Four-body decays $B_{(s)}\to (K\pi {)}_{S/P}(K\pi {)}_{S/P}$ in the perturbative QCD approach

Abstract

We investigate the four-body decays $B_{(s)}\to (K\pi {)}_{S/P}(K\pi {)}_{S/P}$ in the $K\pi$-pair invariant mass region around the $K^{\ast }(892)$ resonance in the perturbative QCD (PQCD) approach, where $(K\pi {)}_{S/P}$ denotes a $S$- or $P$-wave $K\pi$ configuration. The contributions from the $P$-wave resonance $K^{\ast }(892)$ and from possible $S$-wave components are included into the parametrization of the $K\pi$ two-meson distribution amplitudes, which are the nonperturbative inputs in this formalism. We calculate the branching ratio for each resonance and observe sizable $S$-wave contributions to the $B_s$ modes, well consistent with recently available LHCb data. We also deduce the polarization fractions for the $K^{\ast }{\bar{K}}^{\ast }$ final states, and compare our predictions to those in previous PQCD analyses of the corresponding two-body $B\to K^{\ast }{\bar{K}}^{\ast }$ decays. The polarization puzzle associated with the two $U$-spin related channels $B^0\to K^{\ast 0}{\bar{K}}^{\ast 0}$ and $B_s^0\to K^{\ast 0}{\bar{K}}^{\ast 0}$ still exists. In addition to direct $CP$ asymmetries, triple-product asymmetries and $S$-wave induced direct $CP$ asymmetries originating from the interference among various helicity amplitudes, are presented. It is shown that true triple-product asymmetries are rather small, while direct $CP$ asymmetries and $S$-wave-induced direct $CP$ asymmetries are significant in some decays. Our results will be subject to stringent tests with precise data from $B$ factories in the near future.

Comments: 40 pages, 3 figures, 7 tables, matched with the published version in JHEP