Resonance contributions ϕ(1020,1680)→KK¯ for the three-body decays B→KK¯h

We study the contributions for the K + K - and K 0 K ¯ 0 originating from the intermediate states ϕ ( 1020 ) and ϕ ( 1680 ) in the charmless three-body decays B → K K ¯ h , with h = ( π , K ) , in the perturbative QCD approach. The subprocesses ϕ ( 1020 , 1680 ) → K K ¯ are introduced into the distribution amplitudes of K 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 - or K 0 K ¯ 0 are about 6 - 8 % of the corresponding results for the quasi-two-body decays B → ϕ ( 1020 ) h → K + K - h in this work.