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

TL;DR

This work investigates off-shell contributions from the $K^*$ resonance in charmless three-body $B$ decays using perturbative QCD. By modeling the $K^*\to K^*\pi$ and $K^*\to K\rho$ sub-processes with two-meson distribution amplitudes and a Breit–Wigner propagator $1/D_{K^*}(s)$, and relating couplings via SU(3) symmetry, the authors predict CP-averaged quasi-two-body branching fractions in the $10^{-9}$ to $10^{-7}$ range for $B\to K^*[\to K^*\pi]h$ and $B\to K^*[\to K\rho]h$, with the latter typically about half of the former due to phase-space effects. The analysis finds small off-shell contributions overall, discusses their dependence on input parameters (e.g., $\omega_B$, Gegenbauer moments, and $g_{\rho\omega\pi}$), and notes that experimental verification from Belle II and LHCb will shed light on the role of virtual $K^*$ states in these decays. The study also highlights differential branching fractions as functions of the intermediate invariant masses, which peak away from the pole due to phase-space suppression near threshold.

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.

Figures (2)

• Figure 1: The leading order Feynman diagrams for the decays $B\to K^* h$ with the subprocesses $K^*\to K^*\pi$ and $K^*\to K\rho$. The symbol $\otimes$ represents the insertion of the four-fermion vertices in the effective theory and $\times$ denotes the possible attachments of hard gluons.
• Figure 2: Differential branching fractions for the $B^+ \to K^{*0}[\to K^{*+}\pi^-] \pi^+$ and $B^+ \to K^{*0}[\to K^+\rho^-] \pi^+$ decays with the invariant mass $m_{K^*\pi}$ or $m_{K\rho}$ ranging from threshold to $3.5~\rm GeV$.