Studying the localized CP violation and the branching fraction of the decay

In this work, we study the localized violation and the branching fraction of the four-body decay by employing a quasi-two-body QCD factorization approach. Considering the interference of and channels, we predict and , respectively, which shows that the interference mechanism of these two channels can induce the localized violation to this four-body decay. Meanwhile, within the two quark model framework for the scalar mesons and , we calculate the direct CP violations and branching fractions of the and decays, respectively. The corresponding results are , , and , indicating that the violations of these two-body decays are both positive and the branching fractions quite different. These studies provide a new way to investigate the aforementioned four-body decay and can be helpful in clarifying the configuration of the structure of the light scalar meson.

Charge-Parity ( ) violation is one of the most fundamental and important properties of a weak interaction. Nonleptonic decays of hadrons containing a heavy quark play an important role in testing the standard model (SM) picture for the violation mechanism in flavor physics, improving our understanding of nonperturbative and perturbative QCD and exploring new physics beyond the SM.
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]. In additino to the weak phase, a large strong phase is usually needed for a large violation. Generally, this strong phase is provided by QCD loop corrections and some phenomenological models. Recently, theoretical and experimental studies on two-or three-body heavy meson decays have attracted more attention [3][4][5][6][7][8][9][10][11][12], while studies on four-body nonleptonic decays of these heavy mesons have been limited [13][14][15]. Because of the complicated phase spaces and relatively smaller branching fractions, four-body decays of heavy mesons are difficult to be investigated. However, in regard to studying the intermediate resonances, fourbody decays of heavy mesons can provide rich information, especially for the unclear compositions of scalar mesons such as ( ), ( ), and . The descriptions of inner structures of light scal-qqqqqq B D ar states are still unclear and even controversial, which could be, for example, , , meson-meson bound states or even those supplemented with a scalar glueball [16][17][18][19]. 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 considerable development of the large hadron collider beauty (LHCb) and Belle-II experiments, more four-body decay modes involving one or two scalar states in the and 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 decay , i.e., and , where the light scalars and will be considered as lowest-lying and first excited states [20], respectively. We can then explore whether the localized CP violation of the four-body decay can be induced by the interference of these two channels.

B D
Theoretically, to calculate the hadronic matrix elements of or weak decays, some approaches, such as QCD factorization (QCDF) [6,21], the perturbative QCD (pQCD) [22], and the soft-collinear effective theory (SCET) [23], have been fully developed and extensively employed in recent years. Unfortunately, in collinear factorization approximation, the calculation of annihilation corrections always suffers from end-point divergence. In the QCDF approach, such divergence is usually parameterized in a model-independent manner [6,21] and will be explicitly expressed in Sec. 2.
The remainder of this paper is organized as follows. In Sec. 2, we present our theoretical framework. The numerical results are given in Sec. 3, and we summarize our work in Sec. 4. 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.

Theoretical framework 2.1 Kinematics of the four-body decaȳ
The kinematics of the process is described in terms of the five variables displayed in Fig. 1 [24,25] in which

Kπ
(i) the invariant mass squared of the system is ; (ii) the invariant mass squared of the system is ; is the angle of the in the center-of- mass frame with respect to the ' line of flight in the rest frame ; is the angle of the in the center-ofmass system with respect to the line of flight in ; ϕ Kπ ππ (v) is the angle between the and planes. The physical ranges are We consider the localization of violation of the decay when the invariant mass of is near the masses of (including ), and the invariant mass of is near the masses of (including ). We adopt  Instead of the individual momenta , , , , it is more convenient to use the following kinematic variables It follows that X where the function is defined as

B
The effective weak Hamiltonian for nonleptonic weak decays is [6] where represents the Fermi constant, , and are the CKM matrix elements, are Wilson coefficients, are the tree level operators, are the QCD penguin operators, arise from electroweak penguin diagrams, and and are the electromagnetic and chromomagnetic dipole operators, respectively. With the effective Hamiltonian in Eq. (6), 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] and are also listed in Appendix A for convenience. The annihilation contributions ( ) can be simplified as [26] A i where the superscripts and refer to X gluon emission from the initial and final state quarks, respectively. The model-dependent parameter is used to estimate the end-point contributions and expressed as with being a typical scale of order 500 , an unknown real parameter, and the free strong phase in the range . 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 . In our work, when dealing with the endpoint divergences from the hard spectator scattering and weak annihilation contributions, we will follow the assumption for the two-body decays [20].

Four-body decay amplitudes and localized CP viol-
For the decay, we consider the contributions from and channels. For convenience, , , and mesons will be denoted as , , and , respectively. The amplitudes of these two channels are and where  ,  ,  and  are strong Hamiltonians for  ,  ,  , and decays, respectively. , , , and are the reciprocals of the dynamical functions of the corresponding mesons. Since the width of is larger than the other three mesons, we shall adopt the Breit-Wigner function and the Bugg model [27,28] to deal with the distributions of the first three mesons ( , and ) and meson, respectively.
In the Breit-Wigner model, takes the form , corresponding to , and mesons. or when dealing with or systems. σ The Bugg model is used to parameterize the distribution of [27,28], Chinese Physics C Vol. 44, No. 10 (2020) 103104 where with , and where we abbreviate as , related parameters are fixed to be , , , , and , as given in the fourth column of Table I in Ref. [27]. The parameters are the phase-space factors of the decay channels , and , respectively, which are defined as [27] ρ and . When dealing with the final state interactions, unitarized chiral perturbation theory is an effective method; they have been studied in Refs. [29][30][31][32]. Now we will adopt the method in Refs. [7,28] and respectively, where and are the strong coupling constants of the corresponding vector and scalar meson decays, respectively. Generally, these coupling constants can be derived from experiments, which have been listed in Eq. (C4).B Within the QCDF framework in Ref. [6], we can get the decay amplitudes of , which have been listed in Appendix B. Combining Eqs. (35), (15) and (10), (B2), (16) and (11), respectively, the amplitudes of and channels can be written as and  There can be a relative strong phase between the two interference amplitudes, the value of which depends on experimental data and theoretical models. Since little Chinese Physics C Vol. 44, No. 10 (2020) 103104 information about can be provided by experiments, we choose to adopt the same method as that in Refs. [7,33,34], i.e., setting . The total decay amplitude of the including both and channels can be written as The differential CP asymmetry parameter can be defined as

CP
The localized integration asymmetry can be measured by experiments and takes the following form: where represents the phase space given in Eq. (2) with . As for the decay rate, one has [13] with σ(s ππ ) = √ 1 − 4m 2 π /s ππ . (23) This leads to the branching fraction ΓB0B 0 where is the decay width of the meson. Within the QCDF approach, we get the amplitudes of the two-body decays and , where the light scalar and mesons are considered as the lowestlying and first excisted states [20], respectively. As for the parameters for the end-point divergences, we take and arbitrary strong phases . All the form factors are evaluated at due to the smallness of , , and compared with . We also simply set and assign its uncertainty as . With the given parameters, we obtain the violations and branching fractions of the and decays by substituting Eqs. (B1) and (B2) into (20), respectively. The results are , , and , respectively. Obviously, the violations of these two-body decays are

Numerical results
qq both positive, with the violation in decay being smaller than that in . The magnitudes of the branching fractions in these two-body decays are different with the former being about two orders smaller than the latter. When dealing with the the four-body decay , we adopt , , , . Then, substituting Eq. (19) into (21) and (24), we respectively get the localized violation and branching fraction of the four-body decay , with the results and . Compared with the uncertainties from the Gegenbauer moments, we find that those from the divergence parameters are much larger. It is clear that the sign of the localized violation of is positive when the invariant masses of and are near the masses of ( ) and ( ), respectively. This indicates that the interference of and channels can induce the localized violation to the fourbody decay . Our theoretical results shown herein are predictions for ongoing experiments at LHCb and Belle-II. If our predictions are confirmed by experiments in the future, the viewpoint that and have the composition should be well supported. However, to exclude other possible structures, more investigations will be needed due to uncertainties from both theory and experiments. By studying the quasi-two-body decays within the QCDF approach, we predicted the localized violation and branching fraction of the four-body decay due to the interference of the two channels and , with the results and . It is clear that the sign of the localized violation of is positive. In the two quark model for the scalar mesons, we also obtained the violations and branching fractions of the two-body decays and as , , and , respectively. Obviously, the violations of these two-body decays are both positive, and the violation in is smaller than that in . Furthermore, the branching fractions in these two body decays are quite different, with the former being two orders smaller than the latter. Our results will be tested by the precise data from future LH-Chinese Physics C Vol. 44, No. 10 (2020) 103104

Summary
103104-5 f 0 (500)K * 0 (700) qq f 0 (500) K * 0 (700) Cb and Belle-II experiments. In the present work, we assumed that and are dominated by the configuration. Other possible structures of and could affect the results in our interference model, which will need further investigation. If predictions of CP qq violation and branch ratio in our work are both confirmed by a future experiment, we agree and support the view that scalar mesons have the composition. However, there is still a long way ahead to study the structure of scalar mesons.

Appendix A: Explicit expressions of hard spectator-scattering and weak annihilation amplitudes
For the hard spectator terms, we obtain [26] for and for , and , is the twist-2 (twist-3) light-cone distribution amplitude of the meson , and r M i χ and (i = 1,2) are "chirally-enhanced" terms defined as .

(A4)
The twist-2 light-cone distribution amplitudes (LCDA) for the pseudoscalar and vector mesons are respectively [6,35] ) and the twist-3 ones are respectively and are the Gegenbauer and Legendre polynomials in Eq. (A5) and Eq. (A6), respectively, are Gegenbauer moments, which depend on the scale .
The twist-2 light-cone distribution amplitude for a scalar meson is [20,26]  Moreover, a quantity is introduced to parametrize the integral over the meson distribution amplitude through [6] With the asymptotic light-cone distribution amplitudes, the building blocks for the annihilation amplitudes are given by [26] A i 1 =πα , for M 1 M 2 = S V, The values of Gegenbauer moments at are taken from [20,35],