Abstract
In this work, we investigate the resonant contributions of and in the three-body within the perturbative QCD approach. The form factor is adopted to describe the nonperturbative dynamics of the S-wave system. The branching ratios of all concerned decays are calculated and predicted to be in the order of to . The ratio of branching fractions between and is predicted to be 0.0552, which implies the discrepancy for the LHCb measurements. We expect that the predictions in this work can be tested by the future experiments, especially, to resolve ratio discrepancy.
1. Introduction
Decays of the type , where a meson decays to a charmed meson and two light pseudoscalar mesons, have attracted people’s attention in recent years. On the one hand, the studies of these three-body processes have shown the potential to constrain the parameters of the unitarity triangle. For instance, the decay is sensitive to measure the CKM angle [1, 2], while the Dalitz plot analysis of the decays and can further improve the determination of the CKM angle [3]. On the other hand, the decays provide opportunities for probing the rich resonant structure in the final states, including the spectroscopy of charmed mesons and the components in two light meson systems. A series of results in this area have been acquired from the measurements performed by the Belle [4], BaBar [5, 6], and LHCb [3, 7–9] Collaborations.
In theory, a direct analysis of the three-body decays is particularly difficult on account of the entangled resonant and nonresonant contributions, the complex interplay between the weak processes and the low-energy strong interactions [10], and other possible final state interactions [11, 12]. Fortunately, most of the three-body hadronic meson decay processes are considered to be dominated by the low-energy -, -, and -wave resonant states, which could be treated in the quasi-two-body framework. By neglecting the interactions between the meson pair originated from the resonant states and the bachelor particle in the final states, the factorization theorem is still valid as in the two-body case [13, 14], and substantial theoretical efforts for different quasi-two-body meson decays have been made within different theoretical approaches [15, 16]. As well, the contributions from various intermediate resonant state for the three-body decays have been investigated in Ref. [17–20].
The understanding of the scalar mesons is a difficult and long-standing issue [21]. The scalar resonances usually have large decay widths which make them overlap strongly with the background. In the specific regions, such as the and thresholds, cusps in the line shapes of the nearby resonances will appear due to the contraction of the phase space. Moreover, the inner natures of scalars are still not completely clear. Part of them, especially the ones below GeV, have also been interpreted as glueballs, meson-meson bound states, or multiquark states, besides the traditional quark-antiquark configurations [22, 23]. The is perhaps the least controversial of the light scalar mesons and generally believed to be a state [24]. It predominantly couples to the channel and has been studied experimentally in many charmless three-body meson decays [25–27]. Recently, measurements of the charmed three-body decays and involving the resonant state were also presented by LHCb [3, 8]. In addition, the subprocess which often ignored in literatures has also been considered in Ref. [8].
In the framework of the PQCD approach [28–30], the investigation of -wave contributions to the decays was carried out in Ref. [31]. In a more recent work [32], contributions of the resonant states and in the three-body decays () were studied systematically within the same method. The is treated as the lowest lying state in view of the controversy for , and the scalar timelike form factor was also discussed in detail. Motivated by the related results measured by LHCb [3, 8], we shall extent the previous work [32] to the study of the charmed three-body decays and analyse the contributions of the resonances and in the decays in this work.
The rest of this article is structured as follows. In Section 2, we give a brief review of the framework of the PQCD approach. The numerical results and phenomenological discussions are presented in Section 3, and a short summary is given in Section 4, respectively. Finally, the relevant factorization formulae for the decay amplitudes are collected in the appendix.
2. Framework
In the light-cone coordinate system, the meson momentum , the total momentum of the pair, and the meson momentum under the rest frame of meson can be written as
with being the meson mass and the mass ratio . The variable equals to , where is the invariant mass squared of pair in the range from to . We also set the momenta of the light quarks in the meson, the pair, and the meson as , , and and have the definitions as follows: where , , and are the momentum fractions and run from zero to unity.
In the PQCD approach, the decay amplitude for the quasi-two-body decay can be expressed as the convolution [33] where the symbol represents the hard kernel with single hard gluon exchange. and are the distribution amplitudes for the and mesons, respectively. denotes the distribution amplitude for the pair with certain spin in the resonant region. In this work, we use the same distribution amplitudes for the and mesons as in Ref. [18] where one can easily find their expressions and the relevant parameters. Inspired by generalized distribution amplitude [34–37], the generalized LCDA for two-meson system are introduced [33, 38] for three-body -meson decay in the framework of PQCD approach and the heavy-to-light transition form factor in light-cone sum rules, respectively. The nonlocal matrix elements of vacuum to with various spin projector can be written as
The -wave distribution amplitude is chosen as [32] where and are the dimensionless lightlike unit vectors. The twist- and twist- light-cone distribution amplitudes have the form
Here, are the Gegenbauer polynomials, are the Gegenbauer moments, and is the scalar form factor for the pair. In this work, we adopt the same formulae and parameters for the -wave distribution amplitude as them in Ref. [32].
According to the typical Feynman diagrams as shown in Figure 1 and the quark currents for each decays, the decay amplitudes for the considered quasi-two-body decays are given as where is the Fermi constant, is the CKM matrix element, and the combinations of the Wilson coefficients are defined as and . The expressions of individual amplitudes , , , , , , , and from different subdiagrams in Figure 1 are collected in the appendix.
(a)
(b)
(c)
(d)
At last, we give the definition of the differential branching ratio for the considered quasi-two-body decays
In the center-of-mass frame of system, the magnitudes of the momenta and can be expressed as
3. Results
In the numerical calculations, the masses of the involved mesons (GeV), the lifetime of the mesons (ps), the resonance decay widths (GeV), and the Wolfenstein parameters are taken from the Review of Particle Physics [21]
The decay constants of the and mesons are set to the values GeV and GeV [39].
By integrating the differential branching ratio in Equation (8), we obtain the branching ratios for the considered quasi-two-body processes with the intermediate resonances and in Tables 1 and 2, respectively. The first error is induced by the shape parameters GeV in the distribution amplitude for the meson. The second and third errors come from the Gegenbauer moments and in the -wave distribution amplitude, respectively. The decay widths GeV and GeV contribute the fourth error. The last one is due to the parameter in the distribution amplitude for meson. The uncertainties from other parameters are comparatively small and have been neglected.
From the numerical results as listed in Tables 1 and 2, we have the following comments: (1)In the decays, we can extract the two-body branching fractions by using the relation under the quasi-two-body approximationFor the branching fractions of two-body decays with and , we shall apply The values Combined with the results listed in Tables 1 and 2, one can obtain the related two-body branching fractions; for example, and , where the errors are propagated from Equation (13) (2)The PQCD prediction for the branching fraction agrees with LHCb’s data [8] within errors, while the PQCD predicted is much larger than the value measured by LHCb [3] with significant uncertainties. By comparison, one can find that the decay modes and contain the same decay topology when neglecting the differences of hadronic parameters between and . Then, we evaluate the ratiowhich is close to the PQCD prediction by using the results listed in Table 1, but different from the value acquired from the central values of the measured branching ratio by LHCb [3, 8]. One can find that in the Ref. [8], the component receives 20% fit fraction of total , but in Ref. [3, 8], component receives only 5.1% of total . The component is playing a such different role in two different process; however, on the theoretical side, the decay amplitudes are exactly same for and ; if we neglect the SU(3) symmetry breaking effect, ratio will be independent of theoretical framework. More precise measurements and more proper partial wave analysis are needed to resolve the discrepancy (3)For the CKM suppressed decay modes , their branching ratios are much smaller than the corresponding results of decays as predicted by PQCD in this work. The major reason comes from the strong CKM suppression factoras discussed in Ref. [40]. The nonvanishing charm quark mass in the fermion propagator generates the main differences between the and . Similarly, for the decay and decay, there still exists the CKM suppression but much moderate than the previous cases: From Table 1, we have The main differences between the and come from the nonvanishing charm quark mass contributions in the nonfactorizable emission diagram. We also suggest more study on the decay mode because it has a large branching ratio and can be found in future experiments (4) was often parameterized by LASS lineshape [41] in partial wave analysis, which incorporate both cusp resonance and slowly varying nonresonance contribution, and it was applied in LHCb measurements [3, 8]. However, rigorous theoretical calculation for nonresonance contribution in the context of PQCD framework is still absent [32], the comparisons between theoretical calculations and experiment measurements focus only on the -wave contribution. More attempts can be made in future study to parameterize the nonresonance contribution for sake of giving a more reliable result(5)The -averaged branching fraction of the charmless quasi-two-body decay involving the intermediate state is predicted to be about one magnitude smaller than the corresponding process containing in [32]. In quasi-two-body charmed decays, the ratio of branching fractions between Tables 1 and 2 is about few percentages, which are smaller than that of charmless cases mainly due to the absence of amplitude, which receive resonance pole mass enhancement as discussed in [32]. And the more compact phase space can also reduce the branching fractions for the decay mode involving . From the partial wave analysis in [8], the mode is measured to be about 1.5% than that of mode, which is about one-third of our prediction, i.e., 4.6%; more precise measurements and more reliable theoretical predictions are needed in the future study(6)In Figure 2, we show the invariant mass-dependent differential branching fraction for the quasi-two-body decays (solid line) and (dashed line). One can easily find that the main portion of the branching fraction comes from the region around the pole mass of the corresponding resonant states; the contribution from the mass region greater than 3 GeV is evaluated about 0.4% compared with the whole kinematic region (i.e., ) in this work and can be safely neglected
4. Conclusion
Motivated by the phenomenological importance of the charmed three-body hadronic -meson decays, in the present work, we have studied the quasi-two-body decays in the PQCD factorization approach with the help of the scalar form factor as a nonperturbative input. The branching ratios of all concerned decays are calculated and are of the order to ; the corresponding two-body branching fractions can be obtained by using the quasi-two-body approximation relation in Equation (11). Under SU(3) flavor symmetry, we found the theoretical framework independent ratio by neglecting the differences of hadronic parameters between and ; this result is consistent with our PQCD prediction, but inconsistent with LHCb measurements. For the decays and , the great difference in their corresponding branching fractions can be understood by a strong CKM suppression factor , while the moderate difference between and as well as and is mainly due to the . More reliable theoretical predictions are needed in the future study for the nonresonance contribution and -wave contribution. We hope the predictions in this work can be tested by the future experiments, especially, to resolve ratio discrepancy.
Appendix
Decay Amplitudes
The factorization formulae for the individual amplitudes from different subdiagrams in Figure 1 are where the hard functions are written as where ( and ) are the evolution factors, which are given by in which the Sudakov exponents are defined as where the quark anomalous dimension . The explicit form for at one loop can be found in [42]. and () are hard scales which are chosen to be the maximum of the virtuality of the internal momentum transition in the hard amplitudes as where we have
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
We are grateful to Ai-Jun Ma for the helpful comments. This work was supported by the National Natural Science Foundation of China under Grant No. 11947040.