Resonance contributions $\varphi (1020,1680)\to K\bar{K}$ 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 $\varphi (1020)$ and $\varphi (1680)$ in the charmless three-body decays $B\to K\bar{K}h$, with $h=(\pi ,K)$, in the perturbative QCD approach. The subprocesses $\varphi (1020,1680)\to K\bar{K}$ 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 $B\to \varphi (1680)h$ with the intermediate state $\varphi (1680)$ decays into $K^{+}{K}^{-}$ or $K^0{\bar{K}}^0$ are about $6\%$-$8\%$ of the corresponding results for the quasi-two-body decays $B\to \varphi (1020)h\to K^{+}{K}^{-}h$ in this work.

Comments: 13 pages, 2 figures