Studying the localized $CP$ violation and the branching fraction of the $bar{B}^0rightarrow K^-pi^+pi^+pi^-$ decay

In this work, we study the localized $CP$ violation and the branching fraction of the four-body decay $bar{B}^0rightarrow K^-pi^+pi^-pi^+$ by employing a quasi-two-body QCD factorization approach. Considering the interference of $bar{B}^0rightarrow bar{K}_0^*(700)rho^0(770)rightarrow K^-pi^+pi^-pi^+$ and $bar{B}^0rightarrow bar{K}^*(892)f_0(500)rightarrow K^-pi^+pi^-pi^+$ channels, we predict $mathcal{A_{CP}}(bar{B}^0rightarrow K^-pi^+pi^-pi^+)in[0.15,0.28]$ and $mathcal{B}(bar{B}^0rightarrow K^-pi^+pi^-pi^+)in[1.73,5.10]times10^{-7}$, 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 $bar{B}^0rightarrow bar{K}_0^*(700)rho^0(770)$ and $bar{B}^0rightarrow bar{K}^*(892)f_0(500)$ decays, respectively. The corresponding results are $mathcal{A_{CP}}(bar{B}^0rightarrow bar{K}_0^*(700)rho^0(770)) in [0.20, 0.36]$, $mathcal{A_{CP}}(bar{B}^0rightarrow bar{K}^*(892)f_0(500))in [0.08, 0.12]$, $mathcal{B} (bar{B}^0rightarrow bar{K}_0^*(700)rho^0(770)in [6.76, 18.93]times10^{-8}$ and $mathcal{B} (bar{B}^0rightarrow bar{K}^*(892)f_0(500))in [2.66, 4.80]times10^{-6}$, 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.