Advances in High Energy Physics
Volume 2019 (2019), Article ID 8958079, 6 pages
https://doi.org/10.1155/2019/8958079

Research Article

Analysis of the System with QCD Sum Rules

Zun-Yan Di1,2 and Zhi-Gang Wang¹

¹Department of Physics, North China Electric Power University, Baoding 071003, China
²School of Nuclear Science and Engineering, North China Electric Power University, Beijing 102206, China

Correspondence should be addressed to Zhi-Gang Wang; zgwang@aliyun.com

Received 16 January 2019; Accepted 27 February 2019; Published 1 April 2019

Guest Editor: Ling-Yun Dai

Copyright © 2019 Zun-Yan Di and Zhi-Gang Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

Abstract.

In this article, we construct the color singlet-singlet-singlet interpolating current with to study the system through QCD sum rules approach. In calculations, we consider the contributions of the vacuum condensates up to dimension-16 and employ the formula to choose the optimal energy scale of the QCD spectral density. The numerical result indicates that there exists a resonance state lying above the threshold to saturate the QCD sum rules. This resonance state may be found by focusing on the channel of the decay in the future.

1. Introduction

Since the observation of the by the Belle collaboration in 2003 [1], more and more exotic hadrons have been observed and confirmed experimentally, such as the charmonium-like , , states, hidden-charm pentaquarks, etc. [2–4]. Those exotic hadron states, which cannot be interpreted as the quark-antiquark mesons or three-quark baryons in the naive quark model [5], are good candidates of the multiquark states [6, 7]. The multiquark states are color-neutral objects because of the color confinement and provide an important platform to explore the low energy behaviors of QCD, as no free particles carrying net color charges have ever been experimentally observed. Compared to the conventional hadrons, the dynamics of the multiquark states is poorly understood and calls for more works.

Some exotic hadrons can be understood as hadronic molecular states [8], which are analogous to the deuteron as a loosely bound state of the proton and neutron. The most impressive example is the original exotic state, the , which has been studied as the molecular state by many theoretical groups [9–17]. Another impressive example is the and pentaquark states observed by the LHCb collaboration in 2015, which are good candidates for the , , molecular states [8]. In additional to the meson-meson type and meson-baryon type molecular state, there may also exist meson-meson-meson type molecular states; in other words, there may exist three-meson hadronic molecules.

In [18, 19], the authors explore the possible existence of three-meson system molecule according to the attractive interactions of the two-body subsystems , , , , and with the Born-Oppenheimer approximation and the fixed center approximation, respectively. In this article, we study the system with QCD sum rules.

The QCD sum rules method is a powerful tool in studying the exotic hadrons [20–25] and has given many successful descriptions; for example, the mass and width of the have been successfully reproduced as an axial vector tetraquark state [26, 27]. In QCD sum rules, we expand the time-ordered currents into a series of quark and gluon condensates via the operator product expansion method. These quark and gluon condensates parameterize the nonperturbative properties of the QCD vacuum. According to the quark-hadron duality, the copious information about the hadronic parameters can be obtained on the phenomenological side [28, 29].

In this article, the color singlet-singlet-singlet interpolating current with is constructed to study the system. In calculations, the contributions of the vacuum condensates are considered up to dimension-16 in the operator product expansion and the energy-scale formula is used to seek the ideal energy scale of the QCD spectral density.

The rest of this article is arranged as follows: in Section 2, we derive the QCD sum rules for the mass and pole residue of the state; in Section 3, we present the numerical results and discussions; Section 4 is reserved for our conclusion.

2. QCD Sum Rules for the State

In QCD sum rules, we consider the two-point correlation function,

where

and the , , and are color indexes. The color singlet-singlet-singlet current operator has the same quantum numbers as the system.

On the phenomenological side, a complete set of intermediate hadronic states, which has the same quantum numbers as the current operator , is inserted into the correlation function to obtain the hadronic representation [28, 29]. We isolate the ground state contribution from the pole term, and get the result:

where the pole residue is defined by , the is the polarization vector of the vector hexaquark state .

At the quark level, we calculate the correlation function via the operator product expansion method in perturbative QCD. The , , , and quark fields are contracted with the Wick theorem, and the following result is obtained:

where the , , , and are the full , , , and quark propagators, respectively. We give the full quark propagators explicitly in the following, (the denotes the or ),

and ; the is the Gell-Mann matrix [29]. We compute the integrals in the coordinate space for the light quark propagators and in the momentum space for the charm quark propagators and obtain the QCD spectral density via taking the imaginary part of the correlation function: [26]. In the operator product expansion, we take into account the contributions of vacuum condensates up to dimension-16 and keep the terms which are linear in the strange quark mass . We take the truncation for the operators of the order in a consistent way and discard the perturbative corrections. Furthermore, the condensates , , and play a minor important role and are neglected.

According to the quark-hadron duality, we match the correlation function gotten on the hadron side and at the quark level below the continuum threshold and perform Borel transform with respect to the variable to obtain the QCD sum rule:

where the QCD spectral density is

and the subscripts 0, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, and 16 denote the dimensions of the vacuum condensates, the is the Borel parameter, and the lengthy and complicated expressions are neglected for simplicity. However, for the explicit expressions of the QCD special densities, the interested readers can obtain them through emailing us.

We derive (9) with respect to and eliminate the pole residue to extract the QCD sum rule for the mass:

3. Numerical Results and Discussions

In this section, we perform the numerical analysis. To extract the numerical values of , we take the values of the vacuum condensates , , , , , at the energy scale [28–30], choose the masses , from the Particle Data Group [2], and neglect the up and down quark masses, i.e., . Moreover, we consider the energy-scale dependence of the input parameters on the QCD side from the renormalization group equation,

where , , , , , and for the flavors , and , respectively [2].

For the hadron mass, it is independent of the energy scale because of its observability. However, in calculations, the perturbative corrections are neglected, the operators of the orders with or the dimensions are discarded, and some higher dimensional vacuum condensates are factorized into lower dimensional ones; therefore, the corresponding energy-scale dependence is modified. We have to take into account the energy-scale dependence of the QCD sum rules.

In [26, 31–34], the energy-scale dependence of the QCD sum rules is studied in detail for the hidden-charm tetraquark states and molecular states, and an energy-scale formula is come up with to determine the optimal energy scale. This energy-scale formula enhances the pole contribution remarkably, improves the convergent behaviors in the operator product expansion, and works well for the exotic hadron states. In this article, we explore the state through constructing the color singlet-singlet-singlet type current based on the color-singlet substructure. For the two-meson molecular states, the basic constituent is also the color-singlet substructure [33, 34]. Hence, the previous works can be extended to study the state. We employ the energy-scale formula with the updated value of the effective -quark mass to take the ideal energy scale of the QCD spectral density.

At the present time, no candidate is observed experimentally for the hexaquark state with the symbolic quark constituent . However, in the scenario of four-quark states, the and can be tentatively assigned to be the ground state and the first radial excited state of the axial vector four-quark states, respectively [35], while the and can be tentatively assigned to be the ground state and the first radial excited state of the scalar four-quark states, respectively [36, 37]. By comparison, the energy gap is about between the ground state and the first radial excited state of the hidden-charm four-quark states. Here, we suppose the energy gap is also about between the ground state and the first radial excited state of the hidden-charm six-quark states and take the relation as a constraint to obey.

In (11), there are two free parameters: the Borel parameter and the continuum threshold parameter . The extracted hadron mass is a function of the Borel parameter and the continuum threshold parameter . To obtain a reliable mass sum rule analysis, we obey two criteria to choose suitable working ranges for the two free parameters. One criterion is the pole dominance on the phenomenological side, which requires the pole contribution (PC) to be about . The PC is defined as

The other criterion is the convergence of the operator product expansion. To judge the convergence, we compute the contributions of the vacuum condensates in the operator product expansion with the formula:

where the is the dimension of the vacuum condensates.

In Figure 1, we show the variation of the PC with respect to the Borel parameter for different values of the continuum threshold parameter at the energy scale . From the figure, we can see that the value is too tiny to obey the pole dominance criterion and result in sound Borel window for the state . To warrant the Borel platform for the mass , we take the value . In the above Borel window, if we choose the value , the PC is about . The pole dominance condition is well satisfied.

Figure 1: The pole contribution with variation of the Borel parameter .

In Figure 2, we draw the absolute contribution values of the vacuum condensates at central values of the above input parameters. From the figure, we can observe that the contribution of the perturbative term is not the dominant contribution; the contributions of the vacuum condensates with dimensions 3, 6, 8, 9, and 11 are very great. If we take the contribution of the vacuum condensate with dimension 11 as a milestone, the absolute contribution values of the vacuum condensates decrease quickly with the increase of the dimensions , and the operator product expansion converges nicely.

Figure 2: The absolute contributions of the vacuum condensates with dimension in the operator product expansion.

Thus, we obtain the values , and for the state . Considering all uncertainties of the input parameters, we get the values of the mass and pole residue of the state :

which are shown explicitly in Figures 3 and 4. Obviously, the energy-scale formula and the relation are also well satisfied. The central value is about above the threshold , which indicates that the is probably a resonance state. For some exotic resonances, the authors have combined the effective range expansion, unitarity, analyticity, and compositeness coefficient to probe their inner structure in [38, 39]. Their studies indicated that the underlying two-particle component (in the present case, corresponding to three-particle component) plays an important or minor role; in other words, there are the other hadronic degrees of freedom inside the corresponding resonance. Hence, a resonance state embodies the net effect. Considering the conservation of the angular momentum, parity and isospin, we list out the possible hadronic decay patterns of the hexaquark state :

To search for the X(3872), Belle, BaBar, and LHCb have collected numerous data in the decay . Thus, the hexaquark state may be found by focusing on the easiest channel in the experiment.

Figure 3: The mass with variation of the Borel parameter .

Figure 4: The pole residue with variation of the Borel parameters .

4. Conclusion

In this article, we construct the color singlet-singlet-singlet interpolating current operator with to study the system through QCD sum rules approach by taking into account the contributions of the vacuum condensates up to dimension-16 in the operator product expansion. In numerical calculations, we saturate the hadron side of the QCD sum rule with a hexaquark molecular state, employ the energy-scale formula to take the optimal energy scale of the QCD spectral density, and seek the ideal Borel parameter and continuum threshold by obeying two criteria of QCD sum rules for multiquark states. Finally, we obtain the mass and pole residue of the corresponding hexaquark molecular state . The predicted mass, , which lies above the threshold, indicates that the is probably a resonance state. This resonance state may be found by focusing on the channel of the decay in the future.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by National Natural Science Foundation, Grant Number 11775079.

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