Abstract
In the present work, we construct eight six-quark currents to study the baryonium and dibaryon states via the QCD sum rules. For either baryonium or dibaryon states, we construct four currents with the and . Except the current of , we find the Borel windows for the other seven. There are two possible molecular states or compact six-quark states with the and . The results of the current of show a possible molecular state with 20 MeV binding energy. We find two baryonium resonances states with the and two dibaryon resonances states with the , respectively. Pole residues of the seven states are also calculated.
1. Introduction
The dibaryon is composed of two baryons, and the baryonium is formed by a baryon and antibaryon. In 1980s, the research of the interaction between the two baryons caused great attention both theoretically and experimentally [1]. The first known dibaryon state discovered in 1932 has been the deuteron, the bound state of proton and neutron [2]. Except for the deuteron, another genuine dibaryon resonance observed in experiment is the [3–5]. The baryon consists of three quarks, six quarks for the dibaryon or baryonium, it is natural to extend the concept of hexaquarks for which the quarks sitting in a common quark bag. The data of the electromagnetic form factors of the state are possible to differentiate whether it is a molecular type or a compact six-quark entity. For the deuteron, its charge radius is 2.1 fm, the proton and the neutron inside the deuteron are on average 4 fm apart from each other and do not overlap [6, 7], these data depict a clear physical picture of the hadronic molecule for the deuteron. One could also make the differentiation via the decay branching; in Ref. [8], the decay width of is considered via the chiral quark model; it is proposed that spends of its time as a six-quark and the rest as a molecule. Theoretically and experimentally, the ways which may find the difference are there; however, seldom work has been done yet for both the dibaryon and baryonium in this aspect. At present, it is too hard for us to differentiate dibaryon or baryonium from the compact six-quark state. We now could only say: it may be the molecular state or compact six-quark state if the mass is below the threshold of the two baryon constituents (bound state); as for the dibaryon resonance state, its mass is above the threshold of the two baryon constituents.
The oldest prediction of the existence of dibaryon with nearly 100 MeV binding energy based on SU(6) symmetry was done by Dyson and Xuong [9] which is in good agreement with the experimental observations of [3–5]. Although the insufficient experimental data for the solid observations of the dibaryon and baryonium states, lots of theoretical calculations have been done in search of them. In recent years, several QCD-inspired models are applied to study the dibaryons and baryoniums, such as the quark delocalization color screening model [10, 11], solving the Bethe-Salpeter equation via effective Lagrangians [12, 13], constituent quark model [14, 15], lattice QCD [16, 17], and the QCD sum rules [18–21]. Especially, in Refs. [22, 23], the authors study the charmed hadrons to investigate the mutual interactions between them and search for poles by solving the Bethe-Salpeter equation. It is natural to ask: what are the predictions of the interactions of these charmed hadrons in the framework of the QCD sum rules? In Ref. [24], the scalar and axialvector dibaryon states are studied in consideration of two-baryon scattering states. It is proposed that the two-baryon scattering states cannot saturate the QCD sum rules and their contributions can be neglected. We follow this idea and do not consider the two-baryon scattering states. The earlier study of the hidden-charm tetraquark molecular states has previously been published in a preprint [25]. In this article, we construct eight six-quark local interpolating currents to study both the baryoniums with the , and the dibaryons with the , , respectively.
The article is arranged as follows: we derive the QCD sum rules of the dibaryon and baryonium states in Section 2. We present the numerical results and discussions in Section 3. Section 4 is reserved for our conclusions.
2. The QCD Sum Rules for the Baryonium and Dibaryon States
Now, let us write down the corresponding two-point correlation functions from the eight constructed currents: where the superscripts are color indices and represents the charge conjugation matrix. The parities of these currents are shown as where is the parity operator, and . A complete set of intermediate hadronic states are inserted into each in the above correlation functions. The correlation functions at the hadron sides are simplified as where
denote the ground states which have the same as , the are the polarization vectors and are the pole residues which show the coupling of the currents to the states ().
For the QCD sides of the correlation functions, we contract the quark fields via the Wick theorem and express the correlation functions in terms of full quark propagators. After the detailed calculations, we find which means the net contribution of all the Feynman diagrams of this state including the leading order, perturbation terms and all kinds of condensates is zero. Except for this term, the others are given by where and are the full light and heavy quark propagators, respectively, under the operator product expansion at the quark level, they are expressed as where , are the Gell-Mann matrices () and [26–28].
Consider the Feynman graphes of the leading orders of these nonzero correlation functions, each one contains two heavy quark lines and four light quark lines. If each heavy quark line emits a gluon and each light quark line contributes quark-antiquark pair, they form the quark-gluon operator with dimension 16. This operator lead to vacuum condensates and . So, the highest dimension of the considered condensates is up to 16 in the present work. As for the truncation of the order , the contributions of the related vacuum condensates are tiny while ; it is accurate enough for us to calculate terms for [29]. Detailed calculations show that the vacuum condensates and have no contribution for all the considered states in this paper. Thus, the chosen vacuum condensates of the operator product expansions are , , , , , , , , and .
After the calculations of the integrals in the coordinate space for the light quarks and momentum space for heavy quarks of the correlation functions, we perform the Borel transforms for both the hadronic sides and the QCD sides, we get the QCD sum rules, where are the QCD spectral densities of , . As for the continuum threshold parameters , we take the experiential values [24, 29, 30], where are the masses of the ground states.
We differentiate the Equation (8) with respect to and eliminate the pole residues, the extracted masses are shown as
3. Numerical Results and Discussions
We apply the standard values of the vacuum condensates , , , at the energy scale [26, 31], and choose the mass [32]. Consider the energy-scale dependence for the above condensates, where , , , , and MeV, MeV, and MeV for the flavors , respectively [32–34]. In this paper, we choose the flavor number for all the states. Compared with the heavy quarks, the masses of the light quarks are too small to make effective difference. So, we ignore the masses of the light quarks. The masses of the baryon and anti-baryon GeV are from the particle data group [32]. We abandon the expressions of the lengthy spectral densities of these currents which are dozens of pages, one can contact us via the email. Note that Equation (10) depends on the energy scale [19, 35], we apply the energy scale formula to determine the best energy scales of the QCD spectral densities [27], where is the effective charm quark mass, we choose the value GeV [36]. Interestingly, in Ref. [29], for the vector current , if we set GeV, the central value of the mass of with the is GeV with GeV and which indicates a possible bound baryonium state, where GeV is the mass of [32]. As part of the systematic study for the dibaryons formed by the charmed baryons, the detailed results of Ref. [29] are also shown in Table 1. From Equatio (10), we find that relies on the threshold parameter and Borel parameter . Of course, at first, we do not know the suitable energy scale, so we try and adjust the value of via trial and error. From the energy scale formula Equation (12), one can determine that increases with the parameter ; however, numerical results show that mass derived from Equation (10) decreases with the increment of the parameter , so we can always find the best energy scale and determine the mass which satisfies the energy scale formula. As for the values of and , we search for the best Borel window which satisfies the following two principles, firstly, the pole dominance criterion; here, we define the pole contributions (PC) of the QCD sum rules as secondly, the convergence of the contribution of operator product expansion, for which, we define the contributions of the vacuum condensates of dimension as
Due to the uncertainties of the input parameters, the uncertainties of the extracted masses and pole resides are determined by the formula, where represent the masses or the pole resides, are the input parameters , , , denote the central values of these parameters. Since it is difficult to find the analytic expressions of the partial derivatives , we carry out the approximate numerical calculations for the uncertainties via,
Except , we find the Borel windows for the other seven currents. Their numerical results are shown in Figures 1–7, respectively. In each figure, we show the masses, dimensional contributions of vacuum condensates and pole residues of each current on , , and graphs, respectively. Currents , , , and are related to the baryonium structures. For the vector current , the extracted mass of the corresponding state with the is 4.88 GeV with the upper uncertainty being 0.09 GeV. Since the threshold of the two and baryon constituents is 4.91 GeV, so, we could not say for sure whether it is a molecular state formed by the two baryons or not. On the safe side, it is a possible molecular or compact six-quark state. In the same way, with the is also a possible molecular or compact six-quark state. As for axial-vector current and scalar current , their extracted masses of the corresponding states with the and are 5.31 GeV and 5.23 GeV, respectively. They are all above the thresholds of the two and baryon constituents even if we take into the account of the lower bounds of uncertainties of the extracted masses. Both these two are the baryonium resonance states.
Currents , , , and are related to the dibaryon states. For the current , its related correlation function . It indicates that there is no such dibaryon state with the . The extracted mass from is GeV which is 20 MeV below the the threshold of the two baryon constituents. Due to the uncertainties of input parameters, the related uncertainty of the mass of this state is up to 0.1 GeV, it is also a possible molecular or a compact six-quark state with the . We also get the masses of and , they are all above the thresholds of the two baryon constituents. The detailed results of each states are listed in Table 1.
For all these seven currents, it is clear to be seen from the curves in each titled figures, the most important items are the leading order, and . We find that the higher dimensional vacuum condensates play the less important roles. So, the convergence of the operator product expansion holds very well. It is worth mentioning that condensate plays a dominant role for the current , what is more, the leading order of contributes the most important part for both the mass and pole residue of the corresponding state.
In 2017, the LHCb has observed the doubly-charmed baryon with mass of MeV [37]. It is proposed that there is large binding energy MeV of the two heavy c-quarks in a baryon [38]. In Ref. [39], the quark-level analogue of nuclear fusion for the two baryons was discussed. In a similar way, we also consider the quark-rearrangement reaction, the possible decays of dibaryon states are given by
Since the of is unknown yet, thus, it is also impossible for us to determine the of the produced neutron after the decay. We can only estimate the upper values of the exothermic energy of each quark-rearrangement reaction by setting the minimum mass of the neutron as MeV. For with the , it is impossible to have such kind of decay since its mass is less than the sum of and . As for and , the corresponding are 310 MeV and 170 MeV, respectively.
4. Conclusions
In this article, we construct the eight interpolating currents to study the baryonium and dibaryon states via the QCD sum rules. The correlation function of is zero which indicates that there is no such dibaryon state with the . We find two possible molecular or compact six-quark states with the and . Their masses are GeV and GeV, respectively. For the currents , , , and , the extracted masses of these four states are GeV, GeV, GeV, and GeV, respectively. Both the four are above the the thresholds of the two baryon constituents. We conclude that with the and are baryonium resonances states and with the and are dibaryon resonances states. We find the central value of the mass of the state with the is just 20 MeV below the threshold of the two baryon constituents. Considering the uncertainty, dibaryon with the is also a possible molecular state. For all the bound states related to and , we need more data, such as the electromagnetic form factors, to differentiate whether the bound dibaryon or baryonium is a molecular state or a compact six-quark entity. This is also the task of our further work. The possible decays, the quark-rearrangement reactions of each states are discussed. The upper values of the exothermic energy are calculated for these decays which may be experimentally verified in the future.
Data Availability
All data included in this manuscript are available upon request by contacting the corresponding author.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work has been supported by the National Natural Science Foundation of China, Grant Number 12175068 and Youth Foundation of NCEPU, Grant Number 93209703.