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Within the QCD factorization approach, we study the $CP$ violations in$B^-\rightarrow K^-\pi^+\pi^-$ and $B^-\rightarrow K^- f_0(500)$ decays. Wefind the experimental data of the localized $CP$ asymmetry in $B^-\rightarrowK^-\pi^+\pi^-$ decays in the region $m_{K^-\pi^+}^2<15$ $\mathrm{GeV}^2$ and$0.08<m_{\pi^+\pi^-}^2<0.66$ $\mathrm{GeV}^2$ can be explained by theinterference of two intermediate resonances, $\rho^0(770)$ and $f_0(500)$ whenthe parameters in our interference model are in the allowed ranges, i.e. therelative strong phase $\delta\in[0, 1.745]\cup[3.578, 6.266]$ and the end-pointdivergence parameters $\rho_S\in[2.790, 5.290]$ and $\phi_S \in [1.518,5.183]$. With the obtained allowed ranges for $\rho_S$ and $\phi_S$, we obtainthe predictions for the $CP$ asymmetry parameter $A_{CP} \in [-0.259, 0.006]$and the branching fraction $\mathcal{B} \in [0.585, 3.230]\times10^{-5}$ for$B^-\rightarrow K^-f_0(500)$ decay modes.

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