Direct CP Violation in Charmless Three-body Decays of $B$ Mesons

Direct CP violation in charmless three-body hadronic decays of $B$ mesons is studied within the framework of a simple model based on the factorization approach. Three-body decays of heavy mesons receive both resonant and nonresonant contributions. Dominant nonresonant contributions to tree-dominated and penguin-dominated three-body decays arise from the $b\to u$ tree transition and $b\to s$ penguin transition, respectively. The former can be evaluated in the framework of heavy meson chiral perturbation theory with some modification, while the latter is governed by the matrix element of the scalar density $\langle M_1M_2|\bar q_1 q_2|0\rangle$. Strong phases in this work reside in effective Wilson coefficients, propagators of resonances and the matrix element of scalar density. In order to accommodate the branching fraction and CP asymmetries observed in $B^-\to K^-\pi^+\pi^-$, the matrix element $\langle K\pi|\bar sq|0\rangle$ should have an additional strong phase, which might arise from some sort of power corrections such as final-state interactions. We calculate inclusive and regional CP asymmetries and find that nonresonant CP violation is usually much larger than the resonant one and that the interference effect is generally quite significant. If nonresonant contributions are turned off in the $K^+K^-K^-$ mode, the predicted CP asymmetries due to resonances will be wrong in sign when confronted with experiment. In our study of $B^-\to \pi^-\pi^+\pi^-$, we find that ${\cal A}_{C\!P}(\rho^0\pi^-)$ should be positive in order to account for CP asymmetries observed in this decay. However, all theories predict a large and negative CP violation in $B^-\to \rho^0\pi^-$. Measurements of CP-asymmetry Dalitz distributions put very stringent constraints on the theoretical models. We check the magnitude and the sign of violation in some (large) invariant mass regions to test our model.


I. INTRODUCTION
The primary goal and the most important mission of B factories built before millennium is to search for CP violation in the B meson system.BaBar and Belle have measured direct CP asymme-A low CP (π + π − π − ), for m 2 π + π − low < 0.4 GeV 2 , m 2 π + π − high > 15 GeV 2 .Hence, significant signatures of CP violation were found in the above-mentioned low mass regions devoid of most of the known resonances.LHCb has also studied CP asymmetries in the rescattering regions of m π + π − or m K + K − between 1.0 and 1.5 GeV where the final-state π + π − ↔ K + K − rescattering is supposed to be important in this region.The measured CP asymmetries A resc CP for the charged final states are given in Table I.
In two-body B decays, the measured CP violation is just a number.But in three-body decays, one can 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 can be very large and positive in some region and becomes very negative in the other region.The sign of CP asymmetries varies from region to region.A successful model must explain not only the inclusive asymmetry but also regional CP violation.Therefore, the study of three-body CP-asymmetry Dalitz distributions provides a great challenge to the theorists.LHCb has measured the raw asymmetry A raw distributions in the Dalitz plots defined by [8] A B → P 1 P 2 P 3 decays (see Table I) provide a new insight of the underlying mechanism of threebody decays.The observed negative relative sign of CP asymmetries between B − → π − π + π − and B − → K − K + K − and between B − → K − π + π − and B − → π − K + K − is in accordance with what expected from U-spin symmetry which enables us to relate the ∆S = 0 amplitude to the ∆S = 1 one.However, symmetry arguments alone do not tell us the relative sign of CP asymmetries between π − π + π − and π − K + K − and between K − π + π − and K − K + K − .The observed asymmetries (integrated or regional) by 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 [8].
After the LHCb measurement of direct CP violation in three-body charged B decays, there are some theoretical works in this regard [5,[16][17][18][20][21][22][23][24][25][26][27][28].In the literature, almost all the works focus on resonant contributions to the rates and asymmetries.This is understandable in terms of the experimental observation that 90% of the Dalitz plot events has m(h + h − ) 2 < 3.0 GeV 2 [29].The events are concentrated in low-mass regions, implying the dominance of charmless decays by resonant contributions.Nevertheless, in [5] we have examined CP violation in three-body decays and stressed the crucial role played by the nonresonant contributions.Indeed, if the nonresonant term is essential to account for the total rate, it should play some role to CP violation.In this work, we would like to study asymmetries arising from both resonant and nonresonant amplitudes and their interference.This will make it clear the relative weight of both contributions and their interference.
It has been argued in [25] that the amplitude at the Dalitz plot center is expected to be both power-and strong coupling α s -suppressed with respect to the amplitude at the edge.The perturbative regime in the central region gets considerably reduced for realistic value of m B .That is, the Dalitz plot is completely dominated by the edges.Since the nonresonant background arises not just from the central region, the above argument is not inconsistent with the experimental observation of dominant nonresonant signals in penguin-dominated 3-body decays.
There are several competing approaches for describing charmless hadronic two-body decays of B mesons, such as QCD factorization (QCDF) [30], perturbative QCD (pQCD) [31] and soft-collinear effective theory (SCET) [32].Unlike the two-body case, to date we still do not have theories for hadronic three-body decays, though attempts along the framework of pQCD and QCDF have been made in the past [23,25,33].In this work, we shall take the factorization approximation as a working hypothesis rather than a first-principles starting point as factorization has not been proven for three-body B decays.That is, we shall work in the phenomenological factorization model rather than in the established QCD-inspired theories.
The layout of the present paper is as follows.In Sec.II we discuss resonant and nonresonant contributions to three-body B decays.The predicted rates for penguin-dominated B → V P modes are generally too small compared to experiment.We add power corrections induced by penguin annihilation to these modes to render a better agreement with the data.Sec.III is devoted to direct CP violation.We consider inclusive and regional CP asymmetries arising from both resonant and nonresonant mechanisms.The effect of final-state rescattering is discussed.Comparison of our FIG.1: Possible configurations of three-body B → P 1 P 2 P 3 decays where the black lines with arrows denote the momenta of the three energetic quarks q 1 q 2 q3 produced in the b-quark decay and the pink lines with arrows denote the momenta of the spectator quark and the quark-antiquark pair: (a) all three produced mesons are moving energetically, (b) two of the energetic mesons, say P 1 and P 2 , are moving collinearly to each other, recoiling against P 3 , (c) P 2 is formed from q 1 q3 or q 2 q3 , while P 1 contains the spectator quark (denoted by the longer pink line) which becomes hard after being kicked by a hard gluon, and (d) is similar to (c) except that P 2 is soft.
work with others available in the literature is made in Sec/ IV.Sec.V contains our conclusions.

II. THREE-BODY DECAYS
Many three-body B decays have been observed with branching fractions of order 10 −5 for penguin-dominated B → Kππ, KKK decays and of order 10 −6 for tree-dominated B → πππ, KKπ.The charmless three-body channels that have been measured are [1]: In B − and B 0 three-body decays, the b → sq q penguin transitions contribute to the final states with odd number of kaons, namely, KKK and Kππ, while b → uq q tree and b → dq q penguin transitions contribute to final states with even number of kaons, e.g.KKπ and πππ.For B 0 s three-body decays, the situation is the other way around.Consider the 3-body decays B → P 1 P 2 P 3 .The b quark decays into three energetic quarks, q 1 q 2 q3 .There exist four possible physical configurations depicted in Fig. 1: (a) all three produced mesons are moving energetically, (b) two of the energetic mesons, say P 1 and P 2 , are moving collinearly to each other, (c) P 3 is formed from q 1 q3 or q 2 q3 , while P 2 contains the spectator quark which becomes hard after being kicked by a hard gluon, and (d) is the same as (c) except that  c), corresponding to quasi-two-particle decays.Therefore, the Dalitz plot for three-body B decays can be divided into several sub-regions with distinct kinematics and factorization properties, which have been investigated in [25].Especially, the regions containing the configuration (b) or (c) can be described in terms of two-meson distribution amplitudes and B → P 1 P 2 form factors [34][35][36].
With the advent of heavy quark effective theory, nonleptonic B decays can be analyzed systematically within the QCD framework.There are three popular approaches available in this regard: QCDF, pQCD and SCET.Theories of hadronic B decays are based on the "factorization theorem" under which the short-distance contributions to the decay amplitudes can be separated from the process-independent long-distance parts.In the QCDF approach, nonfactorizable contributions to the hadronic matrix elements can be absorbed into the effective parameters a i where a i are basically the Wilson coefficients in conjunction with short-distance nonfactorizable corrections such as vertex, penguin corrections and hard spectator interactions, and M 1 M 2 |O i |B fact is the matrix element evaluated under the factorization approximation.Since power corrections of order Λ QCD /m b are suppressed in the heavy quark limit, nonfactorizable corrections to nonleptonic decays are calculable.In the limits of m b → ∞ and α s → 0, naive factorization is recovered in both QCDF and pQCD approaches.Unlike hadronic 2-body B decays, established theories such as QCDF, pQCD and SCET are still not available for three-body decays, though attempts along the framework of pQCD and QCDF have been made in the past [23,25,33].This is mainly because the aforementioned factorization 34.9 ± 4.2 +8.0 −4.5 [44] theorem has not been proven for three-body decays.Hence, we follow [5,12] to take the factorization approximation as a working hypothesis rather than a first-principles starting point.One of the salient features of three-body B decays is the large nonresonant fraction in penguindominated B decay modes, recalling that the nonresonant signal in charm decays is very small, less than 10% [1].Many of the charmless B to three-body decay modes have been measured at B factories and studied using the Dalitz-plot analysis.The measured fractions and the corresponding branching fractions of nonresonant components are summarized in Table II.We see that the nonresonant fraction is about ∼ (70 − 90)% in B → KKK decays, ∼ (17 − 40)% in B → Kππ decays, and ∼ 35% in the B → πππ decay.Moreover, we have the hierarchy pattern (2.3) Hence, the nonresonant contributions play an essential role in penguin-dominated B decays.This is not unexpected because the energy release scale in weak B decays is of order 5 GeV, whereas the major resonances lie in the energy region of 0.77 to 1.6 GeV.Consequently, it is likely that three-body B decays will receive sizable nonresonant contributions.It is important to understand and identify the underlying mechanism for nonresonant decays.It has been argued in [25] that the Dalitz plot is completely dominated by the edges as the amplitude at the center is both power-and α s -suppressed with respect to the one at the edge.As a result, three-body decays become quasi two-body ones.Nevertheless, this argument is not inconsistent with the experimental observation of dominant nonresonant background in penguindominated 3-body decays because the nonresonant background exists in the whole phase space.That is, the vast phase space of charmless three-body B decays is populated by nonresonant components.
The explicit expressions of factorizable amplitudes of charmless B → P 1 P 2 P 3 decays can be found in [5,12].There are three distinct factorizable terms: (i) the current-induced process with a meson emission, B → P 1 × 0 → P 2 P 3 , (ii) the transition process, B → P 1 P 2 × 0 → P 3 , and (iii) the annihilation process B → 0 × 0 → P 1 P 2 P 3 , where A → B denotes a A → B transition matrix element.There are two different kinds of mechanisms for the production of a meson pair.In 0 → P 2 P 3 , the meson pair is produced from the vacuum through a current, whereas in B → P 1 P 2 the meson pair is produced through a current that induces the transition from the B meson.Hence, we call these as current-induced and transition mechanisms, respectively. 1While the latter process is produced at the b → u tree level, the former one is induced at the b → s or b → d penguin level.Schematically, the decay amplitude is the coherent sum of resonant contributions together with the nonresonant background (2.4) In the following, we will discuss these two contributions separately.

A. Nonresonant background
Consider the transition process induced by the b → u current.The nonresonant contribution to the three-body matrix element P 1 P 2 |(ūb) V −A |B has the general expression [45] where (q 1 q 2 ) V −A = q1 γ µ (1 − γ 5 )q 2 .The form factors r, ω ± and h can be evaluated in the framework of heavy meson chiral perturbation theory (HMChPT) [45].Consequently, the nonresonant amplitude induced by the transition process reads However, as pointed out in [5,12], the predicted nonresonant rates based on HMChPT are unexpectedly too large for tree-dominated decays.For example, the branching fractions of nonresonant B − → π + π − π − and B − → K + K − π − are found to be of order 75×10 −6 and 33×10 −6 , respectively, which are one order of magnitude larger than the corresponding measured total branching fractions of 15.2 × 10 −6 and 5.0 × 10 −6 (see Table III below).The issue has to do with the applicability of HMChPT.In order to apply this approach, two of the final-state pseudoscalars in B → P 1 P 2 transition have to be soft; their momenta should be smaller than the chiral symmetry breaking scale of order 1 GeV.Therefore, it is not justified to apply chiral and heavy quark symmetries to a certain kinematic region and then generalize it to the region beyond its validity.Following [12], we shall assume the momentum dependence of nonresonant amplitudes in an exponential form, namely, so that the HMChPT results are recovered in the soft meson limit of p 1 , p 2 → 0. This is similar to the empirical parametrization of the non-resonant amplitudes adopted in the BaBar and Belle analyses [38,46] A NR = c 12 e iφ 12 e −αs 12 + c 13 e iφ 13 e −αs 13 + c 23 e iφ 23 e −αs 23 . (2.8) We shall use the tree-dominated B − → π + π − π − decay data to fix the unknown parameter α NR as its nonresonant component is predominated by the transition process.Hence, the measurement of nonresonant contributions to B − → π + π − π − provides an ideal place to constrain the parameter α NR , which turns out to be [5] α NR = 0.081 +0.015 −0.009 GeV −2 . (2.9) The phase φ 12 of the nonresonant amplitude will be set to zero for simplicity.Note that A HMChPT transition receives nonresonant contributions from the whole Dalitz plot, including the central regions and regions near and along the edge.Since , it is obvious that the nonresonant signal A transition arises mainly from the small invariant mass region of s 12 .
For penguin-dominated decays B → KKK and B → Kππ, the nonresonant background induced from the b → u transition process yields which are too small compared to experiment (see Table III).This is ascribed to the large CKM suppression ts | associated with the b → u tree transition relative to the b → s penguin process.This implies that the two-body matrix element of scalar densities e.g.KK|ss|0 induced from the penguin diagram should have a large nonresonant component.The explicit expression of the nonresonant component of KK|ss|0 will be shown in Eq. (2.17) below.
For the nonresonant contributions to the 2-body matrix elements P 1 P 2 |qγ µ q |0 and P 1 P 2 |qq |0 , we shall use the measured kaon electromagnetic form factors to extract KK|qγ µ q |0 NR and KK|ss|0 NR first and then apply SU(3) symmetry to relate them to other 2-body matrix elements [12].

B. Resonant contributions
In the experimental analysis of three-body decays, the resonant amplitude associated with the intermediate resonance R takes the form [47] A where T R is usually described by a relativistic Breit-Wigner parametrization, W R accounts for the angular distribution of the decay, F P and F R are the transition form factors of the parent particle and resonance, respectively (see e.g.[47] for details).
In general, vector meson and scalar resonances contribute to the two-body matrix elements P 1 P 2 |V µ |0 and P 1 P 2 |S|0 , respectively.The intermediate vector meson contributions to threebody decays are identified through the vector current, while the scalar meson resonances are mainly associated with the scalar density.Both scalar and vector resonances can contribute to the threebody matrix element P 1 P 2 |J µ |B .Effects of intermediate resonances are described as a coherent sum of Breit-Wigner expressions.More precisely,2 where In general, the decay widths Γ V i and Γ S i are energy dependent.For f 0 (500) and K * 0 (800), they are too broad to use the Breit-Wigner formulism.
Notice that the two-body matrix element P 1 P 2 |V µ |0 can also receive contributions from scalar resonances when q 1 = q 2 .For example, both K * and K * 0 (1430) contribute to the matrix element with

C. Nonresonant contribution from matrix element of scalar density
Consider the nonresonant amplitude in the penguin-dominated B − → K + K − K − decay.In addition to the b → u tree transition which yields a rather small nonresonant fraction, we need to consider the nonresonant amplitudes indcued from the b → s penguin transition for q = u, d, s.The two-kaon matrix element created from the vacuum can be expressed in terms of time-like kaon current form factors as (2.14) The weak vector form factors q can be related to the kaon e.m. form factors em for the charged and neutral kaons, respectively.As shown in [12], the nonresonant components of where the nonresonant terms F NR and F NR can be parameterized as with Λ ≈ 0.3 GeV.The unknown parameters x i and x i are fitted from the kaon e.m. data, see [49] for details.The nonresonant component of the matrix element of scalar density is given by [12] 3 From the measured B 0 → K S K S K S rate and the K + K − mass spectrum measured in B 0 → K + K − K S , the nonresonant σ NR term can be constrained to be [12] σ NR = e iπ/4 3.39 +0.18 −0.21 GeV. (2.19) For the parameter α appearing in Eq. (2.17), we will use the experimental measurement α = (0.14 ± 0.02) GeV −2 [50].Numerically, the nonresonant signal is governed by the σ NR component of the matrix element of scalar density.Owing to the exponential suppression factor e −α s ij in Eq.
(2.17), the nonresonant contribution manifests in the low invariant mass regions.

D. Branching fractions
For numerical calculations we follow [5] for the input parameters except the CKM matrix elements, which we will use the updated Wolfenstein parameters A = 0.8227, λ = 0.22543, ρ = 0.1504 and η = 0.3540 [52].The corresponding CKM angles are sin 2β = 0.710 ± 0.011 and γ = (67.01+0.88 −1.99 ) • [52].In Table III we present updated branching fractions of resonant and nonresonant components in As shown before in [5], the calculated B − → K − φ → K − K + K − rate in the factorization approach is smaller than experiment.In the QCD factorization approach, this rate deficit problem calls for the 1/m b power corrections from penguin annihilation.In this approach, it amounts to replacing the penguin contribution characterized by a p 4 → a p 4 + β p 3 , where p = u, c and β 3 is the annihilation contribution induced mainly from (S − P )(S + P ) operators [55].For our purpose we will use This power correction β p 3 [Kφ] is calculated in [56] for the quasi-two-body decay B − → K − φ.In principle, it should be computed in the 3-body decay B − → K + K − K − with m(K + K − ) low peaked at the φ mass in QCDF.We will assume that β p 3 [Kφ] calculated in either way is similar.From Table III it is clear that the predicted rates for the nonresonant component and for the total branching fraction of B − → K + K − K − are consistent with both BaBar and Belle within errors.

B
We first discuss resonant decays.From Table VI of [5], it is obvious that except for f 0 (980)K, the predicted rates for penguin-dominated channels K * π, K * 0 (1430)π and ρK in B − → K − π + π − within the factorization approach are substantially smaller than the data by a factor of 2 ∼ 5. To overcome this problem, we shall use the penguin-annihilation induced power corrections alculated in our previous work [56].The results are for p = u, c.It is evident the discrepancy between theory and experiment for K * 0 π − and ρ 0 K − is greatly improved (see Table III).
The nonresonant component of B → KKK is governed by the KK matrix element of scalar density KK|ss|0 .By the same token, the nonresonant contribution to the penguin-dominated B → Kππ decays should be also dominated by the Kπ matrix element of scalar density, namely Kπ|sq|0 .When the unknown two-body matrix elements such as K − π + |sd|0 and K 0 π − |su|0 , K − π 0 |su|0 and K 0 π 0 |sd|0 are related to K + K − |ss|0 via SU(3) symmetry, e.g.

Decay mode
BaBar [53] Belle [54] Theory we find too large nonresonant and total branching fractions, namely B(B − → K − π + π − ) NR ∼ 29.7 × 10 −6 and B(B − → K − π + π − ) tot ∼ 68.5 × 10 −6 .Furthermore, Eq. (2.22) will lead to negative asymmetries 4% which are wrong in sign when confronted with the data.To accommodate the rates, it is tempting to assume that K − π + |sd|0 becomes slightly smaller because of SU(3) breaking.However, the predicted CP asymmetry is still not correct in sign.As argued in [5], we assumed that some sort of power corrections such as FSIs amount to giving a large strong phase δ to the nonresonant component of We found that δ ≈ ±π will enable us to accommodate both branching fractions and CP asymmetry simultaneously.In practice, we use Our calculated nonresonant rate in B − → K − π + π − is consistent with the Belle measurement, but larger than that of BaBar.It is of the same order of magnitude as that in Indeed, this is what we will expect.The reason why the nonresonant fraction is as large as 90% in KKK decays, but becomes only (17 ∼ 40)% in Kππ channels (see Table II) can be explained as follows.Since the KKK channel receives resonant contributions only from φ and f 0 mesons, while K * , K * 0 , ρ, f 0 resonances contribute to Kππ modes, this explains why the nonresonant fraction is of order 90% in the former and becomes of order 40% or smaller in the latter.
Finally, we wish to stress again that the predicted total rate of B − → K − π + π − is smaller than the measurements of both BaBar and Belle.This is ascribed to the fact that the calculated K * 0 (1430)π − in naive factorization is too small by a factor of 3.

B
Applying U -spin symmetry to Eq. (2.24) leads to which will be used to describe B → KKπ decays.Contrary to naive expectation, ss resonant contributions to the tree-dominated B − → K + K − π − decay are strongly suppressed.The only relevant factorizable amplitude which involves the ss current is given by (see Eq. (5.1) of [5]) The smallness of the penguin coefficients a 3,5,7,9 indicates negligible ss resonant contributions.Indeed, no clear φ(1020) signature is observed in the mass region m 2 K + K − around 1 GeV 2 [7].The branching fraction of the two-body decay B − → φπ − is expected to be very small, of order 4.3 × 10 −8 .It is induced mainly from B − → ωπ − followed by a small ω − φ mixing [56].
The predicted nonresonant fraction is very sizable about 55% in B − → K + K − π − even it is a tree-dominated mode.This should be checked experimentally.
Note that the BaBar result for K * − 0 (1430)π + in [40], all the BaBar results in [42] and Belle results in [43] are their absolute ones.We have converted them into the product branching fractions, namely, B(B → Rh) × B(R → hh).
Decay mode BaBar [42] Belle [43] Theory The current-induced nonresonant contributions to the tree-dominated B − → π + π − π − decay are suppressed by the smallness of the penguin Wilson coefficients a 6 and a 8 .Therefore, the nonresonant component of this decay is predominated by the transition process, and its measurement provides an ideal place to constrain the parameter α NR .

Other B → Kππ decays
Branching fractions of resonant and nonresonant (NR) contributions to other B → Kππ decays such as IV.Except the first channel, the other three have been studied before in [5].In order to improve the discrepancy between theory and experiment for penguin-dominated VP modes in [5], we shall introduce penguin annihilation given in Eq. (2.21).In general, the predicted K * π and ρK rates are now consistent with experiment.However, the calculated K * 0 (1430)π rates are still too small.This explains why the calculated total branching fractions are smaller than experiment, especially for B − → K 0 π − π 0 due to the presence of two K * 0 (1430)π modes.In [5] we have made predictions for the resonant and nonresonant contributions to even though the former involves a π 0 and has no identical particles in the final state.This is because while the latter is dominated by the ρ 0 pole, the former receives ρ ± and ρ 0 resonant contributions.

III. DIRECT CP ASYMMETRIES
Experimental measurements of inclusive and regional direct CP violation by LHCb for various charmless three-body B decays are collected in Table I.CP asymmetries of the pair π − π + π − and K − K + K − are of opposite signs, and likewise for the pair K − π + π − and π − K + K − .This can be understood in terms of U-spin symmetry, which leads to the relation [16,20] and The predicted signs of the ratios R 1 and R 2 are confirmed by experiment.However, because of the momentum dependence of 3-body decay amplitudes, U-spin or flavor SU(3) symmetry does not lead to any testable relations between A CP (π − K + K − ) and A CP (π − π + π − ) and between A CP (K − π + π − ) and A CP (K + K − K − ).That is, symmetry argument alone does not give hints at the relative sign of CP asymmetries in the pair of ∆S = 0(1) decay.The LHCb data in Table I indicate that decays involving a K + K − pair have a larger CP asymmetry (A incl CP or A resc CP ) than their partner channels.The asymmetries are positive for channels with a π + π − pair and negative for those with a K + K − pair.In other words, when K + K − is replaced by π + π − , CP asymmetry is flipped in sign.This observation appears to imply that final-state rescattering may play an important role for direct CP violation.It has been conjectured that maybe the final rescattering between π + π − and K + K − in conjunction with CPT invariance is responsible for the sign change [16,17,60].However, the implication of the CPT theorem for CP asymmetries at the hadron level in exclusive or semi-inclusive reactions is more complicated and remains mostly unclear [61].
It is well known that one needs nontrivial strong and weak phase differences to produce partial rate CP asymmetries.In this work, the strong phases arise from the effective Wilson coefficients a p i listed in Eq. ( 2.3) of [5], the Breit-Wigner expression for resonances and the penguin matrix elements of scalar densities.It has been established that the strong phase in the penguin coefficients a p 6 and a p 8 comes from the Bander-Silverman-Soni mechanism [62].There are two sources for the phase in the penguin matrix elements of scalar densities: σ NR and δ for Kπ-vacuum matrix elements.
In the literature, most of the theory studies concentrate on the resonant effects on CP violation.For example, the authors of [16,18] considered the possibility of having a large local CP violation in B − → π + π − π − resulting from the interference of the resonances f 0 (500) and ρ 0 (770).A similar mechanism has been applied to the decay B − → K − π + π − [18].
In this work, we shall take into account both resonant and nonresonant amplitudes simultaneously and work out their contributions and interference to branching fractions and CP violation in details.

A. CP asymmetries due to resonant and nonresonant contributions
Following the framework of [5,12] we present in Table V the calculated results of inclusive and regional CP asymmetries in our model.We consider both resonant and nonresonant mechanisms and their interference.For nonresonant contributions, direct CP violation arises solely from the interference of tree and penguin nonresonant amplitudes.For example, in the absence of resonances, CP asymmetry in B − → K − π + π − stems mainly from the interference of the nonresonant tree amplitude It is clear from Table V that nonresonant CP violation is usually much larger than the resonant one and that the interference effect is generally quite significant.If nonresonant contributions are turned off in the K + K − K − mode, the predicted asymmetries will be wrong in sign when compared with experiment.This is not a surprise because B − → K + K − K − is predominated by the nonresonant background.The magnitude and the sign of its CP asymmetry should be governed by the nonresonant term.
Large local CP asymmetries A low CP in three-body charged B decays have been observed by LHCb in the low mass regions specified in Eq. (1.1).If intermediate resonant states are not associated in these low-mass regions, it is natural to expect that the Dalitz plot is governed by nonresonant contributions.It is evident from Table V that except the mode K + K − π − , CP violation in the TABLE V: Predicted inclusive and regional CP asymmetries (in %) for various charmless threebody B decays.Two local regions of interest for regional CP asymmetries are the low-mass regions specified in Eq. (1.1) for A incl CP and the rescattering region of m ππ and m KK between 1.0 and 1.5 GeV for A resc CP .Resonant (RES) and nonresonant (NR) contributions to direct CP asymmetries are considered.
interference between ρ(770) and f 0 (980) has a real component proportional to This gives to two zeros: one at s = m 2 ρ(770) and the other at s = m 2 f 0 (980) .However, we only see a sign change around f 0 (980) but not ρ(770) for cos θ < 0 and do not see any zero for cos θ > 0. It is possible that the zeros are contaminated or washed out by other contributions.We are going to investigate this issue.

Strong phase δ
We now discuss in more details why we need to introduce an additional phase δ to the matrix element of scalar density K − π + |sd|0 given in Eq. (2.23).First, we notice that the calculated integrated CP asymmetries (8.3 +1.7 −1.9 )% for π + π − π − and (−6.0 +2.0 −1.5 )% for K + K − K − (see Table V) are consistent with LHC measurements in both sign and magnitude. 4As discussed in passing and in [5], when the unknown two-body matrix elements of scalar densities Kπ|sq|0 and πK|sq|0 are related to K K|ss|0 via SU(3) symmetry so that the calculated nonresonant and total rates of B − → K − π + π − will be too large compared to experiment [see the discussions after Eq. (2.22)].Moreover, the predicted CP violation A incl CP (K − π + π − ) = (−0.8+0.9 −0.6 )% and A incl CP (K + K − π − ) = (4.9+1.1 −1.0 )% are wrong in sign when confronted with experiment.Since the partial rate asymmetry arises from the interference between tree and penguin amplitudes and since nonresonant penguin contributions to the penguin-dominated decay K − π + π − are governed by the matrix element K − π + |sd|0 , it is thus conceivable that a strong phase δ in K − π + |sd|0 induced from some sort of power corrections might flip the sign of CP asymmetry.
It is clear from Table VI that the reason why the predicted inclusive and regional CP asymmetries (except A low CP (K − π + π − )) all are erroneous in sign when δ is set to zero is ascribed to the nonresonant contributions which are opposite in sign to the experimental measurements.By comparing Tables VI and V, we see that when δ is set to ≈ ±π preferred by the data, CP asymmetries induced from nonresonant components will flip the sign as e ±iπ = −1.Consequently, this in turn will lead to the correct sign for the predicted asymmetries.As stressed in [5], we have implicitly assumed that power corrections will not affect CP violation in π + π − π − and K + K − K − .
Finally we would like to remark that unlike the global weak phases, strong phases such as δ and the Breit-Wigner phase are local ones, namely they are energy and channel dependent.For example, when we study CP-asymmetry Dalitz distributions in some large invariant mass regions (see subsection III.4 below), we find that δ needs to vanish in the large invariant mass region for B − → K + K − π − in order to accommodate the observation.with P = π, K.The unitary S matrix reads where the inelasticity parameter η(s) is given by [64] η with The ππ phase shift has the expression with We shall assume that δ K K ≈ δ ππ in the rescattering region.
To calculate S 1/2 , we note that the S-matrix can be recast to the form with and Hence, Consequently, for P = π, K.
For the numerical results presented in Table VII, we have used the parameters given in Eqs.(2.15b') and (2.16) of [64], namely M = 1.5 GeV, M s = 0.92 GeV, M f = 1.32 GeV, 1 = 2.4, 2 = −5.5 and c 0 = 1.3 .Unfortunately, our results are rather disappointed: In the presence of the specific final-state rescattering, CP asymmetries for both π + π − π − and K + K − π − are heading to the wrong direction.While A CP is decreased for the former, it is increased for the latter, rendering the discrepancy between theory and experiment even worse.We also see that A CP (K + K − K − ) is almost not affected by the rescattering of ππ and K K.
Thus far we have confined ourselves to rescattering between π + π − and K + K − in s-wave configuration.It is known from two-body B decays that this particular rescattering channel (through annihilation and total annihilation diagrams, see Fig. 1 of [65]) cannot be sizeable, or the rescattered B 0 → K + K − rate fed from the B 0 → π + π − mode will easily excess the measured rate, which is highly suppressed [1].In fact, the effect of exchange rescattering is expected to be more prominent [65] and one needs to enlarge the rescattering channels.It is clear that ππ and KK are not confined to the s-wave configuration in the three-body decays.Therefore, rescatterings in other partial wave configurations should also be included.Rescatterings between the third meson and other mesons can be relevant.Moreover, other potentially important coupled channels should not be neglected.For example, the decay B − → π + π − π − can be produced through the weak decay B → D D * π followed by the rescattering of D D * π → πππ and likewise for other three-body decays of B mesons.The intermediate

D( * )
(s) P states have large CKM matrix elements and hence can make significant contributions to CP violation when coupled to three light pseudoscalar states.
A comprehensive study of rescattering effects in three-body B decays is beyond the scope of the present work.At any rate, in this work we shall use the phenomenological phase δ ≈ ±π to describe the decays and CP violation of

CP violation in B
It has been claimed that the observed large localized CP violation in B − → π + π − π − may result from the interference of a light scalar meson f 0 (500) and the vector ρ 0 (770) resonance [16,18], even though the latter one is not covered in the low mass region m within errors.Its CP asymmetry is found to be A CP (ρ 0 π − ) = 0.059 +0.012 −0.010 .At first sight, this seems to be in agreement in sign with the BaBar measurement 0.18 ± 0.07 +0.05 −0.15 from the Dalitz plot analysis of B − → π + π − π − [44].However, theoretical predictions based on QCDF, pQCD and SCET all lead to a negative CP asymmetry of order −0.20 for B − → ρ 0 π − (see Table XIII of [56]).As shown explicitly in Table IV of [56], within the framework of QCDF, the inclusion of 1/m b power corrections to penguin annihilation is responsible for the sign flip of A CP (ρ 0 π − ) to a negative one.Specifically, we shall use TABLE VIII: Predicted inclusive and regional CP asymmetries (in %) in B − → π + π − π − decay when penguin annihilation is added to render A CP (ρ 0 π − ) ≈ −0.21.VIII we see that the inclusive and regional CP asymmetries induced by resonances now become negative.Consequently, the predicted A incl CP is wrong in sign, while A low CP and A resc CP are too small when compared with experiment.Hence, the LHCb data imply positive CP violation induced by the ρ and f 0 resonances.Indeed, LHCb has measured asymmetries in B − → π + π − π − in four distinct regions dominated by the ρ [8]: I: 0.47 < m(π + π − ) low < 0.77 GeV, cos θ > 0, II: 0.77 < m(π + π − ) low < 0.92 GeV, cos θ > 0, III: 0.47 < m(π + π − ) low < 0.77 GeV, cos θ < 0, and IV: 0.77 < m(π + π − ) low < 0.92 GeV, cos θ < 0. It is seen that A CP changes sign at m(π + π − ) ∼ m ρ .Summing over the regions I-IV yields CP asymmetry consistent with zero with slightly positive central value (see Table IV of [8]).

NR
Therefore, we encounter a puzzle here.On one hand, BaBar and LHCb measurements of B − → π + π − π − seem to indicate a positive CP asymmetry in the m(π + π − ) region peaked at m ρ .On the other hand, all theories predict a large and negative CP violation in B − → ρ 0 π − .This issue concerning A CP (ρ 0 π − ) needs to be resolved.

Local CP violation in other invariant mass regions
For regional CP violation, so far we have focused on the small invariant mass region specified in Eq. (1.1) and the rescattering region of m ππ and m K K between 1.0 and 1.5 GeV.As noticed in passing, the magnitude and sign of CP asymmetries in the Dalitz plot vary from region to region.A successful model must explain not only the inclusive asymmetry but also regional CP violation.Therefore, the measured CP-asymmetry Dalitz distributions put stringent constraints on the models.In the following we consider the distribution of A CP in some (large) invariant mass regions to test our model.
We see from Fig. 3(a) that A CP is mostly negative in the Dalitz plot region with m(K + K − ) low between 1 and 1.6 GeV and m(K + K − ) high below 4 GeV, but it can be positive at m(K + K − ) high > 4 GeV (see also Fig. 2 of [11]).We consider two regions with positive A CP : (i) m 2 (K + K − ) low = 3-5 GeV 2 and m 2 (K + K − ) high = 18-22 GeV 2 , and (ii) m 2 (K + K − ) low = 8-9 GeV 2 and m 2 (K + K − ) high = 18-19 GeV 2 .We obtain the values of A CP to be 0.11 and 0.41, respectively, in our model.This is consistent with the data as A CP in region (ii) should be much larger than that in region (i).
shows that A CP is large and negative in the region of (i) 16 < m 2 (K + K − ) < 25 GeV 2 and 5 < m 2 (K + π − ) < 10 GeV 2 .It changes sign in the region of (ii) 5 < m 2 (K + K − ) < 9 GeV 2 and 4 < m 2 (K + π − ) < 13 GeV 2 .Our results A local CP ≈ 0.36 and −0.44 in regions (i) and (ii), respectively, are not consistent with experiment.If the phase δ is set to zero, we will have A local CP ≈ −0.73 and 0.54, respectively, in qualitative agreement with the data.Thus it is possible that the phase δ is energy dependent and it vanishes in the large invariant mass region.This issue is currently under study.
In short, for local CP asymmetries in various (large) invariant mass regions, our model predictions are in qualitative agreement with experiment for K + K − K − and π + π − π − modes and yield a correct sign for K − π + π − .However, it appears that the phase δ needs to vanish in the large invariant mass region for K + K − π − in order to accommodate the observation.

IV. COMPARISON WITH OTHER WORKS
CP violation in three-body decays of the charged B meson has been investigated in Ref. [5,[16][17][18][19][20][21][22][23][24][25][26][27][28].The authors of [16,18] considered the possibility of having a large local CP violation in B − → π + π − π − resulting from the interference of the resonances f 0 (500) and ρ 0 (770).A similar mechanism has been applied to the decay B − → K − π + π − [19].Studies of flavor SU(3) symmetry imposed on the decay amplitudes and its implication on CP violation were elaborated on in [20,24].The observed CP asymmetry in B − → π + π − π − decays changes sign at a value of m(π + π − ) low close to the ρ(770) resonance [8].It was argued in [23] that the sign change is caused by the ρ-ω mixing.In our work, we have taken into account both resonant and nonresonant amplitudes simultaneously and worked out their contributions to branching fractions and CP violation in details.We found that even in the absence of f 0 (500) resonance, local CP asymmetry in π + π − π − can already reach the level of 17% due to nonresonant and other resonant contributions.Moreover, the regional asymmetry induced solely by the nonresonant component can be as large as 58% in our calculation.In our work and also in the work of [17,27] to be discussed below, the sign change is ascribed to the real part of the Breit-Wigner propagator for the ρ(770) resonance.
Based on the constraint of CPT invariance on final-state interactions, the authors of [17,27] have studied CP violation in charmless three-body charged B decays, especially the CP-asymmetry distribution in the mass region below 1.6 GeV.We first recapitulate the main points of this work.Writing the S matrix as S λ λ = δ λ λ + it λ λ and the decay amplitude to the leading order in t as with A λ and B λ being complex amplitudes invariant under CP, it follows that the rate difference reads [17,27] ∆Γ where the first term corresponds to the familiar short-distance contribution to direct CP asymmetry and the second term arises from final-state rescattering (so-called compound CP violation).It is interesting to notice the relation (see [27] for the derivation) is valid irrespective of the short-distance one.When the CPT condition λ Im[B * λ A λ ] = 0 is imposed, the CPT constraint λ ∆Γ λ = 0 follows.
Suppose only the two channels α = π + π − P − and β = K + K − P − (P = π, K) in B − decays are strongly coupled through strong interactions with the third meson P being treated as a bachelor or a spectator, it follows from Eq. (4.3) that ∆Γ FSI α = −∆Γ FSI β (not ∆Γ α = −∆Γ β !).It should be stressed again that this relation is not imposed by hand, rather it is a consequence of the assumption of only two channels coupled through final-state resacttering.As a result, where we have used the branching fractions listed in Table III and the averaged ones: B(B − → π + π − K − ) = (51.0± 2.9) × 10 −6 and B(B − → K + K − K − ) = (33.0± 1.0) × 10 −6 .Experimentally, the ratios in Eq. (4.4) are measured to be of order −2.1 and −1.4,respectively.The coincidence between theory and experiment suggests that the LHCb data of CP asymmetries could be described in terms of final-state rescattering.For three-body B decays, the strong couplings between K + K − and π + π − channels with the CPT constraint were used in [27] to fit the observed asymmetries in some channels and then predict CP violation in other modes.Explicitly, the amplitude Eq.
In short, final-state interactions play an essential role in the work of [17,27].The CPT relation ∆Γ FSI α = −∆Γ FSI β is used to describe CP-asymmetry distributions in B − → K + K − P − decays after a fit to B − → π + π − P − channels.Final-state rescattering of π + π − ↔ K + K − dominates the asymmetry in the mass region between 1 and 1.5 GeV.On the contrary, we performed a dynamical model calculation of partial rates and CP asymmetries without taking into account final-state interactions explicitly.We accentuate the crucial role played by nonresonant contributions.Our predicted inclusive CP asymmetries for π + π − π − and K + K − K − agree with experiment and have nothing to do with π + π − and K + K − final-state rescattering, while the calculated CP asymmetries for K + K − π − and π + π − K − are wrong in sign.Hence, we introduce an additional strong phase δ to flip the sign.

V. CONCLUSIONS
We have presented in this work a study of charmless three-body decays of B mesons using a simple model based on the factorization approach.Our main results are: • Dominant nonresonant contributions to tree-dominated and penguin-dominated three-body decays arise from the b → u tree transition and b → s penguin transition, respectively.The former can be evaluated in the framework of heavy meson chiral perturbation theory supplemented by some energy dependence to ensure that HMChPT results are valid in chiral limit.The latter is governed by the matrix element of the scalar density M 1 M 2 |q 1 q 2 |0 .
• Based on the factorization approach, we have considered the resonant contributions to threebody decays and computed the rates for the quasi-two-body decays B → V P and B → SP .While the calculated branching fractions for the tree-dominated modes such as ρπ and f 0 (980)π are consistent with experiment, the predicted rates for penguin-dominated φK, K * π, ρK and K * 0 (1430)π channels are too small compared to the data.This implies the importance of power corrections.We follow the QCD factorization approach to introduce the penguin annihilation characterized by the parameter β 3 to improve the discrepancy between theory and experiment for penguin-dominated ones.
• We have updated the predictions for the resonant and nonresonant contributions to B − → K 0 π − π 0 , B − → K − π 0 π 0 , B 0 → K 0 π + π − and B 0 → K − π + π 0 .The calculated total branching fractions are smaller than experiment.This is ascribed to the fact that the predicted B → K * 0 (1430)π rates in factorization or QCDF are too small compared to the data and that the K * 0 (1430) has the largest contributions to B → Kππ decays.
• In our study of B − → π − π + π − , we find that A CP (ρ 0 π − ) is positive.Indeed, both BaBar and LHCb measurements of B − → π + π − π − indicate positive CP asymmetry in the m(π + π − ) region peaked at m ρ .On the other hand, all theories predict a large and negative CP violation in B − → ρ 0 π − .We have shown that if we add 1/m b penguin-annihilation induced power correction to render A CP (ρ 0 π − ) negative, A incl CP will be wrong in sign and the predicted regional CP asymmetries will become too small compared to experiment.Therefore, the issue with CP violation in B − → ρ 0 π − needs to be resolved.
• While the calculated direct CP asymmetries for K + K − K − and π + π − π − modes are in good agreement with experiment in both magnitude and sign, the predicted asymmetries in B − → π − K + K − and B − → K − π + π − are wrong in signs when confronted with experiment.This is attributed to the sizable nonresonant contributions which are opposite in sign to the experimental measurements (see Table VI).We have studied final-state inelastic π + π − ↔ K + K − rescattering and found that CP violation for both π + π − π − and K + K − K − is heading to the wrong direction, making the discrepancy even worse.In order to accommodate the branching fraction of nonresonant component and CP asymmetry observed in B − → K − π + π − , the matrix element Kπ|sq|0 should have an extra strong phase δ of order ±π in addition to the phase characterized by the parameter σ NR .This phase δ may arise from some sort of power corrections such as final-state interactions.The matrix element Kπ|qs|0 relevant to the decay B − → π − K + K − is related to Kπ|sq|0 via U -spin symmetry.
• In this work, there are three sources of strong phases: effective Wilson coefficients, propagators of resonances and the matrix element of scalar density M 1 M 2 |q 1 q 2 |0 .There are two sources for the phase in the penguin matrix element of scalar densities: σ NR and δ for Kπ-vacuum matrix elements.
• Nonresonant CP violation is usually much larger than the resonant one and the interference effect between resonant and nonresonant components is generally quite significant.If nonresonant contributions are turned off in the B − → K + K − K − mode, the predicted CP asymmetries due to resonances will be incorrect in sign.Since this decay is predominated by the nonresonant background, the magnitude and the sign of its CP asymmetry should be governed by the nonresonant term.
• We have studied CP-asymmetry Dalitz distributions in some (large) invariant mass regions to test our model.Our model predictions are in qualitative agreement with experiment for K + K − K − and π + π − π − modes and yield a correct sign for K − π + π − .However, it appears that the phase δ needs to vanish in the large invariant mass region for K + K − π − in order to accommodate the observation.

( 4 . 1 )
is fitted to the LHCb data of the distribution of CP asymmetries in m(π + π − ) measured in B − → π + π − P − decays with P = π, K. Then the fit parameters in ∆Γ FSI α are used to predict the ∆Γ FSI β (s) distributions of B − → K + K − P − decays in m(K + K − ) (see Figs. 10 and 12 of [27]).It turns out that the CP-asymmetry distributions of B − → K + K − P − observed by LHCb in the rescattering region are fairly accounted for by the final-state rescattering of π

TABLE II :
The fractions and branching fractions of nonresonant components of various charmless three-body decays of B mesons.

TABLE III :
Branching fractions (in units of 10 −6 ) of resonant and nonresonant (NR) contributions to B −

TABLE IV :
Branching fractions (in units of 10 −6 ) of resonant and nonresonant (NR) contributions to B