Contributions of $K^*_0(1430)$ and $K^*_0(1950)$ in the three-body decays $B\to K\pi h$

Abstract

We study the contributions of the resonant states $K_0^{*}(1430)$ and $K_0^{*}(1950)$ in the three-body decays $B\to K\pi h$ (with $h=\pi, K$) in the perturbative QCD approach. The crucial nonperturbative input $F_{K\pi}(s)$ in the distribution amplitudes of the $S$-wave $K\pi$ system is derived from the matrix element of vacuum to $K\pi$ pair. The $CP$ averaged branching fraction of the quasi-two-body decay process $B\to K^*_0(1950)h\to K\pi h$ is about one order smaller than that of the corresponding decay $B\to K^*_0(1430)h\to K\pi h$. In view of the important contribution from the $S$-wave $K\pi$ system for the $B\to K\pi h$ decays, it is not appropriate to neglect the $K_0^{*}(1950)$ in the theoretical or experimental studies for the relevant three-body $B$ meson decays. The predictions in this work for the relevant decays are consistent with the existing experimental data.