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

Abstract

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