Study of localized $CP$ violation in $B^-\rightarrow \pi^- \pi^+\pi^-$ and the branching ratio of $B^-\rightarrow \sigma(600)\pi^-$ in the QCD factorization approach

In this work, within the QCD factorization approach, we study the localized integrated $CP$ violation in the $B^-\rightarrow \pi^-\pi^+\pi^-$ decay and the branching fraction of the $B^-\rightarrow\sigma\pi^-$ decay. Both the resonance and nonresonance contributions are included when we study the localized $CP$ asymmetry in the $B^-\rightarrow \pi^-\pi^+\pi^-$ decay. The resonance contributions from the scalar $\sigma(600)$ and vector $\rho^0(770)$ mesons are included. For the $\sigma(600)$ meson, we apply both the Breit-Wigner and Bugg models to deal with its propagator, and obtain $\mathcal{B}(B^-\rightarrow \sigma(600)\pi^-)<1.67\times10^{-6}$ and $\mathcal{B}(B^-\rightarrow \sigma(600) \pi^-)<1.946\times10^{-5}$ in these two models, respectively. We find that there is no allowed divergence parameters $\rho_S$ and $\phi_S$ to satisfy the experimental data $\mathcal{A_{CP}}(\pi^-\pi^+\pi^-)=0.584\pm0.082\pm0.027\pm0.007$ in the region $m_{\pi^+\pi^- \mathrm{high}}^2>15$ $\mathrm{GeV}^2$ and $m_{\pi^+\pi^-\mathrm{low}}^2<0.4$ $\mathrm{GeV}^2$ and the upper limit of $\mathcal{B}(B^-\rightarrow \sigma(600)\pi^-)$ in the Breit-Wigner model, however, there exists the region $\rho_S\in[1.70,3.34]$ and $\phi_S \in [0.50,4.50]$ satisfying the data for $\mathcal{A_{CP}}(\pi^-\pi^+\pi^-)$ and the upper limit of $\mathcal{B}(B^-\rightarrow \sigma(600)\pi^-)$ in the Bugg model. This reveals that the Bugg model is more plausible than the Breit-Wigner model to describe the propagator of the $\sigma(600)$ meson even though the finite width effects are considered in both models. The large values of $\rho_S$ indicate that the contributions from weak annihilation and hard spectator scattering processes are both large, especially, the weak annihilation contribution should not be negleted for $B$ decays to final states including a scalar meson.


I. INTRODUCTION
Charge-Parity (CP ) violation is essential to our understanding of both particle physics and the evolution of the early universe. It is one of the most fundamental and important properties of weak interaction, and has gained extensive attentions ever since its first discovery in 1964 [1]. In the Standard Model (SM), CP violation is related to the weak complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which describes the mixing of different generations of quarks [2,3]. Besides the weak phase, a large strong phase is also needed for a large CP asymmetry. Generally, this strong phase is provided by QCD loop corrections and some phenomenological models.
In recent years, the charmless three-body decays of B mesons have attracted much more attention, because by studying them the CKM parameters can be determined and the possible new physics effects beyond the SM can be search for. However, the three-body decays of B mesons are more complicated than the two-body cases, because both resonance and nonresonance contributions are involved in the hadronic matrix elements. For the three-body B decay B → M 1 M 2 M 3 , under the factorization hypothesis, one of the nonresonance contributions arises from the transitions B → M 1 M 2 . The nonresonance background in the charmless three-body B decays due to the transition B → M 1 M 2 has been studied extensively based on the heavy meson chiral perturbation theory (HMChPT). In order to apply this approach, both of the final-state pseudoscalars in the B → M 1 M 2 transition have to be soft [4]. Besides the nonresonance background, the three-body meson decays are generally dominated by intermediate resonances, namely, they proceed via quasi-two-body decays containing resonance states. LHCb has observed the large CP asymmetry in the B − → π − π + π − decay in the localized region of the phase space [5], A CP (π − π + π − ) = 0.584±0.082±0.027±0.007, for m 2 π + π − high > 15 GeV 2 and m 2 π + π − low < 0.4 GeV 2 , which spans the σ(600) and ρ 0 (770) mesons. In 2005, BABAR Collaboration reported the upper limit of B(B − → σπ − , σ → π + π − ) as 4.1 × 10 −6 [6]. Both of these experimental results are important for us to study the B decays including the scalar meson σ(600).
Theoretically, to calculate the hadronic matrix elements of B nonleptonic weak decays, some approaches, including the naive factorization [7,8], the QCD factorization (QCDF) [9,10], the perturbative QCD (PQCD) approach [11], and the soft-collinear effective theory (SCET) [12], have been developed and extensively employed in recent years. In this work, within the framework of QCDF, we will study the decays of B − → π − π + π − and B − → σπ − . The remainder of this paper is organized as follows. In Sect. II, we present the form factors, decay constants and distribution amplitudes of different mesons. In Sect. III, we present the formalism for B decays in the QCDF approach. In Sect. IV, we give the calculations of the localized CP violation and the branching ratio of the B meson decays. The numerical results are given in Sect. V and we summarize our work in Sect VI. Since the form factors for B → P , B → V and B → S (P , V and S represent pseudoscalar, vector and scalar mesons, respectively) weak transitions and light-cone distribution amplitudes and decay constants of P , V and S will be used in treating B decays, we first discuss them in this section.
The form factors of B to a meson weak transition can be decomposed as [7,13] where P µ = (p + p ′ ) µ , q µ = (p − p ′ ) µ ,V µ ,Â µ andŜ µ are the weak vector, axial-vector and scalar currents, . The decay constants are defined as [13] The twist-2 light-cone distribution amplitudes (LCDA) for the pseudoscalar and vector mesons are respectively [9,13] and the twist-3 ones are respectively where C The twist-2 light-cone distribution amplitude for a scalar meson reads [14,15] where B m are Gegenbauer moments,f S is the decay constant of the scalar meson, n denotes the u, d quark component of the scalar meson, n = 1 √ 2 (uū + dd), and s denotes the component ss. As for the twist-3 ones, we shall take the asymptotic forms [14,15] Φ s (x) (n,s) =f n,s S .

III. B DECAYS IN QCD FACTORIZATION
In the SM, the effective weak Hamiltonian for non-leptonic B-meson decays is given by [16] where λ Within the framework of QCD factorization [9,10], the matrix elements of the effective Hamiltonian are written in the form where T p A describes the contribution from naive factorization, vertex correction, penguin amplitude and spectator scattering expressed in terms of the parameters a p i , while T p B contains annihilation topology amplitudes characterized by the annihilation parameters b p i . The flavor parameters a p i are basically the Wilson coefficients in conjunction with short-distance nonfactorizable corrections such as vertex corrections and hard spectator interactions. In general, they have the following expressions [9]: where c ′ i are effective Wilson coefficients which are defined as being the matrix element at the tree level, the upper (lower) signs apply when i is odd (even), is leading-order coefficient, C F = (N 2 c − 1)/2N c with N c = 3, the quantities V i (M 2 ) account for one-loop vertex corrections, H i (M 1 M 2 ) describe hard spectator interactions with a hard gluon exchange between the emitted meson and the spectator quark of the B meson, and P p i (M 1 M 2 ) are from penguin contractions [9].
The expressions of the quantities N i (M 2 ) read When M 1 M 2 = V P, P V , the correction from the hard gluon exchange between M 2 and the spectator quark is given by [9,10] for i = 1 − 4, 9, 10, for i = 5, 7 and H i (M 1 M 2 ) = 0 for i = 6, 8.
In Eqs. (11)(12)(13)(14)x = 1 − x,ȳ = 1 − y, and r M i χ (i = 1, 2) are "chirally-enhanced" terms which are defined as The weak annihilation contributions to B → M 1 M 2 can be described in terms of b i and b i,EW , which have the following expressions: where the subscripts 1, 2, 3 of A i,f n (n = 1, 2, 3) stand for the annihilation amplitudes induced from , and (S − P )(S + P ) operators, respectively, the superscripts i and f refer to gluon emission from the initial-and final-state quarks, respectively. Their explicit expressions are given by [9,[13][14][15]17] When dealing with the weak annihilation contributions and the hard spectator contributions, one has to deal with the infrared endpoint singularity X = 1 0 dx/(1 − x). The treatment of this endpoint divergence is model dependent, and we follow Ref. [9] to parameterize the endpoint divergence in the annihilation and hard spectator diagrams as where Λ h is a typical scale of order 0.5 GeV, ρ M 1 M 2 A(H) is an unknown real parameter and φ M 1 M 2 A(H) is a free strong phase in the range [0, 2π] for the annihilation (hard spectator) process. In our work, we will adopt [13,18,19], but for the B → SP decays, we will follow our earlier work [20,21] to use a further assumption X M 1 M 2 = X M 2 M 1 compared with the B → P V decays.

A. Nonresonance background
In the absence of resonances, the factorizable nonresonance amplitude for the B − → π − π + π − decay has the following expression [4,22]: For the parameters a i which contain effective Wilson coefficients, we take the following values [4,22]: For the current-induced process, the amplitude π can be expressed in terms of three unknown form factors [4,22,23] A HMChPT where r and ω ± are form factors which have the expressions as [23,24] where s ij ≡ (p i + p j ) 2 , g is a heavy-flavor-independent strong coupling which can be extracted from the CLEO measurement of the D * + decay width, |g| = 0.59 ± 0.01 ± 0.07 [25], which sign is fixed to be negative in Ref. [26].
However, the predicted nonresonance results based on the HMChPT are not recovered in the soft meson region and lead to decay rates that are too large and in disagreement with experiment [27]. This issue is related to the applicability of the HMChPT, which requires the two mesons in the final state in the B → M 1 M 2 transition have to be soft and hence an exponential form of the amplitudes is necessary [22,28], where α NR = 0.081 +0.015 −0.009 GeV −2 , and the phase φ 12 of the nonresonance amplitude will be set to zero for simplicity [22,28].
In Eq. (26), S ρ is the reciprocal of the propagator of ρ and takes the form s ππlow − m 2 ρ + im ρ Γ ρ , m ρ and Γ ρ are the mass and width of the ρ meson, respectively. As for the scalar intermediate state σ in Eq.
(27), we shall adopt the Breit-Wigner function and Bugg model to deal with its propagator, respectively.
With the Breit-Wigner form, R σ (s) has the following expression: where Γ σ (s ππlow ) is the width which is a function of s ππlow and has the expression where p ′ (s ππlow ) and p ′ (m 2 σ ) are the magnitude of the c.m. momenta of π + or π − in the π + π − and σ rest frames, respectively, and In the Bugg model, the propagator of σ is given by [29][30][31] R Bugg where The parameters in Eqs. (31,32) are fixed to be M = 0.953GeV, s A = 0.14m 2 π , c 1 = 1.302GeV, c 2 = 0.340GeV −1 , A = 2.426GeV 2 and g 4π = 0.011GeV which are given in the fourth column of Table I in Ref. [29]. The parameters ρ 1,2,3 are the phase-space factors of the decay channels ππ, KK and ηη, respectively, which are defined as [29] with m 1 = m π , m 2 = m K , m 3 = m η and s = s ππlow .
Within the QCDF, we can derive the amplitude contributed from ρ resonance to the B − → π − π + π − decay and obtain for the B − → ρπ − → π + π − π − decay mode, whereŝ ππhigh is the midpoint of the allowed range of s ππhigh , i.e.ŝ ππhigh = (s ππhigh,max + s ππhigh,min )/2, with s ππhigh,max and s ππhigh,min being the maximum and minimum values of s ππhigh for fixed s ππlow .
The amplitude of the B − → σπ − decay can be expressed as If we take the Breit-Wigner (Bugg) model, the amplitude contributed from σ resonance to B − → π − π + π − decay can be expressed as Then the decay amplitudes of B − → ρ + σ BW(Bugg) → π − π + π − can be finally obtained as C. Total results for the amplitudes of B − → π − π + π − in two different models Totally, the decay amplitude for B → π − π + π − is the sum of resonance (R) contributions and the In the QCDF, both the resonance and nonresonance contributions have been considered. The decay amplitude via B − → R BW(Bugg) + N R → π − π + π − can be written as D. Localizd CP violation of B − → π − π + π − and branching ratio of B − → σπ − The differential CP asymmetry parameter can be defined as In this work, we will consider resonances and nonresonance in a certain region Ω which includes m 2 π + π − high > 15 GeV 2 and m 2 π + π − low < 0.4 GeV 2 for the B − → π − π + π − decay. By integrating the denominator and numerator of A CP in this region, we get the localized integrated CP asymmetry, which can be measured by experiments and takes the following form: The branching fraction of the B → M 1 M 2 decay has the following form: where τ B and m B are the lifetime and the mass of B meson, respectively, p c is the magnitude of the three momentum of either final state meson in the rest frame of the B meson which can be expressed as with m 1 and m 2 being the two final state mesons' masses, respectively.
The decay rate of the resonance three-body decay is given by [32] Γ via a scalar resonance, where s = s ππlow , λ is the usual triangular function λ(a, b, c) = a 2 + b 2 + c 2 − 2ab − 2bc − 2ca, and g SM 1 M 2 is the strong coupling constant which can can be determine from the measured width of the scalar resonance.
When we use the Breit-Wigner form to deal with the scalar meason, the decay rate in Eq. (44) becomes If the resonance width Γ S is narrow, the expression of the resonance decay rate can be simplified by applying the so-called narrow width approximation Noting where p = λ 1/2 (m 2 B , m 2 S , m 2 P )/(2m B ) is the magnitude of the c.m. three momentum of final state particles in the B rest frame and p ′ (m 2 S ) = λ 1/2 (m 2 S , m 2 1 , m 2 2 )/(2m S ), Eq. (46) leads to the following "factorization" relation [32]: and hence, In fact, this factorization relation works reasonably well as long as the two body decay B → SP is kinematically allowed and the resonance is narrow. However, when B → SM 3 happens at the margin of the kinematically allowed region or is even not allowed, the off resonance peak effect of the intermediate resonance state will become important and we should consider the finite width effect of S meson. Since the width of the σ resonance is very broad, there is no reason to neglect its finite width effect.
We shall follow Refs. [32,33] to define the parameter η: Then we can easily derive the branching ratio of B → SP : The deviation of η from unity will give a measure of the violation of the factorization relation in Eq.
(48). η has the following expression: Similarly, if we adopt the propagator of the σ resonance in the Bugg model, the decay rate of the resonance three-body decay can be expressed as combining Eqs. (47), (53) and Eq. (50), we can get . (54)

V. NUMERICAL RESULTS
The theoretical results obtained in the QCDF approach depend on many input parameters. The values of the Wolfenstein parameters are given asρ = 0.117 ± 0.021,η = 0.353 ± 0.013 [34].
We shall assign an error of ±0.1 to ρ M 1 M 2 and ±20 0 to φ M 1 M 2 for estimation of theoretical uncertainties.
However, for B → P S and B → SP decays, there is little experimental data so the values of ρ S and φ S are not determined well, we will adopt X P S = X SP = (1 + ρ S e iφ S ) ln m B Λ h as in our previous work [20,21]. We can get the expressions of A CP (B − → R BW + N R → π − π + π − ) and A CP (B − → R Bugg + N R → π − π + π − ) which are the functions of ρ S and φ S . Meanwhile, one can also get the expressions of B(B − → σπ − ) in these two models, which are also the functions of ρ S and φ S . By fitting the Breit-Wigner model theoretical results of A CP (B − → R BW + N R → π − π + π − ) and B(B − → σπ − ) to the experimental data A CP (B − → π − π + π − ) = 0.584 ± 0.082 ± 0.027 ± 0.007 in the aforementioned region and B(B − → σπ − ) < and Belle [40][41][42], which suggest the existence of unexpected large annihilation contributions and have attracted much attention [38,43,44]. Besides, there are many theoretical studies indicating possible unnegligible large weak annihilation contributions within different approaches in different decays [18,20,21,38,39,45,46]. Therefore, larger values of the ρ S are acceptable when dealing with the divergence problems for B → SP (P S) decays. Much more experimental and theoretical efforts are expected to understand the underlying QCD dynamics of annihilation and spectator scattering contributions.

VI. SUMMARY
In this work, we study the localized CP violation in the B − → π − π + π − decay and the branching fraction of the B − → σ(600)π − decay within the QCD factorization approach. Both the resonance and nonresonance contributions are included when we study the localized CP asymmetry for the B − → π − π + π − decay. As for the resonance contributions we consider the scalar meson σ(600) and the vector meson ρ 0 (770). Meanwhile, we adopt the Breit-Wigner and the Bugg models to parameterize the propagator of the σ(600) meson, respectively. Since the width of the σ resonance is broad, it is necessary to take into account its finite width effect and we follow Refs. [32,33] to introduce a parameter η defined in Eq.
(50) to modify the branching fraction relationship between the two-body and three-body decays of the B meson. From our calculations, we get η BW = 3.68 and η Bugg = 0.316 for the Breit-Wigner form and the Bugg model, respectively. Meanwhile, we obtain the upper limits of B(B − → σπ − ) as 1.67 × 10 −6 and 1.946 × 10 −5 in these two different models, respectively. It is worth noting that these two limits are very different, so it is important to study the σ meson distribution effects in the Breit-Wigner and the Bugg models even though the finite width effects are considered in both models. By fitting the theoretical results for A CP (B − → R + N R → π − π + π − ) and B(B − → σπ − ) in these two models to the experimental data A CP (π − π + π − ) = 0.584 ± 0.082 ± 0.027 ± 0.007 and the upper limits of B(B − → σπ − ), we find that there is no allowed ρ S and φ S to be found in the Breit-Wigner model, but there exists the region reveal that the contributions from the weak annihilation and the hard spectator scattering processes are both large for the B − → π − π + π − and the B − → σπ − decays. Especially, the contribution from the weak annihilation processes should not be neglected. In fact, the presence of the large weak annihilation and hard spectator scattering contributions has been supported by recent studies both experimentally and theoretically. So the larger values of ρ S are acceptable when dealing with the divergence problems for the B → SP (P S) decays.