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Chiral Structure of Scalar and Pseudoscalar Mesons

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_bar and 27. We study their chiral transformation properties. Particularly, we use the tetraquark currents belonging to the"non-exotic"[(3_bar,3)+(3,3_bar)] 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. The results are consistent with the experimental values.


I. INTRODUCTION
The quark model is very successful in explaining the hadron spectrum with simply using quark-antiquark mesons and three-quark baryons [4][5][6][7][8]. However, there are always multi-quark components in the Fock-space expansion of hadron states [9][10][11]. Hence, it is useful to properly include these multi-quark 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, multi-quark 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 . Particulary, the light scalar mesons are good candidates due to their tetraquark (or molecular) components.
The light scalar mesons f 0 (500) (or σ(500)), κ(800), a 0 (980) and f 0 (980) compose a flavor nonet whose masses are all below 1 GeV [12]. Although such mesons have been intensively studied for many years, their nature is still not fully understood [42,43]. In the conventional quark model, light scalar mesons have aqq configuration of 3 P 0 . 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 [44], 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 ′ [44]. 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 [34-38, 45, 46]. Considering 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.
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 [45,[72][73][74]. 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 Refs. [51][52][53][54] where there are three independent local baryon fields of flavor octet. Previously we chose some mixed currents which provided good QCD sum rule results [45]. 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 "non-exotic" [(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 "non-exotic" [(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) = ij C ij B i (x)B j (x)), 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 S. Weinberg in his recent reference [1][2][3] 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 Sec. II we investigate local tetraquark currents of flavor singlet and J P = 0 + , and others are listed in Appendix A. In Sec. III we study their chiral transformation properties, and results are partly listed in Appendix B. In Sec. IV we use the method of QCD sum rule to study the light scalar mesons through local scalar tetraquark currents belonging to the "non-exotic" [(3, 3)⊕ (3,3)] chiral multiplets. However, the results depend much on the threshold value s 0 suggesting a large contribution from the meson-meson continuum, and so in Sec. V we use only the connected parts of the two-point correlation function to perform the QCD sum rule analyses. Sec. VI is a summary.

II. 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 + : There are two possibilities to construct a flavor single tetraquark current: both of the diquark and antidiquark have the antisymmetric flavor structure3 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, γ 5 , γ µ , γ µ γ 5 and σ µν , we find the following ten independent local tetraquark currents of flavor singlet and J P = 0 + : η S,S 6 = q aT A Cσ µν q b B (q a A σ µν Cq bT B −q b A σ µν Cq aT B ) , η S,S 7 = q aT A Cγ µ γ 5 q b B (q a A γ µ γ 5 Cq bT B −q b A γ µ γ 5 Cq aT B ) , η S,S 8 = q aT A Cγ µ q b B (q a A γ µ Cq bT B +q b A γ µ Cq aT B ) , η S,S 9 = q aT A Cγ µ γ 5 q b B (q a A γ µ γ 5 Cq bT B +q b A γ µ γ 5 Cq aT B ) , η S,S 10 = q aT A Cγ µ q b B (q a A γ µ Cq bT B −q b A γ µ Cq aT B ) .
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; ǫ ABC is the totally anti-symmetric tensor; S ABC P (P = 1 · · · 10) are the normalized totally symmetric matrices; λ N (N = 1 · · · 8) are the Gell-Mann matrices; S ABCD U (U = 1 · · · 27) are the matrices for the 27 flavor representation, as defined in Ref. [61]. Among the ten currents, five current (η S,S 1,2,3,7,8 ) contain diquarks and antidiquarks both having the antisymmetric flavor structure3 ⊗ 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 I.
The chiral structure of tetraquarks is more complicated than their flavor structure: The full (expanded) expressions are shown in Ref. [61]. Among them, the following multiplets contain flavor singlet tetraquarks currents: To clearly see the chiral structure of Eqs. (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: from which we can quickly find out their chiral structure (representations). For example, η S,S 1+2 ≡ η S,S 1 + η S,S 2 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 The results are listed in Table I 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 "non-exotic" 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 S,S 3 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 Ref. [45] belong to the mixing of [(1, 1)] and exotic [(8, 8) To fully study this multiplet, the chiral partners of Eqs. (4) are also studied, i.e., the scalar and pseudoscalar tetraquark currents of flavor singlet, octet, 10, 10 and 27. The results are shown in Appendix. A. The conventional These chiral transformation equations can be compared to those calculated in Ref. [61] which have the same chirality and chiral representation, but in Ref. [61] only the flavor structure is taken into account. The SU(3) V and SU(3) A equations are similar to those ofqq mesons as well as baryons belonging to the same chiral multiplet [56,61], suggesting 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 Ref. [61] is used in the calculations: as well as several other formulae: The transition matrices T N 8×10 and T N 8×27 have been obtained and listed in Ref. [61]. 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.

IV. QCD SUM RULE ANALYSIS
For the past decades QCD sum rule has proven to be a powerful and successful non-perturbative method [75,76]. In sum rule analyses, we consider two-point correlation functions: where J(x) is an interpolating field (current) coupling to a tetraquark state. Here we shall choose the tetraquark currents studied in Sec. II and Appendix. A. We compute Π(q 2 ) 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 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 function, (22) and (23) Assuming the contribution from the continuum states can be approximated well by the spectral density of OPE above a threshold value s 0 (duality), we arrive at the sum rule equation Differentiating Eq. (26) with respect to 1/M 2 B and dividing it by Eq. (26), finally we obtain .
The tetraquark currents classified in Sec. II 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 "non-exotic" [(3, 3) ⊕ (3,3)] chiral multiplets can couple to the light scalar mesons f 0 (500), κ(800), a 0 (980) and f 0 (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 f 0 (500) meson is calculated through tetraquark currents whose quark contents are udūd; κ + (800) through , whose quark contents are udds; a + 0 (980) through whose quark contents are usds; f 0 (980) through whose quark contents are usūs + dsds. 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 strange current quark mass up to m 2 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, particulary the tree-level term α s qq 4 [78,79].
In Eqs. (32)-(37), many terms cancelled, including condensates qq 2 and qq gqGq , which are usually much larger than others. Moreover, Eq. (32) 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 f 0 (980) are the same as those for a 0 (980), and so we obtain the same mass for a 0 (980) and f 0 (980).
There is a minus sign in the definition of the mixed condensate g sq σGq , 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 [80,86].  We use the current σ 1+2 as an example. First we extract its spectral density from Eq. (32) and show it in Fig. 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 σ 1+2 is shown in Fig. 2, when we take s 0 = 0.4 GeV 2 . 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 Eq. (22): It nearly vanishes for κ(800) meson when using κ 1+2 , as shown in Table II. For the f 0 (500) meson it is also not large. This suggests that the two-meson continuum contributes significantly. Only for the a 0 (980) and f 0 (980) mesons it is acceptable. Mathematically, this is because the continuum term is growing as s 4 and the condensates qq 2 and qq gqGq cancelled. We show masses of light scalar mesons as functions of the Borel mass M B and the threshold value s 0 in Fig. 4 and Fig. 5, using solid curves. The masses of σ, κ and a 0 (f 0 ) are around 600 MeV, 900 MeV and 1100 MeV, respectively. However, these results much depend on threshold value s 0 , especially for σ and a 0 , once more suggesting the contribution of meson-meson continuum can not 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 η 1+2 , which have the antisymmetric color structure3 ⊗ 3 and color interactions between quarks and antiquarks are repulsive (the details expressions are similar to Eqs. (28), (29), (30) and (31)): We note that doing this we introduce a few [(8, 1) ⊕ (1, 8)] components, which are still "non-exotic". 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 mod 1+2 as an example. The comparison between pole and continuum contributions for s 0 = 0.4 GeV 2 is shown in Fig. 3 [87][88][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.
We have also studied the threshold value s 0 dependence. The results are shown in Fig. 5, using dashed lines. We can see that s 0 dependence is still significant suggesting the contribution of meson-meson continuum can not 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 [1][2][3]. Mass MB for f0(500), κ(800) and a0(980) (f0(980)) are chosen to be 0.50, 0.60, 0.80 GeV 2 , respectively. The solid curves are obtained using the tetraquark currents σ1+2, σ3, κ1+2, κ3, a01+2 and a03 (see definitions in Eqs. (28), (29), (30) and (31)), while the dashed curves are obtained using the modified currents σ mod 1+2 , κ mod 1+2 and a0 mod 1+2 (see definitions in Eqs. (40)). The results for f0(980) are the same as those for a0(980).  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: Here c is the speed of light. The quantity d X ≡ ΓX (|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 ∆x∆p ≥ /2 to estimate the theoretical uncertainty of d X : Using these equations we obtain: d f0 (500) 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 f 0 (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 the meson-meson continuum contributes much in the cases of f 0 (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.

V. 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, S. Weinberg suggested in his recent reference [1][2][3]: "A one tetraquark pole can only appear in the final, connected, term": where Q(x) is a color-neutral operator, i.e., a tetraquark current. Using Fierz transformation, it can be written in the form (see Appendix. D for details): and B i (x) are color-neutral quark bilinears: In the previous section we have included both the connected parts (the second term in Eq. (44)) and the disconnected parts (the first term in Eq. (44)) 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 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 where a1, a2, b1, b2 are color indices. Its connected parts are just The tetraquark currents Eq. (28) to (31) lead to the following "connected" spectral densities: The sum rules Eqs. (48), (50) and (52) using tetraquark currents σ 3 , κ 3 and a0 3 (f 0 3 ) do not change significantly, i.e., 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 Eqs. (47), (49) and (51) do change significantly.Although the continuum term proportional to s 4 is negative, the spectral densities are positive in our working region s ∼ 1 GeV 2 , as shown in Fig. 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.
Masses of light scalar mesons are calculated using only the connected parts, and the results are shown in Figs. 7, as functions of the Borel Mass M B and the threshold value s 0 . We clearly see that the Borel Mass dependence is still not much; the mass of σ still grows as the threshold value s 0 increases, suggesting that there is still much two-meson contribution (or related to its broad decay width); but the mass curves of κ and a 0 have minimums around s 0 = 0.8 GeV 2 for κ and s 0 = 1.3 GeV 2 for a 0 , where the s 0 dependence is weak. We use these values as inputs, and calculate the masses of light scalar mesons. 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 f 0 (500) is around 400 ∼ 600 MeV, the mass of κ(800) is around 700 ∼ 900 MeV, and masses of a 0 (980) and f 0 (980) are around 900 ∼ 1100 MeV.

VI. SUMMARY
We systematically studied the chiral structure of light scalar mesons using local scalar tetraquark currents that belong to the "non-exotic" [(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, a 0 and f 0 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 η 1 + η 2 −→ 0.99 × η 1 + η 2 . The masses of σ, κ, a 0 and f 0 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 f 0 (500) and κ(800). However, these results still depend much on the threshold value s 0 .
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 σ 3 , κ 3 , a0 3 and f 0 3 (see Eqs. (28)-(31)) do not change significantly. However, the results obtained using the tetraquark currents κ 1+2 , a0 1+2 and f 0 1+2 are improved: the mass curves of κ and a 0 have minimums around s 0 = 0.8 GeV 2 for κ and s 0 = 1.3 GeV 2 for a 0 , where the s 0 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 f 0 (500) is around 400 ∼ 600 MeV, the mass of κ(800) is around 700 ∼ 900 MeV, and the mass of a 0 (980) and f 0 (980) is around 900 ∼ 1100 MeV. We note that in Ref. [45] we used the same method to calculate masses ofqq 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 predominantlyqq states. We note the finite decay width of light scalar mesons can be taken into account which does not change the final result significantly [45]. 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][91][92][93].
We note that we can also use the Firez transformation to write tetraquark currents in a mesonic-mesonic form. Some relations are shown in Appendix. D, and here we show one example: Consideringq R q L andq L q 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 twoqq 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 oneqq pair can also belong to the [(3,3) ⊕ (3, 3)] chiral multiplet. However, all the scalar tetraquark currents inside this multiplet have the q L q LqRqR + q R q RqLqL chirality, and so they are not direct chiral partners of theseqq mesons addressed by chiral singlet quark-antiquark pairs, which have the (q L q R +q R q L ) ⊗ (q L q L +q R q R ) chirality ("chiral" Fock-space expansion), unless these two types of chirality mix with each other; they are more similar to conventionalqq mesons addressed by quark condensates, i.e., mesons(qΓq) ⊗ condensates qq = Similarly, all the pseudoscalar tetraquark currents inside this multiplet also have the q L q LqRqR +q R q RqLqL chirality, and so they are not (direct) terms in the "chiral" Fock-space expansion of theqq 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 Ref. [1], S. 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.
Appendix A: Other Tetraquark Currents

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: . We note that we can prove 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 structure3 ⊗ 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: . We list their chirality and chiral representations in Table III.

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: 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 structure3 ⊗ 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: We list their chirality and chiral representations in Table IV.

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: The two superscripts PS and O denote pseudoscalar and flavor octet, respectively. Among these ten currents, η PS,O 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 structure3⊗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: We list their chirality and chiral representations in Table V.

Scalar Tetraquark Currents of Flavor 27F
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: 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: We list their chirality and chiral representations in Table VI.

Pseudoscalar Tetraquark Currents of Flavor 27F
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: 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: We list their chirality and chiral representations in Table VII. 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: . These two currents both contain diquarks having the antisymmetric flavor structure and antidiquarks the symmetric flavor structure3 ⊗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: We find that these two currents both belong to the chiral representation [(8, 8) + (8,8)] and their chirality is LRLR + RLRL.

Pseudoscalar Tetraquark Currents of Flavor 10F
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: . 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: We find that these two currents both belong to the chiral representation [ . Its chiral transformation properties are

Appendix C: Transition Matrices
The transition matrices T N 10×10 are

The transition matrices (T
Appendix D: (qq)(qq) Tetraquark Currents In Sec. II and Appendix. A we have investigate 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 Firez 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 Ref. [45,63,72].
We shall separately investigate scalar and pseudoscalar in the following subsections. Since Firez 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 structure3 f (qq) ⊗ 3 f (qq), the symmetric flavor structure 6 f (qq) ⊗6 f (qq) and the mixed flavor structure3 f (qq) ⊗6 f (qq). The other case of mixed flavor structure 6 f (qq) ⊗ 3 f (qq) can be similarly investigated.

Scalar Tetraquark Currents
In this subsection we study the scalar tetraquark currents which contain diquarks and antidiquarks having both the antisymmetric flavor structure3 ⊗ 3. There are altogether five independent scalar tetraquark currents constructed using diquarks and antidiquarks: There are altogether ten scalar tetraquark currents constructed using quark-antiquark pairs: Among these ten currents only five are independent. We can verify the following relations: 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: η 1 = q aT A Cγ 5 q b B (q a C γ 5 Cq bT D +q b C γ 5 Cq aT D ) , η 2 = q aT A Cγ µ γ 5 q b B (q a C γ µ γ 5 Cq bT D +q b C γ µ γ 5 Cq aT D ) , η 3 = q aT A Cσ µν q b B (q a C σ µν Cq bT D −q b C σ µν Cq aT D ) , η 4 = q aT A Cγ µ q b B (q a C γ µ Cq bT D −q b C γ µ Cq aT D ) , η 5 = q aT A Cq b B (q a C Cq bT D +q b C Cq aT D ) .
There are altogether ten scalar tetraquark currents constructed using quark-antiquark pairs: , ψ 3 = (q a C σ µν q a A )(q b D σ µν q b B ) + (q a C σ µν q a B )(q b D σ µν q b A ) , ψ 4 = (q a C γ µ γ 5 q a A )(q b D γ µ γ 5 q b B ) + (q a C γ µ γ 5 q a B )(q b D γ µ γ 5 q b A ) , ψ 5 = (q a C γ 5 q a A )(q b D γ 5 q b B ) + (q a C γ 5 q a B )(q b D γ 5 q b A ) , ψ 6 = (q a C λ ab q b A )(q c D λ cd q d B ) + (q a C λ ab q b B )(q c D λ cd q d A ) , ψ 7 = (q a C γ µ λ ab q b A )(q c D γ µ λ cd q d B ) + (q a C γ µ λ ab q b B )(q c D γ µ λ cd q d A ) , ψ 8 = (q a C σ µν λ ab q b A )(q c D σ µν λ cd q d B ) + (q a C σ µν λ ab q b B )(q c D σ µν λ cd q d A ) , ψ 9 = (q a C γ µ γ 5 λ ab q b A )(q c D γ µ γ 5 λ cd q d B ) + (q a C γ µ γ 5 λ ab q b B )(q c D γ µ γ 5 λ cd q d A ) , ψ 10 = (q a C γ 5 λ ab q b A )(q c D γ 5 λ cd q d B ) + (q a C γ 5 λ ab q b B )(q c D γ 5 λ cd q d A ) .
Among these ten currents only five are independent. We can verify the following relations: In this subsection we study pseudoscalar tetraquark currents which contain diquarks and antidiquarks having both the antisymmetric flavor structure3 ⊗ 3. There are altogether three independent pseudoscalar tetraquark currents constructed using diquarks and antidiquarks: There are altogether six pseudoscalar tetraquark currents constructed using quark-antiquark pairs: , ψ 2 = (q a C γ µ q a A )(q b D γ µ γ 5 q b B ) + (q a C γ µ γ 5 q a A )(q b D γ µ q b B ) − (q a C γ µ q a B )(q b D γ µ γ 5 q b A ) − (q a C γ µ γ 5 q a B )(q b D γ µ q b A ) , ψ 3 = (q a C σ µν q a A )(q b D σ µν γ 5 q b B ) − (q a C σ µν q a B )(q b D σ µν γ 5 q b A ) , Among these six currents only three are independent. We can verify the following relations: 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: There are altogether six pseudoscalar currents constructed using quark-antiquark pairs: ψ 1 = (q a C q a A )(q b D γ 5 q b B ) + (q a C γ 5 q a A )(q b D q b B ) + (q a C q a B )(q b D γ 5 q b A ) + (q a C γ 5 q a B )(q b D q b A ) , ψ 2 = (q a C γ µ q a A )(q b D γ µ γ 5 q b B ) + (q a C γ µ γ 5 q a A )(q b D γ µ q b B ) + (q a C γ µ q a B )(q b D γ µ γ 5 q b A ) + (q a C γ µ γ 5 q a B )(q b D γ µ q b A ) , ψ 3 = (q a C σ µν q a A )(q b D σ µν γ 5 q b B ) + (q a C σ µν q a B )(q b D σ µν γ 5 q b A ) , ψ 4 = λ ab λ cd {(q a C q b A )(q c D γ 5 q d B ) + (q a C γ 5 q b A )(q c D q d B ) + (q a C q b B )(q c D γ 5 q d A ) + (q a C γ 5 q b B )(q c D q d A )} , ψ 5 = λ ab λ cd {(q a C γ µ q b A )(q c D γ µ γ 5 q d B ) + (q a C γ µ γ 5 q b A )(q c D γ µ q d B ) + (q a C γ µ q b B )(q c D γ µ γ 5 q d A ) + (q a C γ µ γ 5 q b B )(q c D γ µ q d A )} , ψ 6 = λ ab λ cd {(q a C σ µν q b A )(q c D σ µν γ 5 q d B ) + (q a C σ µν q b B )(q c D σ µν γ 5 q d A )} .
Among these six currents only three are independent. We can verify the following relations: c. Flavor Structure3 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 structure3 ⊗6. There are altogether two independent pseudoscalar tetraquark currents constructed using diquarks and antidiquarks: There are altogether four pseudoscalar tetraquark currents constructed using quark-antiquark pairs: Among these four currents only two are independent. We can verify the following relations: