Abstract
We review the two- and three-body baryonic decays with the dibaryon () as the final states. Accordingly, we summarize the experimental data of the branching fractions, angular asymmetries, and asymmetries. Using the -boson annihilation (exchange) mechanism, the branching fractions of are shown to be interpretable. In the approach of perturbative QCD counting rules, we study the three-body decay channels. In particular, we review the asymmetries of , which are promising to be measured by the LHCb and Belle II experiments. Finally, we remark the theoretical challenges in interpreting and .
1. Introduction
The baryonic meson decays have been richly measured with the branching fractions, angular asymmetries, and asymmetries in two- and three-body decay channels [1–13], as summarized in Table 1. Typically, is as small as . Nonetheless, it is observed that , due to a sharply rising peak in observed around the threshold area of in the dibaryon invariant mass spectrum [4]. Known as the threshold effect, it enhances as large as . While the production shows the tendency to occur around , proceeds at scale, far from the threshold area. This interprets the suppressed [14, 15].
The partial branching fraction of can be a function of , where is the angle between the baryon and meson moving directions in the dibaryon rest frame. One hence defines the forward-backward angular asymmetry,
In Table 1, [2] indicate that one of the dibaryons favors to move collinearly with the meson.
We search for the theoretical approach to interpret the threshold effect, branching fractions, and angular asymmetries of the baryonic decays. We find that the factorization approach can be useful [16], where one factorizes (decomposes) the amplitude of the decay as two separate matrix elements. In our case, we present where and stand for the quark currents and the matrix element of can be parameterized as the timelike baryonic ( to transition) form factors . Moreover, one derives in perturbative QCD (pQCD) counting rules [17–24], where and accounts for the number of the gluon propagators that attach to the baryon pair. It results in , which shapes a peak around in the spectrum, and then, the threshold effect can be interpreted. In the transition, there exists the term of for [24], which is reduced as in the rest frame. Since , the term for can be used to describe the highly asymmetric . Alternatively, the baryonic decays is studied with the pole model, where the nonfactorizable contributions can be taken into account [25–28].
We have explained [29]. We have studied and explained the branching fractions and asymmetries [30–38]. In addition, we have predicted [37], in excellent agreement with the value of measured by LHCb [8]. This demonstrates that the theoretical approach can be reliable. Therefore, we would like to present a review, in order to illustrate how the approach of pQCD counting rules based on the factorization can be applied to the baryonic decays. We will also review our theoretical results that have explained the branching fractions of and , particularly, the asymmetries, promising to be observed by future measurements.
2. Formalism
To review the two-body baryonic decays, we take as our example. According to Figure 1, is regarded as an -boson exchange process [29, 39–41]. In the factorization, we derive the amplitude as [29] where is the Fermi constant and the Cabibbo-Kobayashi-Maskawa (CKM) matrix element. One has defined for the meson annihilation, where is the decay constant and the four-momentum. For the production, the matrix elements read [22, 23] with the (axial-)vector current , where , , and are the timelike baryonic form factors.
At a very large momentum transfer (), the approach of pQCD counting rules results in [17–21] where is the function of QCD to one loop, the flavor number, and GeV the scale factor. Moreover, is the running coupling constant in the strong interaction [20]. Interestingly, reflects the fact that one needs two hard gluon propagators to attach to the baryons as drawn in Figure 1 [42], whereas is caused by the wave function.
As and are combined as the right- or left-handed chiral current, that is, , one obtains for the spacelike transition. With the right-handed current, the matrix elements can be written as [19, 29] where and are the chiral form factors. With () denoting one of the valence quarks in , known as the chiral charge is able to change the flavor for , such that is transformed as . Note that the chirality is regarded as the helicity at . Since the helicity of can be (anti-)parallel to the helicity of , we define that is responsible for acting on . Thus, the approach of pQCD counting rules leads to [19, 29] with , , and , where the flavor () and spin symmetries are both respected. In the crossing symmetry, the spacelike form factors behave as the timelike ones, such that one can relate and with the chiral form factors in Equations (6) and (7) derived in the spacelike region, leading to . In addition to the momentum dependence of Equation (5), and are presented as [22, 23, 29] where .
For , we obtain [23] where is from , and we have used and [23, 29]. In Ref. [43], the pQCQ calculation causes , indicating that has a suppressed contribution. Taking the form factors as the inputs, we reduce the amplitude of as where has been vanishing, in accordance with the conservation of vector current (CVC) . Using the partial conservation of axial-vector current (PCAC), where , it is obtained that [23, 29, 39–41, 44] by which and cancel each other and then . This seems that the -exchange (annihilation) mechanism based on the factorization fails to explain . As a consequence, one turns to think of the nonfactorizable effects as the main contributions [25, 44–50].
In Equation (11), describes a meson pole, so that can be regarded to receive the contribution from the intermediate process of , which is much suppressed. On the other hand, there might exist a QCD-based contribution to , by which , and PCAC is violated. Here, we choose to parameterize with slightly violated PCAC in the timelike region. To this end, we derive with at the threshold area of , where the meson pole is supposed to be inapplicable [40, 41]. Since the QCD-based calculation of is still lacking, besides suggesting , we are allowed to present for its momentum dependence [29].
To describe the three-body baryonic decays, we take with or as our examples. According to Figure 2, the amplitudes are given by [30–32] with and for , and and for . For , we define
(a)
(b)
(c)
(d)
In Equation (12), one presents that , where and are, respectively, the decay constant and polarization four-vector of the vector meson. The amplitudes are both associated with the matrix elements of the transition, and we parameterize them as [24] where , , and () are the transition form factors. Inspired by pQCD counting rules [17, 18, 21, 23, 24], the momentum dependences for and are given by with to be extracted by the data. According to the gluon lines in Figure 2, in should be , which accounts for two gluon propagators attaching to the valence quarks in and an additional one for kicking (speeding up) the spectator quark in [23]. For the gluon kicking, it is similar to the meson transition form factor derived as in pQCD counting rules [20, 51], where is for a hard gluon to transfer the momentum to the spectator quark in the meson.
Like the case of and , we relate to the chiral form factors, which is in terms of [23, 24] where has been used. In addition, we obtain and , which are similar to in Equation (7). Under the flavor and spin symmetries, together with , it is derived that [23, 24] for and , respectively. We also review the direct asymmetry, defined by where denotes the decay width and the antiparticle decay.
3. Numerical Analysis
For the numerical analysis, we adopt the CKM matrix elements as [1] with , , , and in the Wolfenstein parameterization, where . The decay constants are given by GeV [1, 52]. With the global fit to the data, we obtain [29, 37]
We take for odd (even) with the color number, where the effective Wilson coefficients come from Ref. [16]. For in and , we use to take into account the nonfactorizable QCD corrections. We can hence present our theoretical calculations for in Table 2, along with the other results. Besides, we present our studies of the angular and asymmetries.
4. Discussions and Conclusions
The theoretical results in Table 2 can agree well with the data, which is based on the factorization, pQCD counting rules, and baryonic form factors. In particular, several asymmetries are predicted as large as 10-20%, promising to be measured by LHCb and Belle II [55, 56]. It is reasonable to extend the theoretical approach to and [34, 38, 57], where denotes a baryon (meson) containing a charm quark. As a consequence, can also be expalined (see Table 3). Nonetheless, and we have predicted are not verified by the observations [1, 32, 58, 59], where the amplitude of is given by since and are seen to be associated with the transition form factors, which are inferred to cause the overestimations. Besides, inconsistent with the data can be partly due to the inconsistent determination of between the inclusive and exclusive decays.
As the final remark, since the predictions of and are shown to deviate from the observations by the factors of 6 and 20, respectively, the theoretical approach is facing some difficulties. Therefore, the reexamination should be performed elsewhere.
In summary, to review the baryonic decays, we have summarized the experimental data, which includes branching fractions and angular and asymmetries. We have taken and with or for theoretical illustration. We have also reviewed the asymmetries of , which can be used to compare with future measurements by LHCb and Belle II. With the theoretical results listed in the tables, we have demonstrated that the theoretical approach can be used to interpret most observations. Finally, we have also remarked that the theoretical approach has currently encountered some challenges in interpreting and .
Data Availability
This manuscript has no associated data, or the data will not be deposited. The data used in this paper are publicly available, and they can be found in the corresponding references.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank Professors Xin Liu, Zhen-Jun Xiao, Qin Chang, and Rui-Lin Zhu for inviting us to write a review article on the physics of baryonic B meson decays in the special issue “Heavy Flavor Physics and CP Violation” of Advances in High Energy Physics. YKH was supported in part by National Natural Science Foundation of China (Grant Nos. 11675030 and 12175128). LS was supported in part by NSFC (Grant No. 12061141006) and Joint Large-Scale Scientific Facility Funds of the NSFC and CAS (Grant No. U1932108).