Resonance contributions ϕ(1020,1680)→$K\overline{K}$ for the three-body decays B→$K\overline{K}h$

We study the contributions for the K${}^{+}$K${}^{\text{â}}\text{ˆ}\text{’}$ and $K^0{\overline{K}}^0$ originating from the intermediate states ϕ(1020) and ϕ(1680) in the charmless three-body decays B→$K\overline{K}h$, with h=(π,K), in the perturbative QCD approach. The subprocesses ϕ(1020,1680)→$K\overline{K}$ are introduced into the distribution amplitudes of $K\overline{K}$ system via the kaon electromagnetic form factors with the coefficients taken from the fitted results. The predictions of the branching fractions for the decays B→ϕ(1680)h with the intermediate state ϕ(1680) decays into K${}^{+}$K${}^{\text{â}}\text{ˆ}\text{’}$ or $K^0{\overline{K}}^0$ are about 6−8% of the corresponding results for the quasi-two-body decays B→ϕ(1020)h→K${}^{+}$K${}^{\text{â}}\text{ˆ}\text{’}$h in this work.