Resonance contributions $\phi(1020,1680)\to K\bar K$ for the three-body decays $B\to K\bar K h$

Abstract

We study the contributions for the $K^+K^-$ and $K^0\bar K^0$ originated from the intermediate states $\phi(1020)$ and $\phi(1680)$ in the charmless three-body decays $B\to K\bar K h$, with $h=(\pi, K)$, in the perturbative QCD approach. The subprocesses $\phi(1020,1680)\to K\bar K$ are introduced into the distribution amplitudes of $K\bar K$ system via the kaon electromagnetic form factors with the coefficients taken from the fitted results. The predictions of the branching fractions for the decays $B\to\phi(1680)h$ with the intermediate state $\phi(1680)$ decays into $K^+K^-$ or $K^0\bar K^0$ are about $6\%$-$8\%$ of the corresponding results for the quasi-two-body decays $B\to\phi(1020)h \to K^+ K^- h$ in this work.