Study of $CP$ Violation in $B^-\rightarrow K^- \pi^+\pi^-$ and $B^-\rightarrow K^- \sigma(600)$ decays in the QCD factorization approach

In this work, we study the localized $CP$ violation in $B^-\rightarrow K^-\pi^+\pi^-$ and $B^-\rightarrow K^- \sigma(600)$ decays by employing the quasi two-body QCD factorization approach. Both the resonance and the nonresonance contributions are studied for the $B^-\rightarrow K^-\pi^+\pi^-$ decay. The resonance contributions include those not only from $[\pi\pi]$ channels including $\sigma(600)$, $\rho^0(770)$ and $\omega(782)$ but also from $[K\pi]$ channels including $K^*(892)$, $K_0^*(1430)$, $K^*(1410)$, $K^*(1680)$ and $K_2^*(1430)$. By fitting the experimental data $\mathcal{A_{CP}}(K^-\pi^+\pi^-)=0.678\pm0.078\pm0.0323\pm0.007$ for $m_{K^-\pi^+}^2<15$ $\mathrm{GeV}^2$ and $0.08<m_{\pi^+\pi^-}^2<0.66$ $\mathrm{GeV}^2$, we get the end-point divergence parameters in our model, $\phi_S \in [4.75, 5.95]$ and $\rho_S\in[4.2, 8]$. Using these results for $\rho_S$ and $\phi_S$, we predict that the $CP$ asymmetry parameter $\mathcal{A_{CP}} \in [-0.094, -0.034]$ and the branching fraction $\mathcal{B} \in [1.82, 20.0]\times10^{-5}$ for the $B^-\rightarrow K^-\sigma(600)$ decay. In addition, we also analyse contributions to the localized $CP$ asymmetry $\mathcal{A_{CP}}(B^-\rightarrow K^-\pi^+\pi^-)$ from $[\pi\pi]$, $[K\pi]$ channel resonances and nonresonance individually, which are found to be $\mathcal{A_{CP}}(B^-\rightarrow K^-[\pi^+\pi^-] \rightarrow K^-\pi^+\pi^-)=0.585\pm0.045$, $\mathcal{A_{CP}}(B^-\rightarrow [K^-\pi^+] \pi \rightarrow K^-\pi^+\pi^-)=0.086\pm0.021$ and $\mathcal{A_{CP}}^{NR}(B^-\rightarrow K^-\pi^+\pi^-)=0.061\pm0.0042$, respectively. Comparing these results, we can see that the localized $CP$ asymmetry in the $B^-\rightarrow K^-\pi^+\pi^-$ decay is mainly induced by the $[\pi\pi]$ channel resonances while contributions from the $[K\pi]$ channel resonances and nonresonance are both very small.


I. INTRODUCTION
numerically important for realistic hadronic B decays, particularly for pure annihilation processes and direct CP asymmetries. Unfortunately, in the collinear factorization approximation, the calculation of annihilation corrections always suffers from end-point divergence. In the pQCD approach, such divergence is regulated by introducing the parton transverse momentum k T and the Sudakov factor at the expense of modeling the additional k T dependence of meson wave functions, and large complex annihilation corrections are presented [14]. In the SCET approach, such divergence is removed by separating the physics at different momentum scales and using zero-bin subtraction to avoid double counting the soft degrees of freedom [15,16]. In the QCDF approach, such divergence is usually parameterized in a modelindependent manner [10,11] and will be explicitly expressed in Sect. III.
There are many experimental studies which have been successfully carried out at B factories (BABAR and Belle), Tevatron (CDF and D0) and LHCb and are being continued at LHCb and Belle experiments.
These experiments provide highly fertile ground for theoretical studies and have yielded many exciting and important results, such as measurements of pure annihilation B s → ππ and B d → KK decays reported recently by CDF, LHCb and Belle [17][18][19], which may suggest the existence of unexpected large annihilation contributions and have attracted much attention [20][21][22]. So it is also important to consider the annihilation contributions to B decays.
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 present detailed calculations of CP violation for B − → K − π + π − and B − → K − σ(600) decays. The numerical results are given in Sect. V and we summarize our work in Sect. VI.

II. FORM FACTORS, DECAY CONSTANTS AND LIGHT-CONE DISTRIBUTION AMPLI-TUDES
Since the form factors for B → P , B → V , B → S and B → T (P , V , S and T represent pseudoscalar, vector, scalar and tensor mesons, respectively) weak transitions and light-cone distribution amplitudes and decay constants of P , V , S and T will be used in treating B decays, we first discuss them in this section.
The twist-2 light-cone distribution amplitude for a scalar meson reads [25,26] where B m are Gegenbauer moments,f S is the decay constant of the scalar mesons, n denotes the u, d quark component of the scalar meson, n = 1 √ 2 (uū + dd), and s denotes the components ss. As for the twist-3 ones, we shall take the asymptotic forms [25,26] 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 [27] where λ (D) p = V pb V * pD , V pb and V pD are the CKM matrix elements, G F represents the Fermi constant, c i (i = 1 − 10, 7γ, 8g) are Wilson coefficients, O p 1,2 are the tree level operators, O 3−6 are the QCD penguin operators, O 7−10 arise from electroweak penguin diagrams, and O 7γ andO 8g are the electromagnetic and chromomagnetic dipole operators, respectively.
Within the framework of QCD factorization [10,11], the effective Hamiltonian matrix elements 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 expressions [10] 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 [10].
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 [10,11] for i = 1 − 4, 9, 10, for i = 5, 7 and H i (M 1 M 2 ) = 0 for i = 6, 8.
In Eqs. (12)(13)(14)(15)(16)(17)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 [10,[24][25][26]28] , for M 1 M 2 = T P, P T, 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. [10] to parameterize this 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 follow the assumption X M 1 M 2 [24,29,30], but for the B → SP decays, we will further assume that X M 1 M 2 = X M 2 M 1 compared with the B → P V (P T ) decays.

IV. CALCULATION OF CP VIOLATION
A. FRAMEWORK

Nonresonance background
In the absence of resonances, the factorizable nonresonance amplitude for the B − → K − π + π − decay has the expression [6,31] For the parameters a i which contain effective Wilson coefficients, we take the following values [6,31]: For the current-induced process, the amplitude π can be expressed in terms of three unknown form factors [6,31,32] A HMChPT where r, ω ± , and h are form factors which can be evaluated in the framework of HMChPT and the results read [32,33] 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 [34], which sign is fixed to be negative in Ref. [3].
However, the predicted nonresonance results based on HMChPT are not recovered in the soft meson region and lead to decay rates that are too large which are in disagreement with experiment [35]. For example, the branching fraction is found to be of order 7.5 × 10 −5 , which is one order of magnitude larger than the BaBar result, 5.3 × 10 −6 [36]. The issue is related to the applicability 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 [31,37], where α NR is constrained from the tree dominated decay B − → π + π − π − to be α NR = 0.081 +0.015 −0.009 GeV −2 , and the phase φ 12 of the nonresonant amplitude will be set to zero for simplicity [31,37].
K − π + |sd|0 NR = K + K − |ss|0 NR , we shall adopt Ref. [6] to assume that final state interactions amount to giving a large strong phase δ to the nonresonance component of the matrix element of K − π + |sd|0 NR and a fit to the data of direct CP asymmetries in where the parameter σ NR = (3.39 +0.18 −0.21 )e iπ/4 GeV, and ν = ms−m d characterizes the quarkoperator parameter qq which spontaneously breaks the chiral symmetry and the experimental measurement leads to α = (0.14±0.02)GeV −2 [38]. Motivated by the asymptotic constraints from pQCD, namely, [39], the nonresonance form factors in Eq. (27) can be parameterized as [6] F N R (s 23 whereΛ ≈ 0.3 GeV is the QCD scale parameter, the unknown parameters x i and x ′ i are fitted from the kaon electromagentic data, giving the following best-fit values [40]:
ρ − ω mixing has the dual advantages that the strong phase difference is large and well known [42,43].
Because of its large width, σ can not be modeled by a naive Breit-Wigner distribution. In this paper, we will adopt the Bugg model to parameterize the distribution of σ which is given by [49][50][51] where The parameters in Eqs. (32,33) are fixed to be M = 0.953GeV, s A = 0.14m 2 π , c 1 = 1.302GeV 2 , c 2 = 0.340, A = 2.426GeV 2 and g 4π = 0.011GeV, which are given in the fourth column of Table I in Ref. [49]. The parameters ρ 1,2,3 are the phase-space factors of the decay channels ππ, KK and ηη, respectively, which are defined as [49] with m 1 = m π , m 2 = m K and m 3 = m η . Other resonants in Eq. (30) will be modeled by the naive Breit-Wigner distribution.

Localizd CP violation
Totally, the decay amplitude for B → K − π + π − is the sum of resonant (R) contributions and the

nonresonant (NR) background [6]
The differential CP asymmetry parameter can be defined as In this work, we will consider eight resonances in a certain phase region Ω which includes m 2 K − π + < 15 GeV 2 and 0.08 < m 2 π + π − < 0.66 GeV 2 for the B − → K − 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: A Ω CP = Ω ds 12 ds 13 (|A| 2 − |Ā| 2 ) Ω ds 12 ds 13 (|A| 2 + |Ā| 2 ) .
B. Calculation of differential CP violation and branching fraction of B − → K − σ decay Using Eq. (53), the differential CP asymmetry parameter of B → M 1 M 2 can be expressed as 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 the 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 M 1 and m M 2 being the two final state mesons' masses, respectively.
The amplitude of B − → K − σ has the following form:

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 [53]. The effective Wilson coefficients used in our calculations are taken from Ref. [54]: The strong coupling constants are determined from the measured widths through the relations [6,41,55] g S→M ′ where p c (S, V, T ) are the magnitudes of the three momenta of the final state meson in the rest frame of S, V , and T mesons, respectively.
However, for the 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, to make an estimation about A CP (B − → K − σ) and B(B − → K − σ), we will adopt X P S = X SP = (1 + ρ S e iφ S ) ln m B Λ h as described in Sect. III. Now we are left with only two free parameters with all the above considerations, which are the divergence parameters ρ S and φ S for . By fitting the theoretical result to the experimental data A CP (B − → K − π + π − ) = 0.678 ± 0.078 ± 0.0323 ± 0.007 in the region m 2 K − π + < 15 GeV 2 and 0.08 < m 2 π + π − < 0.66 GeV 2 , and varying φ S and ρ S by 0.
It is noted that the range of ρ S ∈ [4. 2,8] is larger than the previously conservative choice of ρ ≤ 1 [10,11]. Since the QCDF itself cannot give information about the parameters ρ and φ, there is no reason to restrict ρ to the range ρ ≤ 1 [22,29,59,60]. In the pQCD approach, the possible un-negligible large weak annihilation contributions were noticed first in Refs. [14,61]. In fact, there are many experimental studies which have been successfully carried out at B factories (BABAR and Belle), Tevatron (CDF and D0) and LHCb in the past and will be continued at LHCb and Belle experiments. These experiments provide highly fertile ground for theoretical studies and have yielded many exciting and important results, such as measurements of pure annihilation B s → ππ and B d → KK decays reported recently by CDF, LHCb and Belle [17][18][19], which suggest the existence of unexpected large annihilation contributions and have attracted much attention [20][21][22]. Thus larger values of ρ S are acceptable when dealing with the divergence problems for B → SP (P S) decays. With the large values of ρ S , it is certain that both the weak annihilation and the hard spectator scattering processes can make large contributions to B − → K − σ decays. Much more experimental and theoretical efforts are expected to understand the underlying QCD dynamics of annihilation and spectator scattering contributions. In the obtained allowed ranges for ρ S and K − π + < 15 GeV 2 and 0.08 < m 2 π + π − < 0.66 GeV 2 . Inserting Eqs. (49) and (50) into Eq. (54) respectively, the results are A CP (B − → [K − π + ]π − → K − π + π − ) = 0.086 ± 0.021 and A CP (B − → K − [π + π − ] → K − π + π − ) = 0.585 ± 0.045. Comparing these two results, we can see the contribution from the [Kπ] resonances are much smaller than that from the [ππ] resonances. This is because B − → [K − π + ]π decays are mediated by the b → s loop (penguin) transition without the b → u tree component as shown in Eqs. (43,(45)(46)(47)(48) and also because the resonance regions of [Kπ] channel mesons have smaller widths and are further away from [ππ] channel mesons (ρ, ω and σ). Therefore, the contributions from the [Kπ] channel resonances are much smaller than that from [ππ] channel resonances. Furthermore, using Eqs. (22)(23)(24)(25)(26)(27)(28)(29) and Eq. (54), we also get that the nonresonance contribution as A CP N R (B − → K − π + π − ) = 0.061 ± 0.0042 which is also much smaller than that from the [ππ] resonances in our studied region m 2 K − π + < 15 GeV 2 and 0.08 < m 2 π + π − < 0.66 GeV 2 . Since both A CP (B − → [K − π + ]π − → K − π + π − ) and A CP N R (B − → K − π + π − ) are much smaller than A CP (B − → K − [π + π − ] → K − π + π − ). We conclude that the large localized CP asymmetry A CP (B − → K − π + π − ) = 0.678 ± 0.078 ± 0.0323 ± 0.007 is mainly induced by the contributions from the [ππ] channel resonances.

VI. SUMMARY
In this work, within a quasi two-body QCD factorization approach, we study the localized integrated CP violation in the B − → K − π + π − decay in the region m 2 K − π + < 15 GeV 2 and 0.08 < m 2 π + π − < 0.66 GeV 2 by including the contributions from both resonances including σ(600), ρ 0 (770) and ω(782) mesons from [ππ] channel and K * (892), K * (1410), K * 0 (1430), K * (1680) and K * 2 (1430) mesons from [Kπ] channel. By fitting the experimental data A CP (B − → K − π + π − ) = 0.678 ± 0.078 ± 0.0323 ± 0.007 in above experimental region, it is found that there exist ranges of parameters ρ S and φ S which satisfy the above experimental data. Thus, the resonance and nonresonance contributions can indeed induce the data for the localized CP asymmetry in the B − → K − π + π − decay. The allowed ranges for φ S and ρ S are φ S ∈ [4.75, 5.95] and ρ S ∈ [4. 2,8] is larger than the previously conservative choice of ρ ≤ 1. In fact, there is no reason to restrict ρ to the range ρ ≤ 1 because the QCDF itself cannot give information and constraint on the parameter ρ and it can only be obtained through the experimental data. Large values of ρ S reveal that the contributions from the weak annihilation and the hard spectator scattering processes are both large for the B − → K − π + π − decay. Especially, the contribution from the weak annihilation part should not be neglected. In fact, the large weak annihilation contributions have been observed and predicted in experimental and theoretical studies. So the larger values of ρ S are acceptable when dealing with the divergence problems for the B → SP (P S) decays. With the obtained allowed ranges for ρ S and φ S , we predict the CP asymmetry parameter and the branching fraction for B − → K − σ. The results are A CP (B − → K − σ) ∈ [−0.094, −0.034] and B(B − → K − σ) ∈ [1.82, 20.0] × 10 −5 when ρ S and φ S vary in their allowed ranges, respectively. In addition, we also calculate the localized CP asymmetry A CP (B − → K − π + π − ) only considering the [ππ], [Kπ] resonances and nonresonance, respectively, in the same region m 2 K − π + < 15 GeV 2 and 0.08 < m 2 π + π − < 0.66 GeV 2 . The results are A CP (B − → [K − π + ]π → K − π + π − ) = 0.086 ± 0.021, A CP (B − → K − [π + π − ] → K − π + π − ) = 0.585 ± 0.045 and A CP N R (B − → K − π + π − ) = 0.061 ± 0.0042, respectively. Therefore, the large localized CP asymmetry in the B − → K − π + π − is mainly induced by the contributions from the [ππ] channel resonances and the contributions from [Kπ] channel resonances and nonresonance are very small.