Chiral Structure of Scalar and Pseudoscalar Mesons

Abstract

We systematically study the chiral structure of local tetraquark currents of flavor singlet and 𝐽^𝑃=0⁺. We also investigate their chiral partners, including scalar and pseudoscalar tetraquark currents of flavor singlet, octet, 𝟏𝟎, 𝟏𝟎, and 𝟐𝟕. We study their chiral transformation properties. Particularly, we use the tetraquark currents belonging to the “nonexotic” [(𝟑,𝟑)⊕(𝟑,𝟑)] 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.

1. Introduction

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 𝑓₀(500) (or 𝜎(500)), 𝜅(800), 𝑎₀(980), and 𝑓₀(980) compose a flavor nonet whose masses are all below 1 GeV [9]. 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 𝑞𝑞 configuration of³𝑃₀. However, because of their internal 𝑝-wave orbital excitation, their masses should exceed 1 GeV and the ordering should be 𝑚_𝜎∼𝑚_𝑎0<𝑚_𝜅<𝑚_𝑓0 [41], 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, 𝜋, 𝐾, 𝜂, and 𝜂^′ [41]. 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 𝑚_𝜎<𝑚_𝜅<𝑚_𝑓0,𝑎0, 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 [52]. 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 [42]. 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” [(𝟑,𝟑)⊕(𝟑,𝟑)] 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 [(𝟑,𝟑)⊕(𝟑,𝟑)] 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, 𝟏𝟎, 𝟏𝟎, and 𝟐𝟕, which can be useful for further studies. We shall use the left handed quark field 𝐿^𝑎_𝐴≡𝑞^𝑎_𝐿𝐴=((1−𝛾₅)/2)𝑞^𝑎_𝐴 and the right handed quark field 𝑅^𝑎_𝐴≡𝑞^𝑎_𝑅𝐴=((1+𝛾₅)/2)𝑞^𝑎_𝐴 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” [(𝟑,𝟑)⊕(𝟑,𝟑)] chiral multiplets. One tetraquark current can be always written as a combination of meson-meson currents through Fierz transformation (𝒬(𝑥)=∑_𝑖𝑗𝐶_𝑖𝑗ℬ_𝑖(𝑥)ℬ_𝑗(𝑥)), and so the two-point correlation function contain two parts: the disconnected parts, 𝒬(𝑥)𝒬(𝑦)_disconn=_{𝑖𝑗𝑘𝑙}𝐶_𝑖𝑗𝐶_𝑘𝑙ℬ_𝑖(𝑥)ℬ_𝑘𝑦₀ℬ_𝑗(𝑥)ℬ_𝑙𝑦₀, (1) and the connected parts, 𝒬(𝑥)𝒬𝑦_conn=_{𝑖𝑗𝑘𝑙}𝐶_𝑖𝑗𝐶_𝑘𝑙ℬ_𝑖(𝑥)ℬ_𝑗(𝑥)ℬ_𝑘𝑦ℬ_𝑙𝑦_0,conn. (2) 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 𝑁_𝑐 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 𝐽^𝑃=0⁺, 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” [(𝟑,𝟑)⊕(𝟑,𝟑)] chiral multiplets. However, the results depend much on the threshold value 𝑠₀ 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 𝐽^𝑃=0⁺: 𝟑⊗𝟑⊗𝟑⊗𝟑=𝟑⊕𝟔⊗𝟑⊕𝟔=𝟑⊗𝟑⊕𝟑⊗𝟔⊕(𝟔⊗𝟑)⊕𝟔⊗𝟔=(𝟏⊕𝟖)⊕𝟖⊕⊕()⊕().𝟏𝟎𝟖⊕𝟏𝟎𝟏⊕𝟖⊕𝟐𝟕 (3) There are two possibilities to construct a flavor single tetraquark current: both of the diquark and antidiquark have the antisymmetric flavor structure 𝟑_𝐹(𝑞𝑞)⊗𝟑_𝐹(𝑞𝑞)→𝟏_𝐹 or have the symmetric flavor structure 𝟔_𝐹(𝑞𝑞)⊗𝟔_𝐹(𝑞𝑞)→𝟏_𝐹. Together with five sets of Dirac matrices, 1, 𝛾₅, 𝛾_𝜇, 𝛾_𝜇𝛾₅, and 𝜎_𝜇𝜈, we find the following ten independent local tetraquark currents of flavor singlet and 𝐽^𝑃=0⁺: 𝜂^S₁^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾₅ℂ𝑞_𝐵^𝑏𝑇−𝑞^𝑏_𝐴𝛾₅ℂ𝑞_𝐵^𝑎𝑇,𝜂^S₂^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝑞^𝑏_𝐵𝑞^𝑎_𝐴ℂ𝑞_𝐵^𝑏𝑇−𝑞^𝑏_𝐴ℂ𝑞_𝐵^𝑎𝑇,𝜂^S₃^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝜎^𝜇𝜈ℂ𝑞_𝐵^𝑏𝑇+𝑞^𝑏_𝐴𝜎^𝜇𝜈ℂ𝑞_𝐵^𝑎𝑇,𝜂^S₄^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾₅ℂ𝑞_𝐵^𝑏𝑇+𝑞^𝑏_𝐴𝛾₅ℂ𝑞_𝐵^𝑎𝑇,𝜂^S₅^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝑞^𝑏_𝐵𝑞^𝑎_𝐴ℂ𝑞_𝐵^𝑏𝑇+𝑞^𝑏_𝐴ℂ𝑞_𝐵^𝑎𝑇,𝜂^S₆^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝜎^𝜇𝜈ℂ𝑞_𝐵^𝑏𝑇−𝑞^𝑏_𝐴𝜎^𝜇𝜈ℂ𝑞_𝐵^𝑎𝑇,𝜂^S₇^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐵^𝑏𝑇−𝑞^𝑏_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐵^𝑎𝑇,𝜂^S₈^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇ℂ𝑞_𝐵^𝑏𝑇+𝑞^𝑏_𝐴𝛾^𝜇ℂ𝑞_𝐵^𝑎𝑇,𝜂^S₉^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐵^𝑏𝑇+𝑞^𝑏_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐵^𝑎𝑇,𝜂^S,𝕊₁₀=𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇ℂ𝑞_𝐵^𝑏𝑇−𝑞^𝑏_𝐴𝛾^𝜇ℂ𝑞_𝐵^𝑎𝑇. (4) In these expressions the summation is taken over repeated indices (𝑎,𝑏,… for color indices, 𝐴,𝐵,… for flavor indices, and 𝜇,𝜈,… for Lorentz indices). The two superscripts S and 𝕊 denote scalar (𝐽^𝑃=0⁺) and flavor singlet, respectively. In this paper we also need to use the following notations: ℂ is the charge-conjugation operator; 𝜖^𝐴𝐵𝐶 is the totally antisymmetric tensor; 𝑆_𝑃^𝐴𝐵𝐶 (𝑃=1⋯10) are the normalized totally symmetric matrices; 𝜆_𝐍 (𝑁=1⋯8) are the Gell-Mann matrices; 𝑆_𝑈^{𝐴𝐵𝐶𝐷} (𝑈=1⋯27) are the matrices for the 𝟐𝟕 flavor representation, as defined in [58].

Among the ten currents, five currents (𝜂^S,𝕊_1,2,3,7,8) contain diquarks and antidiquarks both having the antisymmetric flavor structure 𝟑⊗𝟑, and the rest (𝜂^S,𝕊_4,5,6,9,10) contain diquarks and antidiquarks both having the symmetric flavor structure 𝟔⊗𝟔. 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.

Table 1 

Flavor singlet tetraquark currents of 𝐽^𝑃=0⁺, showing their chiral representations and chirality. The second and third columns show the flavor and color structures of the diquark and antidiquark inside, respectively.

The chiral structure of tetraquarks is more complicated than their flavor structure: ((𝟑,𝟏)⊕(𝟏,𝟑))²⊗𝟑,𝟏⊕𝟏,𝟑²=𝟑,𝟏⊕𝟏,𝟑⊕()⊕()⊕()⊕(⊗()⊕()⊕𝟔,𝟏𝟏,𝟔𝟑,𝟑𝟑,𝟑)𝟑,𝟏𝟏,𝟑𝟔,𝟏⊕𝟏,𝟔⊕𝟑,𝟑⊕𝟑,𝟑. (5) The full (expanded) expressions are shown in [58]. Among them, the following multiplets contain flavor singlet tetraquarks currents: [(𝟏,𝟏)], [(𝟑,𝟑)⊕(𝟑,𝟑)], [(𝟔,𝟔)⊕(𝟔,𝟔)], and [(𝟖,𝟖)⊕(𝟖,𝟖)], as well as their mirror multiples. Tetraquarks of all flavors can be chiral partners of flavor singlet tetraquarks because of the exotic [(𝟖,𝟖)⊕(𝟖,𝟖)] chiral multiplet.

To clearly see the chiral structure of (4), we use the left-handed quark field 𝐿^𝑎_𝐴≡𝑞^𝑎_𝐿𝐴=((1−𝛾₅)/2)𝑞^𝑎_𝐴 and the right-handed quark field 𝑅^𝑎_𝐴≡𝑞^𝑎_𝑅𝐴=((1+𝛾₅)/2)𝑞^𝑎_𝐴 to rewrite these currents and then combine them properly: 𝜂^S₁^,𝕊−𝜂^S₂^,𝕊=−2𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐵^𝑏𝑇−𝐿^𝑏_𝐴ℂ𝐿_𝐵^𝑎𝑇−2𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐵^𝑏𝑇−𝑅^𝑏_𝐴ℂ𝑅_𝐵^𝑎𝑇,𝜂^S₁^,𝕊+𝜂^S₂^,𝕊=2𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐵^𝑏𝑇−𝑅^𝑏_𝐴ℂ𝑅_𝐵^𝑎𝑇+2𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐵^𝑏𝑇−𝐿^𝑏_𝐴ℂ𝐿_𝐵^𝑎𝑇,𝜂^S₄^,𝕊−𝜂^S₅^,𝕊=−2𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐵^𝑏𝑇+𝐿^𝑏_𝐴ℂ𝐿_𝐵^𝑎𝑇−2𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐵^𝑏𝑇+𝑅^𝑏_𝐴ℂ𝑅_𝐵^𝑎𝑇,𝜂^S₄^,𝕊+𝜂^S₅^,𝕊=2𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐵^𝑏𝑇+𝑅^𝑏_𝐴ℂ𝑅_𝐵^𝑎𝑇+2𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐵^𝑏𝑇+𝐿^𝑏_𝐴ℂ𝐿_𝐵^𝑎𝑇,𝜂^S₃^,𝕊=𝐿_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝐿^𝑏_𝐵𝑅^𝑎_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐵^𝑏𝑇+𝑅^𝑏_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐵^𝑎𝑇+𝑅_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑅^𝑏_𝐵𝐿^𝑎_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐵^𝑏𝑇+𝐿^𝑏_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐵^𝑎𝑇,𝜂^S₆^,𝕊=𝐿_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝐿^𝑏_𝐵𝑅^𝑎_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐵^𝑏𝑇−𝑅^𝑏_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐵^𝑎𝑇+𝑅_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑅^𝑏_𝐵𝐿^𝑎_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐵^𝑏𝑇−𝐿^𝑏_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐵^𝑎𝑇,𝜂^S₇^,𝕊−𝜂^S,𝕊₁₀=−4𝐿_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑅^𝑏_𝐵𝐿^𝑎_𝐴𝛾^𝜇ℂ𝑅_𝐵^𝑏𝑇−𝐿^𝑏_𝐴𝛾^𝜇ℂ𝑅_𝐵^𝑎𝑇,2𝜂^S₇^,𝕊+𝜂^S,𝕊₁₀=3𝜆_𝑁^𝐶𝐴𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑅^𝑏_𝐵×𝐿^𝑎_𝐶𝛾^𝜇ℂ𝑅_𝐷^𝑏𝑇−𝐿^𝑏_𝐶𝛾^𝜇ℂ𝑅_𝐷^𝑎𝑇,𝜂^S₈^,𝕊−𝜂^S₉^,𝕊=4𝐿_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑅^𝑏_𝐵𝐿^𝑎_𝐴𝛾^𝜇ℂ𝑅_𝐵^𝑏𝑇+𝐿^𝑏_𝐴𝛾^𝜇ℂ𝑅_𝐵^𝑎𝑇,2𝜂^S₈^,𝕊+𝜂^S₉^,𝕊=−3𝜆_𝑁^𝐶𝐴𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑅^𝑏_𝐵×𝐿^𝑎_𝐶𝛾^𝜇ℂ𝑅_𝐷^𝑏𝑇+𝐿^𝑏_𝐶𝛾^𝜇ℂ𝑅_𝐷^𝑎𝑇, (6) from which we can quickly find out their chiral structure (representations). For example, 𝜂^S,𝕊₁₊₂≡𝜂^S₁^,𝕊+𝜂^S₂^,𝕊 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 (𝟑,𝟑), and its full chiral representation is just [(𝟑,𝟑)+(𝟑,𝟑)].

The results are listed in Table 1. There are four chiral singlets [(𝟏,𝟏)], two [(𝟑,𝟑)+(𝟑,𝟑)] chiral multiplets, two [(𝟔,𝟔)+(𝟔,𝟔)] chiral multiplets, and two [(𝟖,𝟖)+(𝟖,𝟖)] chiral multiplets. The [(𝟑,𝟑)+(𝟑,𝟑)] chiral representation contains the Nambu-Goldstone bosons, 𝜋, 𝐾, 𝜂, and 𝜂^′ mesons, and it does not contain any meson having exotic flavor structure. Considering 𝜎 and 𝜋 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 𝜂^S₃^,𝕊 has the symmetric color structure 𝟔⊗𝟔, 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 [42] belong to the mixing of [(𝟏,𝟏)] and exotic [(𝟖,𝟖)⊕(𝟖,𝟖)] 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, 𝟏𝟎, 𝟏𝟎, and 𝟐𝟕. The results are shown in Appendix A. The conventional pseudoscalar and scalar mesons made by one 𝑞𝑞 pair can also belong to the [(𝟑,𝟑)⊕(𝟑,𝟑)] chiral multiplet. However, all the scalar tetraquark currents inside this multiplet have the q_𝐿𝑞_𝐿𝑞_𝑅𝑞_𝑅+𝑞_𝑅𝑞_𝑅𝑞_𝐿𝑞_𝐿 chirality, and so they are not direct chiral partners of these 𝑞𝑞 mesons addressed by chiral singlet quark-antiquark pairs, which have the (𝑞_𝐿𝑞_𝑅+𝑞_𝑅𝑞_𝐿)⊗(𝑞_𝐿𝑞_𝐿+𝑞_𝑅𝑞_𝑅) 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 𝑞_𝐿𝑞_𝐿𝑞_𝑅𝑞_𝑅+𝑞_𝑅𝑞_𝑅𝑞_𝐿𝑞_𝐿 chirality, and so they are not (direct) terms in the “chiral” Fock-space expansion of the 𝑞𝑞 pseudoscalar mesons (𝜋, etc.).

3. Chiral Transformations

We can study their chiral transformation properties to verify which currents are chiral partners. Under the 𝐔(𝟏)_𝐕, 𝐔(𝟏)_𝐀, 𝐒𝐔(𝟑)_𝐕, and 𝐒𝐔(𝟑)_𝐀 chiral transformations, the quark field, 𝑞=𝑞_𝐿+𝑞_𝑅, transforms as 𝐔(𝟏)_𝐕∶𝑞⟶exp𝑖𝑎⁰(𝟑)𝑞=𝑞+𝛿𝑞,𝐒𝐔_𝐕∶𝑞⟶exp𝑖𝜆⋅𝑎𝑞=𝑞+𝛿^𝑎𝐔(𝟏)𝑞,_𝐀∶𝑞⟶exp𝑖𝛾₅𝑏⁰𝑞=𝑞+𝛿₅(𝟑)𝑞,𝐒𝐔_𝐀∶𝑞⟶exp𝑖𝛾₅𝜆⋅𝑏𝑞=𝑞+𝛿^𝑏₅𝑞, (7) where 𝜆 are the eight Gell-Mann matrices, 𝑎⁰ an infinitesimal parameter for the 𝐔(𝟏)_𝐕 transformation, 𝑎 the octet of 𝐒𝐔(𝟑)_𝐕 group parameters, 𝑏⁰ an infinitesimal parameter for the 𝐔(𝟏)_𝐀 transformation, and 𝑏 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 [(𝟏,𝟏)] chiral multiplets: 𝜂^S₁^,𝕊−𝜂^S₂^,𝕊,𝜂^PS₁^,𝕊−𝜂^PS₂^,𝕊,𝜂^S₄^,𝕊−𝜂^S₅^,𝕊,𝜂^PS₄^,𝕊−𝜂^PS₅^,𝕊,𝜂^S₇^,𝕊−𝜂^S,𝕊₁₀,𝜂^PS₇^,𝕊−𝜂^PS,𝕊₁₀,𝜂=0_S8,𝕊−𝜂^S₉^,𝕊,𝜂^PS₈^,𝕊−𝜂^PS₉^,𝕊=0; (8) there are two [(𝟑,𝟑)⊕(𝟑,𝟑)] chiral multiplets (or mirror multiplets): 𝜂^S₁^,𝕊+𝜂^S₂^,𝕊,𝜂^PS₁^,𝕊+𝜂^PS₂^,𝕊,𝜂^S,𝕆_1,𝑁+𝜂^S,𝕆_2,𝑁,𝜂^PS,𝕆_1,𝑁+𝜂^PS,𝕆_2,𝑁,𝜂^S₃^,𝕊,𝜂^PS₃^,𝕊,𝜂^S,𝕆_3,𝑁,𝜂^PS,𝕆_3,𝑁; (9) there are two [(𝟔,𝟔)⊕(𝟔,𝟔)] chiral multiplets (or mirror multiplets): 𝜂^S₄^,𝕊+𝜂^S₅^,𝕊,𝜂^PS₄^,𝕊+𝜂^PS₅^,𝕊,𝜂^S,𝕆_4,𝑁+𝜂^S,𝕆_5,𝑁,𝜂^PS,𝕆_4,𝑁+𝜂^PS,𝕆_5,𝑁,𝜂^S,𝕋𝕊_1,𝑈+𝜂^S,𝕋𝕊_2,𝑈,𝜂^PS,𝕋𝕊_1,𝑈+𝜂^PS,𝕋𝕊_2,𝑈,𝜂^S₆^,𝕊,𝜂^PS₆^,𝕊,𝜂^S,𝕆_6,𝑁,𝜂^PS,𝕆_6,𝑁,𝜂^S,𝕋𝕊_3,𝑈,𝜂^PS,𝕋𝕊_3,𝑈; (10) there are two [(𝟖,𝟖)⊕(𝟖,𝟖)] chiral multiplets: 2𝜂^S₇^,𝕊+𝜂^S,𝕊₁₀,2𝜂^PS₇^,𝕊+𝜂^PS,𝕊₁₀=0,5𝜂^S,𝕆_7,𝑁+𝜂^S,𝕆_10,𝑁,𝜂^PS,𝕆_7,𝑁+𝜂^PS,𝕆_10,𝑁,𝜂^S,𝕋𝕊_5,𝑈,𝜂^PS,𝔻_1,𝑃,𝜂^PS,𝔻_2,𝑃,2𝜂^S₈^,𝕊+𝜂^S₉^,𝕊,2𝜂^PS₈^,𝕊+𝜂^PS₉^,𝕊=0,5𝜂^S,𝕆_8,𝑁+𝜂^S,𝕆_9,𝑁,𝜂^PS,𝕆_8,𝑁+𝜂^PS,𝕆_9,𝑁,𝜂^S,𝕋𝕊_4,𝑈,𝜂^PS,𝔻_2,𝑃,𝜂^PS,𝔻_1,𝑃; (11) there are four [(𝟖,𝟏)⊕(𝟏,𝟖)] chiral multiplets (or mirror multiplets): 𝜂^S,𝕆_1,𝑁−𝜂^S,𝕆_2,𝑁,𝜂^PS,𝕆_1,𝑁−𝜂^PS,𝕆_2,𝑁,𝜂^S,𝕆_4,𝑁−𝜂^S,𝕆_5,𝑁,𝜂^PS,𝕆_4,𝑁−𝜂^PS,𝕆_5,𝑁,𝜂^S,𝕆_7,𝑁−𝜂^S,𝕆_10,𝑁,𝜂^PS,𝕆_7,𝑁−𝜂^PS,𝕆_10,𝑁,𝜂^S,𝕆_8,𝑁−𝜂^S,𝕆_9,𝑁,𝜂^PS,𝕆_8,𝑁−𝜂^PS,𝕆_9,𝑁; (12) there is only one [(𝟐𝟕,𝟏)⊕(𝟏,𝟐𝟕)] chiral multiplet: 𝜂^S,𝕋𝕊_1,𝑈−𝜂^S,𝕋𝕊_2,𝑈,𝜂^PS,𝕋𝕊_1,𝑈−𝜂^PS,𝕋𝕊_2,𝑈. (13) Their chiral transformation properties are shown in Appendix B, except those for the two [(𝟑,𝟑)⊕(𝟑,𝟑)] chiral multiplets, which we show here. We use (𝜂^S₍^,𝕊_𝟑,𝟑),𝜂^PS₍^,𝕊_𝟑,𝟑),𝜂^S₍^,𝕆_{𝟑,𝟑),𝑁},𝜂^PS₍^,𝕆_{𝟑,𝟑),𝑁}) to denote these two multiplets, (𝜂^S₁^,𝕊+𝜂^S₂^,𝕊,𝜂^PS₁^,𝕊+𝜂^PS₂^,𝕊,𝜂^S,𝕆_1,𝑁+𝜂^S,𝕆_2,𝑁,𝜂^PS,𝕆_1,𝑁+𝜂^PS,𝕆_2,𝑁) and (𝜂^S₃^,𝕊,𝜂^PS₃^,𝕊,𝜂^S,𝕆_3,𝑁,𝜂^PS,𝕆_3,𝑁), and they have the same chiral transformation properties: 𝛿₅𝜂^S(PS₍^),𝕊_𝟑,𝟑)=4𝑖𝑏𝜂^PS(S₍^),𝕊_𝟑,𝟑),𝛿^𝑎𝜂^S(PS₍^),𝕊_𝟑,𝟑)𝛿=0,_𝑏5𝜂^S(PS),𝕊_𝟑,𝟑=4𝑖𝑏^𝑁𝜂^PS(S₍^),𝕆_{𝟑,𝟑),𝑁},𝛿₅𝜂^S(PS₍^),𝕆_{𝟑,𝟑),𝑁}=4𝑖𝑏𝜂^PS(S₍^),𝕆_{𝟑,𝟑),𝑁},𝛿^𝑎𝜂^S(PS₍^),𝕆_{𝟑,𝟑),𝑁}=2𝑎^𝑁𝑓_𝑁𝑀𝑂𝜂^S(PS₍^),𝕆_{𝟑,𝟑),𝑂},𝛿^𝑏₅𝜂^S(PS₍^),𝕆_{𝟑,𝟑),𝑁}=23𝑖𝑏^𝑀𝜂^PS(S₍^),𝕊_𝟑,𝟑)−2𝑖𝑏^𝑁𝑑_𝑁𝑀𝑂𝜂^PS(S₍^),𝕆_{𝟑,𝟑),𝑂}. (14) These chiral transformation equations can be compared to those calculated in [58] which have the same chirality and chiral representation, but in [58] only the flavor structure is taken into account. The𝐒𝐔(𝟑)_𝐕and 𝐒𝐔(3)_𝐀 equations are similar to those of 𝑞𝑞 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 𝐔(1)_𝐀 equations are different, which may be reasons for the 𝑈(1)_𝐴 anomaly.

The following formula obtained from [58] is used in the calculations:𝜆^𝑁_𝐶𝐴𝜆^𝑀_𝐷𝐵1=−𝛿12^𝑁𝑀𝛿_𝐶𝐴𝛿_𝐷𝐵+14𝛿^𝑁𝑀𝛿_𝐷𝐴𝛿_𝐶𝐵−25𝑑_𝑁𝑀𝑂𝛿_𝐶𝐴𝜆^𝑂_𝐷𝐵+35𝑑_𝑁𝑀𝑂+𝑖3𝑓_𝑁𝑀𝑂𝛿_𝐷𝐴𝜆^𝑂_𝐶𝐵+35𝑑_𝑁𝑀𝑂−𝑖3𝑓_𝑁𝑀𝑂𝛿_𝐶𝐵𝜆^𝑂_𝐷𝐴−25𝑑_𝑁𝑀𝑂𝛿_𝐷𝐵𝜆^𝑂_𝐶𝐴+𝐓^𝑁_8×10^∗_𝑀𝑃𝜖_𝐴𝐵𝐸𝑆^{𝟏𝟎,𝑃}_𝐶𝐷𝐸+𝐓^𝑁_{8×10𝑀𝑃}𝜖_𝐶𝐷𝐸𝑆^{𝟏𝟎,𝑃}_𝐴𝐵𝐸+𝐓^𝑁_{8×27𝑀𝑈}𝑆^{𝟐𝟕,𝑈}_{𝐶𝐷;𝐴𝐵}, (15) as well as several other formulae: 𝜖_𝐴𝐵𝐸𝑆^{𝟏𝟎,𝑃}_𝐶𝐺𝐸𝜆^𝑁_𝐷𝐺=𝟎^𝑁𝑃×𝛿_𝐶𝐴𝛿_𝐷B+𝟎^𝑁𝑃×𝛿_𝐷𝐴𝛿_𝐶𝐵−13𝐓^𝑁_8×10^𝑇_𝑃𝑂𝛿_𝐶𝐴𝜆^𝑂_𝐷𝐵+23𝐓^𝑁_8×10^𝑇_𝑃𝑂𝛿_𝐷𝐴𝜆^𝑂_𝐶𝐵+13𝐓^𝑁_8×10^𝑇_𝑃𝑂𝛿_𝐶𝐵𝜆^𝑂_𝐷𝐴−23𝐓^𝑁_8×10^𝑇_𝑃𝑂𝛿_𝐷𝐵𝜆^𝑂_𝐶𝐴+𝐓^𝑁_10×10^𝑇_𝑃𝑄𝜖_𝐴𝐵𝐸𝑆^{𝟏𝟎,𝑃}_𝐶𝐷𝐸+𝟎^𝑁𝑃𝑄×𝜖_𝐶𝐷𝐸𝑆^{𝟏𝟎,𝑃}_𝐴𝐵𝐸+𝟎^𝑁𝑃𝑈×𝑆^{𝟐𝟕,𝑈}_{𝐶𝐷;𝐴𝐵},𝑆^{𝟏𝟎,𝑃}_𝐴𝐺𝐸𝜖_𝐶𝐷𝐸𝜆^𝑁_𝐺𝐵=𝟎^𝑁𝑃×𝛿_𝐶𝐴𝛿_𝐷𝐵+𝟎^𝑁𝑃×𝛿_𝐷𝐴𝛿_𝐶𝐵−13𝐓^𝑁_8×10^†_𝑃𝑂𝛿_𝐶𝐴𝜆^𝑂_𝐷𝐵+13𝐓^𝑁_8×10^†_𝑃𝑂𝛿_𝐷𝐴𝜆^𝑂_𝐶𝐵+23𝐓^𝑁_8×10^†_𝑃𝑂𝛿_𝐶𝐵𝜆^𝑂_𝐷𝐴−23𝐓^𝑁_8×10^†_𝑃𝑂𝛿_𝐷𝐵𝜆^𝑂_𝐶𝐴+𝟎^𝑁𝑃𝑄𝜖_𝐴𝐵𝐸𝑆^{𝟏𝟎,𝑃}_𝐶𝐷𝐸+𝐓^𝑁_{10×10𝑃𝑄}×𝜖_𝐶𝐷𝐸𝑆^{𝟏𝟎,𝑃}_𝐴𝐵𝐸+𝟎^𝑁𝑃𝑈×𝑆^{𝟐𝟕,𝑈}_{𝐶𝐷;𝐴𝐵},𝜖_𝐴𝐺𝐸𝑆^{𝟏𝟎,𝑃}_𝐶𝐷𝐸𝜆^𝑁_𝐺𝐵=𝟎^𝑁𝑃×𝛿_𝐶𝐴𝛿_𝐷𝐵+𝟎^𝑁𝑃×𝛿_𝐷𝐴𝛿_𝐶𝐵+1𝐓15^{𝑁8×10𝑇𝑃𝑂}𝛿_𝐶𝐴𝜆^𝑂_𝐷𝐵+1𝐓15^{𝑁8×10𝑇𝑃𝑂}𝛿_𝐷𝐴𝜆^𝑂_𝐶𝐵−4𝐓15^{𝑁8×10𝑇𝑃𝑂}𝛿_𝐶𝐵𝜆^𝑂_𝐷𝐴−4𝐓15^{𝑁8×10𝑇𝑃𝑂}𝛿_𝐷𝐵𝜆^𝑂_𝐶𝐴−12𝐓^𝑁_10×10^𝑇_𝑃𝑄𝜖_𝐴𝐵𝐸𝑆^{𝟏𝟎,𝑃}_𝐶𝐷𝐸+𝟎^𝑁𝑃𝑄×𝜖_𝐶𝐷𝐸𝑆^{𝟏𝟎,𝑃}_𝐴𝐵𝐸+𝐓^𝐁_10×27^𝑁_𝑃𝑈𝑆^{𝟐𝟕,𝑈}_{𝐶𝐷;𝐴𝐵},𝑆^{𝟏𝟎,𝑃}_𝐴𝐵𝐸𝜖_𝐶𝐺𝐸𝜆^𝑁_𝐷𝐺=𝟎^𝑁𝑃×𝛿_𝐶𝐴𝛿_𝐷𝐵+𝟎^𝑁𝑃×𝛿_𝐷𝐴𝛿_𝐶𝐵+1𝐓15^{𝑁8×10†𝑃𝑂}𝛿_𝐶𝐴𝜆^𝑂_𝐷𝐵−4𝐓15^{𝑁8×10†𝑃𝑂}𝛿_𝐷𝐴𝜆^𝑂_𝐶𝐵+1𝐓15^{𝑁8×10†𝑃𝑂}𝛿_𝐶𝐵𝜆^𝑂_𝐷𝐴−4𝐓15^{𝑁8×10†𝑃𝑂}𝛿_𝐷𝐵𝜆^𝑂_𝐶𝐴+𝟎^𝑁𝑃𝑄𝜖_𝐴𝐵𝐸𝑆^{𝟏𝟎,𝑃}_𝐶𝐷𝐸−12𝐓^𝑁_{10×10𝑃𝑄}×𝜖_𝐶𝐷𝐸𝑆^{𝟏𝟎,𝑃}_𝐴𝐵𝐸+𝐓^𝐀_10×27^𝑁_𝑃𝑈𝑆^{𝟐𝟕,𝑈}_{𝐶𝐷;𝐴𝐵},𝑆^{𝟐𝟕,𝑈}_{𝐶𝐷;𝐴𝐺}𝜆^𝑁_𝐺𝐵=𝟎^𝑁𝑈×𝛿_𝐶𝐴𝛿_𝐷𝐵+𝟎^𝑁𝑈×𝛿_𝐷𝐴𝛿_𝐶𝐵−1𝐓10^{𝑁8×27†𝑈𝑂}𝛿_𝐶𝐴𝜆^𝑂_𝐷𝐵−1𝐓10^{𝑁8×27†𝑈𝑂}𝛿_𝐷𝐴𝜆^𝑂_𝐶𝐵+25𝐓^𝑁_8×27^†_𝑈𝑂𝛿_𝐶𝐵𝜆^𝑂_𝐷𝐴+25𝐓^𝑁_8×27^†_𝑈𝑂𝛿_𝐷𝐵𝜆^𝑂_𝐶𝐴+32𝐓^𝐁_10×27^𝑁†_𝑈𝑃×𝜖_𝐴𝐵𝐸𝑆^{𝟏𝟎,𝑃}_𝐶𝐷𝐸+𝟎^𝑁𝑈𝑃×𝜖_𝐶𝐷𝐸𝑆^{𝟏𝟎,𝑃}_𝐴𝐵𝐸+𝐓^𝐀_27×27^𝑁_𝑈𝑉𝑆^{𝟐𝟕,𝑈}_{𝐶𝐷;𝐴𝐵},𝑆^{𝟐𝟕,𝑈}_{𝐶𝐺;𝐴𝐵}𝜆^𝑁_𝐷𝐺=𝟎^𝑁𝑈×𝛿_𝐶𝐴𝛿_𝐷𝐵+𝟎^𝑁𝑈×𝛿_𝐷𝐴𝛿_𝐶𝐵−1𝐓10^{𝑁8×27†𝑈𝑂}𝛿_𝐶𝐴𝜆^𝑂_𝐷𝐵+25𝐓^𝑁_8×27^†_𝑈𝑂𝛿_𝐷𝐴𝜆^𝑂_𝐶𝐵−1𝐓10^{𝑁8×27†𝑈𝑂}𝛿_𝐶𝐵𝜆^𝑂_𝐷𝐴+25𝐓^𝑁_8×27^†_𝑈𝑂𝛿_𝐷𝐵𝜆^𝑂_𝐶𝐴+𝟎^𝑁𝑈𝑃×𝜖_𝐴𝐵𝐸𝑆^{𝟏𝟎,𝑃}_𝐶𝐷𝐸+32𝐓^𝐀_10×27^𝑁†_𝑈𝑃×𝜖_𝐶𝐷𝐸𝑆^{𝟏𝟎,𝑃}_𝐴𝐵𝐸+𝐓^𝐁_27×27^𝑁_𝑈𝑉𝑆^{𝟐𝟕,𝑈}_{𝐶𝐷;𝐴𝐵}. (16) The transition matrices 𝐓^𝑁_8×10 and 𝐓^𝑁_8×27 have been obtained and listed in [58]. We list the transition matrices 𝐓_10×10, 𝐓^𝐀_10×27, and 𝐓^𝐁_10×27 in Appendix C. However, the transition matrices 𝐓^𝐀_27×27 and 𝐓^𝐁_27×27 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: Π𝑞²≡𝑖𝑑⁴𝑥𝑒^𝑖𝑞𝑥⟨0|𝑇𝐽(𝑥)𝐽^†(0)|0⟩, (17) where 𝐽(𝑥) 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 Π(𝑞²) 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: Π𝑞²=1𝜋^∞_𝑠<ImΠ(𝑠)𝑠−𝑞²−𝑖𝜀𝑑𝑠, (18) where the integration starts from the mass square of all current quarks. The imaginal part of the two-point correlation function is ImΠ(𝑠)≡𝜋_𝑛𝛿𝑠−𝑀²_𝑛⟨0|𝜂|𝑛⟩⟨𝑛|𝜂^†|0⟩. (19) For the second equation, as usual, we adopt a parametrization of one pole dominance for the ground state 𝑌(⟨0|𝜂(0)|𝑌⟩≡𝑓_𝑌, where 𝑓_𝑌 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): Π^(all)𝑀²_𝐵≡ℬ_𝑀²_𝐵Π𝑝²=1𝜋^∞_𝑠<𝑒^{−𝑠/𝑀2𝐵}ImΠ(𝑠)𝑑𝑠. (20) Assuming the contribution from the continuum states can be approximated well by the spectral density of OPE above a threshold value 𝑠₀ (duality), we arrive at the sum rule equation 𝑓²_𝑌𝑒^{−𝑀2𝑌/𝑀2𝐵}=Π𝑠₀,𝑀²_𝐵≡1𝜋^𝑠0_𝑠<𝑒^{−𝑠/𝑀2𝐵}ImΠ(𝑠)𝑑𝑠. (21) Differentiating (21) with respect to 1/𝑀²_𝐵 and dividing it by (21), finally we obtain 𝑀²_𝑌=𝜕/𝜕−1/𝑀²_𝐵Π𝑠₀,𝑀²_𝐵Π𝑠₀,𝑀²_𝐵. (22)

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” [(𝟑,𝟑)⊕(𝟑,𝟑)] chiral multiplets can couple to the light scalar mesons 𝑓₀(500), 𝜅(800), 𝑎₀(980), and 𝑓₀(980). 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 𝑓₀(500) meson is calculated through tetraquark currents: 𝜎₁₊₂=16𝜂^S₁^,𝕊+𝜂^S₂^,𝕊+1√3𝜂^S,𝕆_1,𝑁=8+𝜂^S,𝕆_2,𝑁=8,𝜎₃=16𝜂^S₃^,𝕊+1√3𝜂^S,𝕆_3,𝑁=8, (23) whose quark contents are 𝑢𝑑𝑢𝑑; 𝜅⁺(800) through 𝜅₁₊₂=12𝜂^S,𝕆_1,𝑁=4+𝜂^S,𝕆_2,𝑁=4−𝑖𝜂^S,𝕆_1,𝑁=5−𝑖𝜂^S,𝕆_2,𝑁=5,𝜅₃=12𝜂^S,𝕆_3,𝑁=4−𝑖𝜂^S,𝕆_3,𝑁=5, (24) whose quark contents are 𝑢𝑑𝑑𝑠; 𝑎⁺₀(980) through 𝑎0₁₊₂=12𝜂^S,𝕆_1,𝑁=1+𝜂^S,𝕆_2,𝑁=1−𝑖𝜂^S,𝕆_1,𝑁=2−𝑖𝜂^S,𝕆_2,𝑁=2,𝑎0₃=12𝜂^S,𝕆_3,𝑁=1−𝑖𝜂^S,𝕆_3,𝑁=2, (25) whose quark contents are 𝑢𝑠𝑑𝑠; 𝑓₀(980) through 𝑓0₁₊₂=13√2𝜂^S₁^,𝕊+𝜂^S₂^,𝕊−1√6𝜂^S,𝕆_1,𝑁=8+𝜂^S,𝕆_2,𝑁=8,𝑓0₃=13√2𝜂^S₃^,𝕊−1√6𝜂^S,𝕆_3,𝑁=8, (26) whose quark contents are 𝑢𝑠𝑢𝑠+𝑑𝑠𝑑𝑠.

They lead to the following QCD sum rules where we have computed the operator product expansion up to the eighth dimension: 𝑓²_𝜎1+2𝑒^{−𝑀2𝜎1+2/𝑀2𝐵}=Π^𝜎1+2𝑠₀,𝑀²_𝐵=^𝑠0_𝑠<𝑒^{−𝑠/𝑀2𝐵}1𝑑𝑠×30720𝜋⁶𝑠⁴+𝑔²𝐺𝐺3072𝜋⁶𝑠²,𝑓²_𝜎3𝑒^{−𝑀2𝜎3/𝑀2𝐵}=Π^𝜎3𝑠₀,𝑀²_𝐵=^𝑠0_𝑠<𝑒^{−𝑠/𝑀2𝐵}1𝑑𝑠×1280𝜋⁶𝑠⁴+11𝑔²𝐺𝐺768𝜋⁶𝑠²,𝑓²_𝜅1+2𝑒^{−𝑀2𝜅1+2/𝑀2𝐵}=Π^𝜅1+2𝑠₀,𝑀²_𝐵=^𝑠0_𝑠<𝑒^{−𝑠/𝑀2𝐵}×1𝑑𝑠30720𝜋⁶𝑠⁴−𝑚²_𝑠1536𝜋⁶𝑠³+𝑔²𝐺𝐺3072𝜋⁶+𝑚_𝑠⟨𝑠𝑠⟩192𝜋⁴𝑠²−𝑚²_𝑠𝑔²𝐺𝐺1024𝜋⁶𝑚𝑠+^𝑠𝑔²𝐺𝐺⟨𝑠𝑠⟩768𝜋⁴,𝑓²_𝜅3𝑒^{−𝑀2𝜅3/𝑀2𝐵}=Π^𝜅3𝑠₀,𝑀²_𝐵=^𝑠0_𝑠<𝑒^{−𝑠/𝑀2𝐵}×1𝑑𝑠1280𝜋⁶𝑠⁴−𝑚²_𝑠64𝜋⁶𝑠³+11⟨𝑔²𝐺𝐺⟩768𝜋⁶+𝑚_𝑠⟨𝑠𝑠⟩8𝜋⁴𝑠²−11𝑚²_𝑠⟨𝑔²𝐺𝐺⟩256𝜋⁶𝑠+11𝑚_𝑠⟨𝑔²𝐺𝐺⟩⟨𝑠𝑠⟩192𝜋⁴,𝑓²_𝑎01+2𝑒^{−𝑀2𝑎01+2/𝑀2𝐵}=Π^𝑎01+2𝑠₀,𝑀²_𝐵=^𝑠0_𝑠<𝑒^{−𝑠/𝑀2𝐵}×1𝑑𝑠30720𝜋⁶𝑠⁴−𝑚²_𝑠768𝜋⁶𝑠³+𝑔²𝐺𝐺3072𝜋⁶+𝑚_𝑠⟨𝑠𝑠⟩96𝜋⁴𝑠²−𝑚²_𝑠𝑔²𝐺𝐺512𝜋⁶𝑠+𝑚_𝑠𝑔²𝐺𝐺⟨𝑠𝑠⟩384𝜋⁴+𝑚²_𝑠4𝑞𝑞²+⟨𝑠𝑠⟩²24𝜋²,𝑓²_𝑎03𝑒^{−𝑀23𝑎0/𝑀2𝐵}=Π^𝑎03𝑠₀,𝑀²_𝐵=^𝑠0_𝑠<𝑒^{−𝑠/𝑀2𝐵}×1𝑑𝑠1280𝜋⁶𝑠⁴−𝑚²_𝑠32𝜋⁶𝑠³+11𝑔²𝐺𝐺768𝜋⁶+𝑚_𝑠⟨𝑠𝑠⟩4𝜋⁴𝑠²−11𝑚²_𝑠𝑔²𝐺𝐺128𝜋⁶𝑠+11𝑚_𝑠𝑔²𝐺𝐺⟨𝑠𝑠⟩96𝜋⁴+𝑚²_𝑠4𝑞𝑞²+⟨𝑠𝑠⟩²𝜋². (27)(28)(29)(30)(31)(32) In this expression we only show terms containing the 𝑠𝑡𝑟𝑎𝑛𝑔𝑒 current quark mass up to 𝑚²_𝑠, while we keep all terms in the calculations. We also keep the terms containing the 𝑢𝑝 and 𝑑𝑜𝑤𝑛 current quark masses in the calculations, although they are quite small and give little contribution [77]. We note that we do not include high dimension terms which can be important, particularly the tree-level term 𝛼_𝑠⟨𝑞𝑞⟩⁴ [78, 79].

In (27)–(32), many terms are cancelled, including condensates ⟨𝑞𝑞⟩² and ⟨𝑞𝑞⟩⟨𝑔𝑞𝐺𝑞⟩, which are usually much larger than others. Moreover, (27) shows that effects of gluons are significant in the OPE of the 𝜎 meson since the up and down current quark masses are quite small. The sum rules for 𝑓₀(980) are the same as those for 𝑎₀(980), and so we obtain the same mass for 𝑎₀(980) and 𝑓₀(980).

To perform the numerical analysis, we use the following values for the condensates and other parameters, which correspond to the energy scale of 1 GeV [9, 59, 80–86]: ⟨𝑞𝑞⟩=−(0.240±0.010)³GeV³,⟨𝑠𝑠⟩=−(0.8±0.1)×(0.240±0.010)³GeV³,⟨𝑔²_𝑠𝐺𝐺⟩=(0.48±0.14)GeV⁴,𝑚_𝑢=2.90±0.20MeV,𝑚_𝑑𝑚=6.35±0.27MeV,_𝑠𝑔=125±20MeV,_𝑠𝑞𝜎𝐺𝑞=−𝑀²₀×,𝑀𝑞𝑞₂₀=(0.8±0.2)GeV². (33) As usual we assume the vacuum saturation for higher dimensional operators such as ⟨0|𝑞𝑞𝑞𝑞|0⟩∼⟨0|𝑞𝑞|0⟩⟨0|𝑞𝑞|0⟩. There is a minus sign in the definition of the mixed condensate ⟨𝑔_𝑠𝑞𝜎𝐺𝑞⟩, which is different from that used in some other QCD sum rule studies. This difference just comes from the definition of coupling constant 𝑔_𝑠 [80, 81].

We use the current 𝜎₁₊₂ as an example. First we extract its spectral density from (27) and show it in Figure 1 as a function of the energy 𝑠. It is almost positive definite, and so we can use it to perform QCD sum rule analyses. Then we need to study its OPE convergence. The Borel transformed correlation function of the current 𝜎₁₊₂ is shown in Figure 2, when we take 𝑠₀=0.4GeV². We can clearly see that the 𝐷=4 terms give large contributions, and the convergence is good in the region 𝑀_𝐵>0.5 GeV, where OPEs are reliable. To fix the upper bound of the Borel window, we need to use the pole contribution, defined as the pole part divided by the sum of the pole and the continuum parts in the two-point correlation function equation (17): ∫Polecontribution≡^𝑠00𝑒^{−𝑠/𝑀2𝐵}𝜌(𝑠)𝑑𝑠∫^∞₀𝑒^{−𝑠/𝑀2𝐵}𝜌(𝑠).𝑑𝑠 (34) It nearly vanishes for 𝜅(800) meson when using 𝜅₁₊₂, as shown in Table 2. For the 𝑓₀(500) meson it is also not large. This suggests that the two-meson continuum contributes significantly. Only for the 𝑎₀(980) and 𝑓₀(980) mesons it is acceptable. Mathematically, this is because the continuum term is growing as 𝑠⁴ and the condensates ⟨𝑞𝑞⟩² and ⟨𝑞𝑞⟩⟨𝑔𝑞𝐺𝑞⟩ cancelled.

Table 2 

Pole contributions of various currents.

Figure 1 

Spectral density for the current 𝜎1+2 as a function of the energy 𝑠.

Figure 2 

Various contributions to the correlation function of the current 𝜎1+2 as functions of the Borel mass 𝑀𝐵 in units of GeV10 at 𝑠0 = 0.4 GeV2. The labels indicate the dimension up to which the OPE terms are included.

We show masses of light scalar mesons as functions of the Borel mass 𝑀_𝐵 and the threshold value 𝑠₀ in Figures 4 and 5, using solid curves. The masses of 𝜎, 𝜅, and 𝑎₀ (𝑓₀) are around 600 MeV, 900 MeV, and 1100 MeV, respectively. However, these results much depend on threshold value 𝑠₀, especially for 𝜎 and 𝑎₀, once more suggesting that the contribution of meson-meson continuum cannot be neglected. In such cases, the use of local quark-hadron duality with one resonance approximation is not valid.

In order to use the quark-hadron duality and obtain more reliable QCD sum rules, we should try to increase the pole contribution. This can be done by slightly changing the mixing parameters of currents 𝜂₁₊₂, which have the antisymmetric color structure 𝟑⊗𝟑 and color interactions between quarks and antiquarks are repulsive (the details expressions are similar to (23), (24), (25), and (26)): 𝜎₁₊₂∼𝜂^S₁^,𝕊(𝕆)+𝜂^S₂^,𝕊(𝕆)⟶𝜎^mod₁₊₂∼0.99×𝜂^S₁^,𝕊(𝕆)+𝜂^S₂^,𝕊(𝕆),𝜅₁₊₂∼𝜂^S₁^,𝕊(𝕆)+𝜂^S₂^,𝕊(𝕆)⟶𝜅^mod₁₊₂∼0.99×𝜂^S₁^,𝕊(𝕆)+𝜂^S₂^,𝕊(𝕆),𝑎0₁₊₂∼𝜂^S₁^,𝕊(𝕆)+𝜂^S₂^,𝕊(𝕆)⟶𝑎0^mod₁₊₂∼0.99×𝜂^S₁^,𝕊(𝕆)+𝜂^S₂^,𝕊(𝕆),𝑓0₁₊₂∼𝜂^S₁^,𝕊(𝕆)+𝜂^S₂^,𝕊(𝕆)⟶𝑓0^mod₁₊₂∼0.99×𝜂^S₁^,𝕊(𝕆)+𝜂^S₂^,𝕊(𝕆). (35) We note that doing this we introduce a few [(𝟖,𝟏)⊕(𝟏,𝟖)] components, which are still “nonexotic.” Mathematically, the condensates ⟨𝑞𝑞⟩² and ⟨𝑞𝑞⟩⟨𝑔𝑞𝐺𝑞⟩ appear and contribute, although the mixing parameters are only slightly modified.

Still we use the current 𝜎^mod₁₊₂ as an example. The comparison between pole and continuum contributions for 𝑠₀=0.4 GeV² is shown in Figure 3 [87–89]. We find that the pole contribution is significantly increased to around 50% when 𝑀_𝐵 is around 0.5 MeV, but it decreases very quickly as the Borel mass increases. Therefore, we obtain a very narrow Borel window around 𝑀_𝐵∼0.5 GeV.

Figure 3 

The solid curve shows the pole contribution and the dashed curve shows the continuum contribution (=1− pole contribution).

Figure 4 

Mass of light scalar mesons 𝑓0(500), 𝜅(800), and 𝑎0(980) (𝑓0(980)) as functions of Borel mass 𝑀𝐵. The threshold values 𝑠0 for 𝑓0(500), 𝜅(800), and 𝑎0(980) (𝑓0(980)) are chosen to be 0.40, 0.90, and 1.80GeV2, respectively. The solid curves are obtained using the tetraquark currents 𝜎1+2, 𝜎3, 𝜅1+2, 𝜅3, 𝑎01+2, and 𝑎03 (see definitions in (23), (24), (25), and (26)), while the dashed curves are obtained using the modified currents 𝜎mod1+2, 𝜅mod1+2, and 𝑎0mod1+2 (see definitions in (35)). The results for 𝑓0(980) are the same as those for 𝑎0(980).

Figure 5 

Mass of light scalar mesons 𝑓0(500), 𝜅(800), and 𝑎0(980) (𝑓0(980)) as functions of the threshold value 𝑠0. The Borel mass 𝑀𝐵 for 𝑓0(500), 𝜅(800), and 𝑎0(980) (𝑓0(980)) are chosen to be 0.50, 0.60, and 0.80GeV2, respectively. The solid curves are obtained using the tetraquark currents 𝜎1+2, 𝜎3, 𝜅1+2, 𝜅3, 𝑎01+2, and 𝑎03 (see definitions in (23), (24), (25), and (26)), while the dashed curves are obtained using the modified currents 𝜎mod1+2, 𝜅mod1+2, and 𝑎0mod1+2 (see definitions in (35)). The results for 𝑓0(980) are the same as those for 𝑎0(980).

Using the modified currents listed in (35) we calculate masses of light scalar mesons. The results are shown in Figure 4 as functions of Borel mass 𝑀_𝐵, but using dashed curves. The masses of 𝜎, 𝜅, and 𝑎₀ (𝑓₀) are around 500 MeV, 700 MeV, and 900 MeV, respectively, better consistent with the experimental results. The pole contributions are significantly increased to be around 50% for 𝑓₀(500) and 𝜅(800), as shown in Table 2. Since there is still 50% continuum and considering that nearly all the continuum comes from the 𝜋-𝜋 contribution, the 𝜎 meson is probably still contributed significantly by its underlying 𝜋-𝜋 continuum.

We have also studied the threshold value 𝑠₀ dependence. The results are shown in Figure 5, using dashed lines. We can see that 𝑠₀ dependence is still significant suggesting the contribution of meson-meson continuum cannot be neglected. To solve this problem, we shall use only the connected parts of the two-point correlation function to perform the QCD sum rule analysis in the next section [72–74].

To investigate this meson-meson continuum we simply use the theory of relativity to estimate how far at most the two final pseudoscalar mesons can travel away from each other in the lifetimes of the initial light scalar mesons. From this distance we shall clearly see the difficulty to separate the meson-meson continuum. Our assumptions are very simple and straightforward: the initial state, an unstable particle 𝑋, is at rest in the beginning; it has mass 𝑚_𝑋 and decay width Γ_𝑋; it decays into two particles 𝐴 and 𝐵, having masses 𝑚_𝐴 and 𝑚_𝐵, respectively; when 𝑋 is decaying into 𝐴 and 𝐵, the mass difference between the initial and final states, 𝑚_𝑋−𝑚_𝐴−𝑚_𝐵, is totally and immediately transferred into kinetic energies of 𝐴 and 𝐵; this makes they have speeds 𝑣_𝐴 and 𝑣_𝐵, in opposite direction. This process can be easily described using the following equations: 𝑚_𝐴𝑐²1−𝑣²_𝐴/𝑐²+𝑚_𝐵𝑐²1−𝑣²_𝐵/𝑐²=𝑚_𝑋𝑐²,𝑚_𝐴1−𝑣²_𝐴/𝑐²𝑣_𝐴+𝑚_𝐵1−𝑣²_𝐵/𝑐²𝑣_𝐵=0. (36) Here 𝑐 is the speed of light. The quantity 𝑑_𝑋≡(ℏ/Γ_𝑋)(|𝑣_𝐴|+|𝑣_𝐵|) is just the farthest distance that 𝐴 and 𝐵 can travel away from each other in the half-life of 𝑋. We can use the uncertainty principle Δ𝑥Δ𝑝≥ℏ/2 to estimate the theoretical uncertainty of 𝑑_𝑋: Δ𝑑_𝑋=ℏ21𝑚_𝐴/1−𝑣²_𝐴/𝑐²𝑣_𝐴+𝑚_𝐵/1−𝑣²_𝐵/𝑐²𝑣_𝐵. (37) Using these equations we obtain: 𝑑_{𝑓0(500)→𝜋𝜋}=0.41−0.85 fm, 𝑑_{𝜅(800)→𝜋𝐾}=0.19−0.37 fm, 𝑑_{𝑎0(980)→𝜋𝜂}=2.76−5.69 fm, and 𝑑_{𝑓0(980)→𝜋𝜋}=3.79−9.50 fm. The relevant theoretical error bars from the uncertainty principle are Δ𝑑_𝑓0(500)=0.34 fm, Δ𝑑_𝜅(800)=0.76 fm, Δ𝑑_𝑎0(980)=0.16 fm, and Δ𝑑_𝑓0(980)=0.11 fm. Moreover, in order to obtain these results we have assumed that the mass difference is totally and immediately transferred into kinetic energies, and so the actual distance that the two final states travel away from each other in the half-life of the initial unstable hadron can be even smaller. Therefore, in the cases of 𝑓₀(500) and 𝜅(800), if the initial hadron is spherical in the beginning and the two final hadrons are both spherical in the end, the two final states may not separate geometrically even after the whole decay process. From this effect we clearly see that the meson-meson continuum contributes much in the cases of 𝑓₀(500) and 𝜅(800), and it is quite difficulty to separate the meson-meson continuum. We note that this distance can be estimated for other hadrons, and this problem is not only for light scalar mesons.

5. QCD Sum Rule Using Only Connected Parts

In this section we use only the connected parts of the two-point correlation function to perform the QCD sum rule analysis. Using the large 𝑁_𝑐 approximation, Weinberg suggested the following in his recent reference [72–74]: “A one tetraquark pole can only appear in the final, connected, term”: 𝒬(𝑥)𝒬𝑦₀=_{𝑖𝑗𝑘𝑙}𝐶_𝑖𝑗𝐶_𝑘𝑙ℬ_𝑖(𝑥)ℬ_𝑘(𝑦)₀ℬ_𝑗(𝑥)ℬ_𝑙𝑦₀+ℬ_𝑖(𝑥)ℬ_𝑗(𝑥)ℬ_𝑘𝑦ℬ_𝑙𝑦_conn, (38) where 𝒬(𝑥) is a color-neutral operator, that is, a tetraquark current. Using Fierz transformation, it can be written in the form (see Appendix D for details) 𝒬(𝑥)=_𝑖𝑗𝐶_𝑖𝑗ℬ_𝑖(𝑥)ℬ_𝑗(𝑥), (39) and ℬ_𝑖(𝑥) are color-neutral quark bilinears: ℬ_𝑖(𝑥)=_𝑎𝑞^𝐴_𝑎(𝑥)Γ_𝑖𝑞^𝐵_𝑎(𝑥). (40)

In the previous section we have included both the connected parts (the second term in (39)) and the disconnected parts (the first term in (39)) to perform the QCD sum rule analysis. In this section we shall use only the connected parts. We shall use the same tetraquark currents. Although these currents are constructed using diquark and antidiquark fields, we do not need to change them to meson-meson form. We can simply select the connected parts in the contracted two-point correlation function. Take the current 𝐽(𝑥)=𝑢^𝑇_𝑎𝐶𝛾₅𝑑_𝑏𝑢_𝑎𝛾₅𝐶𝑑^𝑇_𝑏−𝑢_𝑏𝛾₅𝐶𝑑^𝑇_𝑎, (41) as an example. We use 𝑆^𝑞_𝑎𝑏(𝑥) to denote the quark propagator (𝑞=𝑢 for up quark, and 𝑞=𝑑 for down quark), and the contracted two-point correlation function is ⟨0|𝐽(𝑥)𝐽^†(0)|0⟩=Trℂ𝑆^𝑢𝑇_𝑎1𝑎2(𝑥)ℂ𝛾₅𝑆^𝑑_𝑏1𝑏2(𝑥)𝛾₅×Tr𝑆^𝑢_𝑎2𝑎1(−𝑥)𝛾₅ℂ𝑆^𝑑𝑇_𝑏2𝑏1(−𝑥)ℂ𝛾₅−Trℂ𝑆^𝑢𝑇_𝑎1𝑎2(𝑥)ℂ𝛾₅𝑆^𝑑_𝑏1𝑏2(𝑥)𝛾₅×Tr𝑆^𝑢_𝑏2𝑎1(−𝑥)𝛾₅ℂ𝑆^𝑑𝑇_𝑎2𝑏1(−𝑥)ℂ𝛾₅−Trℂ𝑆^𝑢𝑇_𝑎1𝑎2(𝑥)ℂ𝛾₅𝑆^𝑑_𝑏1𝑏2(𝑥)𝛾₅×Tr𝑆^𝑢_𝑎2𝑏1(−𝑥)𝛾₅ℂ𝑆^𝑑𝑇_𝑏2𝑎1(−𝑥)ℂ𝛾₅+Trℂ𝑆^𝑢𝑇_𝑎1𝑎2(𝑥)ℂ𝛾₅𝑆^𝑑_𝑏1𝑏2(𝑥)𝛾₅×Tr𝑆^𝑢_𝑏2𝑏1(−𝑥)𝛾₅ℂ𝑆^𝑑𝑇_𝑎2𝑎1(−𝑥)ℂ𝛾₅, (42) where 𝑎1,𝑎2,𝑏1, and 𝑏2 are color indices. Its connected parts are just ⟨0|𝐽(𝑥)𝐽^†(0)|0⟩_conn=−Trℂ𝑆^𝑢𝑇_𝑎1𝑎2(𝑥)ℂ𝛾₅𝑆^𝑑_𝑏1𝑏2(𝑥)𝛾₅×Tr𝑆^𝑢_𝑏2𝑎1(−𝑥)𝛾₅ℂ𝑆^𝑑𝑇_𝑎2𝑏1(−𝑥)ℂ𝛾₅−Trℂ𝑆^𝑢𝑇_𝑎1𝑎2(𝑥)ℂ𝛾₅𝑆^𝑑_𝑏1𝑏2(𝑥)𝛾₅×Tr𝑆^𝑢_𝑎2𝑏1(−𝑥)𝛾₅ℂ𝑆^𝑑𝑇_𝑏2𝑎1(−𝑥)ℂ𝛾₅. (43) The tetraquark currents equations (23) to (26) lead to the following “connected” spectral densities: 𝑓²_𝜎1+2𝑒^{−𝑀2𝜎1+2/𝑀2𝐵}=Π^𝜎1+2𝑠₀,𝑀²_𝐵=^𝑠0_𝑠<𝑒^{−𝑠/𝑀2𝐵}−1𝑑𝑠×61440𝜋⁶𝑠⁴+𝑔²𝐺𝐺3072𝜋⁶,𝑓²_𝜎3𝑒^{−𝑀2𝜎3/𝑀2𝐵}=Π^𝜎3𝑠₀,𝑀²_𝐵=^𝑠0_𝑠<𝑒^{−𝑠/𝑀2𝐵}1𝑑𝑠×5120𝜋⁶𝑠⁴+5𝑔²𝐺𝐺768𝜋⁶,𝑓²_𝜅1+2𝑒^{−𝑀2𝜅1+2/𝑀2𝐵}=Π^𝜅1+2𝑠₀,𝑀²_𝐵=^𝑠0_𝑠<𝑒^{−𝑠/𝑀2𝐵}×−1𝑑𝑠61440𝜋⁶𝑠⁴+𝑚²_𝑠3072𝜋⁶𝑠³+𝑔²𝐺𝐺3072𝜋⁶−𝑚_𝑠⟨𝑠𝑠⟩384𝜋⁴𝑠²−𝑚²_𝑠𝑔²𝐺𝐺1024𝜋⁶𝑚𝑠+^𝑠𝑔²𝐺𝐺⟨𝑠𝑠⟩768𝜋⁴,𝑓²_𝜅3𝑒^{−𝑀2𝜅3/𝑀2𝐵}=Π^𝜅3𝑠₀,𝑀²_𝐵=^𝑠0_𝑠<𝑒^{−𝑠/𝑀2𝐵}×1𝑑𝑠5120𝜋⁶𝑠⁴−𝑚²_𝑠256𝜋⁶𝑠³+5𝑔²𝐺𝐺768𝜋⁶+𝑚_𝑠⟨𝑠𝑠⟩32𝜋⁴𝑠²−5𝑚²_𝑠𝑔²𝐺𝐺256𝜋⁶𝑠+5𝑚_𝑠𝑔²𝐺𝐺⟨𝑠𝑠⟩192𝜋⁴,𝑓²_𝑎01+2𝑒^{−𝑀2𝑎01+2/𝑀2𝐵}=Π^𝑎01+2𝑠₀,𝑀²_𝐵=^𝑠0_𝑠<𝑒^{−𝑠/𝑀2𝐵}×−1𝑑𝑠61440𝜋⁶𝑠⁴+𝑚²_𝑠1536𝜋⁶𝑠³+𝑔²𝐺𝐺3072𝜋⁶−𝑚_𝑠⟨𝑠𝑠⟩192𝜋⁴𝑠²−𝑚²_𝑠𝑔²𝐺𝐺512𝜋⁶𝑠+𝑚_𝑠𝑔²𝐺𝐺⟨𝑠𝑠⟩384𝜋⁴−𝑚²_𝑠4𝑞𝑞²+⟨𝑠𝑠⟩²48𝜋²,𝑓²_𝑎03𝑒^{−𝑀23𝑎0/𝑀2𝐵}=Π^𝑎03𝑠₀,𝑀²_𝐵=^𝑠0_𝑠<𝑒^{−𝑠/𝑀2𝐵}×1𝑑𝑠5120𝜋⁶𝑠⁴−𝑚²_𝑠128𝜋⁶𝑠³+5𝑔²𝐺𝐺768𝜋⁶+𝑚_𝑠⟨𝑠𝑠⟩16𝜋⁴𝑠²−5𝑚²_𝑠𝑔²𝐺𝐺128𝜋⁶𝑠+5𝑚_𝑠𝑔²𝐺𝐺⟨𝑠𝑠⟩96𝜋⁴+𝑚²_𝑠4𝑞𝑞²+⟨𝑠𝑠⟩²4𝜋². (44)(45)(46)(47)(48)(49) The sum rules equations (45), (47), and (49) using tetraquark currents 𝜎₃, 𝜅₃, and 𝑎0₃ (𝑓0₃) do not change significantly; that is, the connected and disconnected parts lead to similar results. This suggests that the meson-meson contribution is significant in both the connected and disconnected parts of these currents. We note that they have the symmetry color structure 𝟔⊗𝟔, where color interactions between quarks and antiquarks are repulsive.

The sum rules equations (44), (46), and (48) do change significantly. Although the continuum term proportional to 𝑠⁴ is negative, the spectral densities are positive in our working region 𝑠∼1GeV², as shown in Figure 6 for the spectral densities 𝜌^𝜎1+2, 𝜌^𝜅1+2, and 𝜌^𝑎01+2. We note that the pole contribution is not well defined because these spectral densities are negative when 𝑠 is large.

Figure 6 

Spectral densities of light scalar mesons 𝑓0(500), 𝜅(800), and 𝑎0(980) (𝑓0(980)) as functions of the energy 𝑠, where only the connected parts are taken into account.

Masses of light scalar mesons are calculated using only the connected parts, and the results are shown in Figure 7, as functions of the Borel mass 𝑀_𝐵 and the threshold value 𝑠₀. We clearly see that the Borel mass dependence is still not much; the mass of 𝜎 still grows as the threshold value 𝑠₀ increases, suggesting that there is still much two-meson contribution (or related to its broad decay width); but the mass curves of 𝜅 and 𝑎₀ have minimums around 𝑠₀=0.8GeV² for 𝜅 and 𝑠₀=1.3GeV² for 𝑎₀, where the 𝑠₀ dependence is weak. We use these values as inputs, and calculate the masses of light scalar mesons.

Figure 7 

Masses of light scalar mesons 𝑓0(500), 𝜅(800), and 𝑎0(800) (𝑓0(980)) as functions of the Borel mass 𝑀𝐵 and the threshold value 𝑠0, where only the connected parts, (44), are taken into account. The threshold value 𝑠0 for the left figure is chosen to be 0.4GeV2, 0.8GeV2, and 1.3GeV2, while the Borel mass for the right figure is chosen to be 0.5 GeV, 0.6 GeV, and 1.5 GeV, for 𝑓0(500), 𝜅(800), and 𝑎0(800) (𝑓0(980)), respectively.

Altogether there are two kinds of error bars: one is due to the two-meson continuum and the other is due to the [(𝟖,𝟏)⊕(𝟏,𝟖)] components. This makes our results have large error bars: the mass of 𝑓₀(500) is around 400∼600 MeV, the mass of 𝜅(800) is around 700∼900 MeV, and masses of 𝑎₀(980) and 𝑓₀(980) are around 900∼1100 MeV.

6. Summary

We systematically studied the chiral structure of light scalar mesons using local scalar tetraquark currents that belong to the “nonexotic” [(𝟑,𝟑)⊕(𝟑,𝟑)] chiral multiplets. This chiral representation only contains flavor singlet and octet mesons, and it does not contain any meson having exotic flavor structure. The nine light scalar mesons can just compose one [(𝟑,𝟑)⊕(𝟑,𝟑)] chiral multiplet. To do a systematical study, we investigated both scalar and pseudoscalar tetraquark currents, since they are chiral partners. We also investigated tetraquark currents of flavor singlet, octet, 𝟏𝟎, 𝟏𝟎, and 𝟐𝟕, which can be useful for further studies. Then we used the left handed quark field 𝐿^𝑎_𝐴≡𝑞^𝑎_𝐿𝐴=((1−𝛾₅)/2)𝑞^𝑎_𝐴 and the right handed quark field 𝑅^𝑎_𝐴≡𝑞^𝑎_𝑅𝐴=((1+𝛾₅)/2)𝑞^𝑎_𝐴 to rewrite these currents. After making proper combinations we verified their chiral representations.

We then used the QCD sum rule to calculate their masses. The masses of 𝜎, 𝜅, 𝑎₀, and 𝑓₀ are around 600 MeV, 900 MeV, 1100 MeV, and 1100 MeV, respectively, generally consistent with the experimental values. However, the pole contributions are very small. Then we introduced a few [(𝟖,𝟏)⊕(𝟏,𝟖)] components by slightly changing the mixing parameters from 𝜂₁+𝜂₂→0.99×𝜂₁+𝜂₂. The masses of 𝜎, 𝜅, 𝑎₀, and 𝑓₀ are now around 500 MeV, 700 MeV, 900 MeV, and 900 MeV, respectively, better consistent with the experimental results. The pole contributions are significantly increased to be around 50% for 𝑓₀(500) and 𝜅(800). However, these results still depend much on the threshold value 𝑠₀.

To solve this problem, we use only the connected parts of the two-point correlation function to perform the QCD sum rule analysis. We find that the results obtained using the tetraquark currents 𝜎₃, 𝜅₃, 𝑎0₃, and 𝑓0₃ (see (23)–(26)) do not change significantly. However, the results obtained using the tetraquark currents 𝜅₁₊₂, 𝑎0₁₊₂, and 𝑓0₁₊₂ are improved: the mass curves of 𝜅 and 𝑎₀ have minimums around 𝑠₀=0.8GeV² for 𝜅 and 𝑠₀=1.3GeV² for 𝑎₀, where the 𝑠₀ dependence is weak. We use these values as inputs and calculate the masses.

Altogether there are three kinds of error bars. The dominant one is due to the two-meson continuum. All light scalar mesons couple strongly to it, but we still do not know how to effectively separate them. The second one is due to the mixing of different chiral components. For example, we have included a few [(𝟖,𝟏)⊕(𝟏,𝟖)] components to make our results reliable, but we do not know how much it is contained in light scalar mesons. The third one comes from our QCD sum rule calculations that we did not include the high dimensional terms, such as 𝛼_𝑠⟨𝑞𝑞⟩⁴. Consequently, we obtained masses of light scalar mesons with large error bars: the mass of 𝑓₀(500) is around 400∼600 MeV, the mass of 𝜅(800) is around 700∼900 MeV, and the mass of 𝑎₀(980) and 𝑓₀(980) is around 900∼1100 MeV. We note that in [42] we used the same method to calculate masses of 𝑞𝑞 scalar mesons, which are all above 1 GeV.

We have also used these pseudoscalar tetraquark currents to perform the QCD sum rule analyses. For example, the one containing quark contents 𝑞𝑠𝑞𝑠 has a mass around 1.3–1.6 GeV. This is significantly larger than the masses of the 𝜂 and 𝜂^′ mesons, suggesting that the Nambu-Goldstone bosons, 𝜋, 𝐾, 𝜂, and 𝜂^′, are predominantly 𝑞𝑞 states. We note that the finite decay width of light scalar mesons can be taken into account which does not change the final result significantly [42]. We also note that the contribution of instanton has not been considered in this paper whose effects can be significant since light scalar mesons have the same quantum numbers as vacuum. There are many papers discussing this [90–93].

We note that we can also use the Fierz transformation to write tetraquark currents in a mesonic-mesonic form. Some relations are shown in Appendix D, and here we show one example: 𝜂^S₁^,𝕊+𝜂^S₂^,𝕊1=−2𝑞^𝑎_𝐴𝛾₅𝑞^𝑎_𝐴𝑞^𝑏_𝐵𝛾₅𝑞^𝑏_𝐵+12𝑞^𝑎_𝐴𝛾₅𝑞^𝑎_𝐵𝑞^𝑏_𝐵𝛾₅𝑞^𝑏_𝐴−12𝑞^𝑎_𝐴𝑞^𝑎_𝐴𝑞^𝑏_𝐵𝑞^𝑏_𝐵+12𝑞^𝑎_𝐴𝑞^𝑎_𝐵𝑞^𝑏_𝐵𝑞^𝑏_𝐴+14𝑞^𝑎_𝐴𝜎_𝜇𝜈𝑞^𝑎_𝐴𝑞^𝑏_𝐵𝜎^𝜇𝜈𝑞^𝑏_𝐵−14𝑞^𝑎_𝐴𝜎_𝜇𝜈𝑞^𝑎_𝐵𝑞^𝑏_𝐵𝜎^𝜇𝜈𝑞^𝑏_𝐴=−𝑞^𝑎_𝐿𝐴𝑞^𝑎_𝑅𝐴𝑞^𝑏_𝐿𝐵𝑞^𝑏_𝑅𝐵+𝑞^𝑎_𝑅𝐴𝑞^𝑎_𝐿𝐴𝑞^𝑏_𝑅𝐵𝑞^𝑏_𝐿𝐵+𝑞^𝑎_𝐿𝐴𝑞^𝑎_𝑅𝐵𝑞^𝑏_𝐿𝐵𝑞^𝑏_𝑅𝐴+𝑞^𝑎_𝑅𝐴𝑞^𝑎_𝐿𝐵𝑞^𝑏_𝑅𝐵𝑞^𝑏_𝐿𝐴+14𝑞^𝑎_𝐿𝐴𝜎_𝜇𝜈𝑞^𝑎_𝑅𝐴𝑞^𝑏_𝐿𝐵𝜎^𝜇𝜈𝑞^𝑏_𝑅𝐵+𝑞^𝑎_𝑅𝐴𝜎_𝜇𝜈𝑞^𝑎_𝐿𝐵𝑞^𝑏_𝑅𝐵𝜎^𝜇𝜈𝑞^𝑏_𝐿𝐴−14𝑞^𝑎_𝐿𝐴𝜎_𝜇𝜈𝑞^𝑎_𝑅𝐵𝑞^𝑏_𝐿𝐵𝜎^𝜇𝜈𝑞^𝑏_𝑅𝐴+𝑞^𝑎_𝑅𝐴𝜎_𝜇𝜈𝑞^𝑎_𝐿A𝑞^𝑏_𝑅𝐵𝜎^𝜇𝜈𝑞^𝑏_𝐿𝐵. (50) Considering 𝑞_𝑅𝑞_𝐿 and 𝑞_𝐿𝑞_𝑅 both belong to [(𝟑,𝟑)⊕(𝟑,𝟑)] representation (or its mirror), all local scalar tetraquark currents that belong to [(𝟑,𝟑)⊕(𝟑,𝟑)] chiral multiplets are more similar to the combination of two 𝑞𝑞 mesons that both belong to this same representation. Consequently, the light scalar mesons are more similar to (like) tetraquarks or molecular states consisting two “non-chiral-singlet” 𝑞𝑞 mesons, unless different types of chirality mix with others.

The conventional pseudoscalar and scalar mesons made by one 𝑞𝑞 pair can also belong to the [(𝟑,𝟑)⊕(𝟑,𝟑)] chiral multiplet. However, all the scalar tetraquark currents inside this multiplet have the 𝑞_𝐿𝑞_𝐿𝑞_𝑅𝑞_𝑅+𝑞_𝑅𝑞_𝑅𝑞_𝐿𝑞_𝐿 chirality, and so they are not direct chiral partners of these 𝑞𝑞 mesons addressed by chiral singlet quark-antiquark pairs, which have the (𝑞_𝐿𝑞_𝑅+𝑞_𝑅𝑞_𝐿)⊗(𝑞_𝐿𝑞_𝐿+𝑞_𝑅𝑞_𝑅) chirality (“chiral” Fock-space expansion), unless these two types of chirality mix with each other; they are more similar to conventional 𝑞𝑞 mesons addressed by quark condensates; that is, mesons(𝑞Γ𝑞)⊗condensates⟨𝑞𝑞⟩=(𝑞_𝐿𝑞_𝑅+𝑞_𝑅𝑞_𝐿)⊗(𝑞_𝐿𝑞_𝑅+𝑞_𝑅𝑞_𝐿).

Similarly, all the pseudoscalar tetraquark currents inside this multiplet also have the 𝑞_𝐿𝑞_𝐿𝑞_𝑅𝑞_𝑅+𝑞_𝑅𝑞_𝑅𝑞_𝐿𝑞_𝐿 chirality, and so they are not (direct) terms in the “chiral” Fock-space expansion of the 𝑞𝑞 pseudoscalar mesons (𝜋, etc.). Therefore, in order to write the Fock-space expansion of the conventional pseudoscalar and scalar mesons, we probably need to study the mix of different types of chirality, which will be our next focus. In [72], Weinberg calculated the decay width of tetraquarks using the Large-N method. This can be done also using the method of QCD sum rule, which will be also our next focus.

Appendices

A. Other Tetraquark Currents

A.1. Pseudoscalar Tetraquark Currents of Flavor Singlet

In this subsection we study flavor singlet tetraquark currents of 𝐽^𝑃=0⁻. There are altogether six independent pseudoscalar currents as listed in the following: 𝜂^PS₁^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾₅ℂ𝑞_𝐵^𝑏𝑇−𝑞^𝑏_𝐴𝛾₅ℂ𝑞_𝐵^𝑎𝑇,𝜂^PS₂^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴ℂ𝑞_𝐵^𝑏𝑇−𝑞^𝑏_𝐴ℂ𝑞_𝐵^𝑎𝑇,𝜂^PS₃^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝜎^𝜇𝜈𝛾₅ℂ𝑞_𝐵^𝑏𝑇+𝑞^𝑏_𝐴𝜎^𝜇𝜈𝛾₅ℂ𝑞_𝐵^𝑎𝑇,𝜂^PS₄^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾₅ℂ𝑞_𝐵^𝑏𝑇+𝑞^𝑏_𝐴𝛾₅ℂ𝑞_𝐵^𝑎𝑇,𝜂^PS₅^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴ℂ𝑞_𝐵^𝑏𝑇+𝑞^𝑏_𝐴ℂ𝑞_𝐵^𝑎𝑇,𝜂^PS₆^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝜎^𝜇𝜈𝛾₅ℂ𝑞_𝐵^𝑏𝑇−𝑞^𝑏_𝐴𝜎^𝜇𝜈𝛾₅𝐶𝑞_𝐵^𝑎𝑇. (A.1) We note that we can prove 𝜂^PS₇^,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐵^𝑏𝑇−𝑞^𝑏_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐵^𝑎𝑇𝜂=0,_PS8,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇ℂ𝑞_𝐵^𝑏𝑇+𝑞^𝑏_𝐴𝛾^𝜇ℂ𝑞_𝐵^𝑎𝑇𝜂=0,_PS9,𝕊=𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐵^𝑏𝑇+𝑞^𝑏_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐵^𝑎𝑇𝜂=0,_PS,𝕊10=𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇ℂ𝑞_𝐵^𝑏𝑇−𝑞^𝑏_𝐴𝛾^𝜇ℂ𝑞_𝐵^𝑎𝑇=0. (A.2) The two superscripts PS and 𝕊 denote pseudoscalar (𝐽^𝑃=0⁻) and flavor singlet, respectively. 𝜂^PS,𝕊_1,2,3 contain diquarks and antidiquarks having the antisymmetric flavor structure 𝟑⊗𝟑; 𝜂^PS,𝕊_4,5,6 contain diquarks and antidiquarks having the symmetric flavor structure 𝟔⊗𝟔. From the following combinations we can clearly see their chiral structure, where the left handed quark field 𝐿^𝑎_𝐴≡𝑞^𝑎_𝐿𝐴=((1−𝛾₅)/2)𝑞^𝑎_𝐴 and the right handed quark field 𝑅^𝑎_𝐴≡𝑞^𝑎_𝑅𝐴=((1+𝛾₅)/2)𝑞^𝑎_𝐴 are used: 𝜂^PS₁^,𝕊−𝜂^PS₂^,𝕊=2𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐵^𝑏𝑇−𝐿^𝑏_𝐴ℂ𝐿_𝐵^𝑎𝑇−2𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐵^𝑏𝑇−𝑅^𝑏_𝐴ℂ𝑅_𝐵^𝑎𝑇,𝜂^PS₁^,𝕊+𝜂^PS₂^,𝕊=−2𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐵^𝑏𝑇−𝑅^𝑏_𝐴ℂ𝑅_𝐵^𝑎𝑇+2𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐵^𝑏𝑇−𝐿^𝑏_𝐴ℂ𝐿_𝐵^𝑎𝑇,𝜂^PS₄^,𝕊−𝜂^PS₅^,𝕊=2𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐵^𝑏𝑇+𝐿^𝑏_𝐴ℂ𝐿_𝐵^𝑎𝑇−2𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐵^𝑏𝑇+𝑅^𝑏_𝐴ℂ𝑅_𝐵^𝑎𝑇,𝜂^PS₄^,𝕊+𝜂^PS₅^,𝕊=−2𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐵^𝑏𝑇+𝑅^𝑏_𝐴ℂ𝑅_𝐵^𝑎𝑇+2𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐵^𝑏𝑇+𝐿^𝑏_𝐴ℂ𝐿_𝐵^𝑎𝑇,𝜂^PS₃^,𝕊=−𝐿_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝐿^𝑏_𝐵𝑅^𝑎_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐵^𝑏𝑇+𝑅^𝑏_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐵^𝑎𝑇+𝑅_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑅^𝑏_𝐵𝐿^𝑎_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐵^𝑏𝑇+𝐿^𝑏_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐵^𝑎𝑇,𝜂^PS₆^,𝕊=−𝐿_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝐿^𝑏_𝐵𝑅^𝑎_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐵^𝑏𝑇−𝑅^𝑏_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐵^𝑎𝑇+𝑅_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑅^𝑏_𝐵𝐿^𝑎_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐵^𝑏𝑇−𝐿^𝑏_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐵^𝑎𝑇. (A.3) We list their chirality and chiral representations in Table 3.

Table 3 

Flavor singlet tetraquark currents of 𝐽^𝑃=0⁻ classified in Appendix A.1.

A.2. Scalar Tetraquark Currents of Flavor Octet

In this subsection we study flavor octet tetraquark currents of 𝐽^𝑃=0⁺. There are altogether ten independent scalar currents as listed in the following: 𝜂^S,𝕆_1,𝑁=𝜆_𝑁^𝐷𝐵𝑞_𝐴^𝑎𝑇ℂ𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾₅ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐴𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂^S,𝕆_2,𝑁=𝜆_𝑁^𝐷𝐵𝑞_𝐴^𝑎𝑇ℂ𝑞^𝑏_𝐵𝑞^𝑎_𝐴ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐴ℂ𝑞_𝐷^𝑎𝑇,𝜂^S,𝕆_3,𝑁=𝜆_𝑁^𝐷𝐵𝑞_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝜎^𝜇𝜈ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐴𝜎^𝜇𝜈ℂ𝑞_𝐷^𝑎𝑇,𝜂^S,𝕆_4,𝑁=𝜆_𝑁^𝐷𝐵𝑞_𝐴^𝑎𝑇ℂ𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾₅ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐴𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂^S,𝕆_5,𝑁=𝜆_𝑁^𝐷𝐵𝑞_𝐴^𝑎𝑇ℂ𝑞^𝑏_𝐵𝑞^𝑎_𝐴ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐴ℂ𝑞_𝐷^𝑎𝑇,𝜂^S,𝕆_6,𝑁=𝜆_𝑁^𝐷𝐵𝑞_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝜎^𝜇𝜈ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐴𝜎^𝜇𝜈ℂ𝑞_𝐷^𝑎𝑇,𝜂^S,𝕆_7,𝑁=𝜆_𝑁^𝐷𝐵𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂^S,𝕆_8,𝑁=𝜆_𝑁^𝐷𝐵𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐴𝛾^𝜇ℂ𝑞_𝐷^𝑎𝑇,𝜂^S,𝕆_9,𝑁=𝜆_𝑁^𝐷𝐵𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂^S,𝕆_10,𝑁=𝜆_𝑁^𝐷𝐵𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐴𝛾^𝜇ℂ𝑞_𝐷^𝑎𝑇. (A.4) The two superscripts S and 𝕆 denote scalar and flavor octet, respectively. Five currents 𝜂^S,𝕆_1,2,3,7,8 contain diquarks and antidiquarks having the antisymmetric flavor structure 𝟑⊗𝟑 and other five currents 𝜂^S,𝕆_4,5,6,9,10 contain diquarks and antidiquarks having the symmetric flavor structure 𝟔⊗𝟔. From the following combinations we can clearly see their chiral structure, where the left handed quark field 𝐿^𝑎_𝐴≡𝑞^𝑎_𝐿𝐴=((1−𝛾₅)/2)𝑞^𝑎_𝐴 and the right handed quark field 𝑅^𝑎_𝐴≡𝑞^𝑎_𝑅𝐴=((1+𝛾₅)/2)𝑞^𝑎_𝐴 are used: 𝜂^S,𝕆_1,𝑁−𝜂^S,𝕆_2,𝑁=−2𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐷^𝑏𝑇−𝐿^𝑏_𝐴ℂ𝐿_𝐷^𝑎𝑇−2𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐷^𝑏𝑇−𝑅^𝑏_𝐴ℂ𝑅_𝐷^𝑎𝑇,𝜂^S,𝕆_1,𝑁+𝜂^S,𝕆_2,𝑁=2𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐷^𝑏𝑇−𝑅^𝑏_𝐴ℂ𝑅_𝐷^𝑎𝑇+2𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐷^𝑏𝑇−𝐿^𝑏_𝐴ℂ𝐿_𝐷^𝑎𝑇,𝜂^S,𝕆_4,𝑁−𝜂^S,𝕆_5,𝑁=−2𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐷^𝑏𝑇+𝐿^𝑏_𝐴ℂ𝐿_𝐷^𝑎𝑇−2𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐷^𝑏𝑇+𝑅^𝑏_𝐴ℂ𝑅_𝐷^𝑎𝑇,𝜂^S,𝕆_4,𝑁+𝜂^S,𝕆_5,𝑁=2𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐷^𝑏𝑇+𝑅^𝑏_𝐴ℂ𝑅_𝐷^𝑎𝑇+2𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐷^𝑏𝑇+𝐿^𝑏_𝐴ℂ𝐿_𝐷^𝑎𝑇,𝜂^S,𝕆_3,𝑁=𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝐿^𝑏_𝐵×𝑅^𝑎_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐷^𝑏𝑇+𝑅^𝑏_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐷^𝑎𝑇+𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑅^𝑏_𝐵×𝐿^𝑎_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐷^𝑏𝑇+𝐿^𝑏_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐷^𝑎𝑇,𝜂^S,𝕆_6,𝑁=𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝐿^𝑏_𝐵×𝑅^𝑎_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐷^𝑏𝑇−𝑅^𝑏_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐷^𝑎𝑇+𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑅^𝑏_𝐵×𝐿^𝑎_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐷^𝑏𝑇−𝐿^𝑏_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐷^𝑎𝑇,𝜂^S,𝕆_7,𝑁−𝜂^S,𝕆_10,𝑁=−2𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑅^𝑏_𝐵×𝐿^𝑎_𝐴𝛾^𝜇ℂ𝑅_𝐷^𝑏𝑇−𝐿^𝑏_𝐴𝛾^𝜇ℂ𝑅_𝐷^𝑎𝑇−2𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝛾_𝜇𝐿^𝑏_𝐵×𝑅^𝑎_𝐴𝛾^𝜇ℂ𝐿_𝐷^𝑏𝑇−𝑅^𝑏_𝐴𝛾^𝜇ℂ𝐿_𝐷^𝑎𝑇,5𝜂^S,𝕆_7,𝑁+𝜂^S,𝕆_10,𝑁=6𝑑^𝑁𝑀𝑂𝜆_𝑀^𝐶𝐴𝜆_𝑂^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑅^𝑏_𝐵×𝐿^𝑎_𝐶𝛾^𝜇ℂ𝑅_𝐷^𝑏𝑇−𝐿^𝑏_𝐶𝛾^𝜇ℂ𝑅_𝐷^𝑎𝑇,𝜂^S,𝕆_8,𝑁−𝜂^S,𝕆_9,𝑁=2𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑅^𝑏_𝐵×𝐿^𝑎_𝐴𝛾^𝜇ℂ𝑅_𝐷^𝑏𝑇+𝐿^𝑏_𝐴𝛾^𝜇ℂ𝑅_𝐷^𝑎𝑇+2𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝛾_𝜇𝐿^𝑏_𝐵×𝑅^𝑎_𝐴𝛾^𝜇ℂ𝐿_𝐷^𝑏𝑇+𝑅^𝑏_𝐴𝛾^𝜇ℂ𝐿_𝐷^𝑎𝑇,5𝜂^S,𝕆_8,𝑁+𝜂^S,𝕆_9,𝑁=−6𝑑^𝑁𝑀𝑂𝜆_𝑀^𝐶𝐴𝜆_𝑂^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑅^𝑏_𝐵×𝐿^𝑎_𝐶𝛾^𝜇ℂ𝑅_𝐷^𝑏𝑇+𝐿^𝑏_𝐶𝛾^𝜇ℂ𝑅_𝐷^𝑎𝑇. (A.5) We list their chirality and chiral representations in Table 4.

Table 4 

Flavor octet tetraquark currents of 𝐽^𝑃=0⁺ classified in Appendix A.2.

A.3. Pseudoscalar Tetraquark Currents of Flavor Octet

In this subsection we study flavor octet tetraquark currents of 𝐽^𝑃=0⁻. There are altogether ten independent pseudoscalar currents as listed in the following: 𝜂^PS,𝕆_1,𝑁=𝜆_𝑁^𝐵𝐷𝑞_𝐴^𝑎𝑇ℂ𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾₅ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐴𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂^PS,𝕆_2,𝑁=𝜆_𝑁^𝐵𝐷𝑞_𝐴^𝑎𝑇ℂ𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐴ℂ𝑞_𝐷^𝑎𝑇,𝜂^PS,𝕆_3,𝑁=𝜆_𝑁^𝐵𝐷𝑞_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝜎^𝜇𝜈𝛾₅ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐴𝜎^𝜇𝜈𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂^PS,𝕆_4,𝑁=𝜆_𝑁^𝐵𝐷𝑞_𝐴^𝑎𝑇ℂ𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾₅ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐴𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂^PS,𝕆_5,𝑁=𝜆_𝑁^𝐵𝐷𝑞_𝐴^𝑎𝑇ℂ𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐴ℂ𝑞_𝐷^𝑎𝑇,𝜂^PS,𝕆_6,𝑁=𝜆_𝑁^𝐵𝐷𝑞_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝜎^𝜇𝜈𝛾₅ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐴𝜎^𝜇𝜈𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂^PS,𝕆_7,𝑁=𝜆_𝑁^𝐵𝐷𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂^PS,𝕆_8,𝑁=𝜆_𝑁^𝐵𝐷𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐴𝛾^𝜇ℂ𝑞_𝐷^𝑎𝑇,𝜂^PS,𝕆_9,𝑁=𝜆_𝑁^𝐵𝐷𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐴𝛾^𝜇𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂^PS,𝕆_10,𝑁=𝜆_𝑁^𝐵𝐷𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐴𝛾^𝜇ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐴𝛾^𝜇ℂ𝑞_𝐷^𝑎𝑇. (A.6) The two superscripts PS and 𝕆 denote pseudoscalar and flavor octet, respectively. Among these ten currents, 𝜂^PS,𝕆_1,2,3 contain diquarks and antidiquarks having both the antisymmetric flavor structure 𝟑⊗𝟑; 𝜂^PS,𝕆_4,5,6 contain diquarks and antidiquarks having both the symmetric flavor structure 𝟔⊗𝟔; 𝜂^PS,𝕆_7,8 contain diquarks having the symmetric flavor structure and antidiquarks the antisymmetric flavor structure 𝟔⊗𝟑; 𝜂^PS,𝕆_9,10 contain diquarks having the antisymmetric flavor structure and antidiquarks the symmetric flavor structure 𝟑⊗𝟔. From the following combinations we can clearly see their chiral structure, where the left handed quark field 𝐿^𝑎_𝐴≡𝑞^𝑎_𝐿𝐴=((1−𝛾₅)/2)𝑞^𝑎_𝐴 and the right handed quark field 𝑅^𝑎_𝐴≡𝑞^𝑎_𝑅𝐴=((1+𝛾₅)/2)𝑞^𝑎_𝐴 are used: 𝜂^PS,𝕆_1,𝑁−𝜂^PS,𝕆_2,𝑁=2𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐷^𝑏𝑇−𝐿^𝑏_𝐴ℂ𝐿_𝐷^𝑎𝑇−2𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐷^𝑏𝑇−𝑅^𝑏_𝐴ℂ𝑅_𝐷^𝑎𝑇,𝜂^PS,𝕆_1,𝑁+𝜂^PS,𝕆_2,𝑁=−2𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐷^𝑏𝑇−𝑅^𝑏_𝐴ℂ𝑅_𝐷^𝑎𝑇+2𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐷^𝑏𝑇−𝐿^𝑏_𝐴ℂ𝐿_𝐷^𝑎𝑇,𝜂^PS,𝕆_4,𝑁−𝜂^PS,𝕆_5,𝑁=2𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐷^𝑏𝑇+𝐿^𝑏_𝐴ℂ𝐿_𝐷^𝑎𝑇−2𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐷^𝑏𝑇+𝑅^𝑏_𝐴ℂ𝑅_𝐷^𝑎𝑇,𝜂^PS,𝕆_4,𝑁+𝜂^PS,𝕆_5,𝑁=−2𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝐿^𝑏_𝐵𝑅^𝑎_𝐴ℂ𝑅_𝐷^𝑏𝑇+𝑅^𝑏_𝐴ℂ𝑅_𝐷^𝑎𝑇+2𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝑅^𝑏_𝐵𝐿^𝑎_𝐴ℂ𝐿_𝐷^𝑏𝑇+𝐿^𝑏_𝐴ℂ𝐿_𝐷^𝑎𝑇,𝜂^PS,𝕆_3,N=−𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝐿^𝑏_𝐵×𝑅^𝑎_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐷^𝑏𝑇+𝑅^𝑏_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐷^𝑎𝑇+𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑅^𝑏_𝐵×𝐿^𝑎_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐷^𝑏𝑇+𝐿^𝑏_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐷^𝑎𝑇,𝜂^PS,𝕆_6,𝑁=−𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝐿^𝑏_𝐵×𝑅^𝑎_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐷^𝑏𝑇−𝑅^𝑏_𝐴𝜎_𝜇𝜈ℂ𝑅_𝐷^𝑎𝑇+𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑅^𝑏_𝐵×𝐿^𝑎_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐷^𝑏𝑇−𝐿^𝑏_𝐴𝜎_𝜇𝜈ℂ𝐿_𝐷^𝑎𝑇,𝜂^PS,𝕆_8,𝑁−𝜂^PS,𝕆_9,𝑁=2𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑅^𝑏_𝐵𝐿^𝑎_𝐴𝛾^𝜇ℂ𝑅_𝐷^𝑏𝑇+𝐿^𝑏_𝐴𝛾^𝜇ℂ𝑅_𝐷^𝑎𝑇−2𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝛾_𝜇𝐿^𝑏_𝐵×𝑅^𝑎_𝐴𝛾^𝜇ℂ𝐿_𝐷^𝑏𝑇+𝑅^𝑏_𝐴𝛾^𝜇ℂ𝐿_𝐷^𝑎𝑇,𝜂^PS,𝕆_8,𝑁+𝜂^PS,𝕆_9,𝑁=−2𝑖𝑓^𝑁𝑀𝑂𝜆_𝑀^𝐶𝐴𝜆_𝑂^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑅^𝑏_𝐵×𝐿^𝑎_𝐶𝛾^𝜇ℂ𝑅_𝐷^𝑏𝑇+𝐿^𝑏_𝐶𝛾^𝜇ℂ𝑅_𝐷^𝑎𝑇,𝜂^PS,𝕆_7,𝑁−𝜂^PS,𝕆_10,𝑁=−2𝜆_𝑁^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑅^𝑏_𝐵×𝐿^𝑎_𝐴𝛾^𝜇ℂ𝑅_𝐷^𝑏𝑇−𝐿^𝑏_𝐴𝛾^𝜇ℂ𝑅_𝐷^𝑎𝑇+2𝜆_𝑁^𝐷𝐵𝑅_𝐴^𝑎𝑇ℂ𝛾_𝜇𝐿^𝑏_𝐵×𝑅^𝑎_𝐴𝛾^𝜇ℂ𝐿_𝐷^𝑏𝑇−𝑅^𝑏_𝐴𝛾^𝜇ℂ𝐿_𝐷^𝑎𝑇,𝜂^PS,𝕆_7,𝑁+𝜂^PS,𝕆_10,𝑁=2𝑖𝑓^𝑁𝑀𝑂𝜆_𝑀^𝐶𝐴𝜆_𝑂^𝐷𝐵𝐿_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑅^𝑏_𝐵×𝐿^𝑎_𝐶𝛾^𝜇ℂ𝑅_𝐷^𝑏𝑇−𝐿^𝑏_𝐶𝛾^𝜇ℂ𝑅_𝐷^𝑎𝑇. (A.7) We list their chirality and chiral representations in Table 5.

Table 5 

Flavor octet tetraquark currents of 𝐽^𝑃=0⁻ classified in Appendix A.3.

A.4. Scalar Tetraquark Currents of Flavor 𝟐𝟕_𝐹

In this subsection we study flavor 𝟐𝟕_𝐹 tetraquark currents of 𝐽^𝑃=0⁺. There are altogether five independent scalar currents as listed in the following: 𝜂^S,𝕋𝕊_1,𝑈=𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝑞_𝐴^𝑎𝑇𝐶𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾₅𝐶𝑞_𝐷^𝑏𝑇,𝜂^S,𝕋𝕊_2,𝑈=𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝑞_𝐴^𝑎𝑇𝐶𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝐶𝑞_𝐷^𝑏𝑇,𝜂^S,𝕋𝕊_3,𝑈=𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝑞_𝐴^𝑎𝑇𝐶𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝜎^𝜇𝜈𝐶𝑞_𝐷^𝑏𝑇,𝜂^S,𝕋𝕊_4,𝑈=𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝑞_𝐴^𝑎𝑇𝐶𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾^𝜇𝛾₅𝐶𝑞_𝐷^𝑏𝑇,𝜂^S,𝕋𝕊_5,𝑈=𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝑞_𝐴^𝑎𝑇𝐶𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾^𝜇𝐶𝑞_𝐷^𝑏𝑇. (A.4) All these five currents contain diquarks and antidiquarks having the symmetric flavor structure 𝟔⊗𝟔. From the following combinations we can clearly see their chiral structure, where the left handed quark field 𝐿^𝑎_𝐴≡𝑞^𝑎_𝐿𝐴=((1−𝛾₅)/2)𝑞^𝑎_𝐴 and the right handed quark field 𝑅^𝑎_𝐴≡𝑞^𝑎_𝑅𝐴=((1+𝛾₅)/2)𝑞^𝑎_𝐴 are used: 𝜂^S,𝕋𝕊_1,𝑈−𝜂^S,𝕋𝕊_2,𝑈=−2𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝐿_𝐴^𝑎𝑇𝐶𝐿^𝑏_𝐵𝐿^𝑎_𝐶𝐶𝐿_𝐷^𝑏𝑇−2𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝑅_𝐴^𝑎𝑇𝐶𝑅^𝑏_𝐵𝑅^𝑎_𝐶𝐶𝑅_𝐷^𝑏𝑇,𝜂^S,𝕋𝕊_1,𝑈+𝜂^S,𝕋𝕊_2,𝑈=2𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝐿_𝐴^𝑎𝑇𝐶𝐿^𝑏_𝐵𝑅^𝑎_𝐶𝐶𝑅_𝐷^𝑏𝑇+2𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝑅_𝐴^𝑎𝑇𝐶𝑅^𝑏_𝐵𝐿^𝑎_𝐶𝐶𝐿_𝐷^𝑏𝑇,𝜂^S,𝕋𝕊_3,𝑈=𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝐿_𝐴^𝑎𝑇𝐶𝜎_𝜇𝜈𝐿^𝑏_𝐵𝑅^𝑎_𝐶𝜎_𝜇𝜈𝐶𝑅_𝐷^𝑏𝑇+𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝑅_𝐴^𝑎𝑇𝐶𝜎_𝜇𝜈𝑅^𝑏_𝐵𝐿^𝑎_𝐶𝜎_𝜇𝜈𝐶𝐿_𝐷^𝑏𝑇,𝜂^S,𝕋𝕊_4,𝑈=−2𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝐿_𝐴^𝑎𝑇𝐶𝛾_𝜇𝑅^𝑏_𝐵×𝐿^𝑎_𝐶𝛾^𝜇𝐶𝑅_𝐷^𝑏𝑇+𝐿^𝑏_𝐶𝛾^𝜇𝐶𝑅_𝐷^𝑎𝑇,𝜂^S,𝕋𝕊_5,𝑈=2𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝐿_𝐴^𝑎𝑇𝐶𝛾_𝜇𝑅^𝑏_𝐵𝐿^𝑎_𝐶𝛾^𝜇𝐶𝑅_𝐷^𝑏𝑇−𝐿^𝑏_𝐶𝛾^𝜇𝐶𝑅_𝐷^𝑎𝑇. (A.9) We list their chirality and chiral representations in Table 6.

Table 6 

Flavor 𝟐𝟕_𝐹 tetraquark currents of 𝐽^𝑃=0⁺ classified in Appendix A.4.

A.5. Pseudoscalar Tetraquark Currents of Flavor 𝟐𝟕_𝐹

In this subsection we study flavor 𝟐𝟕_𝐹 tetraquark currents of 𝐽^𝑃=0⁻. There are altogether three independent pseudoscalar currents as listed in the following: 𝜂^PS,𝕋𝕊_1,𝑈=𝑆^{𝐶𝐷;A𝐵}_𝑈𝑞_𝐴^𝑎𝑇𝐶𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾₅𝐶𝑞_𝐷^𝑏𝑇,𝜂^PS,𝕋𝕊_2,𝑈=𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝑞_𝐴^𝑎𝑇𝐶𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝐶𝑞_𝐷^𝑏𝑇,𝜂^PS,𝕋𝕊_3,𝑈=𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝑞_𝐴^𝑎𝑇𝐶𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝜎^𝜇𝜈𝛾₅𝐶𝑞_𝐷^𝑏𝑇. (A.10) All these three currents contain diquarks and antidiquarks having the symmetric flavor structure 𝟔⊗𝟔. From the following combinations we can clearly see their chiral structure, where the left handed quark field 𝐿^𝑎_𝐴≡𝑞^𝑎_𝐿𝐴=((1−𝛾₅)/2)𝑞^𝑎_𝐴 and the right handed quark field 𝑅^𝑎_𝐴≡𝑞^𝑎_𝑅𝐴=((1+𝛾₅)/2)𝑞^𝑎_𝐴 are used: 𝜂^PS,𝕋𝕊_1,𝑈−𝜂^PS,𝕋𝕊_2,𝑈=2𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝐿_𝐴^𝑎𝑇𝐶𝐿^𝑏_𝐵𝐿^𝑎_𝐶𝐶𝐿_𝐷^𝑏𝑇−2𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝑅_𝐴^𝑎𝑇𝐶𝑅^𝑏_𝐵𝑅^𝑎_𝐶𝐶𝑅_𝐷^𝑏𝑇,𝜂^PS,𝕋𝕊_1,𝑈+𝜂^PS,𝕋𝕊_2,𝑈=−2𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝐿_𝐴^𝑎𝑇𝐶𝐿^𝑏_𝐵𝑅^𝑎_𝐶𝐶𝑅_𝐷^𝑏𝑇+2𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝑅_𝐴^𝑎𝑇𝐶𝑅^𝑏_𝐵𝐿^𝑎_𝐶𝐶𝐿_𝐷^𝑏𝑇,𝜂^PS,𝕋𝕊_3,𝑈=−𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝐿_𝐴^𝑎𝑇𝐶𝜎_𝜇𝜈𝐿^𝑏_𝐵𝑅^𝑎_𝐶𝜎_𝜇𝜈𝐶𝑅_𝐷^𝑏𝑇+𝑆_𝑈^{𝐶𝐷;𝐴𝐵}𝑅_𝐴^𝑎𝑇𝐶𝜎_𝜇𝜈𝑅^𝑏_𝐵𝐿^𝑎_𝐶𝜎_𝜇𝜈𝐶𝐿_𝐷^𝑏𝑇. (A.11) We list their chirality and chiral representations in Table 7.

Table 7 

Flavor 𝟐𝟕_𝐹 tetraquark currents of 𝐽^𝑃=0⁻ classified in Appendix A.5.

A.6. Pseudoscalar Tetraquark Currents of Flavor 𝟏𝟎_𝐹

In this subsection we study flavor 𝟏𝟎_𝐹 tetraquark currents of 𝐽^𝑃=0⁻. There are altogether two independent pseudoscalar currents as listed in the following: 𝜂^PS,𝔻_1,𝑃=𝜖^𝐴𝐵𝐸𝑆_𝑃^𝐶𝐷𝐸𝑞_𝐴^𝑎𝑇𝐶𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾^𝜇𝛾₅𝐶𝑞_𝐷^𝑏𝑇,𝜂^PS,𝔻_2,𝑃=𝜖^𝐴𝐵𝐸𝑆_𝑃^𝐶𝐷𝐸𝑞^𝑎T_𝐴𝐶𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾^𝜇𝐶𝑞_𝐷^𝑏𝑇. (A.12) These two currents both contain diquarks having the antisymmetric flavor structure and antidiquarks the symmetric flavor structure 𝟑⊗𝟔. From the following combinations we can clearly see their chiral structure, where the left handed quark field 𝐿^𝑎_𝐴≡𝑞^𝑎_𝐿𝐴=((1−𝛾₅)/2)𝑞^𝑎_𝐴 and the right handed quark field 𝑅^𝑎_𝐴≡𝑞^𝑎_𝑅𝐴=((1+𝛾₅)/2)𝑞^𝑎_𝐴 are used: 𝜂^PS,𝔻_1,𝑃=−2𝜖^𝐴𝐵𝐸𝑆_𝑃^𝐶𝐷𝐸𝐿_𝐴^𝑎𝑇𝐶𝛾_𝜇𝑅^𝑏_𝐵×𝐿^𝑎_𝐶𝛾^𝜇𝐶𝑅_𝐷^𝑏𝑇+𝐿^𝑏_𝐶𝛾^𝜇𝐶𝑅_𝐷^𝑎𝑇,𝜂^PS,𝔻_2,𝑃=2𝜖^𝐴𝐵𝐸𝑆_𝑃^𝐶𝐷𝐸𝐿_𝐴^𝑎𝑇𝐶𝛾_𝜇𝑅^𝑏_𝐵×𝐿^𝑎_𝐶𝛾^𝜇𝐶𝑅_𝐷^𝑏𝑇−𝐿^𝑏_𝐶𝛾^𝜇𝐶𝑅_𝐷^𝑎𝑇. (A.13) We find that these two currents both belong to the chiral representation [(𝟖,𝟖)+(𝟖,𝟖)] and their chirality is 𝐿𝑅𝐿𝑅+𝑅𝐿𝑅𝐿.

A.7. Pseudoscalar Tetraquark Currents of Flavor 𝟏𝟎_𝐹

In this subsection we study flavor 𝟏𝟎_𝐹 tetraquark currents of 𝐽^𝑃=0⁻. There are altogether two independent pseudoscalar currents as listed in the following: 𝜂^PS,𝔻_1,𝑃=𝑆_𝑃^𝐴𝐵𝐸𝜖^𝐶𝐷𝐸𝑞_𝐴^𝑎𝑇𝐶𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾^𝜇𝛾₅𝐶𝑞_𝐷^𝑏𝑇,𝜂^PS,𝔻_2,𝑃=𝑆_𝑃^𝐴𝐵𝐸𝜖^𝐶𝐷𝐸𝑞_𝐴^𝑎𝑇𝐶𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾^𝜇𝐶𝑞_𝐷^𝑏𝑇. (A.14) These two currents both contain diquarks having the antisymmetric flavor structure and antidiquarks the symmetric flavor structure 𝟔⊗𝟑. From the following combinations we can clearly see their chiral structure, where the left handed quark field 𝐿^𝑎_𝐴≡𝑞^𝑎_𝐿𝐴=((1−𝛾₅)/2)𝑞^𝑎_𝐴 and the right handed quark field 𝑅^𝑎_𝐴≡𝑞^𝑎_𝑅𝐴=((1+𝛾₅)/2)𝑞^𝑎_𝐴 are used: 𝜂^PS,𝔻_1,𝑃=−2𝑆_𝑃^𝐴𝐵𝐸𝜖^𝐶𝐷𝐸𝐿_𝐴^𝑎𝑇𝐶𝛾_𝜇𝑅^𝑏_𝐵×𝐿^𝑎_𝐶𝛾^𝜇𝐶𝑅_𝐷^𝑏𝑇−𝐿^𝑏_𝐶𝛾^𝜇𝐶𝑅_𝐷^𝑎𝑇,𝜂^PS,𝔻_2,𝑃=2𝑆_𝑃^𝐴𝐵𝐸𝜖^𝐶𝐷𝐸𝐿_𝐴^𝑎𝑇𝐶𝛾_𝜇𝑅^𝑏_𝐵×𝐿^𝑎_𝐶𝛾^𝜇𝐶𝑅_𝐷^𝑏𝑇+𝐿^𝑏_𝐶𝛾^𝜇𝐶𝑅_𝐷^𝑎𝑇. (A.15) We find that these two currents both belong to the chiral representation [(𝟖,𝟖)+(𝟖,𝟖)] and their chirality is 𝐿𝑅𝐿𝑅+𝑅𝐿𝑅𝐿.

B. Other Chiral Transformations

There are four [(𝟏,𝟏)] chiral multiplets, (𝜂^S₁^,𝕊−𝜂^S₂^,𝕊,𝜂^PS₁^,𝕊−𝜂^PS₂^,𝕊), (𝜂^S₄^,𝕊−𝜂^S₅^,𝕊,𝜂^PS₄^,𝕊−𝜂^PS₅^,𝕊), (𝜂^S₇^,𝕊−𝜂^S,𝕊₁₀,𝜂^PS₇^,𝕊−𝜂^PS,𝕊₁₀=0), and (𝜂^S₈^,𝕊−𝜂^S₉^,𝕊,𝜂^PS₈^,𝕊−𝜂^PS₉^,𝕊=0). We use (𝜂^S,𝕊_(𝟏,𝟏),𝜂^PS,𝕊_(𝟏,𝟏)) to denote them, and their chiral transformation properties are 𝛿₅𝜂^S(PS),𝕊_(𝟏,𝟏)𝛿=0,_𝑎𝜂^S(PS),𝕊_(𝟏,𝟏)𝛿=0,_𝑏5𝜂^S(PS),𝕊_(𝟏,𝟏)=0. (B.1) There are two [(𝟔,𝟔)⊕(𝟔,𝟔)] chiral multiplets, (𝜂^S₄^,𝕊+𝜂^S₅^,𝕊,𝜂^PS₄^,𝕊+𝜂^PS₅^,𝕊,𝜂^S,𝕆_4,𝑁+𝜂^S,𝕆_5,𝑁,𝜂^PS,𝕆_4,𝑁+𝜂^PS,𝕆_5,𝑁,𝜂^S,𝕋𝕊_1,𝑈+𝜂^S,𝕋𝕊_2,𝑈,𝜂^PS,𝕋𝕊_1,𝑈+𝜂^PS,𝕋𝕊_2,𝑈), (𝜂^S₆^,𝕊,𝜂^PS₆^,𝕊,𝜂^S,𝕆_6,𝑁,𝜂^PS,𝕆_6,𝑁,𝜂^S,𝕋𝕊_3,𝑈,𝜂^PS,𝕋𝕊_3,𝑈). We use (𝜂^S,𝕊_(𝟔,𝟔),𝜂^PS,𝕊_(𝟔,𝟔),𝜂^S,𝕆_{(𝟔,𝟔),𝑁},𝜂^PS,𝕆_{(𝟔,𝟔),𝑁},𝜂^S,𝕋𝕊_(𝟔,𝟔),𝜂^PS,𝕋𝕊_(𝟔,𝟔)) to denote them, and their chiral transformation properties are 𝛿₅𝜂^S(PS),𝕊_(𝟔,𝟔)=4𝑖𝑏𝜂^PS(S),𝕊_(𝟔,𝟔),𝛿^𝑎𝜂^S(PS),𝕊_(𝟔,𝟔)𝛿=0,_𝑏5𝜂^S(PS),𝕊_(𝟔,𝟔)=4𝑖𝑏^𝑁𝜂^PS(S),𝕆_{(𝟔,𝟔),𝑁},𝛿₅𝜂^S(PS),𝕆_{(𝟔,𝟔),𝑁}=4𝑖𝑏𝜂^PS(S),𝕆_{(𝟔,𝟔),𝑁},𝛿^𝑎𝜂^S(PS),𝕆_{(𝟔,𝟔),𝑁}=2𝑎^𝑁𝑓_𝑁𝑀𝑂𝜂^S(PS),𝕆_{(𝟔,𝟔),𝑂},𝛿^𝑏₅𝜂^S(PS),𝕆_{(𝟔,𝟔),𝑁}=53𝑖𝑏^𝑀𝜂^PS(S),𝕊_(𝟔,𝟔)+145𝑖𝑏^𝑁𝑑_𝑁𝑀𝑂𝜂^PS(S),𝕆_{(𝟔,𝟔),𝑂}+4𝑖𝑏^𝑁𝐓_8×27^𝑁_𝑀𝑈𝜂^{PS(S),𝕋𝕊}_{(𝟔,𝟔),𝑈},𝛿₅𝜂^{S(PS),𝕋𝕊}_(𝟔,𝟔)=4𝑖𝑏𝜂^{PS(S),𝕋𝕊}_{(𝟔,𝟔),𝑈},𝛿^𝑎𝜂^{S(PS),𝕋𝕊}_(𝟔,𝟔)=2𝑖𝑎^𝑁𝐓^𝐀_27×27−𝐓^𝐁_27×27^𝑁_𝑈𝑉𝜂^{S(PS),𝕋𝕊}_{(𝟔,𝟔),𝑉},𝛿^𝑏₅𝜂^{S(PS),𝕋𝕊}_(𝟔,𝟔)=65𝑖𝑏^𝑁𝐓^†_8×27^𝑁_𝑈𝑂𝜂^PS(S),𝕆_{(𝟔,𝟔),𝑂}+2𝑖𝑏^𝑁𝐓^𝐀_27×27+𝐓^𝐁_27×27^𝑁_𝑈𝑉𝜂^{PS(S),𝕋𝕊}_{(𝟔,𝟔),𝑉}. (B.2) There are two [(𝟖,𝟖)⊕(𝟖,𝟖)] chiral multiplets, (2𝜂^S₇^,𝕊+𝜂^S,𝕊₁₀,2𝜂^PS₇^,𝕊+𝜂^PS,𝕊₁₀=0,5𝜂^S,𝕆_7,𝑁+𝜂^S,𝕆_10,𝑁,𝜂^PS,𝕆_7,𝑁+𝜂^PS,𝕆_10,𝑁,𝜂^S,𝕋𝕊_5,𝑈,𝜂^PS,𝔻_1,𝑃,𝜂^PS,𝔻_2,𝑃), and (2𝜂^S₈^,𝕊+𝜂^S₉^,𝕊,2𝜂^PS₈^,𝕊+𝜂^PS₉^,𝕊=0,5𝜂^S,𝕆_8,𝑁+𝜂^S,𝕆_9,𝑁,𝜂^PS,𝕆_8,𝑁+𝜂^PS,𝕆_9,𝑁,𝜂^S,𝕋𝕊_4,𝑈,𝜂^PS,𝔻_2,𝑃,𝜂^PS,𝔻_1,𝑃). We use (𝜂^S,𝕊_(𝟖,𝟖),𝜂^PS,𝕊_(𝟖,𝟖),𝜂^S,𝕆_{(𝟖,𝟖),𝑁},𝜂^PS,𝕆_{(𝟖,𝟖),𝑁},𝜂^S,𝕋𝕊_{(𝟖,𝟖),𝑈},𝜂^PS,𝔻_{(𝟖,𝟖),𝑃},𝜂^PS,𝔻_{(𝟖,𝟖),𝑃}) to denote them, and their chiral transformation properties are 𝛿₅𝜂^S,𝕊_(𝟖,𝟖)𝛿=0,_𝑎𝜂^S,𝕊_(𝟖,𝟖)𝛿=0,_𝑏5𝜂^S,𝕊_(𝟖,𝟖)=6𝑖𝑏^𝑁𝜂^PS,𝕆_{(𝟖,𝟖),𝑁},𝛿₅𝜂^S,𝕆_{(𝟖,𝟖),𝑁}𝛿=0,_𝑎𝜂^S,𝕆_{(𝟖,𝟖),𝑁}=2𝑎^𝑁𝑓_𝑁𝑀𝑂𝜂^S,𝕆_{(𝟖,𝟖),𝑂},𝛿^𝑏₅𝜂^S,𝕆_{(𝟖,𝟖),𝑁}=6𝑖𝑏^𝑁𝑑_𝑁𝑀𝑂𝜂^PS,𝕆_{(𝟖,𝟖),𝑂}−12𝑖𝑏^𝑁𝐓_8×10^𝑁_𝑀𝑃𝜂^PS,𝔻_{(𝟖,𝟖),𝑃}−12𝑖𝑏^𝑁𝐓^∗_8×10^𝑁_𝑀𝑃𝜂^PS,𝔻_{(𝟖,𝟖),𝑃},𝛿₅𝜂^PS,𝕆_{(𝟖,𝟖),𝑁}𝛿=0,_𝑎𝜂^PS,𝕆_{(𝟖,𝟖),𝑁}=2𝑎^𝑁𝑓_𝑁𝑀𝑂𝜂^PS,𝕆_{(𝟖,𝟖),𝑂},𝛿^𝑏₅𝜂^PS,𝕆_{(𝟖,𝟖),𝑁}=𝑖𝑏^𝑀𝜂^S,𝕊_(𝟖,𝟖)+65𝑖𝑏^𝑁𝑑_𝑁𝑀𝑂𝜂^S,𝕆_{(𝟖,𝟖),𝑂}−4𝑖𝑏^𝑁𝐓_8×27^𝑁_𝑀𝑈𝜂^S,𝕋𝕊_{(𝟖,𝟖),𝑈},𝛿₅𝜂^S,𝕋𝕊_{(𝟖,𝟖),𝑈}𝛿=0,_𝑎𝜂^S,𝕋𝕊_{(𝟖,𝟖),𝑈}=2𝑖𝑎^𝑁𝐓^𝐀_27×27−𝐓^𝐁_27×27^𝑁_𝑈𝑉𝜂^S,𝕋𝕊_{(𝟖,𝟖),𝑉},𝛿^𝑏₅𝜂^S,𝕋𝕊_{(𝟖,𝟖),𝑈}=−𝑖𝑏^𝑁𝐓^†_8×27^𝑁_𝑈𝑂𝜂^PS,𝕆_{(𝟖,𝟖),𝑂}+3𝑖𝑏^𝑁𝐓^𝐁†_10×27^𝑁_𝑈𝑃𝜂^PS,𝔻_{(𝟖,𝟖),𝑃}+3𝑖𝑏^𝑁𝐓^𝐀†_10×27^𝑁_𝑈𝑃𝜂^PS,𝔻_{(𝟖,𝟖),𝑃},𝛿₅𝜂^PS,𝔻_{(𝟖,𝟖),𝑃}𝛿=0,_𝑎𝜂^PS,𝔻_{(𝟖,𝟖),𝑃}=3𝑖𝑎^𝑁𝐓_10×10^𝑁_𝑃𝑄𝜂^PS,𝔻_{(𝟖,𝟖),𝑄},𝛿^𝑏₅𝜂^PS,𝔻_{(𝟖,𝟖),𝑃}2=−5𝑖𝑏^𝑁𝐓^†_8×10^𝑁_𝑃𝑂𝜂^S,𝕆_{(𝟖,𝟖),𝑂}+2𝑖𝑏^𝑁𝐓^𝐀_10×27^𝑁_𝑃𝑈𝜂^S,𝕋𝕊_{(𝟖,𝟖),𝑈},𝛿₅𝜂^PS,𝔻_{(𝟖,𝟖),𝑃}𝛿=0,_𝑎𝜂^PS,𝔻_{(𝟖,𝟖),𝑃}=−3𝑖𝑎^𝑁𝐓^∗_10×10^𝑁_𝑃𝑄𝜂^PS,𝔻_{(𝟖,𝟖),𝑄},𝛿^𝑏₅𝜂^PS,𝔻_{(𝟖,𝟖),𝑃}2=−5𝑖𝑏^𝑁𝐓^†∗_8×10^𝑁_𝑃𝑂𝜂^S,𝕆_{(𝟖,𝟖),𝑂}+2𝑖𝑏^𝑁𝐓^𝐁_10×27^𝑁_𝑃𝑈𝜂^S,𝕋𝕊_{(𝟖,𝟖),𝑈}. (B.3) There are four [(𝟖,𝟏)⊕(𝟏,𝟖)] chiral multiplets, (𝜂^S,𝕆_1,𝑁−𝜂^S,𝕆_2,𝑁,𝜂^PS,𝕆_1,𝑁−𝜂^PS,𝕆_2,𝑁), (𝜂^S,𝕆_4,𝑁−𝜂^S,𝕆_5,𝑁,𝜂^PS,𝕆_4,𝑁−𝜂^PS,𝕆_5,𝑁), (𝜂^S,𝕆_7,𝑁−𝜂^S,𝕆_10,𝑁,𝜂^PS,𝕆_7,𝑁−𝜂^PS,𝕆_10,𝑁), (𝜂^S,𝕆_8,𝑁−𝜂^S,𝕆_9,𝑁,𝜂^PS,𝕆_8,𝑁−𝜂^PS,𝕆_9,𝑁). We use (𝜂^S,𝕆_{(𝟖,𝟏),𝑁},𝜂^PS,𝕆_{(𝟖,𝟏),𝑁}) to denote them, and their chiral transformation properties are 𝛿₅𝜂^S(PS),𝕆_{(𝟖,𝟏),𝑁}𝛿=0,_𝑎𝜂^S(PS),𝕆_{(𝟖,𝟏),𝑁}=2𝑎^𝑁𝑓_𝑁𝑀𝑂𝜂^S(PS),𝕆_{(𝟖,𝟏),𝑂},𝛿^𝑏₅𝜂^S(PS),𝕆_{(𝟖,𝟏),𝑁}=2𝑏^𝑁𝑓_𝑁𝑀𝑂𝜂^PS(S),𝕆_{(𝟖,𝟏),𝑂}. (B.4) There is only one [(𝟐𝟕,𝟏)⊕(𝟏,𝟐𝟕)] chiral multiplet, (𝜂^S,𝕋𝕊_1,𝑈−𝜂^S,𝕋𝕊_2,𝑈,𝜂^PS,𝕋𝕊_1,𝑈−𝜂^PS,𝕋𝕊_2,𝑈). Its chiral transformation properties are 𝛿₅𝜂^{S(PS),𝕋𝕊}_1−2,𝑈𝛿=0,_𝑎𝜂^{S(PS)𝕋𝕊}_1−2,𝑈=2𝑖𝑎^𝑁𝐓^𝐀_27×27−𝐓^𝐁_27×27^𝑁_𝑈𝑉𝜂^{S(PS),𝕋𝕊}_1−2,𝑉,𝛿^𝑏₅𝜂^{S(PS),𝕋𝕊}_1−2,𝑈=2𝑖𝑎^𝑁𝐓^𝐀_27×27−𝐓^𝐁_27×27^𝑁_𝑈𝑉𝜂^{PS(S),𝕋𝕊}_1−2,𝑉. (B.5)

C. Transition Matrices

The transition matrices 𝐓^𝑁_10×10 are𝐓¹_10×10=01√3100000000√3023020000000301√3100000000√3√00000000000023√00000000230√23√0000000023100000000000030100000003,𝐓000000000000_210×10=𝑖0−√3𝑖00000000√30−2𝑖3000000002𝑖3𝑖0−√3𝑖00000000√3𝑖√000000000000−23𝑖√0000000023𝑖√0−23𝑖√0000000023𝑖000000000000−30𝑖00000003,𝐓000000000000_310×10=011000000000310000000000−320000000000−1000000000032000000000000000000000−310000000000310000000000−30,𝐓0000000000_410×10=10000√3√00000000002310000000000310000000000000√3200000030√0023√0000002301003200000000000310000√3√0000023100000000000√3,𝐓00_510×10=𝑖0000−√3𝑖√0000000000−23𝑖0000000000−3𝑖0000000000000√3000000−2𝑖30𝑖√0023𝑖√000000−230𝑖003000000000002𝑖3𝑖0000−√3𝑖√0000023𝑖00000000000√3,𝐓00_610×10=1000000000000003√00000000002310000000000√3010003√000000000023√000023100000√32000030√0000023200000000003100√3100000000√30,𝐓⁷_10×10=𝑖00000000000000−3𝑖√0000000000−23𝑖0000000000−√30𝑖0003𝑖√000000000023𝑖√0000−23𝑖00000√30000−2𝑖30𝑖√000002300000000002𝑖3𝑖00−√3𝑖00000000√30,𝐓⁸_10×10=1√301000000000√310000000000√310000000000√310000000000000000000000000000000000000000000−√310000000000−√302000000000−√3. (C.1) The transition matrices (𝐓^𝐴_10×27)^𝑁† are 𝐓^𝐴_10×27^1†=2000032520−3250000000031000000000301031000000000−3710000000000003√7100000000000−3√01100−32502−150000002011532510000000000000√3100000000√61000000003√210−3√2100000000√6100000000−√30000000000000230√000000023√00000000002300000000−2310000−√6√00000000−2310−√6−1000000√6√0−23010000000√6,𝐓0000000000000000010000000000000000000−1000000000000000000000000000000_{𝐴10×272†}=200003𝑖25023𝑖2500000000𝑖31000000000𝑖3100−𝑖31000000000𝑖310000000000007𝑖3√1000000000007𝑖3√01100−3𝑖250−𝑖215000000−𝑖21150−3𝑖25𝑖0000000000000√3𝑖00000000√6𝑖000000003√20𝑖3√2𝑖00000000−√6𝑖00000000√30000000000000𝑖230𝑖√000000023𝑖√0000000000−2300000000𝑖23𝑖0000−√6𝑖√00000000−230𝑖√6−𝑖000000√60𝑖√23𝑖00000000−√6,𝐓00000000000000000𝑖0000000000000000000𝑖000000000000000000000000000000_{𝐴10×273†}=400000−3√50000000000−3500000000000000000−3570000000000000−3√07100000000−3√21000003250200000003251000000000000000000000000√32000000003100000000−√32√0000000000000000000000000000000002302√0000000−231000000000000000√21000000003√2100000000−3√2100000000√2√0000000000000000000000000000−2,𝐓0000000000000000000000000000000000000000_{𝐴10×274†}=00000000250100−2√11500000000−√30000000−201152√52000000√1150000000−3√510000000032500003410000000−325400000000−3√5√0000000−100000000−32100000000−√210000000020000000000000000000000000−23200000000−3√0000000023100000000000000000000000000000003√21000000003100000000−√610000000000000−√3010000000√600000000000000010000000023010000000√3,𝐓00000000_{𝐴10×275†}=00000000−𝑖250𝑖00−2√𝑖1500000000−√30000000𝑖20𝑖152√5000000−2𝑖√𝑖150000000−3√51000000003𝑖250000−𝑖34100000003𝑖25000000004𝑖3√5𝑖√0000000−𝑖00000000−32𝑖00000000−√2𝑖0000000020000000000000000000000000−𝑖2300000000−2𝑖3𝑖√0000000023𝑖0000000000000000000000000000000−3√2𝑖00000000−3𝑖00000000√6𝑖0000000000000−√30𝑖0000000√6000000000000000−𝑖00000000−𝑖23𝑖00000000−√3,𝐓00000000_{𝐴10×276†}=0000000−25100000−2√520000−√011500−√300000−20115002√11500000000000000−32500−3110000003√540000000003√540000000032510000000000000000002100000000√2√00000000−32√000000001000000000000000000000000023200000000−30000000023100000000000√610000000031000000003√21000000000000000000000000000000000−√6010000000√310000−√300000000−23,𝐓00000000−1000000000_{𝐴10×277†}=0000000𝑖25𝑖00000−2√500002𝑖√0𝑖1500−√300000𝑖20𝑖15002√11500000000000000−3𝑖2500𝑖3𝑖10000003√5000000000−4𝑖3√5400000000−3𝑖25𝑖0000000000000000002𝑖00000000√2𝑖√00000000−32𝑖√00000000𝑖00000000000000000000000002300000000−2𝑖300000000𝑖23𝑖00000000000−√6𝑖00000000−3𝑖00000000−3√2𝑖000000000000000000000000000000000−√60𝑖0000000√3𝑖0000√300000000𝑖23,𝐓00000000𝑖000000000_{𝐴10×278†}=20000000000000000−√5200000000−√5200000000√50000000000000−3010000000031000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003200000000320000000032−0000000032.000000000000000000000000000000000000000000000000000000000000000000000 (C.2) The transition matrices (𝐓^𝐵_10×27)^𝑁† are 𝐓^𝐵_10×27^1†=2000032520−32500000000−310000000003010310000000003−10000020152/5300000000−2/530271500000000000000−3√071000000003√110000000−√310000000000√61000000003√210−3√2100000000√610000000000√301000√6−100000000√6√0−23√00000000−2310−√6100000000−√600000000000000−23√0000000000230√0000000230000000000230,𝐓000000000000000000000000000000000000000−100000000000000000001_{𝐵10×272†}=20000−3𝑖2520−3𝑖2500000000−𝑖31000000000−𝑖3100𝑖31000000000−𝑖3𝑖100000201153𝑖250100000003𝑖250𝑖21500000000000000−7𝑖3√0100000000−7𝑖3√𝑖10000000−√3𝑖0000000000√6𝑖00000000−3√2𝑖0−3√2𝑖00000000−√6𝑖0000000000−√30𝑖000√6𝑖00000000√6𝑖√0−230𝑖√000000023𝑖0−√6𝑖00000000√600000000000000−𝑖23𝑖√0000000000230𝑖√0000000−230000000000−𝑖230,𝐓000000000000000000000000000000000000000−𝑖0000000000000000000−𝑖_{𝐵10×273†}=400000−3√500000000−35000000000000000000000−350200032520000000000325700000000000000−3√7100000000000−3√011000000000000000−√320000000000310000000000√310000000000000√210000000000−3√2100000000003√210000000000√22√00000000000000000000000−232√0000000000230√00000000000000000000000000000000000000000000000000000000000−2,𝐓0000000000_{𝐵10×274†}=0000000025012√52000000√115000−√30000000−2011500−2√415000000000000−3√540000000000−325100000003250000311000000−3√5100000000000000210000000000−√2√0000000000−3210000000000−10000000000−√6100000000003100000000003√2√0000000000000000000000000002320000000000−30000000000−2301000√300000000002310000000000100000000000000000000000√610000000000−√30,𝐓^𝐵_10×27^5†=00000000𝑖250−𝑖2√50000002𝑖√0𝑖1500√30000000−𝑖20𝑖15002√15000000000000−4𝑖3√540000000000−3𝑖2510000000−3𝑖250000𝑖3𝑖10000003√5−𝑖0000000000000020𝑖000000000√2𝑖√000000000032𝑖0000000000𝑖0000000000−√6𝑖00000000003𝑖00000000003√2𝑖√000000000000000000000000000−2300000000002𝑖30000000000𝑖230𝑖000√30000000000𝑖23𝑖0000000000𝑖00000000000000000000000−√6𝑖0000000000√30,𝐓^𝐵_10×27^6†=0000000−2501002√1150000000000−√300000−211500000−2√520000−√04150000325400000000003√510000000003√510000000000−32500−3√1010000000000−3210000000000√210000000000210000000000000000000000000000003√210000000000310000000000√600000002320000000000−3√0000000000230000000000000−10000000000−2310000000000−√3100000000000000√310000000000−√60,𝐓0000000000_{𝐵10×277†}=0000000−𝑖25𝑖000−2√𝑖150000000000√300000−𝑖2𝑖15000002√50000−2𝑖√041500003𝑖2500000000004𝑖3√5𝑖000000000−3√5100000000003𝑖2500−𝑖30𝑖√10−𝑖00000000032𝑖0000000000−√2𝑖0000000000−2𝑖0000000000000000000000000000003√2𝑖00000000003𝑖0000000000√60000000−𝑖2300000000002𝑖3𝑖√0000000000−230000000000000−𝑖0000000000−𝑖23𝑖0000000000−√3𝑖00000000000000−√3𝑖0000000000√60,𝐓0000000000_{𝐵10×278†}=200000000000000√520000000000−√520000000000−√50000000000000000000000000000003100000000000−30−100000000000000000000000000000000000000000000000000032000000000032000000000032000000000032.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 (C.3)

D. (𝑞𝑞)(𝑞𝑞) Tetraquark Currents

In Section 2 and Appendix A we have investigated the chiral structure of local scalar and pseudoscalar tetraquark currents constructed using diquarks and antidiquarks, while they can also be constructed using two quark-antiquark pairs. These two different constructions can be related to each other through Fierz transformations, and so they can equally describe the full space of local tetraquark currents. In this appendix we show these relations. We note that some of these relations have been obtained in [42, 60, 69].

We shall separately investigate scalar and pseudoscalar in the following subsections. Since Fierz transformations can only change the Lorentz structure and not change the flavor symmetry and the color symmetry of diquarks and antidiquarks, we shall fix these two symmetries in the following discussions, and separately study tetraquark currents having the antisymmetric flavor structure 𝟑_𝑓(𝑞𝑞)⊗𝟑_𝑓(𝑞𝑞), the symmetric flavor structure 𝟔_𝑓(𝑞𝑞)⊗𝟔_𝑓(𝑞𝑞), and the mixed flavor structure 𝟑_𝑓(𝑞𝑞)⊗𝟔_𝑓(𝑞𝑞). The other case of mixed flavor structure 𝟔_𝑓(𝑞𝑞)⊗𝟑_𝑓(𝑞𝑞) can be similarly investigated.

D.1. Scalar Tetraquark Currents

D.1.1. Flavor Structure 𝟑_𝑓(𝑞𝑞)⊗𝟑_𝑓(𝑞𝑞)

In this subsection we study the scalar tetraquark currents which contain diquarks and antidiquarks having both the antisymmetric flavor structure 𝟑⊗𝟑. There are altogether five independent scalar tetraquark currents constructed using diquarks and antidiquarks: 𝜂₁=𝑞_𝐴^𝑎𝑇ℂ𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾₅ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐶𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂₂=𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾^𝜇𝛾₅ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐶𝛾^𝜇𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂₃=𝑞_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝜎^𝜇𝜈ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐶𝜎^𝜇𝜈ℂ𝑞_𝐷^𝑎𝑇,𝜂₄=𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾^𝜇ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐶𝛾^𝜇ℂ𝑞_𝐷^𝑎𝑇,𝜂₅=𝑞_𝐴^𝑎𝑇ℂ𝑞^𝑏_𝐵𝑞^𝑎_𝐶ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐶ℂ𝑞_𝐷^𝑎𝑇. (D.1.1) There are altogether ten scalar tetraquark currents constructed using quark-antiquark pairs: 𝜓₁=𝑞^𝑎_𝐶𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝑞^𝑏_𝐵−𝑞^𝑎_𝐶𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝑞^𝑏_𝐴,𝜓₂=𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝛾^𝜇𝑞^𝑏_𝐵−𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝛾^𝜇𝑞^𝑏_𝐴,𝜓₃=𝑞^𝑎_𝐶𝜎_𝜇𝜈𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝜎^𝜇𝜈𝑞^𝑏_𝐵−𝑞^𝑎_𝐶𝜎_𝜇𝜈𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝜎^𝜇𝜈𝑞^𝑏_𝐴,𝜓₄=𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝛾^𝜇𝛾₅𝑞^𝑏_𝐵−𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝛾^𝜇𝛾₅𝑞^𝑏_𝐴,𝜓₅=𝑞^𝑎_𝐶𝛾₅𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝛾₅𝑞^𝑏_𝐵−𝑞^𝑎_𝐶𝛾₅𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝛾₅𝑞^𝑏_𝐴,𝜓₆=𝑞^𝑎_𝐶𝜆_𝑎𝑏𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝜆_𝑐𝑑𝑞^𝑑_𝐵−𝑞^𝑎_𝐶𝜆_𝑎𝑏𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝜆_𝑐𝑑𝑞^𝑑_𝐴,𝜓₇=𝑞^𝑎_𝐶𝛾_𝜇𝜆_𝑎𝑏𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝛾^𝜇𝜆_𝑐𝑑𝑞^𝑑_𝐵−𝑞^𝑎_𝐶𝛾_𝜇𝜆_𝑎𝑏𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝛾^𝜇𝜆_𝑐𝑑𝑞^𝑑_𝐴,𝜓₈=𝑞^𝑎_𝐶𝜎_𝜇𝜈𝜆_𝑎𝑏𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝜎^𝜇𝜈𝜆_𝑐𝑑𝑞^𝑑_𝐵−𝑞^𝑎_𝐶𝜎_𝜇𝜈𝜆_𝑎𝑏𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝜎^𝜇𝜈𝜆_𝑐𝑑𝑞^𝑑_𝐴,𝜓₉=𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝜆_𝑎𝑏𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝛾^𝜇𝛾₅𝜆_𝑐𝑑𝑞^𝑑_B−𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝜆_𝑎𝑏𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝛾^𝜇𝛾₅𝜆_𝑐𝑑𝑞^𝑑_𝐴,𝜓₁₀=𝑞^𝑎_𝐶𝛾₅𝜆_𝑎𝑏𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝛾₅𝜆_𝑐𝑑𝑞^𝑑_𝐵−𝑞^𝑎_𝐶𝛾₅𝜆_𝑎𝑏𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝛾₅𝜆_𝑐𝑑𝑞^𝑑_𝐴. (D.2) Among these ten currents only five are independent. We can verify the following relations: 𝜂₁1=−4𝜓₁−14𝜓₂+18𝜓₃−14𝜓₄−14𝜓₅,𝜂₂=𝜓₁−12𝜓₂+12𝜓₄−𝜓₅,𝜂₃=3𝜓₁+12𝜓₃+3𝜓₅,𝜂₄=𝜓₁+12𝜓₂−12𝜓₄−𝜓₅,𝜂₅1=−4𝜓₁+14𝜓₂+18𝜓₃+14𝜓₄−14𝜓₅,𝜓₆1=−6𝜓₁+12𝜓₂+14𝜓₃−12𝜓₄+12𝜓₅,𝜓₇=2𝜓₁−53𝜓₂−𝜓₄−2𝜓₅,𝜓₈=6𝜓₁−53𝜓₃+6𝜓₅,𝜓₉=−2𝜓₁−𝜓₂−53𝜓₄+2𝜓₅,𝜓₁₀=12𝜓₁−12𝜓₂+14𝜓₃+12𝜓₄−16𝜓₅. (D.3)

D.1.2. Flavor Structure 𝟔_𝑓(𝑞𝑞)⊗𝟔_𝑓(𝑞𝑞)

In this subsection we study the scalar tetraquark currents which contain diquarks and antidiquarks having both the symmetric flavor structure 𝟔⊗𝟔. There are altogether five independent scalar tetraquark currents constructed using diquarks and antidiquarks: 𝜂₁=𝑞_𝐴^𝑎𝑇ℂ𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾₅ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐶𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂₂=𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾^𝜇𝛾₅ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐶𝛾^𝜇𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂₃=𝑞_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝜎^𝜇𝜈ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐶𝜎^𝜇𝜈ℂ𝑞_𝐷^𝑎𝑇,𝜂₄=𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾^𝜇ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐶𝛾^𝜇ℂ𝑞_𝐷^𝑎𝑇,𝜂₅=𝑞_𝐴^𝑎𝑇ℂ𝑞^𝑏_𝐵𝑞^𝑎_𝐶ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐶ℂ𝑞_𝐷^𝑎𝑇. (D.4) There are altogether ten scalar tetraquark currents constructed using quark-antiquark pairs: 𝜓₁=𝑞^𝑎_𝐶𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝑞^𝑏_𝐵+𝑞^𝑎_𝐶𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝑞^𝑏_𝐴,𝜓₂=𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝛾^𝜇𝑞^𝑏_𝐵+𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝛾^𝜇𝑞^𝑏_𝐴,𝜓₃=𝑞^𝑎_𝐶𝜎_𝜇𝜈𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝜎^𝜇𝜈𝑞^𝑏_𝐵+𝑞^𝑎_𝐶𝜎_𝜇𝜈𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝜎^𝜇𝜈𝑞^𝑏_𝐴,𝜓₄=𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝛾^𝜇𝛾₅𝑞^𝑏_𝐵+𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝛾^𝜇𝛾₅𝑞^𝑏_𝐴,𝜓₅=𝑞^𝑎_𝐶𝛾₅𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝛾₅𝑞^𝑏_𝐵+𝑞^𝑎_𝐶𝛾₅𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝛾₅𝑞^𝑏_𝐴,𝜓₆=𝑞^𝑎_𝐶𝜆_𝑎𝑏𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝜆_𝑐𝑑𝑞^𝑑_𝐵+𝑞^𝑎_𝐶𝜆_𝑎𝑏𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝜆_𝑐𝑑𝑞^𝑑_𝐴,𝜓₇=𝑞^𝑎_𝐶𝛾_𝜇𝜆_𝑎𝑏𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝛾^𝜇𝜆_𝑐𝑑𝑞^𝑑_𝐵+𝑞^𝑎_𝐶𝛾_𝜇𝜆_𝑎𝑏𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝛾^𝜇𝜆_𝑐𝑑𝑞^𝑑_𝐴,𝜓₈=𝑞^𝑎_𝐶𝜎_𝜇𝜈𝜆_𝑎𝑏𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝜎^𝜇𝜈𝜆_𝑐𝑑𝑞^𝑑_𝐵+𝑞^𝑎_𝐶𝜎_𝜇𝜈𝜆_𝑎𝑏𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝜎^𝜇𝜈𝜆_𝑐𝑑𝑞^𝑑_𝐴,𝜓₉=𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝜆_𝑎𝑏𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝛾^𝜇𝛾₅𝜆_𝑐𝑑𝑞^𝑑_𝐵+𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝜆_𝑎𝑏𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝛾^𝜇𝛾₅𝜆_𝑐𝑑𝑞^𝑑_𝐴,𝜓₁₀=𝑞^𝑎_𝐶𝛾₅𝜆_𝑎𝑏𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝛾₅𝜆_𝑐𝑑𝑞^𝑑_𝐵+𝑞^𝑎_𝐶𝛾₅𝜆_𝑎𝑏𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝛾₅𝜆_𝑐𝑑𝑞^𝑑_𝐴. (D.5) Among these ten currents only five are independent. We can verify the following relations: 𝜂₁1=−4𝜓₁−14𝜓₂+18𝜓₃−14𝜓₄−14𝜓₅,𝜂₂=𝜓₁−12𝜓₂+12𝜓₄−𝜓₅,𝜂₃=3𝜓₁+12𝜓₃+3𝜓₅,𝜂₄=𝜓₁+12𝜓₂−12𝜓₄−𝜓₅,𝜂₅1=−4𝜓₁+14𝜓₂+18𝜓₃+14𝜓₄−14𝜓₅,𝜓₆7=−6𝜓₁−12𝜓₂−14𝜓₃+12𝜓₄−12𝜓₅,𝜓₇=−2𝜓₁+13𝜓₂+𝜓₄+2𝜓₅,𝜓₈=−6𝜓₁+13𝜓₃−6𝜓₅,𝜓₉=2𝜓₁+𝜓₂+13𝜓₄−2𝜓₅,𝜓₁₀1=−2𝜓₁+12𝜓₂−14𝜓₃−12𝜓₄−76𝜓₅. (D.6)

D.2. Pseudoscalar Tetraquark Currents

D.2.1. Flavor Structure 𝟑_𝑓(𝑞𝑞)⊗𝟑_𝑓(𝑞𝑞)

In this subsection we study pseudoscalar tetraquark currents which contain diquarks and antidiquarks having both the antisymmetric flavor structure 𝟑⊗𝟑. There are altogether three independent pseudoscalar tetraquark currents constructed using diquarks and antidiquarks: 𝜂₁=𝑞_𝐴^𝑎𝑇ℂ𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾₅ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐶𝛾₅𝐶𝑞_𝐷^𝑎𝑇,𝜂₂=𝑞_𝐴^𝑎𝑇𝐶𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐶ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐶ℂ𝑞_𝐷^𝑎𝑇,𝜂₃=𝑞_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝜎^𝜇𝜈𝛾₅ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐶𝜎^𝜇𝜈𝛾₅ℂ𝑞_𝐷^𝑎𝑇. (D.7) There are altogether six pseudoscalar tetraquark currents constructed using quark-antiquark pairs: 𝜓₁=𝑞^𝑎_𝐶𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝛾₅𝑞^𝑏_𝐵+𝑞^𝑎_𝐶𝛾₅𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝑞^𝑏_𝐵−𝑞^𝑎_𝐶𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝛾₅𝑞^𝑏_𝐴−𝑞^𝑎_𝐶𝛾₅𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝑞^𝑏_𝐴,𝜓₂=𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝛾^𝜇𝛾₅𝑞^𝑏_𝐵+𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝛾^𝜇𝑞^𝑏_𝐵−𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝛾^𝜇𝛾₅𝑞^𝑏_𝐴−𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝛾^𝜇𝑞^𝑏_𝐴,𝜓₃=𝑞^𝑎_𝐶𝜎_𝜇𝜈𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝜎^𝜇𝜈𝛾₅𝑞^𝑏_𝐵−𝑞^𝑎_𝐶𝜎_𝜇𝜈𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝜎^𝜇𝜈𝛾₅𝑞^𝑏_𝐴,𝜓₄=𝜆_𝑎𝑏𝜆_𝑐𝑑𝑞^𝑎_𝐶𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝛾₅𝑞^𝑑_𝐵+𝑞^𝑎_𝐶𝛾₅𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝑞^𝑑_𝐵−𝑞^𝑎_𝐶𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝛾₅𝑞^𝑑_𝐴−𝑞^𝑎_𝐶𝛾₅𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝑞^𝑑_𝐴,𝜓₅=𝜆_𝑎𝑏𝜆_𝑐𝑑𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝛾^𝜇𝛾₅𝑞^𝑑_𝐵+𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝛾^𝜇𝑞^𝑑_𝐵−𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝛾^𝜇𝛾₅𝑞^𝑑_𝐴−𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝛾^𝜇𝑞^𝑑_𝐴,𝜓₆=𝜆_𝑎𝑏𝜆_𝑐𝑑𝑞^𝑎_𝐶𝜎_𝜇𝜈𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝜎^𝜇𝜈𝛾₅𝑞^𝑑_𝐵−𝑞^𝑎_𝐶𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝜎^𝜇𝜈𝛾₅𝑞^𝑑_𝐴. (D.2.1) Among these six currents only three are independent. We can verify the following relations: 𝜂₁1=−4𝜓₁−14𝜓₂+18𝜓₃,𝜂₂1=−4𝜓₁+14𝜓₂+18𝜓₃,𝜂₃=3𝜓₁+12𝜓₃,𝜓₄=13𝜓₁+12𝜓₃,𝜓₅8=−3𝜓₂,𝜓₆=6𝜓₁−53𝜓₃. (D.9)

D.2.2. Flavor Structure 𝟔_𝑓(𝑞𝑞)⊗𝟔_𝑓(𝑞𝑞)

In this subsection we study pseudoscalar tetraquark currents which contain diquarks and antidiquarks having both the symmetric flavor structure 𝟔⊗𝟔. There are altogether three independent pseudoscalar tetraquark currents constructed using diquarks and antidiquarks: 𝜂₁=𝑞_𝐴^𝑎𝑇ℂ𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾₅ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐶𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂₂=𝑞_𝐴^𝑎𝑇ℂ𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐶ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐶ℂ𝑞_𝐷^𝑎𝑇,𝜂₃=𝑞_𝐴^𝑎𝑇ℂ𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝜎^𝜇𝜈𝛾₅ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐶𝜎^𝜇𝜈𝛾₅ℂ𝑞_𝐷^𝑎𝑇. (D.10) There are altogether six pseudoscalar currents constructed using quark-antiquark pairs: 𝜓₁=𝑞^𝑎_𝐶𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝛾₅𝑞^𝑏_𝐵+𝑞^𝑎_𝐶𝛾₅𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝑞^𝑏_𝐵+𝑞^𝑎_𝐶𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝛾₅𝑞^𝑏_𝐴+𝑞^𝑎_𝐶𝛾₅𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝑞^𝑏_𝐴,𝜓₂=𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝛾^𝜇𝛾₅𝑞^𝑏_𝐵+𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝛾^𝜇𝑞^𝑏_𝐵+𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝛾^𝜇𝛾₅𝑞^𝑏_𝐴+𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝛾^𝜇𝑞^𝑏_𝐴,𝜓₃=𝑞^𝑎_𝐶𝜎_𝜇𝜈𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝜎^𝜇𝜈𝛾₅𝑞^𝑏_𝐵+𝑞^𝑎_𝐶𝜎_𝜇𝜈𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝜎^𝜇𝜈𝛾₅𝑞^𝑏_𝐴,𝜓₄=𝜆_𝑎𝑏𝜆_𝑐𝑑𝑞^𝑎_𝐶𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝛾₅𝑞^𝑑_𝐵+𝑞^𝑎_𝐶𝛾₅𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝑞^𝑑_𝐵+𝑞^𝑎_𝐶𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝛾₅𝑞^𝑑_𝐴+𝑞^𝑎_𝐶𝛾₅𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝑞^𝑑_𝐴,𝜓₅=𝜆_𝑎𝑏𝜆_𝑐𝑑𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝛾^𝜇𝛾₅𝑞^𝑑_𝐵+𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝛾^𝜇𝑞^𝑑_𝐵+𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝛾^𝜇𝛾₅𝑞^𝑑_𝐴+𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝛾^𝜇𝑞^𝑑_𝐴,𝜓₆=𝜆_𝑎𝑏𝜆_𝑐𝑑𝑞^𝑎_𝐶𝜎_𝜇𝜈𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝜎^𝜇𝜈𝛾₅𝑞^𝑑_𝐵+𝑞^𝑎_𝐶𝜎_𝜇𝜈𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝜎^𝜇𝜈𝛾₅𝑞^𝑑_𝐴. (D.11) Among these six currents only three are independent. We can verify the following relations: 𝜂₁1=−4𝜓₁−14𝜓₂+18𝜓₃,𝜂₂1=−4𝜓₁+14𝜓₂+18𝜓₃,𝜂₃=3𝜓₁+12𝜓₃,𝜓₄5=−3𝜓₁−12𝜓₃,𝜓₅=43𝜓₂,𝜓₆=−6𝜓₁+13𝜓₃. (D.12)

D.2.3. Flavor Structure 𝟑_𝑓(𝑞𝑞)⊗𝟔_𝑓(𝑞𝑞)

In this subsection we study pseudoscalar tetraquark currents which contain diquarks having the antisymmetric flavor structure and antidiquarks the symmetric flavor structure 𝟑⊗𝟔. There are altogether two independent pseudoscalar tetraquark currents constructed using diquarks and antidiquarks: 𝜂₁=𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾^𝜇𝛾₅ℂ𝑞_𝐷^𝑏𝑇+𝑞^𝑏_𝐶𝛾^𝜇𝛾₅ℂ𝑞_𝐷^𝑎𝑇,𝜂₂=𝑞_𝐴^𝑎𝑇ℂ𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑎_𝐶𝛾^𝜇ℂ𝑞_𝐷^𝑏𝑇−𝑞^𝑏_𝐶𝛾^𝜇ℂ𝑞_𝐷^𝑎𝑇. (D.13) There are altogether four pseudoscalar tetraquark currents constructed using quark-antiquark pairs: 𝜓₁=𝑞^𝑎_𝐶𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝛾₅𝑞^𝑏_𝐵−𝑞^𝑎_𝐶𝛾₅𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝑞^𝑏_𝐵−𝑞^𝑎_𝐶𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝛾₅𝑞^𝑏_𝐴+𝑞^𝑎_𝐶𝛾₅𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝑞^𝑏_𝐴,𝜓₂=𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝛾^𝜇𝛾₅𝑞^𝑏_𝐵−𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑎_𝐴𝑞^𝑏_𝐷𝛾^𝜇𝑞^𝑏_𝐵−𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝛾^𝜇𝛾₅𝑞^𝑏_𝐴+𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑎_𝐵𝑞^𝑏_𝐷𝛾^𝜇𝑞^𝑏_𝐴,𝜓₃=𝜆_𝑎𝑏𝜆_𝑐𝑑𝑞^𝑎_𝐶𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝛾₅𝑞^𝑑_𝐵−𝑞^𝑎_𝐶𝛾₅𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝑞^𝑑_𝐵−𝑞^𝑎_𝐶𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝛾₅𝑞^𝑑_𝐴+𝑞^𝑎_𝐶𝛾₅𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝑞^𝑑_𝐴,𝜓₄=𝜆_𝑎𝑏𝜆_𝑐𝑑𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝛾^𝜇𝛾₅𝑞^𝑑_𝐵−𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑏_𝐴𝑞^𝑐_𝐷𝛾^𝜇𝑞^𝑑_𝐵−𝑞^𝑎_𝐶𝛾_𝜇𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝛾^𝜇𝛾₅𝑞^𝑑_𝐴+𝑞^𝑎_𝐶𝛾_𝜇𝛾₅𝑞^𝑏_𝐵𝑞^𝑐_𝐷𝛾^𝜇𝑞^𝑑_𝐴. (D.14) Among these four currents only two are independent. We can verify the following relations: 𝜂₁=𝜓₁−12𝜓₂,𝜂₂=𝜓₁+12𝜓₂,𝜓₃2=−3𝜓₁−𝜓₂,𝜓₄=−4𝜓₁−23𝜓₂. (D.15)

Acknowledgments

The authors would like to thank Professor Shi-Lin Zhu for helpful discussions. This work is partly supported by the National Natural Science Foundation of China under Grant no. 11205011 and the Fundamental Research Funds for the Central Universities.