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

Abstract

In this work, we study the localized $CP$ violation and the branching fraction of the four-body decay ${\bar{B}}^0\to K^{-}{\pi }^{+}{\pi }^{-}{\pi }^{+}$ by employing a quasi-two-body QCD factorization approach. Considering the interference of ${\bar{B}}^0\to {\bar{K}}_0^{\ast }(700){\rho }^0(770)\to K^{-}{\pi }^{+}{\pi }^{-}{\pi }^{+}$ and ${\bar{B}}^0\to {\bar{K}}^{\ast }(892)f_0(500)\to K^{-}{\pi }^{+}{\pi }^{-}{\pi }^{+}$ channels, we predict ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}({\bar{B}}^0\to K^{-}{\pi }^{+}{\pi }^{-}{\pi }^{+})\in [0.15,0.28]$ and $\mathcal{B}({\bar{B}}^0\to K^{-}{\pi }^{+}{\pi }^{-}{\pi }^{+})\in [1.73,5.10]\times 10^{-7}$, 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 ${\bar{K}}_0^{\ast }(700)$, we calculate the direct CP violations and branching fractions of the ${\bar{B}}^0\to {\bar{K}}_0^{\ast }(700){\rho }^0(770)$ and ${\bar{B}}^0\to {\bar{K}}^{\ast }(892)f_0(500)$ decays, respectively. The corresponding results are ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}({\bar{B}}^0\to {\bar{K}}_0^{\ast }(700){\rho }^0(770))\in [0.20,0.36]$, ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}({\bar{B}}^0\to {\bar{K}}^{\ast }(892)f_0(500))\in [0.08,0.12]$, $\mathcal{B}({\bar{B}}^0\to {\bar{K}}_0^{\ast }(700){\rho }^0(770)\in [6.76,18.93]\times 10^{-8}$ and $\mathcal{B}({\bar{B}}^0\to {\bar{K}}^{\ast }(892)f_0(500))\in [2.66,4.80]\times 10^{-6}$, 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.