The virtual contributions from $K^*$ in the $B \to K^*πh$ and $B \to Kρh$ decays

Abstract

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