Phenomenological studies on the $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^+\pi^-$ decay

Within the quasi-two-body decay model, we study the localized $CP$ violation and branching fraction of the four-body decay $\bar{B}^0\rightarrow [K^-\pi^+]_{S/V}[\pi^+\pi^-]_{V/S} \rightarrow K^-\pi^+\pi^-\pi^+$ when $K^-\pi^+$ and $\pi^-\pi^+$ pair invariant masses are $0.35<m_{K^-\pi^+}<2.04 \, \mathrm{GeV}$ and $0<m_{\pi^-\pi^+}<1.06\, \mathrm{GeV}$, with the pairs being dominated by the $\bar{K}^*_0(700)^0$, $\bar{K}^*(892)^0$, $\bar{K}^*(1410)^0$, $\bar{K}^*_0(1430)$ and $\bar{K}^*(1680)^0$, and $f_0(500)$, $\rho^0(770)$ , $\omega(782)$ and $f_0(980)$ resonances, respectively. When dealing with the dynamical functions of these resonances, $f_0(500)$, $\rho^0(770)$, $f_0(980)$ and $\bar{K}^*_0(1430)$ are modeled with the Bugg model, Gounaris-Sakurai function, Flatt$\acute{\mathrm{e}}$ formalism and LASS lineshape, respectively, while others are described by the relativistic Breit-Wigner function. Adopting the end point divergence parameters $\rho_A\in[0,0.5]$ and $\phi_A\in[0,2\pi]$, our predicted results are $\mathcal{A_{CP}}(\bar{B}^0\rightarrow K^-\pi^+\pi^+\pi^-)\in[-0.383,0.421]$ and $\mathcal{B}(\bar{B}^0\rightarrow K^-\pi^+\pi^+\pi^-)\in[7.36,199.69]\times10^{-8}$ based on the hypothetical $q\bar{q}$ structures for the scalar mesons in the QCD factorization approach. Meanwhile, we calculate the $CP$ violating asymmetries and branching fractions of the two-body decays $\bar{B}^0\rightarrow SV(VS)$ and all the individual four-body decays $\bar{B}^0\rightarrow SV(VS) \rightarrow K^-\pi^+\pi^-\pi^+$, respectively. Our theoretical results for the two-body decays $\bar{B}^0\rightarrow \bar{K}^*(892)^0$$f_0(980)$, $\bar{B}^0\rightarrow \bar{K}^*_0(1430)^0$$\omega(782)$, $\bar{B}^0\rightarrow \bar{K}^*(892)^0f_0(980)$, $\bar{B}^0\rightarrow\bar{K}^*_0(1430)^0\rho$,


I. INTRODUCTION
Differences in the behaviour of matter and antimatter (CP violation) have been observed in several processes and, in particular, in charmless B decays. The current understanding of the composition of matter in the Universe indicates that other mechanisms, beyond that proposed within the Standard Model (SM) of particle physics, could exist in order to account for the observed imbalance between the matter and antimatter. The study of CP violating processes may therefore be used to test the corresponding SM predictions and place constraints on extensions of this framework. 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 also needed for direct CP violation in decay processes. Usually, this large phase is provided by the short-distance or long-distance QCD interactions. The short-distance interactions are caused by QCD loop corrections, and the long-distance interaction which is more sensitive to the structure of the final states can be obtained through some phenomenological mechanisms.
Four-body decays of heavy mesons are hard to be investigated because of their complicated phase spaces and relatively smaller branching fractions. This leads to much less research in four-body dacays than that in two-or three-body decays [3][4][5][6][7][8][9][10][11][12][13]. We have discussed localized CP violation and the branching fraction of the four-body decaysB 0 → K − π + π − π + in Ref. [14]. We have focused on the ππ and Kπ invariant masses are near the masses of f 0 (500) and ρ 0 (770) mesons. Here we will further expand the area of our research to predict the study the CP violation and the branching fraction in theB 0 four-body decays, which include more contributions from more different resonances. Specifically, the invariant mass of the K − π + pair lies in the range 0.35 < m K − π + < 2.04 GeV which is dominated by theK * 0 (700) 0 , K * (892) 0 ,K * (1410) 0 ,K * 0 (1430) andK * (1680) 0 resonances, and that of the π − π + pair is in the range 0 < m π − π + < 1.06 GeV which includes the f 0 (500), ρ 0 (770) , ω(782) and f 0 (980) resonances. Meanwhile, studying the multibody decays can provide rich information for their intermediate resonances especially for the unclear compositions of scalar mesons. It is known that the identification of scalar mesons is difficult experimentally and the underlying structures of scalar mesons are not well established [15]. The investigation of their structures can improve our understanding about QCD and the quark confinement mechanism. The first charmless B decay into a scalar meson that has been observed is B → f 0 (980)K. It was first measured by Belle in the charged B decays to K ± π + π ± and a large branching fraction for the f 0 (980)K ± final states was found [16] (updated in [17]), which was subsequently confirmed by BABAR [18]. Studies of the mass spectra of scalar mesons as well as their strong and electromagnetic decays suggest that there exist two typical scenarios for their structures [19,20]. In Scenario 1 (S1), the light scalar mesons with their masses below or near 1 GeV are treated as the lowest-lying qq states forming an SU(3) flavor nonet, including f 0 (500), f 0 (980),K * 0 (700) 0 and a 0 (980), and those with masses near 1.5 GeV are suggested as the first corresponding excited states forming another SU(3) flavor nonet, such as a 0 (1450), K * 0 (1430), f 0 (1370) and f 0 (1500) [21,22]. In Scenario 2 (S2), the heavier nonet mesons are regarded as the ground states of qq, while those lighter nonet ones are not regular mesons and might be four-quark states.
In 2019, LHCb collaboration study the B 0 → ρ(770) 0 K * (892) 0 decay within an quasi-two-body decay [23]. In our work, we will adopt this mechanism to study the four-body where the scalar mesons will be treated in S1 as mentioned above. We can then calculate the localized CP violating asymmetries and branching fractions of the four-body decayB 0 → K − π + π − π + . Besides we can also calculate the CP asymmetries and branching fractions of the two-body decaysB 0 → SV (V S) and all the individual four-body decaysB 0 → SV (V S) → K − π + π − π + , respectively. In fact, with the great development of the large hadron collider beauty (LHCb) and Belle-II experiments, more and more decay modes involving one or two scalar states in the B and D meson decays are expected to be measured with the high precision in the future.
The remainder of this paper is organized as follows. Our theoretical framework are presented in Sect.
II. In Sect. III, we give our numerical results. And we summarize our work in Sect IV. Appendix A collects the explicit formulas for all four-body decay amplitudes. The dynamical functions for the corresponding resonances are summarized in Appendix B. We also consider the f 0 (500) − f 0 (980) mixing in Appendix C. Related theoretical parameters are listed in Appendix D.

A. B decay in QCD factorization
With the operator product expansion, the effective weak Hamiltonian for B meson decays can be written as [6] 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, 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 [24] where α and β are color indices, q ′ = u, d, s, c or b quarks.
Within the framework of QCD factorization [6,24], 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 [6] 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, 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 [6].
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. The explicit expressions for A i,f n can be found in Ref. [25].
In the expressions for the spectator and annihilation corrections, there are end-point divergences . The QCD factorization approach suffers from these end-point divergences, which can be parametrized as [19] with Λ h being a typical scale of order 500 MeV, ρ A,H an unknown real parameter and φ A,H the free strong phase in the range [0, 2π].

B. Four-body decay amplitudes
For the four-body decayB 0 → K − π + π − π + , we consider the two-body cascade decays modeB 0 → Within the QCDF framework in Ref. [6], we can deduce the two-body weak decay amplitudes ofB withK * 0 0i =K * 0 (700) 0 ,K * 0 (1430) 0 corresponding to i = 1, 2, respectively, and are the form factors forB 0 to scalar and vector meson transitions, respectively, f V ,f S , and fB0 are decay constants of vector, scalar, andB 0 mesons, respectively,f s f 0j andf n f 0j are decay constants of f 0j mesons coming from the up and strange quark components, respectively.
In the framework of the two two-body decays, the four-body decay can be factorized into three pieces as the following: and where H ef f is the effective weak Hamiltonian,  Table IV and Appendix C, respectively.

When considering the contributions fromB
channels as listed in Eqs. (10) and (11), the total decay amplitude of theB 0 → K − π + π + π − decay can be written as 1 C. Kinematics of the four-body decay and localized CP violation One can use the five variables to describe the four-body decay kinematics [28,29], which are the invariant mass squared of the Kπ system s Kπ = (p 1 + p 2 ) 2 = m 2 Kπ , the invariant mass squared of the ππ system s ππ = (p 3 + p 4 ) 2 = m 2 ππ , the angles θ π , θ K and φ, where θ π is the angle of the π + in the π − π + center-of-mass frame Σ ππ with respect to the pions' line of flight in theB 0 rest frame ΣB0, 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 and φ is the angle between the Kπ and ππ planes, respectively. Their physical ranges are Instead of the individual momenta p 1 , p 2 , p 3 , p 4 , it is more convenient to use the following kinematic variables It follows that where and the function X is defined as With the decay amplitude, one can get the decay rate of the four-body decay [30], where σ(s ππ ) = 1 − 4m 2 π /s ππ , Ω represents the phase space with dΩ = ds ππ ds Kπ dcosθ π dcosθ K dφ. The differential CP asymmetry parameter and the localized integrated CP asymmetry take the following forms and respectively. V and m 2 S compared with m 2B 0 . We also set FB 0 →κ (0) = 0.3 and assign its uncertainty to be ±0.1 for simplicity. As for the decay constants and Gegenbauer moments of theK * (1410) 0 and theK * (1680) 0 mesons, we assume they have the same central values as that ofK * (892) 0 and assign their uncertainties to be ±0.1 [34].
As for the scalar meson, we adopt Scenario 1 in Ref. [19], in which those with masses below or near 1 GeV (σ, f 0 (980), κ) and near 1.5 GeV (K * 0 (1430)) are suggested as the lowest-lying qq states and the first excited state, respectively. When dealing with the decay constants of f 0j mesons, we consider the f 0 (500) − f 0 (980) mixing with the mixing angle |ϕ m | = 17 0 (see Appendix A for details). With the QCDF approach, we have obtained the amplitudes of the two-body decaysB 0 → SV andB 0 → V S, which are listed in Eqs. (7)- (9). Generally, the end-point divergence parameter ρ A is constrained in the range of [0, 1] and φ A is treated as a free strong phase. The experimental data of B two-body decays can provide important information to restrict the ranges of these two parameters. In fact, compared with the B → P V /V P/P P decays, there are much less experimental data for the B → V S/P S and B → SV /SP decays, so the values of ρ A and φ A have not been determined well in these decays. Thus we adopt ρ A,H < 0.5 and 0 ≤ φ A,H ≤ 2π as in Refs. [19,25]. With more accumulation of experimental data, both of them could be defined in small regions in the future.
Substituting Eqs. (7)-(9) into Eq. (19), we obtain the CP violating asymmetries of the two-body decaysB 0 → SV andB 0 → V S with the parameters given in Table IV and Appendix F, which are listed in Table I. From Table I, one can see our theoretical results for the CP asymmetries ofB 0 →K * (892) 0 f 0 (980) andB 0 →K * 0 (1430) 0 ω are consistent with the data from BABAR Collaboration. However, the predicted central values of the CP asymmetries of theB 0 →K * 0 (1430) 0 ρ andB 0 →K * 0 (1430) 0 ω are larger than those in Ref. [20]. The main difference between our work and Ref. [20] is the structure of theK * 0 (1430) 0 meson, which is explored in S1 in our work and S2 in Ref. [20], respectively. Besides, we predict the CP asymmetries of some other channel decays. We find the signs of the CP asymmetries are negative inB 0 →κρ,B 0 →K * (1410) 0 f 0 (980) andB 0 →K * (1680) 0 f 0 (980) decays, with the first one being one order larger than the other two. For the positive values of the CP asymmetries in our work, those for theB 0 →κω andB 0 →K * (892) 0 σ decays are also one order larger than the others. Moreover, we also calculate the branching fractions of the two-body decaysB 0 → SV andB 0 → V S which are listed in Table II. As can be seen from Table II,   II: Branching fractions (in units of 10 −6 ) of the two-body decaysB 0 → [K − π + ] S/V [π + π − ] V /S . We have used B(f 0 (980) → π + π − ) = 0.5 to obtain the experimental branching fractions for f 0 (980)V . The theoretical errors come from the uncertainties of the form factors, decay constants Gegenbauer moments and divergence parameters.

IV. SUMMARY
In this work, we have revisited the four-body decayB 0 → K − π + π − π + in the framework of the two two-body decays. We consider more contributions from more different resonances. Meanwhile, we update the model when dealing with the dynamical function for the ρ resonance. The most important thing is III: Direct CP asymmetries (in units of 10 −2 ) and branching fractions (in units of 10 −6 ) of the four-body decaysB 0 → [K − π + ] S/V [π + π − ] V /S → K − π + π + π − . The theoretical errors come from the uncertainties of the form factors, decay constants, Gegenbauer moments and divergence parameters.

Appendix A: FOUR-BODY DECAY AMPLITUDES
Considering the related weak and strong decays, one can obtain the four-body decay amplitudes of thē B 0 → [K − π + ] S/V [π + π − ] V /S → K − π + π + π − channels as the following: and  Table I in Ref. [37]. The parameters ρ 1,2,3 are the phase-space factors of the decay channels ππ, KK and ηη, respectively, and have been defined as [37] with m 1 = m π , m 2 = m K and m 3 = m η .

THE GOUNARIS-SAKURAI FUNCTION
For the ρ 0 (770) resonance, an analytic dispersive term is included to ensure unitarity far from the pole mass, known as the Gounaris-Sakurai model. It takes the form [38] T R (m ππ ) where Γ 0 and m 0 are the natural width and the Breit-Wigner mass of the ρ 0 (770) meson, respectively, The concrete form of f (m ππ ) is where q 0 is the value of q = | q| when the invariant mass, m ππ , is equal to the mass of the ρ 0 (770) resonance, The constant parameter D is given by

RELATIVISTIC BREIT-WIGNER
We adopt the relativistic Breit-Wigner function to describe the distributions of theK * 0 (700) 0 ,K * (892) 0 , K * (1410) 0 andK * (1680) 0 resonances [45], with where m Kπ is the invariant mass of the Kπ pair, s Kπ = m 2 Kπ , p Kπ (p R ) is the momentum of either daughter in the Kπ (or R) rest frame, and M R and Γ R 0 are the nominal mass and width, respectively, F R is the Blatt-Weisskopf centrifugal barrier factor [47], which are listed in Table V and depend on a single parameter R r representing the meson radius, for which one can adopt R r = 1.5GeV −1 [48]. Appendix C: f 0 (500) − f 0 (980) MIXING Analogous to the η − η ′ mixing, the scalar f 0 (500) − f 0 (980) mixing can also be parameterized by a 2 × 2 rotation matrix with a single angle ϕ m in the quark-flavor basis, namely, with the quark-flavor states f s ≡ ss and f q ≡ uū+dd √ 2 . Various mixing angle ϕ m measurements have been derived and summarized in the literature with a wide range of values [20,49]. However, it is worth pointing out that, based on the recent measurement and the accompanied discussion performed by the LHCb Collaboration [50], the upper limit |ϕ m | < 31 0 has been set for the first time in the B meson decays with a two-quark structure description of f 0 (500) and f 0 (980). In our calculation, we adopt |ϕ m | = 17 0 [20].