Charmless Quasi-Two-Body <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06580067pt" id="M1" height="11.4652pt" version="1.1" viewBox="-0.0657574 -11.3994 10.4833 11.4652" width="10.4833pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-67" d="M578 512C578 619 494 650 387 650H140L134 622C219 615 223 609 210 542L127 119C112 40 104 34 23 28L17 0H235C317 0 387 7 444 33C515 65 564 122 564 195C564 289 495 335 412 352V354C505 373 578 422 578 512ZM486 510C486 422 423 367 314 367H257L294 565C299 591 305 602 315 608C324 614 339 617 362 617C421 617 486 595 486 510ZM466 200C466 100 393 35 296 35C222 35 198 51 212 127L250 333H303C388 333 466 303 466 200Z"/></g> Decays in Perturbative QCD Approach: Taking <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.993899pt" id="M2" height="17.2456pt" version="1.1" viewBox="-0.0657574 -14.2517 156.3 17.2456" width="156.3pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><use xlink:href="#g113-67"/></g><g transform="matrix(.017,0,0,-0.017,15.107,0)"><path id="g117-149" d="M567 230V280H69V230H567Z"/></g><g transform="matrix(.017,0,0,-0.017,22.732,0)"><path id="g117-148" d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H796L694 83L716 68C790 149 861 220 901 255Z"/></g><!-- irelop --><g transform="matrix(.017,0,0,-0.017,43.998,0)"><path id="g113-76" d="M743 650H503L496 622L527 618C563 613 564 603 532 573C449 495 371 431 323 392C301 374 272 355 246 346L280 522C297 609 300 614 379 622L385 650H135L129 622C209 614 215 609 198 522L124 133C106 39 99 35 23 28L17 0H271L277 28C193 35 192 39 208 133L239 316C264 328 280 325 303 288C368 183 435 90 502 0H652L659 28C602 34 584 43 543 94C495 154 403 283 347 369L574 554C634 603 659 612 735 624L743 650Z"/></g><g transform="matrix(.017,0,0,-0.017,56.957,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,62.894,0)"><path id="g198-19" d="M852 669C820 657 792 643 768 628C705 667 627 687 547 687C310 687 154 548 154 412C154 332 212 282 296 282C434 282 515 426 523 557H503C474 402 410 312 298 312C241 312 195 352 195 408C195 535 341 657 546 657C613 657 683 644 738 609C640 538 574 431 486 271C374 71 314 16 189 16C118 16 69 60 69 95H71C78 84 93 83 101 83C129 83 144 106 144 131C144 160 121 179 93 179C60 179 34 151 34 108C34 42 100 -15 207 -15C364 -15 491 73 603 304C614 297 639 293 650 293C681 293 683 254 683 245C683 181 599 150 599 70C599 21 645 -12 692 -12C747 -12 804 30 868 103L851 119C789 52 741 18 706 18C685 18 674 29 674 52C674 92 750 157 750 226C750 257 727 287 696 310V312C812 315 876 394 876 473C876 525 849 577 793 612C812 627 832 639 859 650L852 669ZM814 486C814 417 765 345 686 337C669 348 649 354 626 354C673 451 708 533 766 588C799 561 814 525 814 486Z"/></g><g transform="matrix(.017,0,0,-0.017,83.887,0)"><use xlink:href="#g117-149"/></g><g transform="matrix(.017,0,0,-0.017,91.512,0)"><use xlink:href="#g117-148"/></g><!-- irelop --><g transform="matrix(.017,0,0,-0.017,107.984,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g><g transform="matrix(.017,0,0,-0.017,113.921,0)"><use xlink:href="#g113-76"/></g><g transform="matrix(.012,0,0,-0.012,126.881,-7.578)"><path id="g54-36" d="M556 236V289H337V504H275V289H56V236H275V-4H337V236H556Z"/></g><g transform="matrix(.017,0,0,-0.017,134.905,0)"><use xlink:href="#g113-76"/></g><g transform="matrix(.012,0,0,-0.012,147.864,-7.578)"><path id="g54-33" d="M556 236V289H56V236H556Z"/></g> as Examples - Liu, Wen-Feng; Zou, Zhi-Tian; Li, Ying (Advances in High Energy Physics, 2022)
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⟶KR⟶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⟶KR⟶π+π−. All predictions are expected to be tested in the ongoing LHCb and Belle-II experiments.