In this work, within the QCD factorization approach, we study the localized integrated $CP$ violation in the $B^-\rightarrow ��^-��^+��^-$ decay and the branching fraction of the $B^-\rightarrow����^-$ decay. Both the resonance and nonresonance contributions are included when we study the localized $CP$ asymmetry in the $B^-\rightarrow ��^-��^+��^-$ decay. The resonance contributions from the scalar $��(600)$ and vector $��^0(770)$ mesons are included. For the $��(600)$ meson, we apply both the Breit-Wigner and Bugg models to deal with its propagator, and obtain $\mathcal{B}(B^-\rightarrow ��(600)��^-)<1.67\times10^{-6}$ and $\mathcal{B}(B^-\rightarrow ��(600) ��^-)<1.946\times10^{-5}$ in these two models, respectively. We find that there is no allowed divergence parameters $��_S$ and $��_S$ to satisfy the experimental data $\mathcal{A_{CP}}(��^-��^+��^-)=0.584\pm0.082\pm0.027\pm0.007$ in the region $m_{��^+��^- \mathrm{high}}^2>15$ $\mathrm{GeV}^2$ and $m_{��^+��^-\mathrm{low}}^2<0.4$ $\mathrm{GeV}^2$ and the upper limit of $\mathcal{B}(B^-\rightarrow ��(600)��^-)$ in the Breit-Wigner model, however, there exists the region $��_S\in[1.70,3.34]$ and $��_S \in [0.50,4.50]$ satisfying the data for $\mathcal{A_{CP}}(��^-��^+��^-)$ and the upper limit of $\mathcal{B}(B^-\rightarrow ��(600)��^-)$ in the Bugg model. This reveals that the Bugg model is more plausible than the Breit-Wigner model to describe the propagator of the $��(600)$ meson even though the finite width effects are considered in both models. The large values of $��_S$ indicate that the contributions from weak annihilation and hard spectator scattering processes are both large, especially, the weak annihilation contribution should not be negleted for $B$ decays to final states including a scalar meson.

arXiv admin note: substantial text overlap with arXiv:1811.02167