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In this work, within the QCD factorization approach, we study the localizedintegrated $CP$ violation in the $B^-\rightarrow \pi^-\pi^+\pi^-$ decay and thebranching fraction of the $B^-\rightarrow\sigma\pi^-$ decay. Both the resonanceand nonresonance contributions are included when we study the localized $CP$asymmetry in the $B^-\rightarrow \pi^-\pi^+\pi^-$ decay. The resonancecontributions from the scalar $\sigma(600)$ and vector $\rho^0(770)$ mesons areincluded. For the $\sigma(600)$ meson, we apply both the Breit-Wigner and Buggmodels 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. Wefind that there is no allowed divergence parameters $\rho_S$ and $\phi_S$ tosatisfy 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 $\mathcal{A_{CP}}(\pi^-\pi^+\pi^-)$ andthe upper limit of $\mathcal{B}(B^-\rightarrow \sigma(600)\pi^-)$ in the Buggmodel. This reveals that the Bugg model is more plausible than the Breit-Wignermodel to describe the propagator of the $\sigma(600)$ meson even though thefinite width effects are considered in both models. The large values of$\rho_S$ indicate that the contributions from weak annihilation and hardspectator scattering processes are both large, especially, the weakannihilation contribution should not be negleted for $B$ decays to final statesincluding a scalar meson.

nuclear physics b

Study of localized $CP$ violation in $B^-\rightarrow \pi^- \pi^+\pi^-$  and the branching ratio of $B^-\rightarrow \sigma(600)\pi^-$ in the QCD  factorization approach