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

Abstract

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 $0.08<m_{\pi^+\pi^-}^2<0.66$ $\mathrm{GeV}^2$ 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 $\delta\in[0, 1.745]\cup[3.578, 6.266]$ and the end-point divergence 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 obtain the 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.