In this work, we study the localized $CP$ violation and the branching fraction of the four-body decay $\bar{B}^0\rightarrow K^-\pi^+\pi^-\pi^+$ by employing a quasi-two-body QCD factorization approach. Considering the interference of $\bar{B}^0\rightarrow \bar{K}_0^*(700)\rho^0(770)\rightarrow K^-\pi^+\pi^-\pi^+$ and $\bar{B}^0\rightarrow \bar{K}^*(892)f_0(500)\rightarrow K^-\pi^+\pi^-\pi^+$ channels, we predict $\mathcal{A_{CP}}(\bar{B}^0\rightarrow K^-\pi^+\pi^-\pi^+)\in[0.15,0.28]$ and $\mathcal{B}(\bar{B}^0\rightarrow K^-\pi^+\pi^-\pi^+)\in[1.73,5.10]\times10^{-7}$, respectively, which shows that this two channels' interference mechanism can induce the localized $CP$ violation to this four-body decay. Meanwhile, within the two quark model framework for the scalar mesons $f_0(500)$ and $\bar{K}_0^*(700)$, we calculate the direct CP violations and branching fractions of the $\bar{B}^0\rightarrow \bar{K}_0^*(700)\rho^0(770)$ and $\bar{B}^0\rightarrow \bar{K}^*(892)f_0(500)$ decays, respectively. The corresponding results are $\mathcal{A_{CP}}(\bar{B}^0\rightarrow \bar{K}_0^*(700)\rho^0(770)) \in [0.20, 0.36]$, $\mathcal{A_{CP}}(\bar{B}^0\rightarrow \bar{K}^*(892)f_0(500))\in [0.08, 0.12]$, $\mathcal{B} (\bar{B}^0\rightarrow \bar{K}_0^*(700)\rho^0(770)\in [6.76, 18.93]\times10^{-8}$ and $\mathcal{B} (\bar{B}^0\rightarrow \bar{K}^*(892)f_0(500))\in [2.66, 4.80]\times10^{-6}$, respectively, indicating the $CP$ violations of these two two-body decays are both positive and the branching fractions are quite different. These studies provide a new way to investigate the aforementioned four-body decay and could be helpful in clarifying the configuration of the structure of light scalar meson.

中文翻译：

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研究局部 $CP$ 违规和 $\bar{B}^0\rightarrow K^-\pi^+\pi^+\pi^-$ 衰减的分支分数

在这项工作中，我们研究了局部 $CP$ 违反和四体衰变 $\bar{B}^0\rightarrow K^-\pi^+\pi^-\pi^+$ 的分支分数一种准二体 QCD 分解方法。考虑$\bar{B}^0\rightarrow \bar{K}_0^*(700)\rho^0(770)\rightarrow K^-\pi^+\pi^-\pi^+$的干扰和 $\bar{B}^0\rightarrow \bar{K}^*(892)f_0(500)\rightarrow K^-\pi^+\pi^-\pi^+$ 通道，我们预测 $\mathcal {A_{CP}}(\bar{B}^0\rightarrow K^-\pi^+\pi^-\pi^+)\in[0.15,0.28]$ 和 $\mathcal{B}(\bar {B}^0\rightarrow K^-\pi^+\pi^-\pi^+)\in[1.73,5.10]\times10^{-7}$，分别说明这两个信道的干扰机制可以将局部的 $CP$ 违反引入到这种四体衰变中。同时，在标量介子 $f_0(500)$ 和 $\bar{K}_0^*(700)$ 的两个夸克模型框架内，我们计算 $\bar{B}^0\rightarrow \bar{K}_0^*(700)\rho^0(770)$ 和 $\bar{B}^0\ 的直接 CP 违规和分支分数rightarrow \bar{K}^*(892)f_0(500)$ 分别衰减。对应的结果是 $\mathcal{A_{CP}}(\bar{B}^0\rightarrow \bar{K}_0^*(700)\rho^0(770)) \in [0.20, 0.36]$ , $\mathcal{A_{CP}}(\bar{B}^0\rightarrow \bar{K}^*(892)f_0(500))\in [0.08, 0.12]$, $\mathcal{B} (\bar{B}^0\rightarrow \bar{K}_0^*(700)\rho^0(770)\in [6.76, 18.93]\times10^{-8}$ 和 $\mathcal{B} (\bar{B}^0\rightarrow \bar{K}^*(892)f_0(500))\in [2.66, 4.80]\times10^{-6}$，分别表示$CP$违反这两个二体衰变都是正的，并且分支分数有很大的不同。