Study of $CP$ Violations in $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$ and $B^{-}\to K^{-}{f}_0(500)$ decays in the QCD factorization approach

Abstract

Within the QCD factorization approach, we study the $CP$ violations in $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$ and $B^{-}\to K^{-}{f}_0(500)$ decays. We find the experimental data of the localized $CP$ asymmetry in $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$ 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 ${\varphi }_S\in [1.518,5.183]$. With the obtained allowed ranges for ${\rho }_S$ and ${\varphi }_S$, we obtain the predictions for the $CP$ asymmetry parameter $A_{CP}\in [-0.259,0.006]$ and the branching fraction $\mathcal{B}\in [0.585,3.230]\times 10^{-5}$ for $B^{-}\to K^{-}{f}_0(500)$ decay modes.

Comments: 13 pages, 4 figures