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Three-body $B$ decays not only significantly broaden the study of $B$ mesondecay mechanisms, but also provide information of resonant particles. Becauseof complicate dynamics, it is very hard for us to study the whole phase spacein 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 movecollinearly with large energy and the bachelor one recoils back. To describethe dynamics of two collinear particles, the ($S$, $P$ and $D$)-wave functionsof kaon-pair with different waves are introduced. By keeping the transversemomenta, all possible diagrams including the hard spectator diagrams andannihilation ones can be calculated in PQCD approach. Most results are wellconsistent with the current measurements from BaBar, Belle and LHCbexperiments. Moreover, under the narrow-width approximation we can extract thebranching fractions of the two-body decays involving the resonant states, andalso 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 inthe ongoing LHCb and Belle-II experiments.

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