  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 $B^-\rightarrow\sigma\pi^-$ 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 $\mathcal{B}(B^-\rightarrow
\sigma(600)\pi^-)<1.67\times10^{-6}$ and $\mathcal{B}(B^-\rightarrow
\sigma(600) \pi^-)<1.946\times10^{-5}$ in these two models, respectively. We
find that there is no allowed divergence parameters $\rho_S$ and $\phi_S$ to
satisfy the experimental data
$\mathcal{A_{CP}}(\pi^-\pi^+\pi^-)=0.584\pm0.082\pm0.027\pm0.007$ 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 $\rho_S\in[1.70,3.34]$ 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.
