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 $B_{(s)}\to h{\chi }_{c0}\to h{\pi }^{+}{\pi }^{-}(K^{+}{K}^{-})$ with $h=\pi ,K$. We compare our predicted branching ratios for the $B_{(s)}\to K{\chi }_{c0}\to K{\pi }^{+}{\pi }^{-}(K^{+}{K}^{-})$ 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 $B_{(s)}\to h{\chi }_{c0}\to h{\pi }^{+}{\pi }^{-}(K^{+}{K}^{-})$ 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 $B^{+}\to {\pi }^{+}{\chi }_{c0}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}$ and $B^{+}\to K^{+}{\chi }_{c0}\to K^{+}{\pi }^{+}{\pi }^{-}$ and the distribution of branching ratios for different decay modes in invariant mass are considered in this work.