Abstract
We study the violation induced by the interference between two intermediate resonances and in the phase space of singly-Cabibbo-suppressed decay . We adopt the factorization-assisted topological approach in dealing with the decay amplitudes of . The asymmetries of two-body decays are predicted to be very tiny, which are and , respectively, for and , while the differential asymmetry of is enhanced because of the interference between the two intermediate resonances, which can reach as large as . For some NPs which have considerable impacts on the chromomagnetic dipole operator , the global asymmetries of and can be then increased to and , respectively. The regional asymmetry in the overlapped region of the phase space can be as large as .
1. Introduction
Charge-Parity () violation, which was first discovered in meson system in 1964 [1], is one of the most important phenomena in particle physics. In the Standard Model (SM), violation originates from the weak phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [2, 3] and the unitary phases which usually arise from strong interactions. One reason for the smallness of violation is that the unitary phase is usually small. Nevertheless, violation can be enhanced in three-body decays of heavy hadrons, when the corresponding decay amplitudes are dominated by overlapped intermediate resonances in certain regions of phase space. Owing to the overlapping, a regional asymmetry can be generated by a relative strong phase between amplitudes corresponding to different resonances. This relative strong phase has nonperturbative origin. As a result, the regional asymmetry can be larger than the global one. In fact, such kind of enhanced violation has been observed in several three-body decay channels of meson [4–7], which was followed by a number of theoretical works [8–19].
The study of violation in singly-Cabibbo-suppressed (SCS) meson decays provides an ideal test of the SM and exploration of New Physics (NP) [20–23]. In the SM, violation is predicted to be very small in charm system. Experimental researches have shown that there is no significant violation so far in charmed hadron decays [24–33]. asymmetry in SCS meson decay can be as small asor even less, due to the suppression of the penguin diagrams by the CKM matrix as well as the smallness of Wilson coefficients in penguin amplitudes. The SCS decays are sensitive to new contributions to the QCD penguin and chromomagnetic dipole operators, while such contributions can affect neither the Cabibbo-favored (CF) nor the doubly-Cabibbo-suppressed (DCS) decays [34]. Besides, the decays of charmed mesons offer a unique opportunity to probe violation in the up-type quark sector.
Several factorization approaches have been wildly used in nonleptonic decays. In the naive factorization approach [35, 36], the hadronic matrix elements were expressed as a product of a heavy to light transition form factor and a decay constant. Based on Heavy Quark Effect Theory, it is shown in the QCD factorization approach that the corrections to the hadronic matrix elements can be expressed in terms of short-distance coefficients and meson light-cone distribution amplitudes [37, 38]. Alternative factorization approach based on QCD factorization is often applied in study of quasi two-body hadronic decays [19, 39, 40], where they introduced unitary meson-meson form factors, from the perspective of unitarity, for the final state interactions. Other QCD-inspired approaches, such as the perturbative approach (pQCD) [41] and the soft-collinear effective theory (SCET) [42], are also wildly used in meson decays.
However, for meson decays, such QCD-inspired factorization approaches may not be reliable since the charm quark mass, which is just above 1 GeV, is not heavy enough for the heavy quark expansion [43, 44]. For this reason, several model-independent approaches for the charm meson decay amplitudes have been proposed, such as the flavor topological diagram approach based on the flavor symmetry [44–47] and the factorization-assisted topological-amplitude (FAT) approach with the inclusion of flavor breaking effect [48, 49]. One motivation of these aforementioned approaches is to identify as complete as possible the dominant sources of nonperturbative dynamics in the hadronic matrix elements.
In this paper, we study the violation of SCS meson decay in the FAT approach. Our attention will be mainly focused on the region of the phase space where two intermediate resonances, and , are overlapped. Before proceeding, it will be helpful to point out that direct asymmetry is hard to be isolated for decay process with -eigen-final-state. When the final state of the decay process is eigenstate, the time integrated violation for , which is defined ascan be expressed as [34]where , , and are the asymmetries in decay, in mixing, and in the interference of decay and mixing, respectively. As is shown in [34, 50, 51], the indirect violation is universal and channel-independent for two-body -eigenstate. This conclusion is easy to be generalized to decay processes with three-body -eigenstate in the final state, such as . In view of the universality of the indirect asymmetry, we will only consider the direct violations of the decay throughout this paper.
The remainder of this paper is organized as follows. In Section 2, we present the decay amplitudes for various decay channels, where the decay amplitudes of are formulated via the FAT approaches. In Section 3, we study the asymmetries of and the asymmetry of induced by the interference between different resonances in the phase space. Discussions and conclusions are given in Section 4. We list some useful formulas and input parameters in the Appendix.
2. Decay Amplitude for
In the overlapped region of the intermediate resonances and in the phase space, the decay process is dominated by two cascade decays, and , respectively. Consequently, the decay amplitude of can be expressed asin the overlapped region, where and are the amplitudes for the two cascade decays and is the relative strong phase. Note that nonresonance contributions have been neglected in (4).
The decay amplitude for the cascade decay can be expressed aswhere and represent the amplitudes corresponding to the strong decay and weak decay , respectively, is the helicity index of , is the invariant mass square of system, and and are the mass and width of , respectively. The decay amplitude for the cascade decay, , is the same as (5) except replacing the subscripts and with and , respectively.
For the strong decays , one can express the decay amplitudes aswhere and represent the momentum for and mesons, respectively, and is the effective coupling constant for the strong interaction, which can be extracted from the experimental data viawithand . The isospin symmetry of the strong interaction implies that .
The decay amplitudes for the weak decays, and , will be handled with the aforementioned FAT approach [48, 49]. The relevant topological tree and penguin diagrams for are displayed in Figure 1, where and denote a light pseudoscalar and vector meson (representing and in this paper), respectively.
(a)
(b)
(c)
(d)
(e)
The two tree diagrams in first line of Figure 1 represent the color-favored tree diagram for transition and the -exchange diagram with the pseudoscalar (vector) meson containing the antiquark from the weak vertex, respectively. The amplitudes of these two diagrams will be, respectively, denoted as and .
According to these topological structures, the amplitudes of the color-favored tree diagrams , which are dominated by the factorizable contributions, can be parameterized asandrespectively, where is the Fermi constant, , with and being the CKM matrix elements, , with and being the scale-dependent Wilson coefficients, and the number of color , and are the decay constant and mass of the vector (pseudoscalar) meson, respectively, and are the form factors for the transitions and , respectively, is the polarization vector of the vector meson, and is the momentum of meson. The scale of Wilson coefficients is set to energy release in individual decay channels [52, 53], which depends on masses of initial and final states and is defined as [48, 49]with the mass ratios , where represents the soft degrees of freedom in the meson, which is a free parameter.
For the -exchange amplitudes, since the factorizable contributions to these amplitudes are helicity-suppressed, only the nonfactorizable contributions need to be considered. Therefore, the -exchange amplitudes are parameterized aswhere is the mass of meson, , , and are the decay constants of the , , and mesons, respectively, and and characterize the strengths and the strong phases of the corresponding amplitudes, with representing the strongly produced quark pair. The ratio of over indicates that the flavor breaking effects have been taken into account from the decay constants.
The penguin diagrams shown in the second line of Figure 1 represent the color-favored, the gluon-annihilation, and the gluon-exchange penguin diagrams, respectively, whose amplitudes will be denoted as , , and , respectively.
Since a vector meson cannot be generated from the scalar or pseudoscalar operator, the amplitude does not include contributions from the penguin operator or . Consequently, the color-favored penguin amplitudes and can be expressed asandrespectively, where with and being the CKM matrix elements, , with being the Wilson coefficients, and is a chiral factor, which takes the formwith being the masses of quark. Note that the quark-loop corrections and the chromomagnetic-penguin contribution are also absorbed into as shown in [49].
Similar to the amplitudes , the amplitudes only include the nonfactorizable contributions as well. Therefore, the amplitudes , which are dominated by and [48], can be parameterized as
For the amplitudes and , the helicity suppression does not apply to the matrix elements of , so the factorizable contributions exist. In the pole resonance model [54], after applying the Fierz transformation and the factorization hypothesis, the amplitudes and can be expressed asandrespectively, where is an effective strong coupling constant obtained from strong decays, e.g., , , and , and is set as [54] in this work, and are the mass and decay constant of the pole resonant pseudoscalar meson , respectively, and and are the strengths and the strong phases of the corresponding amplitudes.
From Figure 1, the decay amplitudes of and in the FAT approach can be easily written downandrespectively, where is the helicity of the polarization vector . In the FAT approach, the fitted nonperturbative parameters, , , , , are assumed to be universal and can be determined by the data [49].
In Table 1, we list the magnitude of each topological amplitude for and by using the global fitted parameters for in [49]. One can see from Table 1 that the penguin contributions are greatly suppressed. is dominant in the penguin contributions of , while is small in , which is even smaller than the amplitude . This difference is because of the chirally enhanced factor contained in (14) while not in (13). The very small do not receive the contributions from the quark-loop and chromomagnetic penguins, since these two contributions to and are canceled with each other in (16). Besides, the relations , , and can be read from Table 1; this is because that the isospin symmetry and the flavor breaking effect have been considered.
Since the form factors are inevitably model-dependent, we list in Table 2 the branching ratios of and predicted by the FAT approach, by various form factor models. The pole, dipole, and covariant light-front (CLF) models are adopted. The uncertainties in Table 2 mainly come from decay constants. The CLF model agrees well with the data for both decay channels, and other models are also consistent with the data. However, the model-dependence of form factor leads to large uncertainty of the branching fraction, as large as . Because of the smallness of the Wilson coefficients and the CKM-suppression of the penguin amplitudes, the branching ratios are dominated by the tree amplitudes. Therefore, there is no much difference for the branching ratios whether we consider the penguin amplitudes or not.
3. Asymmetries for and
The direct asymmetry for the two-body decay is defined aswhere represents the decay amplitude of the conjugate process , such as or . In the framework of FAT approach, we predict very small direct asymmetries of and presented in Table 3. The uncertainties induced by the model-dependence of form factor to the asymmetries of and are about and , respectively.
The differential asymmetry of the three-body decay , which is a function of the invariant mass of and , is defined aswhere the invariant mass . As can be seen from (4), the differential asymmetry depends on the relative strong phase , which is impossible to be calculated theoretically because of its nonperturbative origin. Despite this, we can still acquire some information of this relative strong phase from data. By using a Dalitz plot technique [55, 58, 59], the phase difference between decays to and can be extracted from data. One should notice that is not the same as the strong phase defined in (4). The strong phase is the relative phase between the decay amplitudes of and . On the other hand, the phase is defined throughin the overlapped region of the phase space, where is the phase of the amplitude :Therefore, neglecting the CKM suppressed penguin amplitudes, and can be related bywhere are the phases in tree-level amplitudes of and are equivalent to if the penguin amplitudes are neglected. With the relation of (25), and measured by the BABAR Collaboration [56], we have .
In Figure 2, we present the differential asymmetry of in the overlapped region of and in the phase space, with . Namely, we will focus on the region of the phase space. One can see from Figure 2 that the differential asymmetry of can reach in the overlapped region, which is about 10 times larger than the asymmetries of the corresponding two-body decay channels shown in Table 3.
The behavior of the differential asymmetry of in Figure 2 motivates us to separate this region into four areas, area A (), area B (), area C (), and area D (). We further consider the observable of regional asymmetry in areas A, B, C, and D displayed in Table 4, which is defined bywhere represents a certain region of the phase space.
Comparing with the asymmetries of two-body decays, the regional asymmetries, from Table 4, are less sensitive to the models we have used. We would like to use only the CLF model for the following discussion. The uncertainties in Table 4 come from decay constants as well as the relative phase . In addition, if we focus on the right part of area A, that is, , the regional violation will be .
The energy dependence of the propagator of the intermediate resonances can lead to a small correction to asymmetry. For example, if we replace the Breit-Wigner propagator by the Flatté Parametrization [60], the correction to the regional asymmetry will be about .
Since the asymmetry of is extremely suppressed, it should be more sensitive to the NP. For example, some NPs have considerable impacts on the chromomagnetic dipole operator [34, 61–66]. Consequently, the violation in SCS decays may be further enhanced. In practice, the NP contributions can be absorbed into the corresponding effective Wilson coefficient [67, 68]. For comparison, we first consider a relative small value of (as in [48, 64]) lying within the range and the global asymmetry of are no larger than . Moreover, if we follow [49] taking (while , which is extracted from measured by LHCb [69], is a quite large quantity even for the coefficients corresponding tree-level operators, however, such large contribution can be realized if some NPs effects are pulled in. For example, the up squark-gluino loops in supersymmetry (SUSY) can arise significant contributions to . More details about the squark-gluino loops and other models in SUSY can be found in [34, 62, 70–72]), the global asymmetries of and are then and , respectively.
We further display the asymmetry of in the overlapped region of and in Figures 3(a) and 3(b) for and , respectively. After taking the interference effect into account, the differential asymmetry of can be increased as large as and for and , respectively. The regional ones (in phase space of ) can reach and for and , respectively.
(a)
(b)
4. Discussion and Conclusion
In this work, we studied violations in via the FAT approach. The violations in two-body decay processes and are very small, which are and , respectively. Our discussion shows that the violation can be enhanced by the interference effect in three-body decay . The differential asymmetry can reach when the interference effect is taken into account, while the regional one can be as large as .
Besides, since the chromomagnetic dipole operator is sensitive to some NPs, the inclusion of this kind of NPs will lead to a much larger global asymmetries of and , which are and , respectively, while the regional asymmetry of can be also increased to when considering the interference effect in the phase space. Since the of asymmetry is attributed to the large , which is almost impossible for the SM to generate such large contribution, it will indicate NP if such violation is observed. Here, we roughly estimate the number of needed for testing such kind of asymmetries, which is about . This could be observed in the future experiments at Belle II [73, 74], while the current largest yields are about at BABAR and Belle [75, 76] and at BESIII [77].
Appendix
Some Useful Formulas and Input Parameters
(1) Effective Hamiltonian and Wilson Coefficients. The weak effective Hamiltonian for SCS meson decays, based on the Operator Product Expansion (OPE) and Heavy Quark Effective Theory (HQET), can be expressed as [78]where is the Fermi constant, , is the Wilson coefficient, and , , , and are four-fermion operators which are constructed from different combinations of quark fields. The four-fermion operators take the following form:where and are color indices and . Among all these operators, and are tree operators, are QCD penguin operators, and is chromomagnetic dipole operator. The electroweak penguin operators are neglected in practice. One should notice that SCS decays receive contributions from all aforementioned operators while only tree operators can contribute to CF decays and DCS decays.
The Wilson coefficients used in this paper are evaluated at , which can be found in [48].
(2) CKM Matrix. We use the Wolfenstein parameterization for the CKM matrix elements, which up to order read [79, 80]where , and are the Wolfenstein parameters, which satisfy following relation:Numerical values of Wolfenstein parameters which have been used in this work are as follows:
(3) Decay Constants and Form Factors. In (17) and (18), the pole resonance model was employed for the matrix element in the annihilation diagrams. By considering angular momentum conservation at weak vertex and all conservation laws are preserved at strong vertex, the matrix element is therefore dominated by a pseudoscalar resonance [54],where is a strong coupling constant and and are the mass and decay constant of the pseudoscalar resonance . Therefore, and are the dominant resonances for the final states of , which can be expressed as flavor mixing of and ,where is the mixing angle and and are defined byThe decay constants of and are defined bywhereAccording to [81, 82], the decay constants of and can be expressed aswhere and [81], and the mixing angle [83]. Other decay constants used in this paper are listed in Table 5.
The transition form factors and , based on the relativistic covariant light-front quark model [85], are expressed as a momentum-dependent, 3-parameter form (the parameters can be found in Table 6):
(4) Decay Rate. The decay width takes the formwhere represents the center of mass (c.m.) 3-momentum of each meson in the final state and is given by is the corresponding decay amplitude.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was partially supported by National Natural Science Foundation of China (Project Nos. 11447021, 11575077, and 11705081), National Natural Science Foundation of Hunan Province (Project No. 2016JJ3104), the Innovation Group of Nuclear and Particle Physics in USC, and the China Scholarship Council.