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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 ~ 1 fm, and larger than a constituent quark, R ~ 0.3 fm. We find that the compact diquark with R ~ 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.

Quantum chromodynamics (QCD) is the fundamental gauge theory of the strong interaction, and it is a long important problem to describe hadron structure and properties based on QCD. Quarks and gluons, the basic ingredients of QCD, strongly interact with each other in an infrared region, and they are confined in hadrons. Then, due to their non-perturbative properties, it is fairly difficult to describe hadrons directly from QCD. Also, the non-perturbative dynamics in QCD directly relates to the other important physical subject of “mass generation.”

The origin of mass is one of the most fundamental issues in physics. One famous category of mass generation is the Yukawa interaction with the Higgs field. However, even besides the dark sector, the Higgs-origin mass is only about 1% of the total mass in our universe, where dominant massive particles are nuclei (u,d quarks) and electrons. Actually, the Higgs interaction only gives the electron mass (about 0.5MeV) and a small current quark mass (a few MeV) for u,d quarks [

strong interaction, apart from the dark sector. In fact, a large constituent quark mass of

Such a dynamical fermion-mass generation in the strong interaction was first pointed out by Y. Nambu et al. [

Even without chiral symmetry breaking, however, it is likely that QCD has several dynamical mass generation mechanism. 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

The gluon is more drastic case. While the gluon mass is zero in perturbation QCD, the non-perturbative effect of the self-interaction of gluons seems to generate a large effective mass of 0.6 GeV [

Next, let us consider compositeness of hadrons in terms of quarks. As an infrared effective theory, the constituent quark model has been successful for the description of the hadron spectroscopy. The constituent

quark belongs to the fundamental representation

classified as the color-singlet (

In the theoretical study of these states, the diquark picture [

one-gluon-exchange interaction between two quarks is attractive in the color anti-triplet

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 [

Gürsey symmetry. The quark-hadron matter in two-color system is investigated [

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. Therefore, such 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 [

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 [

interacting with gluons acquires a large extra mass of about 1.5 GeV at the cutoff

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. This effect gives a natural UV cutoff of the theory, and reduces the large radiative correction. Then, we take account of the effective size and investigate the mass of the scalar diquark inside a hadron within the SD formalism.

This paper is organized as follows. In Section 2, we review the SD formalism for the light quark, as the typical fermion mass generation in QCD. In Section 3, we investigate the SD equation for the scalar diquark, where a simple form factor is introduced for the possible size of diquark. In Section 4, we present the numerical result of the diquark self-energy with the dependence of the bare mass and size of diquark, and briefly discuss the dynamical mass generation for the scalar diquark in the SD formalism. Section 5 is devoted to conclusion and discussion.

The chiral symmetry is a fundamental symmetry in the light-quark sector of QCD, and it is an exact global symmetry in the chiral limit. In the low-energy region of QCD, spontaneous chiral-symmetry breaking takes place, which generates a large effective mass of light quarks. Actually, in the theoretical analysis with the Schwinger-Dyson (SD) formalism in QCD, a large self-energy generation of quarks is demonstrated in an infrared region, which breaks the chiral symmetry in the physically stable vacuum [

As a merit of the Lorentz-covariant gauge like the Landau gauge, the dressed quark propagator is generally described as

expressed in

depicted in

where _{c}) color group.

In the most SD studies for quarks, one takes the rainbow-ladder approximation with the renormalization- group improvement of the quark-gluon vertex at the one-loop level. Note that, owing to the iterative structure of the SD equation, a simplified full-order treatment on the coupling

Here, we briefly mention the treatment of quark confinement in the SD approach. In most works of the SD approach, the confinement effect is ignored, which seems problematic for the study of QCD. On this point, several recent studies, both analytical works [

At the one-loop level of renormalization-group improvement, the SD kernel is approximated as

and the Landau-gauge gluon propagator is given as

Then, by taking Dirac trace or the trace after multiplying

with the bare quark mass

We use one-loop level renormalization-group-improved coupling in the case of

with an infrared regularization of a simple cut at

The Higashijima-Miransky approximation is to take the larger value of the argument (Euclidean momenta) in the coupling as

where the Wick rotation has been taken. (For the detail, see, e.g., Appendix in Ref. [

unchanged 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 spontaneous chiral-symmetry breaking [

The scale parameter

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

with

respectively. We have numerically checked that they are stable against the variation of the ultraviolet cutoff

In this section, we investigate the scalar diquark, i.e., an extended colored scalar object, and its mass generation, using the Schwinger-Dyson (SD) formalism.

Diquark is a bound-state-like object of two quarks and decomposed into color anti-triplet

diquark as an effective degree of freedom with a peculiar size, assuming it to be an extended scalar field

(11)

where the bare diquark mass

ed. We note that the scalar diquark has the 4-point interaction term of

Since the diquark is a bound-state-like object confined in a hadron, it must have an effective size and its size should be smaller than the hadron. In order to include the size effect of diquark, we introduce a simple “form factor” in the four-dimensional Euclidean space as

where the momentum cutoff

While the scalar QCD Lagrangian (11) is renormalizable, this theory is an effective cutoff theory with an UV cutoff parameter

single quark case, presented in the previous section. For instance, we will use the same running coupling

We now describe the SD equation for the scalar diquark, as shown in

In the right-hand side of Equation (13), the second term arises from the 4-point vertex and the third term is lead from the 3-point vertex, as shown in

The bare mass

The diquark is originally made of two consistent quarks, and the color-Coulomb interaction is one of the main attractive forces. We here estimate the color-Coulomb interaction between the two massive quarks from the

three-quark (3Q) potential [

with the color-Coulomb coefficient

In this paper, we consider two cases of the bare diquark mass. One is twice of constituent quark mass, i.e.,

is determined by the SD equation for single quark Equation (7). 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

We first show in

The scalar QCD includes both 3-point and 4-point interactions, and the existence of 4-point interaction is diagrammatically different from the ordinary QCD. To see the role of each interaction, we consider the calculation of the artificial removal of 3-point interaction and 4-point interaction, respectively. In fact, we investigate the two cases: (a) removal of 4-point interaction and (b) removal of 3-point interaction. The result is shown in

diquark without 4-point interaction term is analogous to the quark SD equation, the behavior is completely

different from the quark case. The diquark self-energy

SD equation without 3-point interaction just rises the self-energy and keeps constant. The strong dependence of the cutoff

We show in

In this subsection, we discuss the mass and the size of the scalar diquark, with comparing to the chiral quark. One of the most important properties of single quark SD equation (7) is the existence of the trivial solution

On the other hand, the SD equation (13) for scalar diquark has no trivial solution and is a highly non-linear equation, even in the zero bare mass limit

Actually, the scalar diquark self-energy

As a quantitative argument, our calculations show that the “compact diquark” with

Finally, we consider the zero bare-mass case of diquark,

finite and takes a large value even for

of an effective scalar diquark field

original diquark is constructed by two chiral quarks. Nevertheless, the effective mass of diquark emerges by the non-perturbative gluonic effect. In fact, the mechanism of dynamical mass generation seems to work in the

scalar diquark theory, even without chiral symmetry breaking. If we take

We have studied various mass generation of colored particles and gluonic dressing effect in a non-perturbative manner, using the Schwinger-Dyson (SD) formalism in QCD. First, we have briefly reviewed 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 have investigated the scalar diquark, a bound-state-like object of two quarks, and its mass generation, which is clearly non-chiral-origin. Considering the possible size of the diquark inside a hadron, the effect of diquark size R is introduced as a cutoff

parameter

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 (

bare mass

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, spontaneous chiral-symmetry breaking may be a special case of massless (or small mass) fermion. As was conjectured in Ref. [

In this study, we have mainly investigated the diquark properties, and have not calculated physical quantities. It is however desired to describe the color-singlet states such as heavy baryon

The tetra-quark states

S.I. thanks T.M. Doi, H. Iida and N. Yamanaka for useful discussion and comments. This work is in part sup- ported by the Grant for Scientific Research [Priority Areas “New Hadrons” (E01:21105006), (C) No.23540306, No.15K05076] from the Ministry of Education, Culture, Science and Technology of Japan.

Shotaro Imai,Hideo Suganum, (2016) Non-Perturbative Analysis of Various Mass Generation by Gluonic Dressing Effect with the Schwinger-Dyson Formalism in QCD. Journal of Modern Physics,07,790-805. doi: 10.4236/jmp.2016.78073