Colormagnetic confinement in the quark-gluon thermodynamics

TL;DR

This paper investigates nonperturbative effects in quark-gluon thermodynamics using the Vacuum Correlator Method, demonstrating how colormagnetic and colorelectric correlators influence thermodynamic properties and align with lattice data.

Contribution

It introduces a framework linking colormagnetic and colorelectric correlators to thermodynamic behaviors, providing a realistic description consistent with lattice results.

Findings

Colormagnetic and colorelectric correlators determine key thermodynamic properties.

The model reproduces the temperature dependence of pressure and entropy.

Results agree well with numerical lattice data.

Abstract

Nonperturbative effects in the quark-gluon thermodynamics are studied in the framework of Vacuum Correlator Method. It is shown, that two correlators: colorelectric $D_1^E(x)$ and colormagnetic $D^H(x)$, provide the Polyakov line and the colormagnetic confinement in the spatial planes respectively. As a result both effects produce the realistic behavior of $P(T)$ and $s(T)$, being in good agreement with numerical lattice data.

Equations

$$\frac{g^2}{N_c}⟨⟨\mathrm{Tr}{E}_i(x)\Phi E_j(y){\Phi }^{†}⟩⟩={\delta }_{ij}\left(D^E(u)+D_1^E(u)+u_4^2\frac{\partial D_1^E}{\partial u^2}\right)+u_i{u}_j\frac{\partial D_1^E}{\partial u^2},$$

$$\frac{g^{2}}{N_{c}}\langle\langle{\rm Tr}H_{i}(x)\Phi H_{j}(y)\Phi^{\dagger}\rangle\rangle=\delta_{ij}\left(D^{H}(u)+D_{1}^{H}(u)+\mbox{\boldmath${\rm u}$}^{2}\frac{\partial D_{1}^{H}}{\partial\mbox{\boldmath${\rm u}$}^{2}}\right)-u_{i}u_{j}\frac{\partial D_{1}^{H}}{\partial u^{2}},$$

$${\sigma }^{E(H)}=\frac{1}{2}\int D^{E(H)}(z)d^{2}z.$$

$$V_D(R)=2c_a{\int }_0^r(r-\lambda )d\lambda {\int }_0^{\infty }d\nu D^E(\lambda ,\nu )=V_D^{(\mathrm{lin})}(r)+V_D^{(\mathrm{sat})}(r)$$

$$V_1(r)=c_a{\int }_0^r\lambda d\lambda {\int }_0^{\infty }d\nu D_1^E(\lambda ,\nu ),c_{\mathrm{fund}}=1,c_{\mathrm{adj}}=9/4.$$

$$⟨F_0^{gl}(B){⟩}_B=-T{\int }_0^{\infty }\frac{ds}{s}\xi (s)d^{4}x(Dz{)}_{xx}^w{e}^{-K}\left[\frac{1}{2}tr⟨{\tilde{\Phi }}_F(x,x){⟩}_B-⟨tr\tilde{\Phi }(x,x)⟩\right].$$

$${\tilde{\Phi }}_F(x,y)=P_{F}P\exp (2ig{\int }_0^t\hat{F}(z(\tau ))d\tau )\exp (ig{\int }_y^x{\tilde{B}}_{\mu }d{z}_{\mu }).$$

$$(Dz{)}_{xy}^w$$

$$P_{gl}=(N_c^2-1){\int }_0^{\infty }\frac{ds}{s}\sum _{n\ne 0}{G}^{(n)}(s),G^{(n)}(s)=\int (Dz{)}_{on}^w{e}^{-K}{\widehat{tr}}_a⟨W_{\sum }(C_n)⟩.$$

$$P_q=2N_c{\int }_0^{\infty }\frac{ds}{s}{e}^{-m_q^{2}s}\sum _{n=1}^{\infty }(-{)}^{n+1}[S^{(n)}(s)+S^{(-n)}(s)],$$

$$S^{(n)}(s)=\int (Dz{)}_{on}^w{e}^{-K}\frac{1}{N_c}\widehat{tr}⟨W_{\sigma }(C_n)⟩.$$

$$V_1(r,T)={\int }_0^{\beta }d\nu (1-\nu T){\int }_0^r\xi d\xi D_1^E(\sqrt{{\xi }^2+{\nu }^2}).$$

$$G^{(n)}(s)=\int (D{z}_4{)}_{on}^w{e}^{-K_4-J_n^E}{G}_3(s),S^{(n)}(s)=\int (\overline{D{z}_4}{)}_{on}^w{e}^{-K_4-J_n^E}{S}_3(s).$$

$$G_3(s)=\int (D^{3}z{)}_{xx}{e}^{-K_{3d}}⟨{\widehat{tr}}_a{W}_3^a⟩,S_3(s)=\int (Dz{)}_{xx}{e}^{-K_{3d}}⟨{\widehat{tr}}_f{W}_3^f⟩.$$

$$J_n^E=\frac{n\beta }{2}{\int }_0^{n\beta }d\nu \left(1-\frac{\nu }{n\beta }\right){\int }_0^{\infty }\xi d\xi D_1^E(\sqrt{{\xi }^2+{\nu }^2}),L_{\mathrm{adj}}^{(n)}=\exp \left(-\frac{9}{4}{J}_n^E\right).$$

$$P_{gl}=\frac{N_c^2-1}{\sqrt{4\pi }}{\int }_0^{\infty }\frac{ds}{s^{3/2}}{G}_3(s)\sum _{n=\pm 1,\pm 2,..}{e}^{-\frac{n^2}{4T^{2}s}}{L}_{\mathrm{adj}}^{(n)}.$$

$$P_q=\frac{4N_c}{\sqrt{4\pi }}{\int }_0^{\infty }\frac{ds}{s^{3/2}}{e}^{-m_q^{2}s}{S}_3(s)\sum _{n=1,2,..}(-{)}^{n+1}{e}^{-\frac{n^2}{4T^{2}s}}{L}_f^{(n)}.$$

$$G_3^{OSC}(s)=\frac{1}{(4\pi {)}^{3/2}\sqrt{s}}\frac{M_{\mathrm{adj}}^2}{sh{M}_{\mathrm{adj}}^{2}s}$$

$$G_3^{\mathrm{lin}}(s)=\frac{1}{(4\pi s{)}^{3/2}}{\left(\frac{M_{\mathrm{adj}}^{2}s}{sh(M_{\mathrm{adj}}^{2}s)}\right)}^{1/2},S_3^{\mathrm{lin}}(s)=G_3^{\mathrm{lin}}(s){|}_{M_{\mathrm{adj}}\to M_f},$$

$$L_i^{(n)}\simeq \exp \left(-c_{i}n\frac{V_1^{np}(\infty ,T)}{2T}\right),c_{adj}=\frac{9}{4},c_f=1.$$

$$L^{\rm bare}=\frac{1}{3}\left\langle Tr\prod^{N_{c}-1}_{x_{0}=0}U_{0}(\mbox{\boldmath${\rm x}$},x_{0})\right\rangle_{\rm vac,\mbox{\boldmath${\rm x}$}}.$$

$$V_1^{np}(\infty ,T)=\frac{0.175\mathrm{GeV}}{1.35\frac{T}{T_c}-1}.$$

Full text

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 $T_{c}$ [8]. At present the high accuracy lattice data are obtained for $n_{f}=2+1$ QCD in the wide temperature region [9, 10, 11].

An important progress was made at large $T$ in the framework of the perturbation theory (Hard Thermal Loop (HTL) theory) [12], where terms up to $O(g^{6})$ have been taken into account.

However, in the region 150 MeV$<T<$ 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 $T$ is connected with the vacuum gluonic fields, appearing in the form of gluon vacuum correlators. In FCM the confinement follows from nonzero quadratic correlator $D^{E}(x-y)$ of colorelectric (CE) fields $E_{i}^{a}(x)$, which produce scalar linear confining interaction $V_{D}^{(\rm lin)}(r)$, while correlators $D^{H}(x-y)$ of colormagnetic (CM) fields $H_{i}^{a}(x)$, are responsible for confinement in spatial surfaces.

[TABLE]

[TABLE]

The confining correlators $D^{E},D^{H}$ generate the nonzero values of CE and CM string tensions,

[TABLE]

The CE correlators $D^{E}$ and $D^{E}_{1}$ produce the scalar confining interaction $V_{D}(R)$ and the vector-like nonperturbative interaction $V_{1}(R)$ respectively.

[TABLE]

[TABLE]

At the beginning of nineties a new theory of temperature transition in QCD was suggested in [15, 16], where at the critical temperature $T_{c}$ the correlator $D^{E}$, 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 $V_{1}(r,T)$, produced by the CE field correlator $D^{E}_{1}(x-y)$.

Recently the approach of FCM for QCD at $T>0$ was reconsidered with the aim to take into account the most important np contributions: vector CE interaction $V_{1}(r,T)$ at all $T$ and CM confinement at $T\geq T_{c}$.

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 $O(g^{6})$.

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 $P(T)$ and entropy $s(T)$ 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 $n_{f}>0$ at $T\geq T_{c}$, 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 $P(T)$, $s(T)$ are obtained. In section 4 the main dynamical input is defined with respect the $V_{1}(r,T)$ and Polyakov lines $L(T)$. 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

[TABLE]

Here $B_{\mu}$ refers to the VBF, $K=\frac{1}{4}\int^{s}_{0}d\tau\left(\frac{dz_{\mu}(\tau)}{d\tau}\right)^{2},$ $\xi(s)$ is a regularizing factor at $s=0$, and

[TABLE]

Here $P_{F},P$ are the ordering operators, and $\hat{F}$ is the field strength of the field $B_{\mu}$, also $\tilde{\Phi}(x,y)$ is obtained from (6) with $\hat{F}=0$. The winding path measure is

[TABLE]

As one can see in (5), there enters the adjoint gluon loop $tr\tilde{\Phi}(x,x)$, which will be a major point of our investigation.

Using relation $P_{gl}V_{3}=-\langle F_{0}(B)\rangle_{B}$, one can rewrite (5) as

[TABLE]

Here $W_{\sum}$ is the adjoint Wilson loop with the contour $C_{n}$, and $\hat{tr}_{a}$ 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)

[TABLE]

[TABLE]

At this point it is important to look into the details of the vacuum dynamics at $T\geq T_{c}$, where the main contribution is given by the correlators $D_{1}^{E}$ and $D^{H}$. The first is acting in the temporal surfaces $(i4)$, via the interaction $V_{1}(r,T)$, which can be written, according to [21] as

[TABLE]

Separating, as in [19, 20] the constant term $V_{1}(\infty,T)$, one obtains in (8), (10) a factorization of the space 3d, where $D^{H}$ is acting, and the temporal direction, which yields [15, 19, 20]

[TABLE]

Here $G_{3}(s),S_{3}(s)$ are 3 d closed loop Green’s functions

[TABLE]

As shown in [21], $V_{1}$ enters in $J_{n}^{E}$, which contributes to PL

[TABLE]

One can see in (14), that for T\mathrel{\mathchoice{\lower 3.6pt\vbox{\halign{\mathsurround=0pt\displaystyle\hfil#\hfil\cr<\crcr\sim\crcr}}}{\lower 3.6pt\vbox{\halign{\mathsurround=0pt\textstyle\hfil#\hfil\cr<\crcr\sim\crcr}}}{\lower 3.6pt\vbox{\halign{\mathsurround=0pt\scriptstyle\hfil#\hfil\cr<\crcr\sim\crcr}}}{\lower 3.6pt\vbox{\halign{\mathsurround=0pt\scriptscriptstyle\hfil#\hfil\cr<\crcr\sim\crcr}}}}M_{glp}=1.5 GeV, (the gluelump mass) and $n<n^{*}=\frac{M_{glp}}{T}$, $J^{E}_{n}\approx nJ^{E}_{1}$, and hence $L_{\rm adj}^{(n)}\approx(L_{\rm adj})^{n},~{}~{}L_{\rm adj}\equiv L_{\rm adj}^{(1)}$.

Here ${M_{glp}}^{-1}$ is the range of $D^{E}_{1}$, as was discussed in [22].

Inserting over $(Dz_{4})$ in (12), one obtains $\int(Dz_{4})^{w}_{on}e^{-K_{4}}=\frac{1}{\sqrt{4\pi s}}e^{-\frac{n^{2}\beta^{2}}{4s}}$ and one has the following form for gluon pressure [19]

[TABLE]

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

[TABLE]

In the next section we analyze the 3d loop CM contributions in $S_{3}(s),G_{3}(s)$.

3 Colormagnetic confinement contribution to $S_{3}(s),G_{3}(s)$

As one can see in (13), $G_{3}(s)$ and $S_{3}(s)$ contain the contribution of the adjoint and fundamental loops respectively, which are subject to the area law, $\langle\hat{tr}_{i}W_{3}\rangle=\exp(-\sigma_{i}{\rm are}a(W))$ $i=$ fund, adj. Kinetic term is in $K_{3d}$ in (13), so both $G_{3}(s)$ and $S_{3}(s)$ are proportional to the Green’s functions of two color charges, connected by confining string, from one point $x$ on the loop to another (arbitrary) point, e.g. the point $u$ 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

[TABLE]

and $S^{OSC}_{3}(s)$ is obtained from (17), replacing $M_{\rm adj}$, by $M_{f}$. Here $M_{\rm adj}=2\sqrt{\sigma_{s}}=m_{D}(T)$, where $m_{D}(T)$ 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 $\sigma_{s}r\to\frac{\sigma_{s}}{2}\left(\frac{r^{2}}{\gamma}+\gamma\right)$, varying the parameter $\gamma$ in the final expressions, imitating in this way linear interaction by an oscillator potential. Following [19, 20] one obtains

[TABLE]

where we have taken into account as in (13), that $S_{3}$ obtains from $G_{3}$ replacing adjoint loop $W_{3}$ by the fundamental one. Finally, substituting these expressions in (15), (16), one obtains the equations for $P_{gl}^{\rm lin},P_{q}^{\rm lin}$, containing the effects of CM confinement, which will be used in what follows.

4 $V_{1}(r,T)$ and the Polyakov lines

In this section we analyze Polyakov lines (PL) $L_{i},i=adj,f,$ and the $np$ interaction $V_{1}^{np}(r,T)$, which generates those as functions of temperature. It is fundamentally important, that $V_{1}^{np}(r,T)$ has a finite nonzero limit at large $r$ , as it is seen in (11), and it is exactly this value which enters in $L_{i}$ at not large $n$,

[TABLE]

On the lattice $PL$ can be measured in two ways, from the correlator of two $PL$ at the distance $r$, which yields the singlet free energy $F_{Q\bar{Q}}^{s}(r,T)$ [25], which is equivalent to $V_{1}(r,T)$, and includes also the perturbative contributions.

On the other hand, $F_{q}(T)$ can be found together with $L_{f}$ from the direct measurement of the fundamental line

[TABLE]

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

In what follows we shall take our $L_{f}$ using our $V_{1}^{np}$ from [21], which are in agreement with data from [26]. More explicitly we are writing for $V_{1}$ as in [17]

[TABLE]

We shall be using these values of $V_{1}^{np}$ and the corresponding values of $L_{i}(T)$, Eq. (19), in our Eqs. (15), (16), where $S_{3}$ and $G_{3}$ are given in (18) to obtain $P(T),s(T)$ and compare to lattice data.

5 Results and discussion

In this section we present our results for $P(T)$, $s(T)$ in the temperature region 150 MeV $\leq T\leq$ 1000 MeV. For $P(T)=\sum_{m_{q}}P_{q}^{(m_{q})}(T)+P_{g}(T)$ we are using Eqs. (15), (16) with $G_{3}(s),S_{3}(s)$ from (18). The Polyakov lines are obtained from (19), (21). We are using $m_{q}=3,5$ and 100 MeV for $m_{q}=m_{u},m_{d}$ and $m_{s}$ respectively.

We compare in Fig. 1 our results for $\frac{P(T)}{T^{4}}$ with the lattice results from Table 3 (right column) of [11]. In the following Fig.2 we report our results for $s(T)$ in comparison with lattice data from [11].

One can see in both Figs.1 and 2 a good agreement of our results with lattice data. Comparing this with our $P(T)$ from [17], where the same $V_{1}^{np}$ and Polyakov loops were exploited, but CM confinement in $S_{3},G_{3}$ was absent, one can deduce, that the CM contribution is very important in the whole interval of $T$ up to 1 GeV. The same is true also for the pure $SU(3)$ 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 $T<600$ 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 $T<150$ MeV, where the correlator $D^{E}(x)$ is acting, since the interesting mechanisms of deconfinement and mutual replacements of $V_{D}$ and $V_{1}^{np}$ 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.