Branching Fractions and CP Violation in $B^-\to K^+K^-\pi^-$ and $B^-\to \pi^+\pi^-\pi^-$ Decays

We present in this work a study of tree-dominated charmless three-body decays of $B$ mesons, $B^-\to K^+K^-\pi^-$ and $B^-\to\pi^+\pi^-\pi^-$, within the factorization approach. The main results are: (i) There are two distinct sources of nonresonant contributions: one arises from the $b\to u$ tree transition and the other from the nonresonant matrix element of scalar densities $\langle M_1M_2|\bar q_1 q_2|0\rangle^{\rm NR}$. It turns out that even for tree-dominated three-body decays, dominant nonresonant contributions originate from the penguin diagram rather than from the $b\to u$ tree process, as implied by the large nonresonant component observed recently in the $\pi^- K^+$ system which accounts for one third of the $B^-\to K^+K^-\pi^-$ rate. (ii) The calculated branching fraction of $B^-\to f_2(1270)\pi^-\to K^+K^-\pi^-$ is smaller than the LHCb by a factor of $\sim 7$ in its central value, but the predicted $\B(B^-\to f_2(1270)\pi^-\to\pi^+\pi^-\pi^-)$ is consistent with the data. Branching fractions of $B^-\to f_2(1270)\pi^-$ extracted from the LHCb measurements of these two processes also differ by a factor of seven! Therefore, it is likely that the $f_2(1270)$ contribution to $B^-\to K^+K^-\pi^-$ is largely overestimated experimentally. Including $1/m_b$ power corrections from penguin annihilation inferred from QCD factorization (QCDF), a sizable CP asymmetry of 32\% in the $f_2(1270)$ component agrees with experiment. (iii) A fraction of 5\% for the $\rho(1450)$ component in $B^-\to\pi^+\pi^-\pi^-$ is in accordance with the theoretical expectation. However, a large fraction of 30\% in $B^-\to K^+K^-\pi^-$ is entirely unexpected. This issue needs to be clarified in the future.


I. INTRODUCTION
In 2013 and 2014 LHCb has measured direct CP violation in charmless three-body decays of B mesons [1][2][3] and found evidence of inclusive integrated CP asymmetries A incl CP in B + → π + π + π − (4.2σ), B + → K + K + K − (4.3σ) and B + → K + K − π + (5.6σ) and a 2.8σ signal of CP violation in B + → K + π + π − . The study of three-body decays allows to measure the distribution of CP asymmetry in the Dalitz plot. Hence, the Dalitz-plot analysis of A CP distributions can reveal very rich information about CP violation. Besides the integrated CP asymmetry, local asymmetry varies in magnitude and sign from region to region. Indeed, LHCb has also observed large asymmetries in localized regions of phase space, such as the low invariant mass region and the rescattering regions of m π + π − or m K + K − between 1.0 and 1.5 GeV.
Recently LHCb has analyzed the decay amplitudes of B + → π + π − π + and B + → K + K − π + decays in the Dalitz plot [4][5][6]. Previously, the only amplitude analysis available at B factories was performed by BaBar for B + → π + π − π + [7]. In the LHCb analysis of the B ± → π ± K + K − decay amplitudes, three contributions were considered in the π ± K ∓ system, namely, K * (892) and K * 0 (1430) resonances plus a nonresonant contribution, and four contributions in the K + K − system: ρ 0 (1450), f 2 (1270), φ(1020) and an amplitude accounting for the ππ ↔ KK rescattering [4]. The largest contribution with a fit fraction of 32% comes the nonresonant amplitude in the π ± K ∓ system. A surprise comes from the quasi-two-body decay B + → ρ(1450)π + which accounts for 31% of the K + K − π + decays. This seems to imply an enormously large coupling of ρ(1450) with K + K − . Another very interesting feature of this analysis is that almost all the observed CP asymmetry in this channel is observed in the rescattering amplitude, which is the largest CP violation effect observed from a single amplitude.
The LHCb analysis of the B − → π + π − π − decay amplitude [5,6] showed some highlights: (i) Instead of a large nonresonant S-wave contribution observed by BaBar [7], the isobar model Swave amplitude was presented by the LHCb as the coherent sum of contributions from the σ (i.e. f 0 (500)) meson and a ππ ↔ KK rescattering amplitude within the mass range 1.0 < m π + π − < 1.5 GeV. A significant CP violation of 15% in B + → σπ + and a large CP asymmetry of order 45% in the rescattering amplitude were found by LHCb. (ii) CP asymmetries for B ± → π ± π + π − were measured in both low and high invariant-masss regions, see Fig. 1. The peak in the low-m low region around 1.3 GeV is due to the resonance f 2 (1270). Indeed, the mode with f 2 (1270) exhibited a CP violation of 40%. It is very interesting to notice a large CP asymmetry also observed in the high-m high region. (iii) CP violation in the quasi-two-body decay B + → ρ 0 (770)π + is measured to be consistent with zero in all three different S-wave approaches, contrary to the existing model calculations. Nevertheless, a significant CP asymmetry in the ρ 0 region can be seen in Fig. 2 where the data are separated by the sign of the value of cos θ hel with θ hel being the helicity angle, evaluated in the π + π − rest frame, between the pion with opposite charge to the B and the third pion from the B decay (see Fig. 3 below). This feature which was already noticed previously in [2] indicates that CP violation close to the ρ(770) resonance is proportional to (m 2 ρ − m 2 low ) cos θ hel . Hence, CP asymmetry in the ρ(770) region arises from the interference between the ρ(770) and S-wave contributions. The interference pattern observed in Fig. 2 will be destroyed by the CP violation FIG. 1: CP asymmetries for B ± → π ± π + π − measured in the low invariant-masss m(π + π − ) low region (left panel) and high invariant-masss m(π + π − ) high region (right panel). This plot is taken from [6].
FIG. 2: The difference of N B − and N B + , the number of B − and B + events respectively, for B ± → π ± π + π − measured in the low-m low region for (a) cos θ hel > 0 and (b) cos θ hel < 0 with the helicity angle θ hel being defined in Fig. 3. This plot is taken from [6].
We have explored three-body B decays in [8][9][10] under the factorization approximation. In this work we shall update the analysis of three-body decays B + → π + π − π + and B − → K + K − π + as the LHCb has presented the new amplitude analyses of them. Attention will be paid to integrated and regional CP violation. We take the factorization approximation as a working hypothesis rather than a first-principles starting point as factorization has not been proved for three-body B decays. Unlike the two-body case, to date we still do not have QCD-inspired theories for hadronic threebody decays, though attempts along the framework of pQCD and QCDF have been made in the past [11][12][13]. I: Experimental results of the Dalitz plot fit for B ± → π ± K + K − decays taken from [4].
II. B ± → π ± K + K − DECAYS The charmless 3-body decays B − → π − K + K − has been studied at B factories by BaBar [14] and Belle [15] only for its branching fraction and direct CP asymmetry. On the theoretical side, this three-body decay mode was analyzed in [8][9][10] in which contributions from K * (892), K * 0 (1430), f 0 (980) and a nonresonant amplitude were considered. The recent LHCb amplitude analysis takes into account a total of seven contributions: K * (892) and K * 0 (1430), ρ 0 (1450), f 2 (1270), φ(1020), a nonresonant amplitude and an amplitude accounting for the ππ ↔ KK rescattering. The results of the Dalitz plot analysis are shown in Table I [4]. The phases of B ± decay amplitudes shown in the table include both weak and strong phases. Nonresonant contributions from both π ± K ∓ and K + K − systems account for almost half of B ± → π ± K + K − rates. A very interesting feature is that the recattering amplitude, acting in the region 0.95 < m K + K − < 1.42 GeV, produced a large and negative CP asymmetry of (−66 ± 4 ± 2)%, which is the largest CP violation effect observed from a single amplitude.

A. Resonant contributions
The explicit expression of the factorizable tree-dominated B − → π − (p 1 )K + (p 2 )K − (p 3 ) decay amplitude can be found in Eq. (5.1) of [9]. It can be decomposed as the coherent sum of resonant contributions together with the nonresonant background (2.1) In general, vector, scalar and tensor resonances all can contribute to the three-body matrix element P 1 P 2 |J µ |B , while only the scalar resonance contributes to P 1 P 2 |S|0 . Effects of intermediate resonances are described as a coherent sum of Breit-Wigner expressions. More precisely, 1 In practice, we shall only keep the leading resonances . We shall follow [16] for the definition of B → P and B → V transition form factors, [17] for form factors in B → S transitions and [18] for B → T transition form factors. 2 In the following we show the amplitudes from various resonances: In [9,10] an additional minus sign was wrongly put in the Breit-Wigner propagator of the scalar resonance. 2 The B → T transition form factors defined in [18] and [19] are different by a factor of i. We shall use the former as they are consistent with the normalization of B → S transition given in [17].
Notice two different types of the decay constant for It has the similar expression as the amplitude of B − → σ/f 0 (500)π − → π + π − π − as will discussed in detail in the next section. Here we write down the amplitude and The order of the arguments of the a p i (M 1 M 2 ) coefficients is dictated by the subscript M 1 M 2 given in Eq. (2.5). The superscript u of the form factor F Bf u 0 0 reminds us that it is the uū quark content that gets involved in the B to f 0 form factor transition. Likewise, the superscript d of the scalar decay constantf d f 0 refers to the d quark component of the f 0 (980).
3. φ(1020) Since contributions from the matrix elements φ|(ūb to the φ production are very suppressed, their effects will not be taken into account.
has been made.
In the approach of QCD factorization (QCDF) [20], the decay amplitue of B − → f 2 (1270)π − receives an additional contribution proportional to (see Eq. (B.8) of [19]) The reader is referred to [19] for the definition of the decay constant f f 2 and the chiral factor r f 2 χ . As stressed in [19], the factorizable amplitude f 2 |J µ |0 π − |J ′ µ |B − vanishes in the factorization approach as the tensor meson cannot be produced through the V −A or tensor current. Nevertheless, beyond the factorization approximation, contributions proportional to the decay constant f f 2 can be produced from vertex, penguin and spectator-scattering corrections.
Using the relation with M µν = g µν − P µ P ν /m 2 f 2 and P = p 2 + p 3 , it is straightforward to show that [21] λ ε * µν (λ)ε αβ (λ)p 2µ p 3ν p α B p β 1 = and where p 1 and p 2 are the momenta of the π − (p 1 ) and K + (p 2 ) in the rest frame of the dikaon K + (p 2 ) and K − (p 3 ). However, the predicted CP asymmetry is of order −0.01 which is wrong in sign and magnitude compared to experiment, especially a large CP violation of 40% observed in the decay B − → π − f 2 (1270) → π − π + π − . We thus follow the QCDF calculation in [19] to include 1/m b power corrections arising from penguin annihilation (see Eq. (B.8) in [19]). This amounts to adding the penguin annihilation contributions β p 2 δ pu + β p 3 + β p 3,EW to the [. . .] term in Eq. (2.11). Therefore, the amplitude A f 2 (1270) reads × a 1 δ pu + a p 4 + a p 10 − (a p 6 + a p 8 )r π χ + β p 2 δ pu + β p 3 + β p 3,EW . (2.17) Numerically, we shall follow [19] to use It should be remarked that the angular momentum distribution for the vector or tensor intermediate state is not put by hand. It will come out automatically in the factorization approach. For example, the decay amplitude of ρ(1450) or φ production contains a term (s 12 − s 13 ) which is proportional to p 1 · p 2 = |p 1 ||p 2 | cos θ 12 (see Eq. (2.16)). Likewise, the angular distribution of a tensor meson decaying into two spin-zero particles is governed by (3 cos 2 θ 12 − 1) [cf. Eq. (2.14)]. In general, the angular momentum distribution is described by the Legendre polynomial P J (cos θ).

B. Nonresonant contributions
The nonresonant contributions arise from the 3-body matrix element K + (p 2 )K − (p 3 )|(ūb) V −A |B − NR in the K + K − system and the 2-body matrix element of scalar density π − (p 1 )K + (p 2 )|ds|0 NR in the π − K + system. The nonresonant contribution to the three-body matrix element can be parameterized in terms of four unknown form factors. The general expression of the nonresonant amplitude in the K + K − system induced from the b → u tree transition reads where the form factors r and ω ± can be calculated using heavy meson chiral perturbation theory (HMChPT) [22,23]. However, HMChPT is applicable only when the two scalars K + and K − in B → K + K − transition are soft. Indeed, the predicted nonresonant rate, of order 33 × 10 −6 in branching fraction, based on HMChPT will be one order of magnitude larger than the world average of the total branching fraction ∼ 5.2 × 10 −6 . Hence, we shall assume the momentum dependence of nonresonant amplitudes in an exponential form [8] so that the HMChPT results are recovered in the soft meson limit p 2 , p 3 → 0. For the parameter α NR we shall use α NR = 0.160 GeV −2 . 3 The nonresonant contribution in the π − K + system is given by The parameter α NR = 0.081 +0.015 −0.009 GeV −2 used in [9,10] was originally constrained from the BaBar's measurement of the nonresonant contribution to B − → π + π − π − [7]. However, a substantial part of the nonresonant amplitude is now replaced by the scalar σ meson in the LHCb analysis based on the isobar model. This leads to a larger α NR .
where the nonresonant matrix element of scalar density has the expression [9,10] π − (p 1 )K + (p 2 )|ds|0 NR = σ NR e −αs 12 where the phase φ πK will be specified later. As stressed in [10], the nonresonant signal in the π − K + system is governed by the nonresonant component of the matrix element of scalar density.
Owing to the exponential suppression factor e −αs 12 , the nonresonant contribution manifests in the low invariant mass regions. Note that in the LHCb analysis, the nonresonant amplitude is parameterized in terms of a simple single-pole form factor of the type (1 + m 2 π ± K ∓ /Λ 2 ) −1 with Λ ∼ 1 GeV. We prefer to use the exponential form for nonresonant amplitudes.
C. Final-state rescattering CP asymmetries (integrated or regional) measured by the LHCb are positive for h − π + π − and negative for h − K + K − with h = π or K. The former usually has a larger CP asymmetry in magnitude than the latter. This has led to the conjecture that π + π − ↔ K + K − rescattering may play an important role in the generation of the strong phase difference needed for such a violation to occur [3]. The CP T theorem requires that ∆Γ FSI λ ≡ Γ(B → λ) FSI − Γ(B →λ) FSI be vanished when summing over all the possible states allowed by final-state interactions; that is, λ ∆Γ FSI λ = 0. However, in the LHCb analysis, only the two channels α = π + π − P − and β = K + K − P − (P = π, K) in B − decays are assumed to be strongly coupled through finalstate interactions with the third meson P being treated as a bachelor or a spectator. It follows that ∆Γ FSI β = −∆Γ FSI α . It was found that final-state rescattering of π + π − ↔ K + K − dominates the asymmetry in the mass region between 1 and 1.5 GeV. In reality, the consideration of only rescattering between π + π − and K + K − in the S-wave configuration is too restrictive and simplified [25]. For example, π + π − is allowed to rescatter into K + K − with charge neutral multi-pion states. Nevertheless, below we shall follow the work of [26] (also the same framework adapted in [27]) to describe the inelastic ππ ↔ KK rescattering process and consider this final-state rescattering effect on inclusive and local CP violation.
Neglecting possible interactions with the third meson under the so-called '2+1' assumption, the S-wave π + π − ↔ K + K − rescattering through final-state interactions is described by [28,29] where the inelasticity parameter η(s) is given by [26] η(s) for rescattering to a pair of kaons (pions). The ππ phase shift has the expression We shall assume that δ KK ≈ δ ππ in the rescattering region. We have shown in [10] that the matrix S 1/2 can be expressed as For numerical calculations we shall use the parameters given in Eqs. (2.15b') and (2.16) of [26], The rescattering amplitude reads from Eqs. (2.24) and (2.30) to be The S-wave amplitudes involved in rescattering are given by The nonresonant amplitude A π + π − NR and the amplitude with the scalar resonance σ(500) will be discussed in Sec.III.
Eq. (2.24) is sometimes expressed in the literature in terms of the S matrix instead of S 1/2 . For example, writing the decay amplitude as A ± = A λ + B λ e ±iγ , it has been shown in [27] that the lowest-order (LO) effect due to FSI in the decay amplitude is given by (see Eq. (18) of [27]) where use of S ij = S ji has been made. However, the reader can check that the above amplitude does not satisfy Eq. (12) of [27] up to the leading order in t λ ′ ,λ , namely, . The correct answer should read Hence, Eq. (2.24) gives the correct description of π + π − ↔ K + K − final-state rescattering. 6 However, the LHCb analysis of ππ ↔ KK rescattering is based on the model described in [27,32].

D. Numerical results and discussions
The total decay amplitude of The strong coupling constants such as g K * 0 →π − K + and g f 0 (980)→K + K − ,· · · etc., are determined from the measured partial widths through the relations 7 (2.39) 6 In Eq. (2.24) we have used the factorized amplitude A fac in the place of A 0λ ′ + e ±iγ B 0λ ′ . They are, however, not exactly the same. In fact, we are using a time evolution picture [30,31] and the rescattering of ππ → KK happens at a much later stage of time-evolution. The full amplitude should read A = S 1/2 A 0 with A 0 being free from any strong phase, and the S-matrix S 1/2 corresponds to a time-evolution operator U (∞, 0) [30] (see Appendix C for details). Then we separate the time-evolution operator into U (∞, 0) = U (∞, τ )U (τ, 0) with τ being short enough to treat quarks and gluons as good degrees of freedom. Consequently, the strong phase in U (τ, 0)A 0 can be calculated in the factorization approach giving A fac = U (τ, 0)A 0 [28,31]. Hence, the full amplitude becomes A = U (∞, τ )A fac , which corresponds to Eq. (2.24) with ππ → KK rescattering contained in U (∞, τ ) = S 1/2 . 7 In the literature, the tensor width is sometimes expressed as [33] 38) for scalar, vector and tensor mesons, respectively, where p c is the c.m. momentum. Numerically, they are given by |g ρ(770)→π + π − | = 6.00, |g K * (892)→K + π − | = 4.59, [34], Γ σ = 350 MeV and m σ = 563 MeV obtained in the isobar model fit by the LHCb [6]. Note that the strong coupling constant is determined up to a strong phase ambiguity, for example, the strong coupling g σ→π + π − has the expression In the below we will use this freedom of the strong phase φ σ to accommodate a large negative CP asymmetry through π + π − → K + K − rescattering.
As for the ρ(1450) meson, there is no any experimental information for its decays to K + K − and π + π − except for the ratio measured by BaBar through the decay J/ψ → h + h − π 0 [35]. Nevertheless, we can use the measured fractions of B − → ρ(1450)π − → π + π − π − and K + K − π − by LHCb and the partial widths of B − → π + π − π − and B − → K + K − π − to extract the strong couplings. Assuming the same Bρ(1450) transition form factors as that of B-ρ(770) ones, we obtain Contrary to the naive expectation, ρ(1450) couples more strongly to K + K − than π + π − . This is not consistent with the BaBar's measurement given in Eq. (2.42). Since As we will see in the next section, the decay B − → ρ(1450) 0 π − → π + π − π − is well described by the pQCD approach. Hence, the issue has to do with the enormously large coupling of ρ(1450) with KK. Indeed, a recent study in [36] showed that the pQCD prediction for the branching fraction of B + → π + ρ(1450) 0 → π + K + K − is about 18 times smaller than experiment. Note that both where the factor of α T P 1 P 2 takes into account the average over spin of the initial state and sum over final isospin states with averaging over initial isospin states. The reader can check that both Eqs. (2.38) and (2.39) lead to the same tensor coupling such as g f2(1270)→π + π − . II: Branching fractions (in units of 10 −6 ) and CP violation of various contributions to B ± → π ± K + K − decays. The experimental branching fraction of each contribution is inferred from the measured fit fraction [4] together with the world average B(B ± → π ± K + K − ) = (5.24±0.42)×10 −6 [39], for example, B(B − → K * (890) 0 For rescattering contributions, we consider two cases for the S-wave ππ → KK transition amplitudes: Eq. (2.51) for case (i) and Eq. (2.52) for case (ii).
BaBar and Belle used to see a broad scalar resonance f X (1500) in B → K + K + K − , K + K − K S and K + K − π + decays at energies around 1.5 GeV. However, the nature of f X (1500) is not clear as it cannot be identified with the well known scaler meson f 0 (1500). An angular-momentum analysis of the above-mentioned three channels by BaBar [37] showed that the f X (1500) state is not a single scalar resonance, but instead can be described by the sum of the well-established resonances f 0 (1500), f 0 (1710) and f ′ 2 (1525). Since ρ(1450) is very board with a width 400 ± 60 MeV [38], a broad vector resonance ρ X (1500) instead of the scalar one f X (1500) is an interesting possibility to describe the broad resonance observed at energies ∼ 1.5 GeV in B → KKK and KKπ decays.
The calculated branching fractions of resonant and nonresonant contributions to Table II. The theoretical errors arise from the uncertainties in (i) form factors and the strange quark mass m s , (ii) the unitarity angle γ and (iii) the parameter σ NR [see Eq. (2.23)] which governs the nonresonant matrix elements of scalar densities.

K * 0 (1430) contribution
We see from Table II that the K * 0 (1430) 0 contribution to B − → K + K − π − is larger than experiment by a factor of 3. Under the narrow width approximation is obtained by the PDG [38]. This mode has been studied in both pQCD and QCDF approaches with the predictions (2.47) in pQCD [42] and (2.48) in QCDF [43], where S1 and S2 denote two different scenarios for the quark content of the scalar meson. All scalar mesons are made of qq quarks in scenario 1, while in scenario 2 the scalar mesons above 1 GeV are lowest-lying qq scalar states and the light scalar mesons are four-quark states. As discussed in [44,45], scenario 2 is preferable. It appears that the current theoretical predictions for B(B − → K * 0 (1430) 0 K − ) are too large compared to experiment. This issue needs to be resolved. It is interesting to notice that the predicted K * 0 π rates in B → Kππ decays are usually smaller than the results obtained by BaBar and Belle, see Table VI of [9]. For example, the calculated branching fraction of K * 0 0 (1430)π − in B − → K − π + π − is smaller than the BaBar measurement by a factor of two and the Belle result by a factor of three. As discussed in detail in [9], BaBar and Belle have different definitions for the K * 0 (1430) and nonresonant components.
The calculated branching fraction for f 2 (1270) is smaller than experiment by a factor of ∼ 7 in its central value. We have used the form factor A Bf 2 (1270) 0 (0) = 0.20 ± 0.04 derived from light-cone sum rules [46]. Notice that the same form factor leads to a prediction of B(B − → f 2 (1270)π − → π + π − π − ) consistent with the experimental value (see Table VI). Using the narrow width approximation (2.45) and the branching fractions of f 2 (1270) [38] MeV is not so narrow, the narrow width approximation is not fully justified and presumably finite-width effects need to be taken into account to extract the branching fraction of it is straightforward to obtain Tables II and VI, respectively. Evidently, B(B − → f 2 (1270)π − ) extracted from two different processes differs by a factor of seven! This implies that the f 2 (1270) contribution to B − → K + K − π − is probably largely overestimated experimentally. Indeed, B − → f 2 (1270)π − is predicted to have the branching fraction of (2.7 +1.4 −1.2 )×10 −6 in the QCDF approach [19]. This issue needs to be clarified in the Run II experiment. (iv) The predicted CP asymmetry of 32% in the f 2 (1270) component agrees with the measured value, though the experimental signature for CP violation is only 2.4σ. Nevertheless, a large CP asymmetry is clearly observed in the process of B − → f 2 (1270)π − → π + π − π − to be discussed in Sec.III.

Nonresonant contributions
Although the nonresonant contribution in the K + K − system was not considered by the LHCb, our calculation shown in Table II indicates that it is very suppressed relative to the nonresonant one in the π − K + system. This is contrary to the previous expectation that the dominant nonresonant contributions for tree-dominated three-body decays arise from the b → u tree transition rather than from the penguin amplitude process. We have identified the nonresonant contribution in the π ± K ∓ system with the matrix element of scalar density π − K + |ds|0 NR . The values of the NR parameters α NR , σ NR and α in Eqs. (2.20) and (2.23) have been modified in this work.

CP violation via rescattering
From Eqs. (2.32) and (2.33), the S-wave π + π − → K + K − transition amplitude reads (2.51) Recall that the phase φ σ of the coupling g σ→π + π − is unknown [see Eq. (2.41)]. By varying φ σ or the relative phase between A σ and A π + π + NR , we find that a large CP asymmetry of −66% can be accommodated at φ σ ≈ 134 • . The branching fraction is (0.20 +0.06 −0.05 ) × 10 −6 as shown in Table II. Since the LHCb analysis of ππ ↔ KK rescattering is based on the model described in [27,32], the S-wave transition amplitude in this case is given by (2.52) The observed CP asymmetry is fitted with the same phase φ σ = 134 • , but the corresponding branching fraction becomes (0.75 +0.21 −0.18 ) × 10 −6 . This is consistent with the experimental value of (0.85 ± 0.10) × 10 −6 . Note that the calculated rate for rescattering differs by a factor of ∼ 4 as the transition amplitude is different by a factor of two to the leading order of t λ,λ ′ (see Eqs. (2.34) and (2.36)). Nevertheless, we have stressed in passing that one should use Eq. (2.24) to describe ππ ↔ KK final-state rescattering. Therefore, the branching fraction of the rescattering contribution seems to be overestimated by the LHCb by a factor of 4! TABLE III: Direct CP asymmetries (in %) and branching fractions of B ± → π ± K + K − decays with the superscripts denoting "incl", "resc" and "low" for CP asymmetries measured in full phase space, in the rescattering regions with 1.0 < m K + K − < 1.5 GeV and in the low invariant mass region where m K + K − < 1.22 GeV, respectively. We consider two cases for the phase of the matrix element π − K + |ds|0 NR : (i) φ πK = 0 and (ii) φ πK = 250 • . Data are taken from [2] for A low CP , [3] for

Inclusive and local CP asymmetries
The inclusive CP asymmetry A incl CP in B − → K + K − π − has been measured at B factories and LHCb with the results: 0.00 ± 0.10 ± 0.03 by BaBar [14], (−17.0 ± 7.3 ± 1.7)% by Belle [15] and (−12.3 ± 1.7 ± 1.2 ± 0.7)% by LHCb [2]. The world average is A incl CP = −0.122 ± 0.021 [39]. Regional CP asymmetries were also measured by Belle and LHCb. The LHCb measurements read [2] A low while Belle found [15] A local CP = −0.90 ± 0.17 ± 0.04, 0.8 < m K + K − < 1.1 GeV, −0.16 ± 0.10 ± 0.01, 1.1 < m K + K − < 1.5 GeV, (2.54) and hence a 4.8σ evidence of a negative CP asymmetry in the region m KK < 1.1 GeV. Note that Belle and LHCb results for local CP violation are consistent with each other. In Table III we show the calculated inclusive and regional CP asymmetries in the presence of final-state rescattering of S-wave π + π − to K + K − and compare with experiment. Consider the phase φ πK of the matrix element π − K + |ds|0 NR defined in Eq. (2.22). If φ πK = 0 is set to zero, the predicted CP asymmetries A resc CP and A low CP will be positive, while experimentally they are negative. At first sight, this appears to be a surprise in view of a large and negative CP violation coming from rescattering. However, since the branching fraction of π + π − → K + K − transition is very small, of order 0.2 × 10 −6 , its effect can be easily washed out by the presence of various resonances. Indeed, in our previous work [9,10] we have considered the case with φ πK = (5/4)π. As shown in Table III, the agreement between theory and experiment is greatly improved for φ πK ≈ 250 • . It should be stressed that although CP violation produced by rescattering alone is quite large, of order −66%, the regional CP asymmetry A resc CP will not be the same as the latter does receive contributions from other resonances. → π ± π + π − decays analyzed in the isobar model [5,6].
As mentioned in the Introduction, BaBar has carried out the amplitude analysis of B − → π + π − π − before [7]. The nonresonant S-wave fraction was measured to be (34.9 ± 4.2 +8.0 −4.5 )%. In the recent LHCb analysis [5,6], the S-wave component of B − → π + π − π − was studied using three different approaches: the isobar model, the K-matrix model and a quasi-model-independent (QMI) binned approach. In the isobar model, the S-wave amplitude was presented by LHCb as a coherent sum of the σ meson contribution and a ππ ↔ KK rescattering amplitude in the mass range 1.0 < m π + π − < 1.5 GeV. The fit fraction of the S-wave is about 25% and predominated by the σ resonance (see Table IV). A large and positive CP asymmetry of 45% was found in the rescattering amplitude of B − → π + π − π − , while the corresponding CP violation in B − → K + K − π − was of order −0.66.
Contrary to the decay B − → K + K − π − where CP violation is observed only in the rescattering amplitude, a clear CP asymmetry was seen in the B − → π + π − π − decay in the following places: (i) the S-wave amplitude at values of m π + π − below the mass of the ρ(770) resonance, see the left panel of Fig. 1, (ii) the f 2 (1270) contribution, see Fig. 1 at values of m π + π − in the f 2 (1270) mass region, and (iii) the interference between S-and P -waves which is clearly visible in Fig. 2 where the data are split according to the sign of cos θ hel . In the isobar model, the S-wave amplitude is predominated by the σ meson. Hence, a significant CP violation of 15% in B − → σπ − is implied in this model. The significance of CP violation in B − → f 2 (1270)π − was found to be 20σ, 15σ and 14σ for the isobar, K-matrix and QMI approaches, respectively. Therefore, CP asymmetry in the f 2 (1270) component was firmly established. As for the significance of CP violation in the interference between S-and P -waves exceeds 25σ in all the S-wave models.
In contrast to the above-mentioned CP -violating observables, CP asymmetry for the dominant quasi-two-body decay mode B − → ρ 0 π − was found to be consistent with zero in all three S-wave approaches (see Table V), which was already noticed by the LHCb previously in 2014 [3]. However, TABLE V: CP asymmetries in the quasi-two-body decay B − → ρ 0 (770)π − measured by the LHCb for each S-wave approach [5,6].
all the existing theoretical predictions lead to a negative CP asymmetry ranging from −7% to −45%. This is a long-standing puzzle [10]. In this section, we will discuss the observed CP violation in various modes and address the CP puzzle with B − → ρ 0 π − .

A. Resonant contributions
The explicit expression of the factorizable tree-dominated B − → π − (p 1 )π + (p 2 )π − (p 3 ) decay amplitude can be found in Eq. (2.4) of [9]. Amplitudes from various resonances are listed below: with ρ i = ρ(770), ρ(1450). Since there are two identical π − mesons π − (p 1 ) and π − (p 3 ) in this decay, one should take into account the identical particle effects. As a result, a factor of 1 2 should be put in the decay rate.
The strong decay of ω(892) to π + π − is isospin-violating and it can occur through ρ-ω mixing. In this work we shall use the measured rate of ω → π + π − to fix the coupling of ω with ππ.
4. σ/f 0 (500) In the approach of QCD factorization [44,45], the decay amplitude of B − → σπ − has the expression  4). This is because one has to consider the convolution with the light-cone distribution amplitude of the σ in QCDF. As a consequence, the amplitude for σ emission does not vanish in QCDF. Those subtitles are beyond the simple factorization approach adapted here.
Since the naive amplitude given by Eq. (3.4) leads to a negative CP asymmetry −0.015, while experimentally A CP (σπ − ) = (16.0 ± 2.8)%, we shall follow QCDF to keep those terms missing in the σ-emission amplitude, The numerical values of the flavor operators a p i (M 1 M 2 ) for M 1 M 2 = σπ and πσ at the scale µ = m b (m b ) are exhibited in Appendix B. It is clear that a p i (πσ) and a p i (σπ) can be very different numerically except for a p 6,8 , for example, a 1 (σπ) ≈ 1 ≫ a 1 (πσ).

B. Nonresonant contributions
Just as the decay B − → π − K + K − , the nonresonant amplitude in the π + π − system coming from the current-induced process through the b → u transition reads Besides the current-induced one, an additional nonresonant contribution can also arise from the penguin amplitude through the nonresonant matrix element of scalar density π + π − |dd|0 NR . In our previous work, we have argued that this nonresonant background from the penguin amplitude is suppressed by the smallness of the penguin Wilson coefficients a 6 and a 8 . This is no longer true in view of the very large nonresonant contribution in the π − K + system of the decay B − → K + K − π − . The nonresonant amplitude is the one we used in Eq. (2.33) for describing final-state π + π − → K + K − rescattering.

D. Numerical results and discussions
Using the input parameters summarized in Appendices A and B and the amplitudes given in Sec.III.A, we show the calculated results in Table VI. In the following we shall discuss each contribution in order.

Nonresonant component
Although nonresonant contributions were not specified in the LHCb analysis, the theoretical calculations are similar to that of B − → K + K − π − . We find that the nonresonant background denoted by NR(π + π − ) in Table VI constitutes about 14% of the B − → π + π − π − rate and is dominated by the matrix element of scalar density π + π − |dd|0 . This together the σ resonance accounts for 35% of the total rate. Indeed, the nonresonant fraction was found to be 35% in the earlier BaBar measurement [7]. As discussed in Sec.II.D.5, a large and negative CP asymmetry in the rescattering amplitude of B − → π − K + K − cannot be accommodated unless the amplitude A σ interferes with A π + π − NR .

CP violation via rescattering
The S-wave K + K − → π + π − transition amplitude reads from Eqs. (3.13) and (2.33) to be Since both nonresonant contribution in the K + K − system and the f 0 (980) contribution to B − → K + K − π − have not been studied by the LHCb yet, we have to rely on the theoretical evaluation of these two amplitudes. The LHCb measurement of the rescattering contribution to B − → π + π − π − corresponds to the following transition amplitude Here we shall adapt a strategy different from that in the decay B − → π + π − π − . We first vary the phase of the f 0 (980)K + K − coupling to fit the "measured" branching fraction and then figure out the CP asymmetry induced by rescattering. It turns out at φ f 0 (980) ≈ 20 • , the phase of g f 0 (980)→K + K − , the K + K − → π + π − transition amplitude (3.16) yields B(rescattering) = (0.22 ± 0.03) × 10 −6 and a CP asymmetry of (16.3 +0.8 −0.9 )% (see Table VI). For the transition amplitude of Eq. (3.15), the branching fraction becomes smaller by a factor of 4, namely, (0.05 ± 0.01) × 10 −6 . Therefore, the branching fraction of the rescattering contribution seems to be overestimated experimentally by a factor of 4!

Inclusive and local CP asymmetries
In Table VII we show inclusive and regional CP asymmetries in B ± → π ± π + π − decays. The calculated A incl CP and A resc CP are too large compared to experiment. For a consideration of ρ-ω mixing effect on local CP violation, see [49].

CP asymmetry induced by interference
Before proceeding to discuss the CP asymmetry induced by interference, we follow [27] to define the quantity θ being the angle between the pions with the same-sign charge. For example, in B − → π − π + π − decay, it is the angle between the momenta of the two π − pions measured in the rest frame of the dipion system (i.e. the resonance). This angle is related to the helicity angle θ hel defined by the LHCb [6] through the relation θ hel + θ = π (see Fig. 3). Hence, cos θ hel = − cos θ. . 3: The angle θ between the momenta of the two π − pions measured in the rest frame of the dipion system in the decay B − → π − (p 1 )π + (p 2 )π − (p 3 ). It is related to the helicity angle θ hel defined by the LHCb through the relation θ + θ hel = π.
Consider the decay B − → π − (p 1 )π + (p 2 )π − (p 3 ) and define s 23 = (p 2 + p 3 ) 2 = m 2 π + π − low . The angular distribution of the vector resonance is governed by the term s 12 − s 13 (see, for example, Eq. (3.1)). From Eq. (2.16) we have in the rest frame of π + (p 2 ) and π − (p 3 ). As noticed in passing, | p 1 | has the same expression as that in Eq. (2.15), but | p 2 | and | p 3 | are replaced by 1 2 s 23 − 4m 2 π . Furthermore, it follows from Eq. with [27] a(s) = 1 For CP violation induced by the interference between different resonances, let us consider the low π + π − invariant mass region of the Dalitz plot which is divided into four zones as shown in Fig.  4. The vertical line dividing zones I and III from zones II and IV is at the ρ(770) mass, while the horizontal line separating zones I and II from zones III and IV is at the position where cos θ = 0, corresponding to s 12 = −b/a. The cosine of the angle θ varies from −1 to 0 in zones III and IV, corresponding to (s 12 ) min = −(1 + b)/a and s 12 = −b/a, respectively. Likewise, The cosine of the angle θ varies from 0 to 1 in zones I and II, corresponding to s 12 = −b/a and (s 12 ) max = (1 − b)/a, respectively. Hence, I, II : The low π + π − invariant mass region of the B + → π + π + π − Dalitz plot of CP asymmetries divided into four zones. This plot is taken from [50].
In short, zones I and II are delimited by cos θ > 0 or cos θ hel < 0, while zones III and IV are delimited by cos θ < 0 or cos θ hel > 0. The difference in the number of B − and B + events measured in the low-m low region for (a) cos θ < 0 (or cos θ hel > 0) and (b) cos θ > 0 (or cos θ hel < 0) is depicted in Fig. 2. In Fig. 2(a) we see that A CP which is proportional to N B − − N B + is negative below the ρ(770) mass (zone III) and positive above it (zone IV) with a zero at m low = m ρ , while in Fig. 2(b) A CP is positive below the ρ(770) mass (zone I) and negative above it (zone II). The sum of CP asymmetries of cos θ > 0 and cos θ < 0 gives rise to the CP violation shown in the left panel of Fig. 1. It is clear that CP asymmetry at m low below the ρ mass is of order 20%, which is the sum of zone I and zone III. From Fig. 4 it is evident that the local CP asymmetry is largest in zone I. Indeed, LHCb has measured A low CP (π + π − π − ) to be 0.584 ± 0.082 ± 0.027 ± 0.007 in the region specified by m 2 π − π − low < 0.4 GeV 2 and m 2 π + π − high > 15 GeV 2 [2]. In [27] the CP asymmetry of the B − → π + π − π − decay in the low-mass region with s 23 < 1 GeV 2 shown in Fig. 2 is described by the interference between the ρ and the nonresonant amplitude and the interference between the ρ(770) and f 0 (980) mesons. Writing for the B + and B − decays, where F BW ρ is the Breit-Wigner propagator of the ρ(770) it follows that CP asymmetry has the expression The rate asymmetry ∆Γ in units of Γ = 1/τ (B ± ) for B ± → π ± π + π − in the low-m low region induced by the interference between ρ(770) and the σ meson for (a) cos θ < 0 or cos θ hel > 0 and (b) cos θ > 0 or cos θ hel < 0. The interference between ρ(770) and the nonresonant amplitude is added to (a) and (b) and shown in (c) and (d), respectively.
The terms (s 23 − m 2 ρ ) cos θ and m ρ Γ ρ arise from the imaginary and real parts, respectively, of the Breit-Wigner propagator F BW ρ . It was argued in [27] that the first two terms violate the CPT constraint locally and will be set to zero. Assuming c ρ ± and c NR ± are complex constants, the parameters Re(c * ρ − c NR − −c * ρ + c NR + ) and Im(c * ρ − c NR − −c * ρ + c NR + ) were obtained in [27] by fitting them to the data. The observed interference pattern in the ρ region is mainly described by the (s 23 − m 2 ρ ) cos θ term.
Instead of fitting the unknown parameters to the data, we would like to predict the interference pattern in our approach. Since the fit fraction of the broad scalar meson σ is about 25% in the isobar model, it is natural to consider the interference between the ρ(770) and σ(500) mesons (or the broad S-wave in the other models) where the identical particle effect has been taken care of by the factor of 1/2, and the amplitudes QCDF [40] QCDF [51] pQCD [52] SCET [53] TDA [54] FAT [ A ρ(770) and A σ are given by Eqs. (3.1) and (3.7), respectively. The rate asymmetry ∆Γ ρ−σ ≡ Γ B − →π − π + π − − Γ B + →π + π + π − due to the ρ(770) and σ interference is shown in Figs. 5(a) and 5(b) for cos θ < 0 and cos θ > 0, respectively. It is evident that the sign of CP asymmetry is flipped below and above the ρ(770) peak and that the interference term is proportional to cos θ. Our calculation indicates that CP asymmetry is positive in zones I and IV, negative in zones II and III, in agreement with the data (see Fig. 2). The interference between ρ and the nonresonant amplitude exhibits a similar feature. This interference effect is included in Figs. 5(c) and 5(d) with the rate asymmetry ∆Γ ρ−σ,ρ−NR . Note that CP violation no longer vanishes exactly at s 23 = m 2 ρ due to the contributions from the imaginary part of F BW ρ . In short, the rate asymmetry depicted in Fig. 2 is the first observation of CP violation mediated by interference between resonances with significance exceeding 25σ, though it vanishes in the ρ(770) region when integrating over the angle.
It has been argued in [56] that in B → P V decays with m V < 1 GeV, for example, V = ρ(770) or K * (892), CP asymmetry induced from a short-distance mechanism is suppressed by the CP T constraint. Under the the '2+1' approximation that the resonances produced in heavy meson decays do not interact with the third particle, there do not exist other states which can be connected to ππ or πK through final-state interactions. Hence, the absence of final-state interactions implies the impossibility to observe CP asymmetry in those processes. However, if we take this argument seriously to explain the approximately vanishing CP asymmetry in B + → ρ 0 π + , it will be at odd with the CP violation seen in other P V modeds. For example, CP violation in the decay B 0 → K * + π − with A CP = −0.308 ± 0.062 was clearly observed by the LHCb [57]. Therefore, it appears that the smallness of A CP (B + → ρ 0 π + ) has nothing to do with the CP T constraint.
As elucidated in [58], the nearly vanishing CP violation in B − → ρ 0 π − is understandable in the QCD factorization approach. There are two kinds of 1/m b corrections in QCDF: penguin annihilation to the penguin amplitude and hard spectator interactions to the flavor operator a 2 . Power corrections in QCDF often involve endpoint divergences which are parameterized in terms of the parameters ρ A , φ A for penguin annihilation and ρ H , φ H for hard spectator interactions (see Eq. (B4) in Appendix B). In the heavy quark limit, A CP (ρ 0 π − ) is of order 6.3%. Power corrections induced from hard spectator interactions will push it up further, say A CP (ρ 0 π − ) ∼ 15%, whereas penguin annihilation will pull it to the opposite direction (see Table III of [58]). Owing to the destructive contributions from these two different 1/m b power corrections, a nearly vanishing A CP (ρ 0 π − ) can be accommodated in QCDF. For example, B(ρ 0 π − ) ≈ 8.4 × 10 −6 and [39] and A CP (ρ 0 π − ) = (0.7 ± 1.9)% in the isobar model.

CP violation at high m high
An inspection of Fig. 1 for CP asymmetries measured in the high invariant-masss region, the peak in the high-m high region could be ascribed to the χ c0 (1P ) resonance with a mass 3414.71 ± 0.30 MeV and a width 10.8 ± 0.6 MeV. As stressed in [59], although LHCb has not yet found the contribution from the B − → π − χ c0 amplitude in B − → π + π − π − decay, the Mirandizing distribution for Run I data has already shown a clear and huge CP asymmetry around the χ c0 invariant mass. We also see from Fig. 1 that CP asymmetry in the high-m high region changes sign at around 4 GeV, near the DD threshold. In analog to the ππ ↔ KK rescattering in the low mass region, final-state rescattering DD → P P could provide the strong phases necessary for CP violation in the high-m high region [59,60]. However, we will not address this issue in this work.

IV. CONCLUSIONS
We have presented in this work a study of charmless three-body decays of B mesons B − → K + K − π − and B − → π + π − π − based on the factorization approach. Our main results are: • There are two distinct sources of nonresonant contributions: one arises from from the b → u tree transition and the other from the nonresonant matrix element of scalar densities M 1 M 2 |q 1 q 2 |0 NR . It turns out that even for tree-dominated three-body decays B → πππ and KKπ, nonresonant contributions are dominated by the penguin mechanism rather than by the b → u tree process, as implied by the large nonresonant component observed in the π − K + system which accounts for one third of the B − → K + K − π − rate. We have identified the nonresonant contribution to the π − K + system with the matrix element π − K + |ds|0 NR .
• The calculated branching fraction of B − → f 2 (1270)π − → K + K − π − is smaller than experiment by a factor of ∼ 7 in its central value. Nevertheless, the same form factor for B → f 2 (1270) transition leads to a prediction of B(B − → f 2 (1270)π − → π + π − π − ) in agreement with the experimental value. Branching fractions of B − → f 2 (1270)π − extracted from the measured rates of B − → f 2 (1270)π − → K + K − π − and B − → f 2 (1270)π − → π + π − π − by the LHCb also differ by a factor of seven! This together with the theoretical predictions of B(B − → f 2 (1270)π − ) leads us to conjecture that the f 2 (1270) contribution to B − → K + K − π − is largely overestimated experimentally . This needs to be clarified in the Run II experiment. Including 1/m b power corrections from penguin annihilation inferred from QCDF, a sizable CP asymmetry of 32% in the f 2 (1270) component are in accordance with the LHCb measurement.
• A fraction of 5% for the ρ(1450) component in B − → π + π − π − is in accordance with the theoretical expectation. However, a large fraction of 30% in B − → K + K − π − is entirely unexpected. If this feature is confirmed in the future, it is likely that the broad vector resonance ρ(1450) may play the role of the s-called f X (1500) broad resonance observed in B → KKK and KKπ decays.
• The contribution of K * 0 (1430) 0 to B − → K + K − π − was found to be too large by a factor of 3 when confronted with experiment. The current theoretical predictions based on both QCDF and pQCD for B(B − → K * 0 (1430) 0 K − ) are also too large compared to experiment. This issue needs to be resolved.
• By varying the relative phase between A σ and A π + π + NR , we find that a large and negative CP asymmetry of −66% through the S-wave π + π − → K + K − rescattering can be accommodated at φ σ ≈ 134 • . However, the predicted branching fraction is less than the LHCb value by a factor of 4! This is ascribed to the fact that one should use Eq. (2.24) to describe ππ ↔ KK final-state rescattering. By the same token, the branching fraction of the rescattering contribution to B − → π + π − π − also seems to be overestimated experimentally by a factor of 4.
• CP asymmetry for the dominant quasi-two-body decay mode B − → ρ 0 π − was found by the LHCb to be consistent with zero in all three S-wave models. In the QCD factorization approach, the 1/m b power corrections, namely penguin annihilation and hard spectator interactions, contribute destructively to A CP (B − → ρ 0 π − ) to render it consistent with zero.
• While CP violation in B − → ρ 0 π − is consistent with zero, a significant CP asymmetry has been seen in the ρ 0 (770) region where the data are separated by the sign of the value of cos θ with θ being the angle between the pions with the same-sign charge. Considering the low π + π − invariant mass region of the B + → π + π + π − Dalitz plot of CP asymmetries divided into four zones as depicted in Fig. 4, we have predicated the sign of CP violation in each zone correctly which arises from the interference between the ρ(770) and σ as well as the nonresonant background.

Acknowledgments
We are very grateful to Zhi-Tian Zou for helpful discussions. This research was supported in part by the Ministry of Science and Technology of R.O.C. under Grant No. 106-2112-M-033-004-MY3.

Appendix A: Input parameters
Many of the input parameters for the decay constants of pseudoscalar and vector mesons and form factors for B → P, V transitions can be found in [40] where uncertainties in form factors are shown. The reader is referred to [44] for decay constants and form factors related to scalar mesons. For reader's convenience, we list the scalar decay constants relevant to this work defined at µ = 1 GeV and expressed in units of MeV. The vector decay constant of K * 0 (1430) is related to the scalar one via The form factors used in this work are A Bρ 2 (0) = 0.221 ± 0.023, A Bω 2 (0) = 0.198 ± 0.023. The B → f 2 (1270) transition form factor is taken from [19], while the form factors for B → V transition are from [61]. There is an updated light-cone sum-rule analysis of B → V transition form factors in [62] in which one has However, we will not use this new analysis in this study for two reasons. First, it will lead to too large B − → ρ 0 π − and B − → ωπ − rates compared to experiment. Second, the parameters (ρ A , φ A ) and (ρ H , φ H ), which govern 1/m b power corrections from penguin annihilation and hard spectator interactions, respectively, have been extracted from the data using B → V from factors given by [61], see Appendix B below. Note that for the σ meson, the Clebsch-Gordon coefficient 1/ √ 2 is already included inf u σ and F Bσ u 0 . For the f 0 (980), one needs to multiple a factor of sin θ/ √ 2 to get its decay constant and form factor, for example,f u f 0 (980) =f f 0 (980) sin θ/ √ 2 with the mixing angle θ ≈ 20 • . For the CKM matrix elements, we use the updated Wolfenstein parameters A = 0.8235, λ = 0.224837,ρ = 0.1569 andη = 0.3499 [63]. The corresponding CKM angles are sin 2β = 0.7083 +0.0127 . Among the quarks, the strange quark gives the major theoretical uncertainty to the decay amplitude. Hence, we will only consider the uncertainty in the strange quark mass given by m s (2 GeV) = 92.0 ± 1.1 MeV [64].

Appendix B: Flavor operators
In our previous works [8][9][10], we have employed the values of the flavor operators a p i given in [8] at the renormalization scale µ = m b /2 = 2.1 GeV. Since then, there is a substantial progress in the determination of 1/m b power corrections to a p i . In the QCD factorization approach, flavor operators have the expressions [20,65] where i = 1, · · · , 10, the upper (lower) signs apply when i is odd (even), c i are the Wilson coefficients, In the QCD factorization approach, there are two kinds of 1/m b corrections: penguin annihilation to the penguin amplitude and hard spectator interactions to a 2 : and Power corrections in QCDF often involve endpoint divergences. We shall follow [20] to model the endpoint divergence X ≡ 1 0 dx/(1 − x) in the penguin annihilation and hard spectator scattering diagrams as with Λ h being a typical hadronic scale of 0.5 GeV, where the superscripts 'i' and 'f ' refer to gluon emission from the initial and final-state quarks, respectively. A fit of the four parameters (ρ i,f A , φ i,f A ) with the first order approximation of ρ H ≈ ρ i A and φ H ≈ φ i A to the B → P P and P V data yields [51,66] (ρ i A , ρ f A ) PP = (2.98 +1.12 −0.86 , 1.18 +0.20 −0.23 ), There are two different sources for the strong phases of a p i : (i) vertex corrections, hard spectator interactions and penguin contractions which are perturbatively calculable in the QCD factorization approach [20] and (ii) 1/m b power corrections.
It should be stressed that the flavor operators given in Eqs. (B7) and (B8) are not applicable to the quasi-two-body B → SP decays owing to the different behavior of light-cone distribution amplitudes (LCDAs) of scalar and pseudoscalar mesons. While the symmetric pion LCDA peaks at x = 1/2, the antisymmetric LCDA of the light scalar such as σ peaks at x = 0.25 and 0.75. As a consequence, a i (SP ) and a i (P S) can be quite different except for a p 6,8 . Numerical values of the flavor operators a p i (M 1 M 2 ) for M 1 M 2 = σπ and πσ are shown in Table IX. We see that, for example, a 1 (σπ) = 0.95 + 0.014i is very different from a 1 (πσ) = 0.015 − 0.004i. In practice, we also use the same set of flavor operators to work out B → f 0 (980)π decays. which can be expressed as ( i; out|O q |B ) * = k i; in| k; out k; out|O q |B = k S † ik k; out|O q |B , where S ik ≡ i; out| k; in denotes the strong interaction S-matrix element. Note that we have used U T (|out (in) ) * = |in (out) to fix the phase convention, which also leads to S * ij = ( i; out|) * U † T U T (|j; in ) * = i; in| j; out = S * ji .
From the following identity k S † ik S 1/2 where use of Eq. (C3) has been made, it is clear that the solution of Eq. (C2) is simply [29] i; out|O q |B = j S 1/2 where A q 0j is a real amplitude. The weak decay amplitude picks up strong scattering phases [67] and finally we have [28] i; where we have defined A 0 ≡ q λ q A q 0 and, consequently, it is free of any strong phase. It will be useful to give an equivalent expression to the above results in terms of time evolution operator [30]. It is well known that the so-called 'in' and 'out' states can be expressed as |i; in = lim with U I (t 2 , t 1 ) the time evolution operator in the interaction picture given by U I (t 2 , t 1 ) = e iH 0 t 2 e −iH(t 2 −t 1 ) e −iH 0 t 1 = e iH 0 t 2 U (t 2 , t 1 )e −iH 0 t 1 , where H 0 is the free Hamiltonian and H is the full strong Hamiltonian. The time evolution operator satisfies U † T U * I (t 2 , t 1 )U T = U I (−t 2 , −t 1 ) and U † I (t 2 , t 1 ) = U I (t 1 , t 2 ), as H 0 and H are time-invariant and hermitian.
The amplitude i; out|O q |B can now be expressed as which is equivalent to Eq. (C2). Furthermore, using Eq. (C8) and the fact that |i, free and |j, free are degenerate eigenstates of H 0 , we are led to i; free|U I (T, 0)|j; free = i; free|U I (0, −T )|j; free = i; free|U I (T /2, −T /2)|j; free , which justifies the following definition, The above expression clearly shows the time evolution nature of rescattering [31] and the rescattering of ππ → KK is considered to happen at a much later stage of time-evolution contained in i; free|e iH 0 T U (T, τ )e −iH 0 τ |j, free , while all the violent and rapid interactions have already happened and are contained in j; free|e iH 0 τ U (τ, 0)H W |B .