Study of $CP$ Violations in $B^-\rightarrow K^- \pi^+\pi^-$ and $B^-\rightarrow K^- f_0(500)$ decays in the QCD factorization approach

Within the QCD factorization approach, we study the $CP$ violations in $B^-\rightarrow K^-\pi^+\pi^-$ and $B^-\rightarrow K^- f_0(500)$ decays. We find the experimental data of the localized $CP$ asymmetry in $B^-\rightarrow K^-\pi^+\pi^-$ decays in the region $m_{K^-\pi^+}^2<15$ $\mathrm{GeV}^2$ and $0.08<m_{\pi^+\pi^-}^2<0.66$ $\mathrm{GeV}^2$ can be explained by the interference of two intermediate resonances, $\rho^0(770)$ and $f_0(500)$ when the parameters in our interference model are in the allowed ranges, i.e. the relative strong phase $\delta\in[0, 1.745]\cup[3.578, 6.266]$ and the end-point divergence parameters $\rho_S\in[2.790, 5.290]$ and $\phi_S \in [1.518, 5.183]$. With the obtained allowed ranges for $\rho_S$ and $\phi_S$, we obtain the predictions for the $CP$ asymmetry parameter $A_{CP} \in [-0.259, 0.006]$ and the branching fraction $\mathcal{B} \in [0.585, 3.230]\times10^{-5}$ for $B^-\rightarrow K^-f_0(500)$ decay modes.


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, prompted by a large number of experimental measurements, three-body hadronic B meson decays have been studied by using different theoretical frameworks [4][5][6][7][8]. Strong dynamics contained in three-body hadronic B meson decays is much more complicated than that in two-body cases, e.g. how to factorize B to three-body final states matrix elements. Both BABAR [9] and Belle [10] Collaborations claimed evidence of partial rate asymmetries in the channels B ± → ρ 0 (770)K ± in the Dalitz plot analysis of B − → K − π + π − . LHCb also observed the large CP asymmetry in the localized region of the phase space [11,12], A CP (K − π + π − ) = 0.678 ± 0.078 ± 0.0323 ± 0.007, for m 2 K − π + < 15 GeV 2 and 0.08 < m 2 π + π − < 0.66 GeV 2 , which spans the π + π − masses around the ρ 0 (770) resonance. Such threebody decays in this region have been studied in Refs. [13,14] using a simple model based on the framework of the factorization approach. The authors of Refs [15,16] considered the possibility of obtaining a large local CP violation in B − → π + π − π − decay from the interference of the resonances ρ 0 (770) and f 0 (500).
In this work, we will apply this mechanism to study CP violation in B − → K − π + π − decay with the interference of ρ 0 (770) and f 0 (500) in the region of m 2 K − π + < 15 GeV 2 and 0.08 < m 2 π + π − < 0.66 GeV 2 . In contrast to vector and tensor mesons, the identification of scalar mesons is a long-standing puzzle, because some of them have large decay widths which cause strong overlaps between resonances and backgrounds in experiments [17]. Up to now, there have been some progresses in the study of charmless hadronic B decays with scalar mesons in the final states both experimentally and theoretically. On the experimental side, measurements of B decays to the scalar mesons such as f 0 (980), f 0 (1370), f 0 (1500), f 0 (1710), a 0 (980), a 0 (1450), and K * 0 (1430) have been reported by BABAR and Belle Collaborations, but the decays to f 0 (500) have not been reported and the CP violation and the branching fractions have not been measured for such processes. So it is important to predict the values of A CP (B − → K − f 0 (500)) and B(B − → K − f 0 (500)). Although the light scalar mesons are widely perceived as primarily the 4-quark bound states, in practice it is difficult to make quantitative predictions based on the 4-quark picture for the light scalar mesons, hence, predictions are made in the 2-quark model for the f 0 (500) meson [18].
Theoretically, to calculate the hadronic matrix elements of B nonleptonic weak decays, some approaches, including the naive factorization [19,20], the QCD factorization (QCDF) [21][22][23], the pertur-bative QCD (PQCD) approach [24][25][26], and the soft-collinear effective theory (SCET) [27,28], have been fully developed and extensively employed in recent years. In this work, within the framework of QCDF [29,30], we will study the decays of B − → K − π + π − via the interference of ρ 0 (770) and f 0 (500) and The remainder of this paper is organized as follows. In Sect. II, we briefly present the formalism of the QCD factorization approach. In Sect. III, we present the formalisms for CP violation of B − → K − f 0 (500) and B − → K − π + π − . The numerical results are given in Sect. IV and we summarize our work in Sect V.

II. QCD FACTORIZATION
With the operator product expansion, the effective weak Hamiltonian for B meson decays can be written as [31] where G F represents the Fermi constant, λ (D) p = V pb V * pD , V pb and V pD are the CKM matrix elements, C i (i = 1, 2, · · · , 10) are the Wilson coefficients, Q p 1,2 are the tree level operators and Q 3−10 are the penguin ones, and Q 7γ and Q 8g are the electromagnetic and chromomagnetic dipole operators, respectively. The explicit forms of the operators Q i are [32] where α and β are color indices, q ′ = u, d, s, c or b quarks.
In dealing with the charmless B decay into two mesons M 1 and M 2 , the decay amplitude is usually divided into the emission part and the annihilation part in terms of the structures of the topological diagrams. In the heavy quark limit, the former part can be written as the product of the decay constant and the form factor, while for the latter part, it is always regarded as being power suppressed. With the standard procedure of the QCDF, the emission part of the decay amplitude has the following form: where α p i (µ) are flavour parameters which can be expressed in terms of the effective parameters a p i , which can be calculated perturbatively, with the expressions given by [31] 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), 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 [31]. Similarly, weak annihilation contributions are described by the terms 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, and the superscripts i and f refer to gluon emission from the initial-and final-state quarks, respectively. The explicit expressions for A i,f n can be found in Ref. [31]. When dealing with the weak annihilation contributions and the hard spectator contributions, one suffers from the infrared endpoint singularity X = 1 0 dx/(1 − x). The treatment of the endpoint divergence is model dependent, and we follow Ref. [22] to parameterize the endpoint divergence in the annihilation diagrams as where Λ h is a typical scale of order 500 MeV, ρ is an unknown real parameter, φ is the free strong phase in the range [0, 2π]. The QCDF approach itself cannot give information or constraints on the phenomenological parameters ρ and φ, both of them should be fixed by experimental data such as branching fractions and CP asymmetries.
C. Calculation of differential CP violation and branching fraction of B − → K − f 0 (500) Using Eq. (11), the differential CP asymmetry parameter of B → M 1 M 2 can be expressed as The branching fraction of B → M 1 M 2 decay has the following form: where τ B is the lifetime of B meson, m B is the mass of B meson, |p c | is the norm of a hadron's three momentum in the final state which can be expressed as where m M 1 and m M 2 are the two final state mesons' masses, respectively.
In general, the twist-2 LCDA of a scalar meson, Φ S , has the following form [18] : wheref S are the decay constants of the scalar meson S, B m are Gegenbauer moments. Based on the QCD sum rule methods [35,36], we can derive the decay constantsf q f 0 (500) (q = u, s) with the f 0 (980)- As for the twist-3 distribution amplitudes, we use [18] For the form factors of mesons, we neglect corrections quadratic in the light meson masses and we adopt the values at q 2 = 0 in Ref. [31] (At this kinematic point, the form factors F + and F 0 coincide.), [34]. Since most of the models indicate that the B meson to a light meson form factor at zero recoil q 2 lies around 0.3, we simply set F B→f 0 (m 2 K ) ≈ F B→f 0 (0) = 0.3 and assign its uncertainty to be δF BS (0) = ±0.03 [18]. The decay constants used in our calculations are f K = 0.156 ± 0.7GeV [17], f ρ = 0.216 ± 0.003GeV, and f B = 0.21 ± 0.02GeV [34].
A general fit of ρ and φ to the B → V P and B → P V data indicates X P V = X V P , i.e. ρ P V ≈ 0.87, ρ V P ≈ 1.07, φ V P ≈ −30 0 and φ P V ≈ −70 0 , we shall assign an error of ±0.1 to ρ P V (V P ) and ±20 0 to φ P V (V P ) for the estimation of theoretical uncertainties [34]. On the other hand, 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 very well, to make an estimation about A CP (B − → K − f 0 (500)) and B(B − → K − f 0 (500)), we adopt X P S = Λ h . With all the above considerations, we only have three free parameters, which are the relative strong phase δ, and the divergence parameters ρ S and φ S for A CP (B − → K − ρ 0 (770)(f 0 (500)) → K − π + π − ).

V. SUMMARY
In this work, within the QCD factorization approach, we study the localized CP violation in B − → K − π + π − decays in the region m 2 K − π + < 15 GeV 2 and 0.08 < m 2 π + π − < 0.66 GeV 2 by including the interference of ρ 0 (770) and f 0 (500). By fitting the experimental data of A CP (B − → K − π + π − ) in this of ρ S indicate that the weak annihilation and the hard spectator scattering processes can make large contributions and we should take more efforts to investigate these contributions in B nonleptonic weak 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 − f 0 (500) decay modes. We find A CP (B − → K − f 0 (500)) ∈ [−0.259, 0.006] and B(B − → K − f 0 (500)) ∈ [0.585, 3.230]×10 −5 in the allowed ranges of φ S and ρ S . These predictions can hopefully be tested in future experiments. In our analysis, the uncertainties coming from the CKM matrix elements, form factors, decay constants, s quark masses and Gegenbauer moments are all considered.