times 10^{-5}\) for \(B^-\rightarrow K^-f_0(500)\) decay modes.