We analyze the quasi-two-body decays B → Kρ → Kππ in the perturbative QCD (PQCD) approach, in which final-state interactions between the pions in the resonant regions associated with the P -wave states ρ(770) and ρ ′ (1450) are factorized into two-pion distribution amplitudes. Adopting experimental inputs for the time-like pion form factors involved in two-pion distribution amplitudes, we calculate branching ratios and direct CP asymmetries of the B → Kρ(770), Kρ ′ (1450) → Kππ modes. It is shown that agreement of theoretical results with data can be achieved, through which Gegenbauer moments of the P -wave two-pion distribution amplitudes are determined. The consistency between the three-body and two-body analyses of the B → Kρ(770) → Kππ decays supports the PQCD factorization framework for exclusive hadronic B meson decays. PACS numbers: 13.20.He, 13.25.Hw, 13.30.Eg
I. INTRODUCTIONStrong dynamics contained in three-body hadronic B meson decays is much more complicated than in two-body cases, because of entangled nonresonant and resonant contributions, and significant final-state interactions [1]. Nonresonant contributions may not be negligible in these decays, as indicated by the observations made in Refs [2-7]. Quasi-two-body channels through intermediate scalar, vector and tensor resonances, which produce hadron pairs with final-state interactions, usually dominate total branching fractions. An amplitude for a three-body hadronic B meson decay, as a coherent sum of nonresonant and resonant contributions, leads to nonuniform distributions of events described by differential branching fractions [2-16] and of direct CP asymmetries [17][18][19][20] in a Dalitz plot [21]. Dalitzplot analyses of abundant three-body hadronic B meson decays from different collaborations (11)(12)(13)(14)(15)(16), Belle [5,6,[8][9][10] and LHCb [17][18][19]) have revealed valuable information on involved strong and weak dynamics.On the theoretical side, substantial progress on three-body hadronic B meson decays by means of symmetry principles and factorization theorems has been made, although rigorous justification of these approaches is not yet available. Isospin, U-spin and flavor SU(3) symmetries were adopted in [22][23][24][25][26][27][28][29][30][31], and the role of the CP T invariance in three-body B meson decays was discussed in Refs [32,33]. The QCD factorization [34,35] has been widely applied to studies of three-body charmless hadronic B meson decays [36][37][38][39][40][41][42][43][44][45][46][47][48], including, for instance, detailed investigation on factorization properties of the B + → π + π + π − mode in various regions of phase space [49]. The perturbative QCD (PQCD) approach based on the k T factorization theorem [50,51] has been employed in Refs. [52][53][54][55][56], where strong dynamics between two final-state hadrons in resonant regions are factorized into a new nonperturbative input, the two-hadron distribution amplitudes. An advantage of the PQCD factorization approach is that both nonresonant and resonant...