Resonance contributions from $\chi_{c0}$ in the charmless three-body hadronic $B$ meson decays

Within the framework of perturbative QCD factorization, we investigate the nonfactorizable contributions to these factorization-forbidden Quasi-two-body decays $B_{(s)}\rightarrow h\chi_{c0}\rightarrow h\pi^+\pi^-(K^+K^-)$ with $ h=\pi, K$. We compare our predicted branching ratios for the $B_{(s)}\rightarrow K\chi_{c0}\rightarrow K\pi^+\pi^-(K^+K^-)$ decay with available experiment data as well as predictions by other theoretical studies. The branching ratios of these decays are consistent with data and other theoretical predictions. In the Cabibbo-suppressed decays $B_{(s)}\rightarrow h\chi_{c0}\rightarrow h\pi^+\pi^-(K^+K^-)$ with $h=\bar{K}^0,\pi$, however, the values of the branching ratios are the order of $10^{-7}$ and $10^{-8}$. The ratio $R_{\chi_{c0}}$ between the decay $B^+\rightarrow \pi^+\chi_{c0}\rightarrow \pi^+\pi^+\pi^-$ and $B^+\rightarrow K^+\chi_{c0}\rightarrow K^+\pi^+\pi^-$ and the distribution of branching ratios for different decay modes in invariant mass are considered in this work.