Resonant contributions to three-body

In this work, we study the (S, P and D)-wave $$K^+K^-$$ contributions to $$B\rightarrow KKK$$ decays in the perturbative QCD approach at leading order. Within the two-meson wave functions describing the nonperturbative dynamics in the kaon-pair for different waves, we calculate the branching fractions and the direct CP asymmetries of these decay modes in the corresponding resonance regions. Most of our numerical results are well consistent with the current measurements. We note that the narrow-width approximation is invalid in the quasi-two-body decays $$B\rightarrow Kf_0(980)\rightarrow KKK$$. For other decays, under the narrow-width approximation we can extract the branching fractions of the corresponding two-body decays involving the intermediate resonant states, and the related branching fractions agree with the current experimental data well. Furthermore, we also predict the branching fractions of the corresponding quasi-two-body decays $$B\rightarrow K\pi ^+\pi ^-$$, which are expected to be tested in the ongoing LHCb and Belle-II experiments