Research Article | Open Access
Hua-Xing Chen, "Chiral Structure of Scalar and Pseudoscalar Mesons", Advances in High Energy Physics, vol. 2013, Article ID 750591, 44 pages, 2013. https://doi.org/10.1155/2013/750591
Chiral Structure of Scalar and Pseudoscalar Mesons
We systematically study the chiral structure of local tetraquark currents of flavor singlet and . We also investigate their chiral partners, including scalar and pseudoscalar tetraquark currents of flavor singlet, octet, , , and . We study their chiral transformation properties. Particularly, we use the tetraquark currents belonging to the “nonexotic” chiral multiplets to calculate the masses of light scalar mesons through QCD sum rule. The two-point correlation functions are calculated including all terms and only the connected parts (Weinberg (2013), Coleman (1985), and Page (2003)). The results are consistent with the experimental values.
The quark model is very successful in explaining the hadron spectrum with simply using quark-antiquark mesons and three-quark baryons [1–5]. However, there are always multiquark components in the Fock-space expansion of hadron states [6–8]. Hence, it is useful to properly include these multiquark components if we want to use Quantum Chromodynamics (QCD), the theory of strong interactions, to investigate hadrons in an exact way. Besides these “exotic” components, multiquark states themselves are also important in order to understand the low-energy behavior of QCD. These subjects have been studied for more than thirties years by lots of theoretical and experimental physicists [9–38]. Particularly, the light scalar mesons are good candidates due to their tetraquark (or molecular) components.
The light scalar mesons (or ), , , and compose a flavor nonet whose masses are all below 1 GeV . Although such mesons have been intensively studied for many years, their nature is still not fully understood [39, 40]. In the conventional quark model, light scalar mesons have a configuration of. However, because of their internal -wave orbital excitation, their masses should exceed 1 GeV and the ordering should be , which is inconsistent with the experiments. In chiral models, light scalar mesons are very important because they are chiral partners of the Nambu-Goldstone bosons, , , , and . Their masses are expected to be less than those of the quark model because of their collective nature. Light scalar mesons are also considered as tetraquark states or molecular states or containing large tetraquark components [31–35, 42, 43]. Considering that the diquark (antidiquark) inside has strong attraction, their masses are expected to be less than 1 GeV and the ordering is expected to be , which is consistent with the experiments.
To study the multiquark components of the light scalar mesons, we can use group theoretical methods, which have been applied to study quark-antiquark mesons and three-quark baryons [44–58]. Cohen and Ji studied the chiral structure of meson currents constructed using one quark and one antiquark fields and baryon currents constructed using three quark fields . In this paper we shall follow their approaches and study the chiral structure of local scalar and pseudoscalar tetraquark currents. These tetraquark currents can be used in the QCD sum rule analyses [43, 59–61] as well as the Lattice QCD calculations [6, 62–68].
In our previous references, we have applied the method of the QCD sum rule to calculate masses of light scalar mesons using local tetraquark currents [42, 69–71]. We systematically classified the scalar tetraquark currents and found that there are altogether as many as five independent scalar local tetraquark currents for each flavor structure. Therefore, right currents should be used in order to study light scalar mesons. This is also closely related to the internal structure of light scalar mesons. A similar question for the baryon case has been studied in [48–51] where there are three independent local baryon fields of flavor octet. Previously we chose some mixed currents which provided good QCD sum rule results . Although we did not know the relation of these currents with the internal structure of light scalar mesons at that time, we found that studying the chiral structure of scalar tetraquark currents can be useful to answer this question.
In this paper we shall try to answer this question (which currents should be used in order to study light scalar mesons). We shall systematically study the chiral structure of light scalar mesons through local tetraquark currents which belong to the “nonexotic” chiral multiplets. This chiral multiplet only contains flavor singlet and octet mesons, and it does not contain any meson having exotic flavor structure. Since there are no experimental signals observing scalar mesons having exotic flavors, we assume that all the nine light scalar mesons (or their dominant components) belong to this multiplet. Moreover, these nine light scalar mesons can together compose one chiral multiplet. To do a systematical study, we shall investigate both scalar and pseudoscalar tetraquark currents, since they are chiral partners. We shall also investigate tetraquark currents of flavor singlet, octet, , , and , which can be useful for further studies. We shall use the left handed quark field and the right handed quark field to rewrite these currents. After making proper combinations we can clearly see their chiral structures.
In this paper we shall use the method of QCD sum rule to calculate the masses of light scalar mesons through local scalar tetraquark currents belonging to the “nonexotic” chiral multiplets. One tetraquark current can be always written as a combination of meson-meson currents through Fierz transformation (), and so the two-point correlation function contain two parts: the disconnected parts, and the connected parts, In this paper we shall use both of them to perform QCD sum rule analysis. However, as suggested by Weinberg in his recent reference [72–74] using the large approximation: “A one tetraquark pole can only appear in the final, connected, term,” we shall also use (only) the connected parts to perform QCD sum rule analysis.
This paper is organized as follows. In Section 2 we investigate local tetraquark currents of flavor singlet and , and others are listed in Appendix A. In Section 3 we study their chiral transformation properties, and results are partly listed in Appendix B. In Section 4 we use the method of QCD sum rule to study the light scalar mesons through local scalar tetraquark currents belonging to the “nonexotic” chiral multiplets. However, the results depend much on the threshold value suggesting a large contribution from the meson-meson continuum, and so in Section 5 we use only the connected parts of the two-point correlation function to perform the QCD sum rule analyses. Section 6 is a summary.
2. Scalar Tetraquark Currents of Flavor Singlet
We write the flavor structure of tetraquarks and study local tetraquark currents of flavor singlet and : There are two possibilities to construct a flavor single tetraquark current: both of the diquark and antidiquark have the antisymmetric flavor structure or have the symmetric flavor structure . Together with five sets of Dirac matrices, , , , , and , we find the following ten independent local tetraquark currents of flavor singlet and : In these expressions the summation is taken over repeated indices ( for color indices, for flavor indices, and for Lorentz indices). The two superscripts S and denote scalar () and flavor singlet, respectively. In this paper we also need to use the following notations: is the charge-conjugation operator; is the totally antisymmetric tensor; () are the normalized totally symmetric matrices; () are the Gell-Mann matrices; () are the matrices for the flavor representation, as defined in .
Among the ten currents, five currents () contain diquarks and antidiquarks both having the antisymmetric flavor structure , and the rest () contain diquarks and antidiquarks both having the symmetric flavor structure . We note that after fixing the flavor and Lorentz structure of the internal diquarks and antidiquarks, their color structure is also fixed through Pauli’s exclusion principle, as shown in Table 1.
The chiral structure of tetraquarks is more complicated than their flavor structure: The full (expanded) expressions are shown in . Among them, the following multiplets contain flavor singlet tetraquarks currents: , , , and , as well as their mirror multiples. Tetraquarks of all flavors can be chiral partners of flavor singlet tetraquarks because of the exotic chiral multiplet.
To clearly see the chiral structure of (4), we use the left-handed quark field and the right-handed quark field to rewrite these currents and then combine them properly: from which we can quickly find out their chiral structure (representations). For example, partly contains two left-handed quarks that have an antisymmetric flavor structure and two right-handed antiquarks that also have an antisymmetric flavor structure; therefore, this part has the chiral representation , and its full chiral representation is just .
The results are listed in Table 1. There are four chiral singlets , two chiral multiplets, two chiral multiplets, and two chiral multiplets. The chiral representation contains the Nambu-Goldstone bosons, , , , and mesons, and it does not contain any meson having exotic flavor structure. Considering and are believed to be chiral partners in chiral models, we assume that light scalar mesons belong to this “nonexotic” multiplet, and we shall concentrate on it in our subsequent analysis. Based on this assumption, we do not need to study other possible tetraquark states having exotic flavor structure. Moreover, the current has the symmetric color structure , where color interactions between quarks and antiquarks are repulsive. Therefore, it is questionable to use this current, but we shall still use it to perform the QCD sum rule analysis for comparison. We note that mixed currents used in  belong to the mixing of and exotic multiplets.
To fully study this multiplet, the chiral partners of (4) are also studied, that is, the scalar and pseudoscalar tetraquark currents of flavor singlet, octet, , , and . The results are shown in Appendix A. The conventional pseudoscalar and scalar mesons made by one pair can also belong to the chiral multiplet. However, all the scalar tetraquark currents inside this multiplet have the chirality, and so they are not direct chiral partners of these mesons addressed by chiral singlet quark-antiquark pairs, which have the chirality (“chiral” Fock-space expansion), unless these two types of chirality mix with each other. Similarly, all the pseudoscalar tetraquark currents inside this multiplet have the same chirality, and so they are not (direct) terms in the “chiral” Fock-space expansion of the pseudoscalar mesons (, etc.).
3. Chiral Transformations
We can study their chiral transformation properties to verify which currents are chiral partners. Under the , , , and chiral transformations, the quark field, , transforms as where are the eight Gell-Mann matrices, an infinitesimal parameter for the transformation, the octet of group parameters, an infinitesimal parameter for the transformation, and the octet of the chiral transformations.
The chiral transformation equations for these tetraquark currents can be calculated straightforwardly, and we only show the final results. The local scalar and pseudoscalar tetraquark currents have been classified in Section 2 and Appendix A. We find that there are four chiral multiplets: there are two chiral multiplets (or mirror multiplets): there are two chiral multiplets (or mirror multiplets): there are two chiral multiplets: there are four chiral multiplets (or mirror multiplets): there is only one chiral multiplet: Their chiral transformation properties are shown in Appendix B, except those for the two chiral multiplets, which we show here. We use to denote these two multiplets, and , and they have the same chiral transformation properties: These chiral transformation equations can be compared to those calculated in  which have the same chirality and chiral representation, but in  only the flavor structure is taken into account. Theand equations are similar to those of mesons as well as baryons belonging to the same chiral multiplet [53, 58], suggesting that chiral transformation properties are closely related to chiral representations, while equations are different, which may be reasons for the anomaly.
The following formula obtained from  is used in the calculations: as well as several other formulae: The transition matrices and have been obtained and listed in . We list the transition matrices , , and in Appendix C. However, the transition matrices and are omitted due to their long expressions.
4. QCD Sum Rule Analysis
For the past decades QCD sum rule has proven to be a powerful and successful nonperturbative method [75, 76]. In sum rule analyses, we consider two-point correlation functions: where is an interpolating field (current) coupling to a tetraquark state. Here we shall choose the tetraquark currents studied in Section 2 and Appendix A. We compute in the operator product expansion (OPE) of QCD up to certain order in the expansion, which is then matched with a hadronic parametrization to extract information about hadron properties. At the hadron level, we express the correlation function in the form of the dispersion relation with a spectral function: where the integration starts from the mass square of all current quarks. The imaginal part of the two-point correlation function is For the second equation, as usual, we adopt a parametrization of one pole dominance for the ground state , where is the decay constant) and a continuum contribution. The sum rule analysis is then performed after the Borel transformation of the two expressions of the correlation functions (17) and (18): Assuming the contribution from the continuum states can be approximated well by the spectral density of OPE above a threshold value (duality), we arrive at the sum rule equation Differentiating (21) with respect to and dividing it by (21), finally we obtain
The tetraquark currents classified in Section 2 and Appendix A can couple to mesons that belong to (or partly belong to) the same representation. Here, we assume that the scalar ones belonging to the “nonexotic” chiral multiplets can couple to the light scalar mesons , , , and . Using these currents, we can calculate the mass of the light scalar mesons through the method of QCD sum rule. In the calculations, we assume an ideal mixing. Hence, the mass of the meson is calculated through tetraquark currents: whose quark contents are ; through whose quark contents are ; through whose quark contents are ; through whose quark contents are .
They lead to the following QCD sum rules where we have computed the operator product expansion up to the eighth dimension: In this expression we only show terms containing the current quark mass up to , while we keep all terms in the calculations. We also keep the terms containing the and current quark masses in the calculations, although they are quite small and give little contribution . We note that we do not include high dimension terms which can be important, particularly the tree-level term [78, 79].
In (27)–(32), many terms are cancelled, including condensates and , which are usually much larger than others. Moreover, (27) shows that effects of gluons are significant in the OPE of the meson since the up and down current quark masses are quite small. The sum rules for are the same as those for , and so we obtain the same mass for and .