Studying the localized $CP$ violation and the branching fraction of the $\bar{B}^0\rightarrow K^-\pi^+\pi^+\pi^-$ decay

In this work, we study the localized $CP$ violation and the branching fraction of the four-body decay $\bar{B}^0\rightarrow K^-\pi^+\pi^-\pi^+$ by employing a quasi-two-body QCD factorization approach. Considering the interference of $\bar{B}^0\rightarrow \bar{K}_0^*(700)\rho^0(770)\rightarrow K^-\pi^+\pi^-\pi^+$ and $\bar{B}^0\rightarrow \bar{K}^*(892)f_0(500)\rightarrow K^-\pi^+\pi^-\pi^+$ channels, we predict $\mathcal{A_{CP}}(\bar{B}^0\rightarrow K^-\pi^+\pi^-\pi^+)\in[0.15,0.28]$ and $\mathcal{B}(\bar{B}^0\rightarrow K^-\pi^+\pi^-\pi^+)\in[1.73,5.10]\times10^{-7}$, respectively, which shows that this two channels' interference mechanism can induce the localized $CP$ violation to this four-body decay. Meanwhile, within the two quark model framework for the scalar mesons $f_0(500)$ and $\bar{K}_0^*(700)$, we calculate the direct CP violations and branching fractions of the $\bar{B}^0\rightarrow \bar{K}_0^*(700)\rho^0(770)$ and $\bar{B}^0\rightarrow \bar{K}^*(892)f_0(500)$ decays, respectively. The corresponding results are $\mathcal{A_{CP}}(\bar{B}^0\rightarrow \bar{K}_0^*(700)\rho^0(770)) \in [0.20, 0.36]$, $\mathcal{A_{CP}}(\bar{B}^0\rightarrow \bar{K}^*(892)f_0(500))\in [0.08, 0.12]$, $\mathcal{B} (\bar{B}^0\rightarrow \bar{K}_0^*(700)\rho^0(770)\in [6.76, 18.93]\times10^{-8}$ and $\mathcal{B} (\bar{B}^0\rightarrow \bar{K}^*(892)f_0(500))\in [2.66, 4.80]\times10^{-6}$, respectively, indicating the $CP$ violations of these two two-body decays are both positive and the branching fractions are quite different. These studies provide a new way to investigate the aforementioned four-body decay and could be helpful in clarifying the configuration of the structure of light scalar meson.


I. INTRODUCTION
Charge-Parity (CP ) violation is one of the most fundamental and important properties of the weak interaction. Nonleptonic decays of hadrons containing a heavy quark play an important role in testing the Standard Model (SM) picture for the CP violation mechanism in flavor physics, improving our understanding of nonperturbative and perturbative QCD and exploring new physics beyond the 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 [1,2]. Besides the weak phase, a large strong phase is usually also needed for a large CP violation. Generally, this strong phase is provided by QCD loop corrections and some phenomenological models.
Recently, more attentions have been focused on the studies of the two-or three-body heavy meson decays both theoretically and experimentally [3][4][5][6][7][8][9][10][11][12], while for the four-body nonleptonic decays of these heavy mesons there are limited studies [13][14][15]. Because of the complicated phase spaces and relatively smaller branching fractions, four-body decays of heavy mesons are hard to be investigated. However, in the aspect of studying the intermediate resonances, four-body decays of heavy mesons can provide rich information, especially for the unclear compositions of scalar mesons like f 0 (500) (σ), K * (700) (κ), a 0 (980) and f 0 (980). Up to now, the descriptions of the inner structures for the light scalar states are still unclear and even controversial, which could be, for example, qq,qqqq, meson-meson bound states or even those supplemented with a scalar glueball. Studying four-body decays of heavy mesons in addition to two-or three-body decays can provide useful information for clarifying configurations of light scalar mesons. In fact, with the great development of the large hadron collider beauty (LHCb) and Belle-II experiments, more and more four-body decay modes involving one or two scalar states in the B and D meson decays are expected to be measured with good precision in the future.
As mentioned above, four-body meson decays are generally dominated by intermediate resonances, which means that they proceed through quasi-two-body or quasi-three-body decays. In our work, we will adopt the quasi-two-body decay mechanism to study the four-body decayB 0 → K − π + π − π + , i.e.
where the light scalars f 0 (500) and K * (700) will be considered as lowest-lying and first excisted qq states [16], respectively. We can then explore whether the localized CP violation of the four-body decayB 0 → K − π + π − π + can be induced by these two channels' interference.
Theoretically, to calculate the hadronic matrix elements of B or D weak decays, some approaches, such as QCD factorization (QCDF) [6,17], the perturbative QCD(pQCD) [18] and the soft-collinear effective theory(SCET) [19], have been fully developed and extensively employed in recent years. Unfortunately, in the collinear factorization approximation, the calculation of annihilation corrections always suffers from the end-point divergence. In the QCDF approach, such divergence is usually parameterized in a model-independent manner [6,17] and will be explicitly expressed in Sect. II.
The remainder of this paper is organized as follows. In Sect. II, we present our theoretical framework.
The numerical results are given in Sect. III and we summarize our work in Sect IV. Appendix A recapitulates explicit expressions of hard spectator-scattering and weak annihilation amplitudes. The factorizable amplitudes of two-body decays are summarized in Appendix B. Related theoretical parameters are listed in Appendix C.
(i) the invariant mass squared of the Kπ system s Kπ = (p 1 + p 2 ) 2 = m 2 Kπ ; (ii) the invariant mass squared of the ππ system s ππ = (p 3 + p 4 ) 2 = m 2 ππ ; (iii) θ π is the angle of the π + in the π − π + center-of-mass frame Σ ππ with respect to the πs' line of flight in theB 0 rest frame ΣB0; (iv) θ K is the angle of the K − in the Kπ center-of-mass system Σ Kπ with respect to the Kπ line of flight in ΣB0; (v) φ is the angle between the Kπ and ππ planes.

B. B decay in QCD factorization
The effective weak Hamiltonian for nonleptonic B weak decays is [6] where G F represents the Fermi constant, λ With the effective Hamiltonian in Eq. (7), the QCDF method has been fully developed and extensively employed to calculate the hadronic two-body B decays. The spectator scattering and annihilation amplitudes are expressed with the convolution of scattering functions and the light-cone wave functions of the participating mesons [6]. The explicit expressions for the basic building blocks of the spectator scattering and annihilation amplitudes have been given in Ref. [6], which are also listed in Appendix A for convenience. The annihilation contributions A i,f n (n = 1, 2, 3) can be simplified to [22]: for for M 1 M 2 = SV , where the superscripts i and f refer to gluon emission from the initial and final state quarks, respectively. The model-dependent parameter X A is used to estimate the end point contributions, and expressed as with Λ h being a typical scale of order 500 MeV, ρ A an unknown real parameter and φ A the free strong phase in the range [0, 2π]. For the spectator scattering contributions, the calculation of twist-3 distribution amplitudes also suffers from the end point divergence, which is usually dealt with in the same manner as in Eq. (9) and labeled by X H . In our work, when dealing with the end-point divergences from the hard spectator scattering and weak annihilation contributions, we will follow the assumption X H = X A for the B two-body decays [16]. Moreover, a quantity λ B is introduced to parametrize the integral over the B meson distribution amplitude through [6] 1 C. Four-body decay amplitudes and localized CP violation For theB 0 → K − π + π − π + decay, we consider the contributions fromB 0 →K * 0 (700)ρ 0 (770) → K − π + π − π + andB 0 →K * (892)f 0 (500) → K − π + π − π + channels. For convenience, f 0 (500), ρ 0 (770), K * 0 (700) andK * (892) mesons will be denoted as σ, ρ,κ andK * , respectively. The amplitudes of these two channels are and respectively, where H ρπ + π − , H σπ + π − , Hκ K − π + and HK * K − π + are strong Hamiltonians for ρ → π − π + , σ → π − π + ,κ → K − π + andK * → K − π + decays, respectively. Sκ, S ρ , SK * and S σ are the reciprocal of the propagators of the corresponding mesons. We shall adopt the Breit-Wigner function and the Bugg model [23,24] to deal with the distributions of the first three mesons (κ, ρ andK * ) and σ meson, respectively.
This leads to the branching fraction where ΓB0 is the decay width of theB 0 meson.

III. NUMERICAL RESULTS
Within the QCDF approach, we get the amplitudes of the two-body decaysB 0 →κρ andB 0 →K * σ, where the light scalar σ andκ mesons are considered as the lowest-lying and first excisted qq states [16], respectively. As for the parameters for the end-point divergences, we take ρ H(A) ≤ 0.5 and arbitrary strong phases φ A(H) . All the form factors are evaluated at q 2 = 0 due to the smallness of m 2 ρ , m 2 κ , m 2 σ and m 2K * compared with m 2B 0 . We also simply set FB 0 →κ (0) = 0.3 and assign its uncertainty to be ±0.1. With the given parameters, we obtain the CP violations and branching fractions of thē .80] × 10 −6 , respectively. Obviously, the CP violations of these two-body decays are both positive, with the CP violation inB 0 →K * σ decay being smaller than that inB 0 →κρ.
The following related decay constants (in GeV) are used [16,27]: As for the form factors, we use [16,27]: