arXiv2021
Within the quasi-two-body decay model, we study the localized $CP$ violation and branching fraction of the four-body decay $\bar{B}^0\rightarrow [K^-π^+]_{S/V}[π^+π^-]_{V/S} \rightarrow K^-π^+π^-π^+$ when $K^-π^+$ and $π^-π^+$ pair invariant masses are $0.35<m_{K^-π^+}<2.04 \, \mathrm{GeV}$ and $0<m_{π^-π^+}<1.06\, \mathrm{GeV}$, with the pairs being dominated by the $\bar{K}^*_0(700)^0$, $\bar{K}^*(892)^0$, $\bar{K}^*(1410)^0$, $\bar{K}^*_0(1430)$ and $\bar{K}^*(1680)^0$, and $f_0(500)$, $ρ^0(770)$ , $ω(782)$ and $f_0(980)$ resonances, respectively. When dealing with the dynamical functions of these resonances, $f_0(500)$, $ρ^0(770)$, $f_0(980)$ and $\bar{K}^*_0(1430)$ are modeled with the Bugg model, Gounaris-Sakurai function, Flatt$\acute{\mathrm{e}}$ formalism and LASS lineshape, respectively, while others are described by the relativistic Breit-Wigner function. Adopting the end point divergence parameters $ρ_A\in[0,0.5]$ and $ϕ_A\in[0,2π]$, our predicted results are $\mathcal{A_{CP}}(\bar{B}^0\rightarrow K^-π^+π^+π^-)\in[-0.383,0.421]$ and $\mathcal{B}(\bar{B}^0\rightarrow K^-π^+π^+π^-)\in[7.36,199.69]\times10^{-8}$ based on the hypothetical $q\bar{q}$ structures for the scalar mesons in the QCD factorization approach. Meanwhile, we calculate the $CP$ violating asymmetries and branching fractions of the two-body decays $\bar{B}^0\rightarrow SV(VS)$ and all the individual four-body decays $\bar{B}^0\rightarrow SV(VS) \rightarrow K^-π^+π^-π^+$, respectively. Our theoretical results for the two-body decays $\bar{B}^0\rightarrow \bar{K}^*(892)^0$$f_0(980)$, $\bar{B}^0\rightarrow \bar{K}^*_0(1430)^0$$ω(782)$, $\bar{B}^0\rightarrow \bar{K}^*(892)^0f_0(980)$, $\bar{B}^0\rightarrow\bar{K}^*_0(1430)^0ρ$,