Charmless Quasi-two-body $B$ Decays in Perturbative QCD Approach: Taking $B\to K({\cal R}\to) K^+K^-$ As Examples

Three-body $B$ decays not only significantly broaden the study of $B$ meson decay mechanisms, but also provide information of resonant particles. Because of complicate dynamics, it is very hard for us to study the whole phase space in a specific approach. In this review, we take $B\to K({\cal R}\to) K^+K^-$ decays as examples and show the application of the perturbative QCD (PQCD) approach in studying the quasi-two-body $B$ decays, where two particles move collinearly with large energy and the bachelor one recoils back. To describe the dynamics of two collinear particles, the ($S$, $P$ and $D$)-wave functions of kaon-pair with different waves are introduced. By keeping the transverse momenta, all possible diagrams including the hard spectator diagrams and annihilation ones can be calculated in PQCD approach. Most results are well consistent with the current measurements from BaBar, Belle and LHCb experiments. Moreover, under the narrow-width approximation we can extract the branching fractions of the two-body decays involving the resonant states, and also predict the branching fractions of the corresponding quasi-two-body decays $B\to K(\cal{R}\to )\pi^+\pi^-$. All prediction are expected to be tested in the ongoing LHCb and Belle-II experiments.