Analysis of three-body charmless $B$-meson decays under the factorization-assisted topological-amplitude approach

We analyze quasi-two-body charmless $B$ decays $B_{(s)} \to P_1 V \to P_1 P_2 P_3$ with $V$ representing a vector resonant, and $P_{1,2,3}$ as a light pseudo-scalar meson, pion, kaon or $\eta^{(\prime)}$. The intermediate processes $B_{(s)} \to P_1 V $ are calculated in the factorization-assisted topological-amplitude approach and the vector resonant effects are described by the Breit-Wigner propagator, which successively decay to $P_1 P_2$ via strong interaction. Taking into account of all vector resonances in ground state, $\rho, K^*, \omega, \phi$, we present the related branching fractions, and calculate the virtual effects for $B_{(s)} \to \pi, K (\rho ,\omega \to) KK$. We also predict direct $\it{CP}$ asymmetries of three body B decay modes with $\rho, K^*$ resonances as intermediate states. Our predicted branching fractions of decay modes dominated by the color-favored tree diagram or the color-favored penguin diagram are consistent with the perturbative QCD approach's predictions as well as QCD factorization approach. While for those nonperturbative contribution dominated decay modes, the branching ratios in this work are in better agreement with current experimental data than the PQCD predictions and the QCD factorization results due to their shortage of the nonperturbative contributions or $1/m_b$ power corrections. Many of the decays channels, especially for direct $\it{CP}$ asymmetries, are waiting for the future experiments.


I. INTRODUCTION
Three-body nonleptonic B meson decays constitute a large portion of B decay channels.
Contrary to two-body B meson decays, they have nontrivial kinematics and phase space distributions, which can provide more opportunities for the study of CP asymmetries and factorization issues in QCD for multibody nonleptonic decays. Generally, one utilize the show up, so that the analysis on these decays enables one to study the properties of these resonances.
Experimentally, a large amount of branching ratios and CP asymmetry parameters of three-body charmless B decays have already been measured by BABAR, Belle and LHCb collaborations [1][2][3][4][5], while more are expected from the upgrade of LHCb and Belle II experiments. On the theoretical side, three-body nonleptonic B meson decays are more complicated than two-body cases as they involve three-body decay matrix elements. In a phenomenological factorization model [6][7][8], the three-body matrix elements of charmless B decays were factorized naively, with the resulting local correlaters studied extensively based on heavy meson chiral perturbation theory for nonresonant contributions and the usual Breit-Wigner formalism for resonant effects. Besides these phenomenological descriptions of three-body charmless B decays in the whole Dalitz plot, theoretical analysis in the context of QCD factorization mainly concentrated on the edges of the Dalitz distribution, since the contribution in the center regions of phase space is argued to be both 1/m b power and α s suppressed [9,10].
Since the vector (scalar) mesons usually decay dominantly to two pseudoscalar mesons with large decay width, the three-body B decays are found to be dominantly happened by B decaying to a vector (scalar) meson and a pseudoscalar meson, and the vector (scalar) meson subsequently decaying to two pseudoscalar mesons. In this case, two of the three final particles are collinear, generating a small invariant mass in the edges of the Dalitz distribution, and recoil against the third meson in three-body decays. Then one expects a similar factorization theorem as for two-body decays, with the difference that one of the mesons in the two-body case is substituted by a pair of collinear moving mesons with small invariant mass. A series of works [10][11][12][13][14][15][16][17][18][19][20][21][22] using perturbative QCD approach concentrated on this situation that the pair of collinear mesons are described collectively through intermediate resonances, that is, the three-body decays are processed as quasi-two-body decays. The CP asymmetry of B → πππ decays has been studied within the QCD factorization framework at leading order in α s [23,24], using naive factorization for three-body matrix elements, which involve generalized form factor of B → ππ transition and dimeson light cone distribution amplitudes in [25][26][27][28]. The calculation of the nonleptonic three-body B decays with heavyto-heavy transitions, up to the precision of next-next-to-leading order in α s has also been done in QCD factorization [29]. In all these quasi-two-body calculations of the three-body B decays, the theoretical calculations rely heavily on the precision of two-body theoretical calculation methods.
In the case of two-body nonleptonic B decays, power corrections in the heavy-quark expansion, such as weak annihilation effects and chirally enhanced power corrections, are difficult to estimate and remain the sources of uncertainty in QCD factorization approach [30].
A complete next-to-leading order calculation of two-body B decays in perturbative QCD factorization approach has not been finished [31]. The problem about precision predictions in the calculation of two-body nonperturbative matrix elements also exist in three-body case. As discussed in Ref [8], the predicted branch ratios of quasi-two-body decays such as B → φ(K + K − )K, K * (K − π + )π, ρ(π + π − )K are smaller than experimental data due to the absence of 1/m b power corrections. To include the nonperturbative and nonfactorization contribution in two-body B decays, a model independent framework of topological diagram approach [32] has been introduced. The equivalence of the topological decay amplitudes and the SU(3) irreducible amplitudes has been approved [33,34]. In this approach, one classifies the decay amplitudes of two-body charmless B decays into different electroweak topological In order to include the SU(3) breaking effects, the so-called factorization-assisted topological-amplitude (FAT) approach [35][36][37][38][39][40][41][42] has been introduced. They can give the most precise decay amplitudes of the two-body charmless B-meson decays. For instance, the amplitude of the color-suppressed topological diagram dominated by the nonfactorizable QCD effect, was larger in the FAT approach than that in other perturbative approaches so that the long-standing B → ππ branching ratio puzzle and B → πK CP asymmetry puzzle were solved simultaneously. Encouraged by the success in two-body B decays, we will systematically analyze three-body charmless B decays using the FAT approach at the edges of the Dalitz distribution with only vector resonance contribution, i.e., quasi-two-body decays with two of the pseudoscalar mesons decaying from the vector resonance. The vector resonant effects are described in terms of the usual Breit-Wigner formalism, and a strong coupling accounts for the subsequent vector meson two-body decay. We will study the branching ratios and CP violations of these three-body B decays, and discuss the virtual effects of intermediate resonance ρ, K * , ω , and φ on quasi-two-body decays. This paper is organized as follows. In Sec. II, the theoretical framework is introduced.
The numerical results and discussions are collected in Sec. III. Section V is a summary.

II. FACTORIZATION AMPLITUDES FOR TOPOLOGICAL DIAGRAMS
As a quasi-two-body process, the B (s) → P 1 P 2 P 3 decay (with P denoting a light pseudoscalar meson) is divided into two stages. First, B (s) meson decays to P 1 V , and the intermediate vector resonance V decays to P 2 P 3 subsequently. The first decay B (s) → P 1 V is a weak decay induced by b → uū d(s) at quark level in leading order (tree diagram) and b → d(s) qq (q = u, d, s) in next-to-leading order (penguin loop diagram). The secondary decay V → P 2 P 3 proceeds via strong interaction. In Fig. 1 we show the topological diagrams of B (s) → P 1 P 2 P 3 under the framework of quasi-two-body decay at tree level, including (i) color-favored tree emission diagram T , (ii) color-suppressed tree emission diagram C, (iii) W -exchange tree diagram E, and (iv) W -annihilation tree diagram A, which are specified by topological structures of the weak interaction. Likewise, one loop penguin diagram can also be grouped into four categories: (i) QCD-penguin emission diagram P , (ii) flavor-singlet QCD-penguin diagram P C or EW-penguin emission diagram P EW , (iii) time-like QCD-penguin diagram P E , and (iv) space-like QCD-penguin annihilation diagram P A , which are shown in Fig. 2. In the two kinds of figures, we only show one case that the intermediate resonance (labeled by gray ovals) is produced as a recoiling particle in emission diagrams.
The electroweak B decays at tree level have been proven to be factorizable at high pre-  cision [43]. Therefore, we can calculate the decay amplitude of color-favored tree diagram T in perturbative QCD order by order. Large nonfactorizable contributions, such as soft corrections to the color-suppressed tree amplitude and 1/m b power corrections from annihilation type diagram, are found to be non-negligible [38]. Similar to the FAT approach for two-body B decays [38], we have to introduce unknown parameters for nonfactorizable contributions for color-suppressed tree diagram C, and W -exchange tree diagram E (Wannihilation tree diagramA is negligible) to be fitted from experimental data. The formulas of B (s) → P 1 V, V P 1 decays are given as [38] The QCD-penguin emission diagram P and the electroweak penguin emission diagram P EW are also proved to be factorizable, and thus are calculable in QCD factorization. The flavorsinglet QCD-penguin diagram P C and spacelike QCD-penguin annihilation diagram P A contain large nonfactorizable contribution which will be determined by χ 2 fit from experimental data. The timelike QCD-penguin diagram P E is found to be smaller than other contribution, which can be ignored in the modes not dominated by it in the fit as discussed in [38]. The formulas can be expressed as [38] In these equations, we use the superscripts P 1 V, V P 1 to distinguish cases in which the recoiling meson (the first particle of P 1 V, V P 1 ) is a pseudoscalar or a vector meson. The quark in the Cabibbo-Kobayashi-Maskawa (CKM) matrix element is q ′ = d, s. ε * V is the polarization vector of vector meson V . r χ = m 2 P 1 m q ′ m B is the chiral factor of pseudo-scalar meson in the "chiral enhanced" term of P V P 1 amplitude. f P 1 , f V are the decay constants of the corresponding meson P 1 and V .
where F i represents form factor F 1 or A 0 , and m pole is the mass of the corresponding pole state, e.g., B for A 0 , and B * for F 1 . The effective Wilson coefficients a i (µ) can be calculated perturbatively. χ C (′) (E, P, P (′) C , P A ) and φ C (′) (E, P, P (′) C , P A ) denote the magnitude and associate phase of C (E, P , P C , and P A ) diagram, which are universal to be fitted globally from the experimental data.
We adopt the relativistic Breit-Wigner line shape to describe ρ, K * , ω, and φ resonances, which is widely used in the experimental data analyses [44][45][46]. The explicit expression of relativistic Breit-Wigner line shape is in the following form: where the two-body invariant mass square is s = m 2 23 = (p 2 +p 3 ) 2 with p 2 and p 3 denoting the 4-momenta of the collinearly moving mesons P 2 and P 3 , respectively. The energy-dependent width of vector resonance Γ V (s) is defined by The Blatt-Weisskopf barrier factor X (q r BW ) is given as [47] X where of the final state P 2 or P 3 in the rest frame of resonance V , and q 0 is the value of q when s = m 2 V . When a pole mass locates outside the kinematics region, i.e., m V < m P 2 + m P 3 , m V will be replaced with an effective mass m eff V , given by the ad hoc formula [48,49], where m max and m min are the upper and lower boundaries of the kinematics region, respectively. The barrier radius r BW = 4.0(GeV) −1 is for all resonances [46]. The full widths of the resonances Γ 0 , together with their masses m V , are taken from Particle Data Group [50] and listed in Table I.
The matrix element of strong decay P 2 (p 2 ) P 3 (p 3 ) |V (p V ) is parametrized as a strong coupling constant g V P 2 P 3 , which can be extracted from measured the partial decay widths where p c is the magnitude of pseudoscalar meson momentum in the rest frame of vector meson. The numerical results of g ρ→π + π − , g K * →K + π − , and g φ→K + K − have already been determined from experimental data [8], The other strong coupling constants can be derived from the relationships with the results in Eq.(10 ) by using the quark model result [51], Combing them together for quasi-two-body decay B (s) → P 1 V → P 1 P 2 P 3 , we express the decay amplitudes of tree topological diagrams shown in Fig.1 as where q ′ = d, s and p V = p 2 + p 3 = √ s. Similarly, the amplitudes of penguin diagrams shown in Fig.2 can be expressed as where the form factor F BP 1 1 is dependent on invariant mass s not a fixed value as in two-body decays.
The matrix element of B (s) → P 1 P 2 P 3 can be written in the following form: where A(s) represents the summation of amplitudes in Eqs.(11)-(13) and Eq. (14) with the where |p 1 | and |p 2 | denote the magnitudes of the momentum p 1 and p 2 , respectively. In the rest frame of the vector resonance, their expressions are where |p 2 | = q in Eq.(6).

III. NUMERICAL RESULTS OF BRANCHING RATIOS
In the numerical calculations, we need input parameters, such as m V , Γ 0 , and g V P 1 P 2 involved in strong interaction decay of vector mesons, which are listed in the previous section.
For the CKM matrix elements needed in the weak transition calculation, we use Wolfenstein parametrization with parameters from global fit [50] as follows: The decay constants of light pseudoscalar mesons and vector mesons, and transition form factors of B-meson decays at recoil momentum square Q 2 = 0 are listed in Tables II and   III, respectively. The decay constants of pseudoscalar mesons π, K, and B are from Particle Data Group [50]. The decay constants of B s and the vector mesons, and all form factors are not measured by experiments and only calculated theoretically, such as in constitute quark model and light cone quark model [52][53][54][55], covariant light front approach [56][57][58], light-cone sum rules , PQCD [80][81][82][83][84][85][86][87][88], and lattice QCD [89][90][91]. We will use the same theoretical values as in the charmless two-body decays [38], with 5% uncertainty kept for decay constants and 10% for form factors. The dipole model parameters α 1 , α 2 applied to describe the Q 2 dependence of form factors are also listed in TableIII [54,81]. .   The effective Wilson coefficients a i (µ) calculated at next-to-leading order can be found in [92], a 1 (m b /2) = 1.054, a 4 (m b /2) = −0.04, and a 9 (m b /2) = −0.009. The 14 nonpertubative parameters of topological diagrams χ C (′) , χ E , χ P , χ P (′) C , χ P A , and associated phases φ C (′) , φ E , φ P , φ P (′) C , φ P A , extracted from experimental data by a global fit performed in Ref. [38], together with uncertainty are χ C = 0.48 ± 0.06, φ C = −1.58 ± 0.08, With all the inputs, the branching fractions of B (s) → P 1 V → P 1 P 2 P 3 can be obtained by integrating the differential width in Eq.(16) over the whole kinematics region. Our numerical results for the branching ratios of B (s) decays are collected in Tables IV-VII, for decays Comparing the results shown in Table IV with the corresponding ones calculated in the previous work [38], we find that the branching ratios of quasi-two-body decay B (s) → P 1 ρ → P 1 ππ are a little smaller than that of the direct two-body decays B (s) → P 1 ρ. It can be attributed to the not narrow resonance width Γ ρ . We cannot apply the narrow width approximation to factorize quasi-two-body decay formula together with B(ρ → ππ) ∼ 100% to obtain the precise results of two-body decays.
From Table IV, it is easy to see that our results are well consistent with the experimental data shown in the last column of the Table. For the decay modes B → π(ρ →)ππ, B → η (′) (ρ →)ππ with color favored T diagram dominated contribution, and the penguin diagram P dominated modes B (s) → K(ρ →)ππ, the perturbation calculation is reliable, since these kinds of diagrams are proved to be factorizable to all orders of α s . Our results are in good agreement with those calculated under the QCD factorization [8] and that in PQCD approach [13] within errors.
For the decay modes with dominated color suppressed diagrams C (′) contributions, such   [50]. The characters T , C (′) , E, P , P C , P A and P EW representing the corresponding topological diagram contributions are also listed in the second column.
B. Branching ratios of B (s) → P 1 K * → P 1 Kπ In Table V, we only list quasi-two-body decays B (s) → P 1 K * → P 1 Kπ through strong decays K * 0 → K + π − ,K * 0 → K − π + , and K * − → K − π 0 with a pair of uū from the sea. The branching ratios for B (s) → P 1 K * → P 1 Kπ via sea quark pair dd strong decay K * 0 → K 0 π 0 , K * 0 →K 0 π 0 , and K * − →K 0 π − can be calculated under the narrow width approximation according to the isospin conservation relationship for strong interaction, The experimental data listed in the last column are measured by BABAR [44,[94][95][96] and Belle [97,98] experiments. As we have considered the power corrections from "chiral enhanced" term, penguin annihilation contribution and EW-penguin diagram for the penguindominated modes B u,d → πK * → πKπ, our results are well consistent with experimental data, and larger by a factor of 2 ∼ 4 than QCD factorization approach [8].
The decay channels B u,d → KK * → KKπ listed in Table V  We also give the numerical results of branching ratios of decay modes B (s) → ηK * → ηKπ, shown in Table V we can obtain the branching ratios for two-body decays B → ηK * , These results agree well with the experimental measurements of these two-body decays and the ones given in FAT approach [38].
We show the branching fraction results of B (s) → P 1 (φ →)KK decays in Table VI together with the results from PQCD approach [16] in the last column. For penguin diagram  [100]. The decay modes B s → K(φ →)KK are CKM suppressed (|V td /V ts |) comparing with B u,d → K(φ →)KK decays, so that the branching ratios of the former are one order smaller than the latter. It is easy to see from Table VI that branching ratios of all the CKM suppressed decay modes from the FAT approach are larger that that from PQCD approach. It can be attribute to the missing power corrections and next-to-leading order corrections to the color suppressed penguin diagram P ′ C in PQCD approach. Summing over the two quasi-two-body decay modes with same weak decay but different subsequent strong decay, i.e. B (s) → P 1 (φ →)K + K − and B (s) → P 1 (φ →)K 0K 0 , we should obtain results of the corresponding two-body decays B (s) → P 1 φ by applying the narrow width approximation. For instance, the summation of the results of B(B − → π − (φ →  and that from PQCD approach. The characters C, P , P (′) C , P A and P EW representing the corresponding topological diagram contributions are also listed in the second column.

D. The virtual effects of B (s) → π, K(ρ, ω →)KK
The decay modes B (s) → π, K(ρ, ω →)KK represent a category of decays, whose pole masses of the resonances ρ, ω smaller than the threshold mass of producing KK pair. They can only happen with the virtual effect, which is also called the Breit-Wigner tail effect.
In Table VII, we collect these virtual effects in FAT approach, together with the results in PQCD approach [17] for comparison. Here we only show subprocesses ρ 0 , ω → K + K −  without ρ 0 , ω → K 0K 0 due to the tiny mass difference between K + K − and K 0K 0 . The virtual effects of B (s) → π, K(ρ →)KK are approximately 1 ∼ 2 order smaller than the dominated contribution of B (s) → π, K(ρ →)ππ in Table IV. Unlike the on-shell resonance contributions that mostly give the similar contributions between the FAT and PQCD approaches, the virtual contributions give quite different results between the two approaches.
Since the color suppressed tree diagram C (′) is calculated in the PQCD approach only to the leading order, its size is significantly smaller that that fitted from experimental data in the FAT approach [38]. As a result, the off-shell effects in C (′) dominated decay modes,B 0 → π 0 (ρ 0 , ω →)K + K − ,B 0 s → K 0 (ρ 0 , ω →)K + K − using FAT approach are nearly one order magnitude larger than that of PQCD approach. For another example, the hierarchy between the magnitude of virtual contributions for quasi-two-body decays B − → π 0 (ρ − →)K − K 0 and B − → π − (ρ 0 →)K + K − is opposite in the PQCD and the FAT approaches, which need further experimental measurements to check.
Comparing the results between Tables IV and VII, one can see that the off-shell effects in B → π, K(ρ →)KK are small, that is only about 1% of the on-resonance contribution.
Comparing the virtual effect contributions of ρ with ω resonance in Table VII, for instance,  Table I. It means that the virtual contributions of these decays are not very sensitive to the full decay width of ρ and ω meson, which is also confirmed by the PQCD approach [17]. This can be explained by the behavior of the Breit-Wigner propagators in Eq. (5) in the kinematics regions of these decays, where the imaginary part i m ρ, ω Γ ρ, ω (s) becomes unimportant when the invariant mass square s is larger than 1GeV 2 .

IV. CP ASYMMETRY
The study of three-body B decays attracts a lot of interests because of its large CP asymmetry. In this work, we will concentrate on direct CP violation and will not discuss mixing-induced CP violation. We also do not consider integrated CP asymmetry of the whole phase space of three-body B decays, but we only consider the direct CP asymmetry of quasi-two-body decay with one resonant as the intermediate state. Consequently this CP asymmetry will arise from the interference between tree and penguin amplitudes of the  [50].
It is well known that direct CP asymmetries are induced by the interference between difference of strong phases and different CKM phases of different contributions. The strong phase is also the major source of direct CP violation uncertainty as it is mostly from nonperturbative QCD dynamics. As illustrated in Ref. [38] for the two-body B decays, the strong phases extracted from experimental data are sufficient to induce correct direct CP asymmetry, while QCD factorization approaches and soft-collinear effect theory usually make wrong  IX: The same as Table VIII, but for the quasi-two-body of quasi-two-body decays B (s) → P 1 K * → P 1 Kπ.

Modes Amplitudes
Quasi-two-body results Experiment predictions or no prediction for the direct CP asymmetries due to absence of nonperturbative strong phases, such as the well-known Kπ puzzle. The problem in two-body decays will also exist in three-body decays. For example, the signs of CP asymmetry of B − → π − K + K − and B − → K − π + π − decay were conflict with experimental data under the QCD factorization approach [8]. The authors need to consider final-state rescattering effect by introducing an unknown strong phase δ to account for the sign flip of CP asymmetry. Therefore possible 1/m b power corrections, such as final-state interactions, or other nonperturbative contri-butions are necessary to correctly describe CP asymmetry. For the quasi-two-body decays under the framework of FAT, the strong phases result in from the nonperturbative contribution parameters in Eq. (18) and the Breit-Wigner formalism for resonance in Eq.(5). These strong phases have already contained all perturbative and nonperturbative QCD effects, for instance, final-state interactions, color suppressed contribution, penguin annihilation contribution, and so on. They explain well the measured branching ratios, and thus are expected to give the right prediction of CP asymmetry parameters in three-body B decays.
We show the CP asymmetries of B (s) → P 1 ρ → P 1 ππ and B (s) → P 1 K * → P 1 Kπ in Tables VIII and IX, respectively. As the theoretical uncertainty from hadronic parameters (form factors and decay constants ) mostly cancel, the major theoretical uncertainties for CP asymmetry parameters are from the nonperturbative contribution parameters in Eq. (18) and weak phases. Here we only list the uncertainty arise from the strong phase in Eq. (18) due to small uncertainties of CKM matrix elements. We also list the experimental data in the last column of the two tables. Actually only three decay modes with the intermediate process K − ρ 0 , π + K * − , and ηK * 0 are well measured in experiments with more than 3σ signal significance. Most CP asymmetries listed here are predictions to be tested by the future experiments. For example, the CP asymmetry of B − → π + ρ 0 → π + π + π − is predicted quite large in this work, which is consistent with the theoretical predictions using QCDF [30] and the PQCD approach [13] but not yet measured by the experiments. We did not list the CP asymmetries of B → P 1 φ → P 1 KK decays in our table, since most of them are from pure penguin diagram contributions, P C , P A , whose CP asymmetry is expected to be zero with only one single CKM phase, at leading order approximation. We also do not list the CP asymmetries of B (s) → P 1 ρ(ω) → P 1 KK decays contributed by the virtual effects.
Most of the quasi-two-body decays are extracted from the Dalitz-plot analysis of threebody ones in experimental analysis. For example, B − → K − ρ 0 and B − → π −K * 0 decays are extracted from B − → K − π + π − decays. Comparing the CP asymmetries of quasi-two-body decay by integrating out the invariant mass square (s) distribution of CP asymmetry over the kinematics region in the Dalitz plot with the corresponding ones of two-body decay with s fixed as m 2 V as in Ref [38], we find that the size of CP asymmetries of B (s) → P 1 ρ → P 1 ππ in Table VIII are slightly smaller than that of the B (s) → P 1 ρ. The CP asymmetry of B (s) → P 1 K * → P 1 Kπ in Table IX are slightly larger than that of B (s) → P 1 K * decay.
These very small changes of CP asymmetry between two-body and quasi-two-body decays arise from the finite decay width of vector resonances.

V. CONCLUSION
We systematically analyze the three-body charmless B meson decays through the intermediate resonance, i.e., they proceed via quasi-two-body decays as B (s) → P 1 V → P 1 P 2 P 3 , with the vector resonant state V including all ground states ρ, K * , ω, φ. The first step of two-body B decays to P 1 V intermediate states is induced by flavor changing weak decays b → uū d(s) at leading order and b → d(s) qq (q = u, d, s) at one loop level. The second step is that the intermediate vector resonant state V , described by the Breit-Wigner propagator, decays into two light pseudoscalar mesons via strong interaction. In order to include all possible perturbative and nonperturbative QCD corrections, the two-body weak B decays are described by the FAT approach with the decay amplitudes extracted from experimental data.
We compare results of the branching fractions from FAT approach with the PQCD approach's predictions and the ones of QCD factorization, as well as the experimental data.
For a color suppressed tree diagram and color suppressed or annihilation penguin diagram dominated decay modes, the branching ratios in FAT approach are larger than that of the PQCD approach and the QCD factorization approach. The reason is that the decay magnitudes and phases extracted from experimental data in FAT approach are larger than that in the other two approaches due to the shortage of nonperturbative contribution and 1/m b power corrections. We have also considered the virtual effects from ρ and ω resonance tail, which are usually ignored by the experimental analysis. These virtual effects from ground state ρ(770) 0 are about 1 ∼ 2 order magnitude smaller than the dominated contribution of B (s) → π, K(ρ →)ππ decays. However, these virtual effects contributions are at the same order as the higher resonance contributions, such as ρ(1450). We also find that the virtual contributions of these decays are not very sensitive to the decay widths of ρ and ω. The branching ratios of B (s) → P 1 (ρ →)ππ, B (s) → P 1 (K * →)Kπ are consistent with experimental data, and the others are predictions waiting for future experiments to test.
For CP asymmetry, theoretical predictions of three-body decays are mostly based on quasi-two-body decays via intermediate resonance, where the strong phases extracted from experimental data in FAT approach are sufficient to induce correct direct CP asymmetries.
In this work, we utilize the same strong phases of two-body decays together with the phases from Breit-Wigner formalism to predict the direct CP asymmetries of quasi-two-body decays, B (s) → P 1 (ρ →)ππ and B (s) → P 1 (K * →)Kπ. Our results agree well with the existing experimental measurements. But many of the CP asymmetry predictions are waiting for future experiments.