Colormagnetic confinement in the quark-gluon thermodynamics

Colormagnetic confinement in the quark-gluon thermodynamics

M.S. Lukashov   and Yu.A. Simonov

Alikhanov Institute for Theoretical and Experimental Physics,
Moscow 117218, Russia
Moscow Institute of Physics and Technology,
Dolgoprudny 141700, Moscow Region, Russia
July 26, 2019

Nonperturbative effects in the quark-gluon thermodynamics are studied in the framework of Vacuum Correlator Method. It is shown, that two correlators: colorelectric and colormagnetic , provide the Polyakov line and the colormagnetic confinement in the spatial planes respectively. As a result both effects produce the realistic behavior of and , being in good agreement with numerical lattice data.

1 Introduction

The idea of a new phase of the QCD matter above some critical temperature has appeared soon after the discovery of QCD, namely in [1, 2, 3] were formulated the first principles of weak interacting quark-gluon medium, named the quark-gluon plasma (QGP).

The first lattice studies [4, 5, 6] have supported this idea, and it was soon realized that high-energy ion collisions can be used to create QGP, see [7] for a recent review and references.

The subsequent lattice studies of QGP and thermal transitions has discovered a variety of sudden and complicated features of the QGP behavior, especially near the transition temperature [8]. At present the high accuracy lattice data are obtained for QCD in the wide temperature region [9, 10, 11].

An important progress was made at large in the framework of the perturbation theory (Hard Thermal Loop (HTL) theory) [12], where terms up to have been taken into account.

However, in the region 150 MeV 600 MeV the nonperturbative (np) effects are most important, which can be taken into account in the framework of the Vacuum Correlator Method (VCM), to be used below.

This method was suggested at the end of 80’s in [13, 14], stating, that the basic origin of the nonperturbative dynamics in QCD at zero or nonzero is connected with the vacuum gluonic fields, appearing in the form of gluon vacuum correlators. In FCM the confinement follows from nonzero quadratic correlator of colorelectric (CE) fields , which produce scalar linear confining interaction , while correlators of colormagnetic (CM) fields , are responsible for confinement in spatial surfaces.


The confining correlators generate the nonzero values of CE and CM string tensions,


The CE correlators and produce the scalar confining interaction and the vector-like nonperturbative interaction respectively.


At the beginning of nineties a new theory of temperature transition in QCD was suggested in [15, 16], where at the critical temperature the correlator , and hence CE confinement disappears, while the CM vacuum fields survive.

The advanced form of the np theory of the thermalized QCD was given in [17], where the Polyakov lines have been derived from the vector CE potential , produced by the CE field correlator .

Recently the approach of FCM for QCD at was reconsidered with the aim to take into account the most important np contributions: vector CE interaction at all and CM confinement at .

It was shown in [18], that the latter phenomenon resolves the old Linde problem, since it produces the effective CM Debye mass and eliminates IR divergence of perturbative theory, however justifying the necessity of summing up the infinite series of diagrams in the order .

In [19, 20] the CM confinement was taken into account together with exact treatment of Polyakov lines in the SU(3) theory. The resulting pressure and entropy are in good agreement with lattice data [11].

It is a purpose of the present paper to apply the same method, as in [19, 20], to the analysis of the QCD matter with at , taking into account accurate values of Polyakov lines and the CM confinement.

Below we explain the general formalism in section 2.

In section 3 the notation of the CM confinement and its dynamics is treated and the resulting for formulas for , are obtained. In section 4 the main dynamical input is defined with respect the and Polyakov lines . In section 5 the numerical results are shown and discussed.

2 General formalism

In this section we are using thermodynamics of quarks and gluons in the vacuum background fields (VBF), as formulated in [15]. For the gluon contribution one obtains


Here refers to the VBF, is a regularizing factor at , and


Here are the ordering operators, and is the field strength of the field , also is obtained from (6) with . The winding path measure is


As one can see in (5), there enters the adjoint gluon loop , which will be a major point of our investigation.

Using relation , one can rewrite (5) as


Here is the adjoint Wilson loop with the contour , and is the normalized adjoint trace.

Note, that we have disregarded so far all perturbative contributions except those possible inside the gluon loop.

We now turn to the quark contribution, which according to [17], can be written in a form, similar to (8)


At this point it is important to look into the details of the vacuum dynamics at , where the main contribution is given by the correlators and . The first is acting in the temporal surfaces , via the interaction , which can be written, according to [21] as


Separating, as in [19, 20] the constant term , one obtains in (8), (10) a factorization of the space 3d, where is acting, and the temporal direction, which yields [15, 19, 20]


Here are 3 d closed loop Green’s functions


As shown in [21], enters in , which contributes to PL


One can see in (14), that for GeV, (the gluelump mass) and , , and hence .

Here is the range of , as was discussed in [22].

Inserting over in (12), one obtains and one has the following form for gluon pressure [19]


In a similar way from (9)–(13) one obtains the quark pressure for one quark flavor with the mass .


In the next section we analyze the 3d loop CM contributions in .

3 Colormagnetic confinement contribution to

As one can see in (13), and contain the contribution of the adjoint and fundamental loops respectively, which are subject to the area law, fund, adj. Kinetic term is in in (13), so both and are proportional to the Green’s functions of two color charges, connected by confining string, from one point on the loop to another (arbitrary) point, e.g. the point on the same loop.

There are two ways, how the CM confinement can be taken into account, suggested in [19]. Considering the oscillator interaction between the charges, one obtains


and is obtained from (17), replacing , by . Here , where is the Debye mass, calculated in [23] in good agreement with lattice data [24].

A more realistic form obtains, when one replaces the linear interaction , varying the parameter in the final expressions, imitating in this way linear interaction by an oscillator potential. Following [19, 20] one obtains


where we have taken into account as in (13), that obtains from replacing adjoint loop by the fundamental one. Finally, substituting these expressions in (15), (16), one obtains the equations for , containing the effects of CM confinement, which will be used in what follows.

4 and the Polyakov lines

In this section we analyze Polyakov lines (PL) and the interaction , which generates those as functions of temperature. It is fundamentally important, that has a finite nonzero limit at large , as it is seen in (11), and it is exactly this value which enters in at not large ,


On the lattice can be measured in two ways, from the correlator of two at the distance , which yields the singlet free energy [25], which is equivalent to , and includes also the perturbative contributions.

On the other hand, can be found together with from the direct measurement of the fundamental line


The resulting values of the renormalized are strongly dependent on the type of lattice quark operator used.

In what follows we shall take our using our from [21], which are in agreement with data from [26]. More explicitly we are writing for as in [17]


We shall be using these values of and the corresponding values of , Eq. (19), in our Eqs. (15), (16), where and are given in (18) to obtain and compare to lattice data.

5 Results and discussion

In this section we present our results for , in the temperature region 150 MeV  1000 MeV. For we are using Eqs. (15), (16) with from (18). The Polyakov lines are obtained from (19), (21). We are using and 100 MeV for and respectively.

We compare in Fig. 1 our results for with the lattice results from Table 3 (right column) of [11]. In the following Fig.2 we report our results for in comparison with lattice data from [11].

Figure 1: The pressure , as obtained from Eqs. (15), (16), – solid line, in comparison with lattice data from [11] – filled dots.
Figure 2: for the pressure given in Fig. 1 – solid line, and the lattice data from [11] – filled dots.

One can see in both Figs.1 and 2 a good agreement of our results with lattice data. Comparing this with our from [17], where the same and Polyakov loops were exploited, but CM confinement in was absent, one can deduce, that the CM contribution is very important in the whole interval of up to 1 GeV. The same is true also for the pure theory, studied in [19, 20]. Moreover, in [18] it was shown, that CM confinement solves the old Linde problems, preventing the accurate perturbative calculations in the region MeV.

Our results show that the FC method can be successfully applied to the quark-gluon thermodynamics and in particular it is planned to extend our analysis to the case of nonzero chemical potential.

We specifically excluded from our analysis the region MeV, where the correlator is acting, since the interesting mechanisms of deconfinement and mutual replacements of and in this region, discussed in [20], require more space and planned for the future.

The authors are grateful for useful discussions to B.O. Kerbikov and M.A. Andreichikov.

This work was done in the framework of the scientific project, supported by the Russian Science Foundation grant number 16-12-10414.


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