Contributions of the subprocesses $ρ(770,1450,1700)\to K \bar{K}$ and $ω(782,1420,1650)\to K \bar{K}$ for the three-body decays $B\to η^{(\prime)} K\bar{K}$

Abstract: As an extension of our prior work, we analyze the resonance contributions for the kaon pair originating from the intermediate $ρ(770)$, $ω(782)$ and their excited states in the three-body decays $B\to η^{(\prime)} K\bar{K}$ within the perturbative QCD approach. The information of subprocesses $ρ(770,1450,1700)\to K\bar K$ and $ω(782,1420,1650)\to K\bar K$ are included in the distribution amplitudes for $K\bar K$ system by using the kaon vector time-like form factors. We calculate the $CP$ averaged branching fractions and the direct $CP$ asymmetries for the relevant quasi-two-body $B$ meson decays. The branching fractions of the virtual contributions for $K\bar K$ from the Breit-Wigner formula tails of $ρ(770)$ and $ω(782)$ for these decays are found comparable to the corresponding contributions from the resonances $ρ(1450,1700)$ and $ω(1420,1650)$. Consequently, they constitute a significant component that should be accounted for in the considered three-body decays. All the predictions in this work are expected to be tested by the LHCb and Belle-II experiments in the future.