In this work, within the QCD factorization approach, we study the localized integrated $CP$ violation in the $B^-rightarrow pi^-pi^+pi^-$ decay and the branching fraction of the $B^-rightarrowsigmapi^-$ decay. Both the resonance and nonresonance contributions are included when we study the localized $CP$ asymmetry in the $B^-rightarrow pi^-pi^+pi^-$ decay. The resonance contributions from the scalar $sigma(600)$ and vector $rho^0(770)$ mesons are included. For the $sigma(600)$ meson, we apply both the Breit-Wigner and Bugg models to deal with its propagator, and obtain $mathcal{B}(B^-rightarrow sigma(600)pi^-)<1.67times10^{-6}$ and $mathcal{B}(B^-rightarrow sigma(600) pi^-)<1.946times10^{-5}$ in these two models, respectively. We find that there is no allowed divergence parameters $rho_S$ and $phi_S$ to satisfy the experimental data $mathcal{A_{CP}}(pi^-pi^+pi^-)=0.584pm0.082pm0.027pm0.007$ in the region $m_{pi^+pi^- mathrm{high}}^2>15$ $mathrm{GeV}^2$ and $m_{pi^+pi^-mathrm{low}}^2<0.4$ $mathrm{GeV}^2$ and the upper limit of $mathcal{B}(B^-rightarrow sigma(600)pi^-)$ in the Breit-Wigner model, however, there exists the region $rho_Sin[1.70,3.34]$ and $phi_S in [0.50,4.50]$ satisfying the data for $mathcal{A_{CP}}(pi^-pi^+pi^-)$ and the upper limit of $mathcal{B}(B^-rightarrow sigma(600)pi^-)$ in the Bugg model. This reveals that the Bugg model is more plausible than the Breit-Wigner model to describe the propagator of the $sigma(600)$ meson even though the finite width effects are considered in both models. The large values of $rho_S$ 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.
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