The virtual contributions from $K^{\ast }$ in the $B\to K^{\ast }\pi h$ and $B\to K\rho h$ decays

Abstract

Inspired by the significant virtual contributions of the subprocesses $D^{\ast }/B^{\ast }\to D\pi$ and $\rho \to \omega \pi /K\bar{K}$ in the three-body hadronic $B$ meson decays, we study the off-shell effects of the resonance $K^{\ast }$ decaying into the $K^{\ast }\pi$ or $K\rho$ system in the decays $B\to K^{\ast }\pi h$ and $B\to K\rho h$ (with $h=\pi ,K$) within the perturbative QCD approach. The strong coupling constants $g_{K^{\ast }{K}^{\ast }\pi }$ and $g_{K^{\ast }K\rho }$ involved in this work are derived from the coupling constant $g_{\rho \omega \pi }$ under the flavor SU$(3)$ symmetry. The $CP$ averaged branching fractions for the quasi-two-body decays $B\to K^{\ast }[\to K^{\ast }\pi ]h$ and $B\to K^{\ast }[\to K\rho ]h$ are predicted to be on the order of $10^{-9}$ to $10^{-7}$ in this study. And the branching fractions for the decays $B\to K^{\ast }[\to K\rho ]h$ are found to be around half of the corresponding results for the $B\to K^{\ast }[\to K^{\ast }\pi ]h$ channels, mainly due to the phase space difference between the $K\rho$ and $K^{\ast }\pi$ pairs originating from the intermediate state $K^{\ast }$. Experimental data for the $K^{\ast }$ decaying into the final states $K^{\ast }\pi$ and $K\rho$ in the $B$ meson decays is expected to be obtained by the LHCb and Belle-II experiments in the near future.

Comments: 13 pages, 2 figures and 5 tables.Matching the published version in PRD