Non-perturbative Analysis of Various Mass Generation by Gluonic Dressing Effect with the Schwinger-Dyson Formalism in QCD

As a topic of"quantum color dynamics", we study various mass generation of colored particles and gluonic dressing effect in a non-perturbative manner, using the Schwinger-Dyson (SD) formalism in (scalar) QCD. First, we review dynamical quark-mass generation in QCD in the SD approach as a typical fermion-mass generation via spontaneous chiral-symmetry breaking. Second, using the SD formalism for scalar QCD, we investigate the scalar diquark, a bound-state-like object of two quarks, and its mass generation, which is clearly non-chiral-origin. Here, the scalar diquark is treated as an extended colored scalar field, like a meson in effective hadron models, and its effective size $R$ is introduced as a form factor. As a diagrammatical difference, the SD equation for the scalar diquark has an additional 4-point interaction term, in comparison with the single quark case. The diquark size $R$ is taken to be smaller than a hadron, $R\sim 1$ fm, and larger than a constituent quark, $R\sim 0.3$ fm. We find that the compact diquark with $R\simeq 0.3$ fm has a large effective mass of about 900 MeV, and therefore such a compact diquark is not acceptable in effective models for hadrons. We also consider the artificial removal of 3- and 4-point interaction, respectively, to see the role of each term, and find that the 4-point interaction plays the dominant role of the diquark self-energy. From the above two different cases, quarks and diquarks, we guess that the mass generation of colored particles is a general result of non-perturbative gluonic dressing effect.

In terms of the structure of exotic hadrons and heavy hadronic states, diquarks have been considered as important effective degrees of freedom. The diquark is composed of two quarks with gluonic interaction, and it still strongly interacts with gluons because of its non-zero color charge. We investigate the gluonic dressing effect for a scalar diquark using the Schwinger-Dyson formalism in the Landau gauge. The scalar diquark is treated as an extended fundamental field like a meson in effective hadron models. The Schwinger-Dyson equation for the scalar diquark diagrammatically has an additional 4-point interaction term, in comparison with the single quark case. Here, we introduce an effective size of the diquark inside a hadron, since it is a bound-state-like object of two quarks. The effective size and the bare mass of the diquark are free parameters in this scalar theory. The diquark effective size R can be taken to be smaller than a hadron, R ∼ 1 fm, and larger than a constituent quark, R ∼ 0.3 fm. The bare mass is considered as twice of constituent quark mass or twice of running quark mass, which is determined by the Schwinger-Dyson equation for single quark. We find that the compact diquark with R ≃ 0.3 fm has a large effective mass about 1 GeV in both cases of bare mass, and therefore such a compact diquark is not acceptable in effective models for hadrons. We also consider the artificial removal of 3-and 4-point interaction, respectively, to see the role of each term, and find that the 4-point interaction plays the dominant role of the diquark self-energy. Finally, we discuss the non-chiral origin mass generation mechanism of scalar diquark by non-perturbative gluonic dressing effect.

Introduction
The description of hadron structure and its properties are fundamental problems of quantum chromodynamics (QCD). The quarks and gluons, which are ingredients of QCD, are confined and strongly interact with each other in a hadron. Due to the non-perturbative property of QCD, it is difficult to identify that they are effective degrees of freedom of components of hadrons. The constituent quark model gives us the hadrons are constructed by some of valence quarks and antiquarks with a sea of gluons and quark/antiquark pairs, which are called constituent valence quarks and antiquarks. The constituent quarks belong to the fundamental representation 3 c in the SU(3) color group, and the color singlet states 1 c can be observed. Three quarks compose baryons and quark/antiquark pair compose mesons. The quark model has been successfully used to describe the hadron spectroscopy.
However, QCD allows the existence of other color singlet states, which are not classified into ordinary baryons and mesons, such as glueballs, hybrids and multi-quark states, called exotic hadrons. Recent experiments have reported the candidates for these exotic states [1]. The heavy hadron which includes one or more heavy (anti)quarks is also a recent hot topic in hadron spectroscopy [1][2][3].
In the theoretical study of these states, the diquark picture [4,5] is discussed as an important effective degree of freedom. The diquark is composed of two quarks with strong correlation, where the one gluon exchange interaction between two quarks is attractive in the color anti-triplet3 c channel [6,7]. In SU(3) flavor case, the flavor antisymmetric and spin singlet with even parity is the most attractive channel in diquark, which is called scalar diquark. If the diquark correlation is developed in a hadron, this scalar diquark channel would be favored. The diquark correlation in a hadron is discussed in various situations, such as tetra-quarks, heavy baryons and other exotic states [8,9]. The tetra-quark states as the bound state of the diquark/antidiquark is suggested in early day [10], and X(3872) [11] and X(1576) [12] are considered as tetra-quark states. Light flavor mesons as tetra-quark [13][14][15][16][17][18][19][20][21][22] and mixing with qq state [23][24][25] are discussed. There are various studies the heavy baryons focused on diquark [26][27][28][29][30], e.g., single heavy quark/light diquark (Qqq) picture [31][32][33][34][35][36]. The other exotic states including heavy quark(s) are studied [37][38][39][40][41][42][43][44]. The ordinary baryon properties focused on the diquarks have been also discussed [45][46][47][48][49][50]. The diquark correlation is found in the lattice QCD simulation [51][52][53][54]. It is also considered that the diquark condensation is occurred in an extremely high density system, which is called the color superconductivity [55]. The diquarks belong to the fundamental representation3 c , which corresponds to the antiquarks. We note that diquark properties strongly depend on the color number N c . If we consider the two-color QCD, the diquarks compose the color singlet (baryons). The strength of correlation between two quarks is same as quark/antiquark channel, and the (diquark-)baryons correspond to the mesons. This fact is known as the Pauli-Gürsey symmetry. The quark-hadron matter in two-color system is investigated [56][57][58][59][60][61][62]. For the N c = 4 case, the diquarks belong to 6 c or 10 c . As an interesting fact for N c = 4, the diquark contents must be quite different between baryons and tetra-quarks. In fact, the diquark qq in an N c = 4 baryon qqqq belongs to 6 c , which is self-adjoint. On the other hand, the diquark qq in a tetra-quark qqqq belongs to 10 c , which is different color of the diquark in a baryon. From this viewpoint, the N c = 3 case is rather special, because the diquarks belong to the same color3 c in both cases of baryon qqq and tetra-quark qqqq.
However, the properties of diquarks such as the mass and size are not understood well, although the diquarks have been discussed as important object of hadron physics. While the diquark is made by two quarks with gluonic interaction, it still strongly interacts with gluons additionally because of its non-zero color charge. The dressing effect of gluons for diquark should be considered in a non-perturbative way. The dynamics of diquark and gluons may affect the structure of hadrons.
In the quark-hadron physics, the Schwinger-Dyson (SD) formalism is often used to evaluate the non-perturbative effect based on QCD [63][64][65][66][67][68][69][70][71][72][73]. In this paper, we apply the SD formalism to scalar diquark to investigate the effective mass of scalar diquark, which reflects a non-perturbative dressing effect by gluons. The scalar diquark is treated as an extended fundamental field like a meson in effective hadron models, and interacts with the gluons [32,74].
For the argument of the scalar diquark, it would be important to consider its effective size. For, point scalar particles generally have large radiative corrections even in the perturbation theory [75,76]. As an example, in the framework of the grand unified theory (GUT), the Higgs scalar field suffers from a large radiative correction of the GUT energy scale, and therefore severe "fine-tuning" is inevitably required to realize the low-lying Higgs mass of about 126GeV [77], which leads to the notorious hierarchy problem [75,76]. A similar large radiative correction also appears for pointlike scalar-quarks, which correspond to compact scalar diquarks, in scalar lattice QCD calculations [74]. In fact, the pointlike scalar-quark interacting with gluons acquires a large extra mass of about 1.5 GeV at the cutoff a −1 ≃ 1 GeV, where a is the lattice spacing. Such a large-mass acquirement would be problematic in describing hadrons with scalar diquarks. However, since it is a bound-state-like object inside a hadron, the diquark must have an effective size. Therefore, we have to consider an effective size of the diquark, particularly for the argument of the effective diquark mass.
As for the mass, it is considered that the origin of masses of light sector quarks are generated by the spontaneous chiral symmetry breaking except for a few MeV current mass. We can see that QCD has several dynamical mass generation mechanism, even without chiral symmetry breaking. For example, while the charm quark has no chiral symmetry, some difference seems to appear between current and constituent masses for charm quarks: the current mass is m c ≃ 1.2 GeV at renormalization point µ = 2 GeV [1], and the constituent charm quark mass is M c ≃ 1.6 GeV in the quark model. The gluon is more drastic case. While the gluon mass is zero in perturbation QCD, the non-perturbative effect of the selfinteraction of gluons seems to generate a large effective mass of 0.6 GeV [78][79][80], and the lowest glueball mass is about 1.6GeV [81,82]. Furthermore, the dynamical mass generation for scalar-quark have been studied in the scalar lattice QCD calculation [74]. This paper is organized as follows. In Sec. 2, we explain the SD formalism for the single quark to set up the formalism for the scalar diquark case. The SD equation for the scalar diquark with above technique is discussed in Sec. 3, we introduce a simple form factor to consider the size of diquark. In Sec. 4, we present the numerical result of the diquark selfenergy with the dependence of the bare mass and size of diquark. The dynamical mass generation for the scalar diquark in the SD formalism is discussed in Sec. 5. Section 6 is devoted to conclusion and discussion.

The Schwinger-Dyson Equation for the Single Quark (basic formalism)
In this section, we explain the Schwinger-Dyson (SD) formalism for the single quark. The basic formalism, which is discussed in this section, will be used in the scalar diquark case. The dressed quark propergator is obtained by solving the SD equation as S(p 2 ) = iZ(p 2 )(/ p − Σ q (p 2 )) −1 with the wave function renormalization Z(p 2 ) and the self-energy of quark Σ q (p 2 ) in the Minkowski space. We assume the rainbow-ladder approximation, where the non-perturbative effect of gluons is included in the dressed quark-gluon vertex Γ µ a (p, k). By improving the momentum dependence of the quark-gluon vertex in the one-loop level renormalization-group, the quark-gluon vertices in the diagram Fig. 1 is rewritten as where Z g ((p − k) 2 ) is the gluon dressing function and T a (a = 1, · · · , 8) is the generator of color group. The Higashijima-Miransky approximation, which is to take the maximum Fig. 1 The Schwinger-Dyson equation for the single quark. The shaded blob denotes the self-energy of quark Σ q (p 2 ), the black dot the bare quark-gluon vertex, the shaded triangle the dressed vertex Γ µ a (p, k), the solid line the quark propagator and the curly line the gluon propagator.
value of the argument in the coupling as ) in the Landau gauge is used in this study. Here, p E denotes the Euclidean space momentum. We use a renormalization-group-improved coupling in the case of N c = 3 and N f = 3, with an infrared regularization of a simple cut at p IR ≃ 640 MeV which leads to ln(p 2 IR /Λ 2 QCD ) = 1/2, and the QCD scale parameter Λ QCD = 500 MeV [67,68,72]. The infrared regularization has been introduced to avoid the divergent pole at p = Λ QCD . The scale parameter Λ QCD is chosen to reproduce chiral properties for quarks in the SD formalism with the Higashijima-Miransky approximation in the Landau gauge (see Eqs. (5)-(7)), while the ordinary QCD scale parameter is around Λ QCD ∼ 200 − 300 MeV. The behavior of the coupling is shown in Fig. 2 in the Euclidean space. We will show all the figures are in the Euclidean space, except for the figures 1, 4, 5 and 10. In the Landau gauge, the gluon propagator is written as and the wave function renormalization can be taken as Z(p 2 ) = 1. p IR Fig. 2 The behavior of the running coupling of our model α s (p 2 ) as a function of the momentum p in the Euclidean space. The thin line is the one-loop renormalization group improved running coupling. We introduce a simple cut at p IR as an infrared regularization.
The quark self-energy Σ q (p 2 E ) is determined by solving the SD equation, where the Wick rotation has been taken and the Casimir operator C 2 (3) = 8 a=1 T a T a = 4/3 obtained by the color trace. The result of the SD equation is shown in Fig. 3 in the chiral limit m q = 0. There is a small cusp structure at p IR due to the coupling behavior Eq. (2). The ultraviolet cutoff Λ UV is taken as 5 GeV. The self-energy Σ q (p 2 E ) is unchange even the cutoff is taken 10 GeV. The quark mass is large at the infrared region and monotonously goes to zero with the momentum, which reflects the spontaneous chiral symmetry breaking [66,67,83,84].  Fig. 3 The quark self-energy Σ q (p 2 ) as a function of the momentum p in the chiral limit. The self-energy is large in the low momentum region and goes to zero monotonously with the momentum.
The self-energy leads to the pion decay constant with the Pagels-Stokar approximation [85]: and the (unrenormalized) chiral condensate: Since the pion decay constant is a physical value, its renormalization is not required and it does not depend on the ultraviolet cutoff Λ UV . Hence, the upper limit of the integration has been taken as Λ UV → ∞. On the other hand, the chiral condensate depends on the renormalization point. We adopt a standard renormalization point µ = 2 GeV [1], and consider the chiral condensate qq µ=2GeV according to the renormalization-group formula [68,69,72,73]: with 3C2(Nc) 16π 2 β0 = 4/9 and β 0 = 11Nc−2Nf 48π 2 corresponding to the lowest coefficient of the β function of the renormalization group. Taking the scale parameter Λ QCD as 500 MeV [67,68,72] and the ultraviolet cutoff Λ UV as 5 GeV, the pion decay constant and the chiral condensate are fixed as f π ≃ 90 MeV and − qq 1/3 µ=2GeV ≃ 242 MeV, respectively. We have numerically checked that they are stable against the variation of the ultraviolet cutoff Λ UV . The SD formalism with the approximations in the Landau gauge reproduces these chiral properties well.

The Schwinger-Dyson Equation for the Scalar Diquark
Diquark is a bound-state-like object of two quarks and decomposed into color anti-triplet3 c and sextet 6 c and flavor anti-triplet3 f and sextet 6 f in SU(3) flavor case. The most attractive channel for diquark is the color and flavor anti-triplet3 c,f and spin singlet with even parity 0 + by one gluon exchange [6,7] and by instanton interactions [86,87], which is called scalar diquark. If the diquark correlation is developed in a hadron such as a heavy baryon (Qqq), this scalar diquark channel would be favored. We consider the scalar diquark as an effective degree of freedom with a peculiar size, assuming it to be an extended fundamental field φ(x) [32,74] like a meson in the effective hadron models. The scalar diquark is composed of two quarks with the gluonic interaction, and still affected by non-perturbative gluonic effects since it has non-zero color charge as shown Fig. 4. The dynamics of the scalar diquark field φ is described by the gauge invariant Lagrangian: where the bare diquark mass m φ and the gauge field A µ a (gluon) with the generator T a have been introduced. We note that the scalar diquark has the 4-point interaction term different from the quark. In general, such gauged scalar fields accompany the 4-point interaction [75,88].
Since our diquark is confined and located in a hadron, the diquark must have an effective size smaller than the hadron. In order to include the size effect of diquark, we introduce a (a) The gluonic interaction between two quarks (b) The gluonic dressing for a diquark Fig. 4 The two types of gluonic interaction for a diquark: (a) inter-two-quarks gluonic interaction to form a diquark and (b) gluonic dressing for the diquark due to its non-zero color charge. The single line denotes a quark, the double line a diquark and the curly line a gluon.
simple "form factor" in the 4D Euclidean space as where the momentum cutoff Λ corresponds to the inverse of the diquark size R. In this paper, we set R ≡ Λ −1 . Since the radiative correction for the scalar particle is generally large, this form factor has also a role of the convergence factor. As for the form factor f Λ (p 2 E ), it has the roles of introducing an effective size and convergence of the SD equation, so one can use arbitrary function such as the step function θ(Λ 2 − p 2 E ), the exponential function exp(−p 2 E /Λ 2 ) and so on. In this study, we take Eq. (9) with ν = 2 to simple analysis and the convergence of the SD equation. The size effect of the diquark can be included in the vertex as α s (p 2 ) → α s (p 2 )f Λ (p 2 ).
We now describe the SD equation for the scalar diquark. Basically, we use the same framework as the single quark case, which is discussed in the previous section such as the running coupling and parameters, for the scalar diquark case. It is notable that we can use the same form of the running coupling for quark/gluon coupling even for the scalar diquark/gluon [89,90]. The origin of the running coupling comes from the renormalizationgroup analysis. In the heavy mass limit of interacting particles, the interaction depends on only color. Since the scalar diquark corresponds to the antiquark in the color group, we may use same form of the running coupling even for the scalar diquark case. The difference between quark and scalar diquark is just the coefficient [89]. The SD equation for the scalar diquark is diagrammatically expressed as Fig. 5 and is written by .
The second term corresponds to the 4-point interaction term and the third term is the 3-point interaction term in Fig. 5. Here, we do not consider the wave functional renormalization, since the diquark is confined in a hadron and do not produce any physical quantities. Fig. 5 The Schwinger-Dyson equation for the scalar diquark. The shaded blob is the selfenergy Σ(p 2 ), the dashed line denotes the scalar diquark propagator and the curly line the gluon propagator. The last term (4-point interaction) is the peculiar term of scalar theory, which does not appear in the single quark case.
The physical quantities will be discussed in the color singlet state such as heavy baryons, which is our future work.

The Parameters in the Theory
The bare mass m φ and cutoff Λ (inverse of the size R) are free parameters of the diquark theory. We have no physical value to fix these parameters, but we can restrict them from the physical analysis. Since the diquark is originally made of two quarks, the bare mass of diquark may be simply considered as the twice of the quark mass. In this paper, we consider two cases of the bare diquark mass. One is twice of constituent quark mass, i.e., m φ = 600 MeV. The other is twice of the running quark self-energy, i.e., m φ (p 2 E ) = 2Σ q (p 2 E ), where Σ q (p 2 E ) is determined by the SD equation for single quark Eq. (4). This means that the diquark is constructed by the two dressing quarks. The constant bare mass case is based on the constituent quark model like picture and the running bare mass case is the SD formalism with omitting the effect of the gluonic attraction force between two quarks. The diquark should be dressed by gluon furthermore because of its non-zero color charge.
The cutoff Λ corresponds to the diquark size in a hadron, R, i.e., Λ ≡ R −1 , so the diquark should be smaller than the hadron. We also consider two cases of the size. One is the typical size of a baryon, R = 1 fm, i.e., Λ = 200 MeV, which gives the upper limit of the size (the lower limit of the cutoff). The diquark covers the baryon in this case. The second is the typical size of a constituent quark, R ≃ 0.3 fm, i.e., Λ = 600 MeV, which gives the lower limit of the size (the upper limit of the cutoff).

The Constant Bare Mass Case
We first show in Fig. 6 the case of the constant bare mass m φ = 600 MeV with dependence on the cutoff Λ. The diquark self-energy Σ(p 2 ) is always larger than the bare mass m φ and almost constant except for a small bump structure at an infrared region. The value of the self-energy is strongly depends on the cutoff Λ, e.g., the "compact diquark" with R ≃ 0.3 fm has a large mass.
The  in the case of Λ = 200 MeV. The bump structure appears in the case without the 4-point interaction term as shown in Fig. 7(a). Although the diagrammatic expression of the SD equation for the scalar diquark without 4-point interaction term is analogous to the quark SD equation, the behavior is completely different from the quark case. The diquark selfenergy Σ(p 2 E ) starts from the bare mass m φ = 600 MeV at zero momentum, then decreases at low momentum and rises up to the original value 600 MeV. On the other hand, the quark self-energy Σ q (p 2 E ) starts from a large value and goes to zero monotonously with the momentum. The SD equation without 3-point interaction just rises the self-energy and keeps constant. The strong dependence of the cutoff Λ (or the size R) mainly comes from the 4-point interaction term.

The Running Bare Mass Case
We show in Fig. 8 the case of the running bare mass m φ (p 2 E ) = 2Σ q (p 2 E ) with dependence on the cutoff Λ. The diquark self-energy Σ(p 2 E ) also strongly depends on the cutoff Λ. In the low-momentum region, the behavior of Σ(p 2 E ) reflects the running property of the bare mass, especially in the Λ = 200 MeV case, the gluonic effect seems to be small, because of Σ(p 2 E ) ≈ 2Σ q (p 2 E ). In the high-momentum region, the diquark self-energy keeps a large value, while the bare mass m φ (p 2 E ) goes to zero. This suggests the mass generation of the scalar diquark by gluonic radiative correction.

Discussion on the Diquark Property
The scalar diquark self-energy Σ(p 2 ) strongly depends on the diquark size R ≡ Λ −1 in the both cases of the bare mass. In an extreme case of the pointlike limit R → 0, i.e., Λ → ∞, the diquark effective mass diverges. This means that the treatment of pointlike diquarks is not suitable in hadron models and the diquark must have an effective size. As a quantitative argument, our calculations show that the "compact diquark" with R ≃ 0.3 fm has a large effective mass more than 1 GeV in both cases. One may be worried about the uncertainty of the bare diquark mass m φ . However, even in the zero bare-mass case of m φ = 0, we find a large value of about 1GeV for the diquark effective mass in the compact case of R ≃ 0.3 fm, as will be discussed in Sec. 5. Such a large diquark mass is not acceptable in effective models for hadrons. In fact, the appropriate diquark is not compact as R ≃ 0.3 fm but fairly extended as R ≃ 1 fm.

Mass Generation for Scalar Diquark
Finally, we consider the zero bare-mass case of diquark, m φ ≡ 0. The bare mass of diquark would be zero even a finite mass of quark, if the attraction between two quarks sufficiently strong. The result is shown in Fig. 9 for the two cases: (a) Λ = 200 MeV and (b) Λ = 600 MeV on the cutoff. The self-energy Σ(p 2 E ) is always finite and takes a large value even for m φ ≡ 0. The mass generation mechanism in QCD is usually considered in the context of the spontaneous chiral symmetry breaking. On the other hand, our scalar diquark theory is composed of an effective scalar diquark field φ(x) and does not have the chiral symmetry explicitly, although the original diquark is constructed by two quarks. Nevertheless, the effective mass of diquark emerges by the non-perturbative gluonic effect. The dynamical mass generation mechanism seems to work in the scalar diquark theory even without chiral symmetry. If we take Λ = 1 GeV, the diquark self-energy is Σ ∼ 1.3 GeV. This result is consistent with the scalar lattice QCD calculation [74].  Fig. 9 The scalar diquark self-energy Σ(p 2 ) as a function of the momentum p in the massless case of m φ = 0. The self-energy Σ(p 2 ) is finite in both cases.

Conclusion and Discussion
We have investigated the gluonic effect to the scalar diquark, considering the size effect of the diquark in a hadron. The non-perturbative effect is evaluated in the Schwinger-Dyson (SD) formalism in the Landau gauge. The basic technology of scalar SD formalism is imported from the single quark case, such as the running coupling, the approximations and so on.
Since the diquark is located in and construct of a hadron, the size should be smaller than the hadron (R ∼ 1 fm) and larger than the constituent quark (R ∼ 0.3 fm). We have considered the two cases of the constant bare mass m φ = 600 MeV and the running bare mass m φ (p 2 E ) = 2Σ q (p 2 E ). The diquark self-energy strongly depends on the size R = Λ −1 in both cases, especially the small diquark (R ≃ 0.3 fm) has a large effective mass by the gluonic dressing effect.
We find that the effective diquark mass is finite and large even for the zero bare-mass case, and the value strongly depends on the size R, which is an example of dynamical mass generation by the gluonic effect, without chiral symmetry breaking. The mass difference between current and constituent charm quark mass and the large glueball mass are also examples of this type of mass generation. In this sense, the spontaneous chiral symmetry breaking may be a special case of massless (or small mass) fermion. It may be a general property of strong interacting theory that all colored particles acquire a large effective mass by the dressing effect as shown in Fig. 10.
→ Fig. 10 The schematic picture for dynamical mass generation of the colored particle. The colored particle (solid line) interacting with the gluons (curly line). The effective mass emerges by the non-perturbative interaction even without the chiral symmetry.
In this study, we do not calculate any physical value, since the diquark is not observable. We must describe the color singlet states such as heavy baryon Qqq based on the scalar theory. One of description of diquark based on QCD is the Bethe-Salpeter (BS) formalism for two quarks [91][92][93][94][95]. However, the treatment of the scalar diquark as an explicit degree of freedom φ(x) is a good approximation for the structure of the heavy baryons. The constituent scalar-quark(diquark)/quark picture in the scalar lattice QCD [74] and the structure of Λ h (h = s, c, b quarks) with explicit diquark degree of freedom using QCD sum rule [32] have been discussed. The description of the heavy baryon as heavy quark/diquark (Qφ) using the BS equation will be investigated as our future work.
The tetra-quark states qqqq may include diquark/antidiquark components. Although the two mesons molecular states may dominate in the tetra-quark due to the strong correlation between quark and antiquark, the diquark/antidiquark would be also important components [11][12][13][14][15]. The tetra-quark states would be described as the linear combination of two mesons and diquark/antidiquark states based on the BS formalism. The structure of sigma meson (light scalar mesons) is also applicable subject. The sigma meson is considered as a chiral partner of the pion in the context of the chiral symmetry, which structure is quark/antiquark bound state. The possibility of the light scalar mesons as four-quark states have been discussed [10,[13][14][15][16][17][18][19][20][21][22][23][24][25]. The structure of the sigma meson (light scalar mesons) can be described as the linear combination of quark/antiquark, diquark/antidiquark and two mesons in the context of the BS formalism.