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

Abstract

In this work, within the QCD factorization approach, we study the localized integrated $CP$ violation in the $B^{-}\to {\pi }^{-}{\pi }^{+}{\pi }^{-}$ decay and the branching fraction of the $B^{-}\to \sigma {\pi }^{-}$ decay. Both the resonance and nonresonance contributions are included when we study the localized $CP$ asymmetry in the $B^{-}\to {\pi }^{-}{\pi }^{+}{\pi }^{-}$ 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 $\mathcal{B}(B^{-}\to \sigma (600){\pi }^{-})<1.67\times 10^{-6}$ and $\mathcal{B}(B^{-}\to \sigma (600){\pi }^{-})<1.946\times 10^{-5}$ in these two models, respectively. We find that there is no allowed divergence parameters ${\rho }_S$ and ${\varphi }_S$ to satisfy the experimental data ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}({\pi }^{-}{\pi }^{+}{\pi }^{-})=0.584\pm 0.082\pm 0.027\pm 0.007$ 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 $\mathcal{B}(B^{-}\to \sigma (600){\pi }^{-})$ in the Breit-Wigner model, however, there exists the region ${\rho }_S\in [1.70,3.34]$ and ${\varphi }_S\in [0.50,4.50]$ satisfying the data for ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}({\pi }^{-}{\pi }^{+}{\pi }^{-})$ and the upper limit of $\mathcal{B}(B^{-}\to \sigma (600){\pi }^{-})$ 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.

Comments: arXiv admin note: substantial text overlap with arXiv:1811.02167