Gauge--invariant field correlators in QCD at finite temperature

TL;DR

This paper investigates how gauge-invariant two-point correlation functions of gauge fields behave across the deconfinement transition in QCD using lattice simulations.

Contribution

It provides new numerical data on gauge-invariant correlators in QCD at finite temperature, focusing on the deconfinement phase transition.

Findings

Correlation functions change significantly across the transition

Results support models of deconfinement involving gauge field correlations

Provides benchmarks for theoretical models of QCD at finite temperature

Abstract

We study by numerical simulations on a lattice the behaviour of the gauge--invariant two--point correlation functions of the gauge field strengths across the deconfinement phase transition.

Tables

Table 1. Table 1: The values of the correlator plotted in Fig. 1.

$T/T_c$	$d_{\mathrm{phys}}$	correlator	error
0.956	4.19942108E-01	7.39002927E+07	2.73704788E+06
0.956	5.59922811E-01	3.07028346E+07	3.07917886E+05
0.956	6.99903514E-01	1.38699901E+07	2.32649070E+05
0.956	8.39884217E-01	5.95307914E+06	1.36852394E+05
0.956	9.79864920E-01	1.71065492E+06	1.36852394E+05
0.978	4.10624802E-01	7.55998203E+07	2.24553922E+06
0.978	5.47499737E-01	3.18043204E+07	3.29345752E+05
0.978	6.84374671E-01	1.40046796E+07	2.76949837E+05
0.978	8.21249605E-01	5.94319379E+06	1.72158007E+05
0.978	9.58124539E-01	1.33983840E+06	1.64672876E+05
1.011	3.97031115E-01	3.90523140E+07	1.88410287E+06
1.011	5.29374819E-01	1.07736428E+07	8.30718082E+05
1.011	6.61718524E-01	2.72338505E+06	3.85384677E+05
1.034	3.88217074E-01	1.01557232E+07	5.80862394E+05
1.034	5.17622766E-01	1.82690592E+06	1.68637469E+05
1.070	3.75357913E-01	7.16100968E+06	3.10882157E+05
1.070	5.00477217E-01	1.22530450E+06	1.06128736E+05
1.131	3.54856502E-01	7.59600890E+06	2.81830719E+05
1.131	4.73142002E-01	1.42257411E+06	9.39435730E+04

Table 2. Table 2: The values of the correlator plotted in Fig. 2.

$T/T_c$	$d_{\mathrm{phys}}$	correlator	error
0.956	4.19942108E-01	1.00586510E+08	4.78983379E+06
0.956	5.59922811E-01	3.96598238E+07	4.10557182E+05
0.956	6.99903514E-01	1.89266861E+07	1.91593352E+05
0.956	8.39884217E-01	1.00997067E+07	1.23167155E+05
0.956	9.79864920E-01	5.63147601E+06	1.09481915E+05
0.956	1.11984562E+00	3.18181816E+06	1.02639295E+05
0.978	4.10624802E-01	1.09282908E+08	3.74256536E+07
0.978	5.47499737E-01	4.22161373E+07	7.48513072E+05
0.978	6.84374671E-01	2.01649422E+07	2.09583660E+05
0.978	8.21249605E-01	1.11004489E+07	1.42217484E+05
0.978	9.58124539E-01	6.38481650E+06	1.27247222E+05
0.978	1.09499947E+00	3.75753562E+06	1.12276961E+05
1.011	3.97031115E-01	1.18184634E+08	2.56923118E+07
1.011	5.29374819E-01	3.93948781E+07	4.28205197E+06
1.011	6.61718524E-01	1.56294897E+07	3.51128262E+05
1.011	7.94062229E-01	7.23838065E+06	1.74707720E+05
1.011	9.26405934E-01	3.76820573E+06	1.37025663E+05
1.011	1.05874964E+00	2.23523113E+06	1.11333351E+05
1.034	3.88217074E-01	1.17109354E+08	2.81062449E+07
1.034	5.17622766E-01	3.56012435E+07	1.59268721E+07
1.034	6.47028457E-01	1.26478102E+07	2.15481211E+06
1.034	7.76434149E-01	5.49477087E+06	1.40531224E+05
1.034	9.05839840E-01	2.44992768E+06	8.43187346E+04
1.034	1.03524553E+00	1.22543228E+06	5.52756149E+04
1.070	3.75357913E-01	1.28640892E+08	3.21602231E+07
1.070	5.00477217E-01	4.04146804E+07	1.60801116E+07
1.070	6.25596522E-01	1.45471409E+07	1.50081041E+06
1.070	7.50715826E-01	6.02468180E+06	2.25121562E+05
1.070	8.75835130E-01	2.66822651E+06	1.39360967E+05
1.070	1.00095443E+00	1.32928922E+06	1.07200744E+05
1.131	3.54856502E-01	1.52993819E+08	4.02615313E+07
1.131	4.73142002E-01	4.79112222E+07	1.87887146E+07
1.131	5.91427503E-01	1.65072278E+07	1.47625615E+06
1.131	7.09713003E-01	6.41500398E+06	2.68410209E+05
1.131	8.27998504E-01	2.75120464E+06	9.39435730E+04
1.131	9.46284005E-01	1.31923617E+06	7.24707563E+04

Table 3. Table 3: The values of the correlator plotted in Fig. 3.

$T/T_c$	$d_{\mathrm{phys}}$	correlator	error
0.956	4.19942108E-01	7.21896378E+07	1.36852394E+06
0.956	5.59922811E-01	3.04428150E+07	2.12121211E+05
0.956	6.99903514E-01	1.34046920E+07	1.77908112E+05
0.956	8.39884217E-01	6.09472137E+06	1.02639295E+05
0.956	9.79864920E-01	2.50371455E+06	9.57966758E+04
0.956	1.11984562E+00	9.85337237E+05	8.21114364E+04
0.978	4.10624802E-01	7.40279428E+07	1.34732353E+06
0.978	5.47499737E-01	3.18342609E+07	2.69464706E+05
0.978	6.84374671E-01	1.45286387E+07	1.72158007E+05
0.978	8.21249605E-01	7.08093366E+06	1.04791830E+05
0.978	9.58124539E-01	3.14375490E+06	9.73066993E+04
0.978	1.09499947E+00	1.31438895E+06	8.98215686E+04
1.011	3.97031115E-01	7.46961146E+07	2.82615430E+06
1.011	5.29374819E-01	3.34256977E+07	2.99743638E+05
1.011	6.61718524E-01	1.61827308E+07	1.76420541E+05
1.011	7.94062229E-01	8.65830908E+06	1.06194889E+05
1.011	9.26405934E-01	4.48245200E+06	9.50615537E+04
1.011	1.05874964E+00	2.17100035E+06	8.64974498E+04
1.034	3.88217074E-01	7.55196062E+07	1.87374966E+06
1.034	5.17622766E-01	3.53763936E+07	5.62124898E+05
1.034	6.47028457E-01	1.79130467E+07	2.24849959E+05
1.034	7.76434149E-01	9.78097322E+06	1.03056231E+05
1.034	9.05839840E-01	5.35236590E+06	1.12424980E+05
1.034	1.03524553E+00	2.69351513E+06	1.40531224E+05
1.070	3.75357913E-01	8.00146351E+07	2.35841636E+06
1.070	5.00477217E-01	3.78633027E+07	5.36003718E+05
1.070	6.25596522E-01	1.93711744E+07	2.46561710E+05
1.070	7.50715826E-01	1.10309565E+07	1.34000930E+05
1.070	8.75835130E-01	6.22836321E+06	1.17920818E+05
1.070	1.00095443E+00	3.24282250E+06	1.39360967E+05
1.131	3.54856502E-01	9.55942958E+07	2.68410209E+06
1.131	4.73142002E-01	4.62336584E+07	5.36820417E+05
1.131	5.91427503E-01	2.41166572E+07	2.81830719E+05
1.131	7.09713003E-01	1.36218181E+07	1.61046125E+05
1.131	8.27998504E-01	7.80805297E+06	1.34205104E+05
1.131	9.46284005E-01	4.16035823E+06	2.68410209E+05

Table 4. Table 4: The values of the correlator plotted in Fig. 4.

$T/T_c$	$d_{\mathrm{phys}}$	correlator	error
0.956	4.19942108E-01	1.01236558E+08	5.47409576E+06
0.956	5.59922811E-01	3.92971649E+07	2.80547408E+05
0.956	6.99903514E-01	1.80439881E+07	2.32649070E+05
0.956	8.39884217E-01	9.08699896E+06	1.43695014E+05
0.956	9.79864920E-01	4.83499508E+06	1.23167155E+05
0.956	1.11984562E+00	2.54545453E+06	1.09481915E+05
0.978	4.10624802E-01	1.12247020E+08	3.44316013E+07
0.978	5.47499737E-01	4.26502748E+07	5.98810457E+05
0.978	6.84374671E-01	1.96709235E+07	2.61979575E+05
0.978	8.21249605E-01	1.01648075E+07	1.52696667E+05
0.978	9.58124539E-01	5.45740881E+06	1.19762091E+05
0.978	1.09499947E+00	2.95138704E+06	1.12276961E+05
1.011	3.97031115E-01	1.31030790E+08	3.59692365E+07
1.011	5.29374819E-01	4.90980079E+07	1.88410287E+06
1.011	6.61718524E-01	2.31659012E+07	2.99743638E+05
1.011	7.94062229E-01	1.19259429E+07	1.71282079E+05
1.011	9.26405934E-01	6.54922721E+06	1.33600021E+05
1.011	1.05874964E+00	3.69541085E+06	1.11333351E+05
1.034	3.88217074E-01	1.44559786E+08	3.74749932E+07
1.034	5.17622766E-01	5.48165463E+07	1.31162476E+06
1.034	6.47028457E-01	2.63477308E+07	4.68437415E+05
1.034	7.76434149E-01	1.38001662E+07	1.87374966E+05
1.034	9.05839840E-01	7.85663232E+06	1.31162476E+05
1.034	1.03524553E+00	4.70779602E+06	1.02119356E+05
1.070	3.75357913E-01	1.64445941E+08	3.64482529E+07
1.070	5.00477217E-01	6.17476284E+07	2.25121562E+06
1.070	6.25596522E-01	2.97696465E+07	3.85922677E+05
1.070	7.50715826E-01	1.58121097E+07	1.82241264E+05
1.070	8.75835130E-01	9.24392013E+06	1.39360967E+05
1.070	1.00095443E+00	5.68163942E+06	1.11488773E+05
1.131	3.54856502E-01	2.04354112E+08	4.16035823E+07
1.131	4.73142002E-01	7.81744732E+07	7.38128073E+06
1.131	5.91427503E-01	3.80203060E+07	1.20784594E+06
1.131	7.09713003E-01	2.00234016E+07	2.08017912E+05
1.131	8.27998504E-01	1.16269934E+07	1.67756380E+05
1.131	9.46284005E-01	7.08810968E+06	1.28836900E+05

Table 5. Table 5: The values of the correlator plotted in Fig. 5.

$T/T_c$	$d_{\mathrm{phys}}$	correlator	error
0.956	4.19942108E-01	8.24169594E+06	4.78983379E+05
0.956	5.59922811E-01	3.94446234E+06	1.78678249E+05
0.956	6.99903514E-01	1.87118963E+06	1.32152198E+05
0.956	8.39884217E-01	8.86735087E+05	1.08077810E+05
0.956	9.79864920E-01	4.00217984E+05	1.02933186E+05
0.956	1.11984562E+00	1.38701270E+05	9.77759035E+04
0.978	4.10624802E-01	8.57417981E+06	5.98810457E+05
0.978	5.47499737E-01	4.10971102E+06	2.17131666E+05
0.978	6.84374671E-01	1.96096577E+06	1.41702507E+05
0.978	8.21249605E-01	9.86734842E+05	1.07524651E+05
0.978	9.58124539E-01	4.82581348E+05	1.02201226E+05
0.978	1.09499947E+00	2.38981511E+05	9.60873715E+04
1.011	3.97031115E-01	4.46635149E+06	5.13846236E+05
1.011	5.29374819E-01	2.03499810E+06	1.92022626E+05
1.011	6.61718524E-01	8.89660527E+05	1.18007786E+05
1.011	7.94062229E-01	4.74053127E+05	9.08201812E+04
1.011	9.26405934E-01	2.64646227E+05	8.28624159E+04
1.011	1.05874964E+00	1.57385964E+05	7.80386843E+04

Equations

$$N_{\tau }\cdot a(\beta )=\frac{1}{T},$$

$$a(\beta )=\frac{1}{{\Lambda }_L}f(\beta ),$$

$$f(\beta )={\left(\frac{8}{33}{\pi }^2\beta \right)}^{51/121}\exp \left(-\frac{4}{33}{\pi }^2\beta \right)\left[1+\mathcal{O}(1/\beta )\right],$$

$${\mathcal{D}}_{\mu \rho ,\nu \sigma }(x)=⟨0|\mathrm{Tr}\left\{G_{\mu \rho }(x)S(x,0)G_{\nu \sigma }(0)S^{†}(x,0)\right\}|0⟩,$$

$$S(x,0)=\mathrm{P}\exp \left(i{\int }_0^{1}dt{x}^{\mu }{A}_{\mu }(xt)\right),$$

$${\mathcal{D}}_{\mu \rho ,\nu \sigma }(x)=(g_{\mu \nu }{g}_{\rho \sigma }-g_{\mu \sigma }{g}_{\rho \nu })\left[\mathcal{D}(x^2)+{\mathcal{D}}_1(x^2)\right]$$

$$⟨0|\mathrm{Tr}\left\{E_i(x)S(x,y)E_k(y)S^{†}(x,y)\right\}|0⟩$$

$$⟨0|\mathrm{Tr}\left\{B_i(x)S(x,y)B_k(y)S^{†}(x,y)\right\}|0⟩$$

$$⟨0|\mathrm{Tr}\left\{E_i(x)S(x,y)B_k(y)S^{†}(x,y)\right\}|0⟩=-\frac{1}{2}{\epsilon }_{ikn}{u}_n\frac{\partial D_1^{BE}}{\partial u_4}.$$

$${\mathcal{D}}_{\parallel }^E({\vec{u}}^2,0)$$

$${\mathcal{D}}_{\perp }^E({\vec{u}}^2,0)$$

$${\mathcal{D}}_{\parallel }^B({\vec{u}}^2,0)$$

$${\mathcal{D}}_{\perp }^B({\vec{u}}^2,0)$$

Full text

March 1996          

IFUP–TH 14/96          

UCY–PHY–96/6

**Gauge–invariant field correlators in QCD

at finite temperature ***Partially supported by MURST (Italian Ministry of the University and of Scientific and Technological Research) and by the EC contract CHEX–CT92–0051.

** A. Di Giacomo, E. Meggiolaro

Dipartimento di Fisica,

Università di Pisa,

and INFN, Sezione di Pisa,

I–56100 Pisa, Italy.

H. Panagopoulos

Department of Natural Sciences,

University of Cyprus,

1678 Nicosia, Cyprus.

Abstract

We study by numerical simulations on a lattice the behaviour of the gauge–invariant two–point correlation functions of the gauge field strengths across the deconfinement phase transition.

1. Introduction

Gauge–invariant correlation functions of the field strenghts in the QCD vacuum play an important role in high–energy phenomenology and in models of stochastic confinement [1, 2, 3]. Recently, a determination of such correlators at zero temperature has been done on the lattice, in a range of distances from $0.1$ to 1 fm [4]. The technique used to make the computation feasible is a local cooling of the configurations: this procedure freezes local fluctuations, leaving long–range correlations unchanged. In this paper we determine the behaviour of the correlators at finite temperature for the pure–gauge theory with $SU(3)$ colour group and in particular we study their behaviour across the deconfining phase transition. The motivations to do that stem from Refs. [5, 6, 7].

2. Notation

To simulate the system at finite temperature, a lattice is used of spatial extent $N_{\sigma}\gg N_{\tau}$, $N_{\tau}$ being the temporal extent, with periodic boundary conditions. The temperature $T$ corresponding to a given value of $\beta=6/g^{2}$ is given by

[TABLE]

where $a(\beta)$ is the lattice spacing. From renormalization group arguments,

[TABLE]

where $\Lambda_{L}$ is the fundamental constant of QCD in the lattice renormalization scheme: its value, extracted from the string tension [8], turns out to be about 5 MeV. At large enough $\beta$, $f(\beta)$ is given by the usual two–loop expression:

[TABLE]

for gauge group $SU(3)$ and in the absence of quarks.

The gauge–invariant two–point correlators of the field strengths in the QCD vacuum are defined as

[TABLE]

where $G_{\mu\rho}=gT^{a}G^{a}_{\mu\rho}$ is the field–strength tensor and $T^{a}$ are the generators of the colour gauge group in the fundamental representation. Moreover,

[TABLE]

with $A_{\mu}=gT^{a}A^{a}_{\mu}$, is the Schwinger phase operator needed to parallel–transport the tensor $G_{\nu\sigma}(0)$ to the point $x$, so to make ${\cal D}_{\mu\rho,\nu\sigma}(x)$ a gauge–invariant quantity. At zero temperature, that is on a symmetric lattice $N_{\sigma}=N_{\tau}$, they are expressed in terms of two independent invariant functions of $x^{2}$, called ${\cal D}(x^{2})$ and ${\cal D}_{1}(x^{2})$, as follows [1, 2, 3]:

[TABLE]

At finite temperature, the $O(4)$ space–time symmetry is broken down to the spatial $O(3)$ symmetry and in principle the bilocal correlators are now expressed in terms of five independent functions [5, 6, 7]; two of them are needed to describe the electric–electric correlations:

[TABLE]

where $E_{i}=G_{i4}$ is the electric field operator and $u_{\mu}=x_{\mu}-y_{\mu}$ [$\vec{u}^{2}=(\vec{x}-\vec{y})^{2}$]. We are considering the Euclidean theory.

Two further functions are needed for the magnetic–magnetic correlations:

[TABLE]

where $B_{k}={1\over 2}\varepsilon_{ijk}G_{ij}$ is the magnetic field operator.

Finally, one more function is necessary to describe the mixed electric–magnetic correlations:

[TABLE]

In Eqs. (2.7), (2.8) and (2.9), the five quantities $D^{E}$, $D_{1}^{E}$, $D^{B}$, $D_{1}^{B}$ and $D_{1}^{BE}$ are all functions of $\vec{u}^{2}$, due to rotational invariance, and of $u_{4}^{2}$, due to time–reversal invariance.

From the conclusions of Refs. [5, 6, 7], one expects that $D^{E}$ is related to the string tension and should have a drop just above the deconfinement critical temperature $T_{c}$. In other words, $D^{E}$ is expected to be the order parameter of the confinement; on the contrary, $D^{E}_{1}$ does not contribute to the area law of the temporal Wilson loop. (Similarly, $D^{B}_{1}$ does not contribute to the area law of the spatial Wilson loop.)

3. Results

We have determined the following four quantities

[TABLE]

by measuring appropriate linear superpositions of the correlators (2.7) and (2.8) at equal times ($u_{4}=0$), on a $16^{3}\times 4$ lattice (so that, in our notation, $N_{\tau}=4$). The critical temperature $T_{c}$ for such a lattice corresponds to $\beta_{c}\simeq 5.69$. The behaviour of ${\cal D}_{\parallel}^{E}$ and ${\cal D}_{\perp}^{E}$ is shown in Figs. 1 and 2 respectively, on three–dimensional plots versus $T/T_{c}$ and the physical distance in the range $0.4$ up to 1 fm. Due to the logarithmic scale, the errors are comparable with the size of the symbols and the lines connecting the points are drawn as an eye–guide. A clear drop is observed for ${\cal D}_{\parallel}^{E}$ and ${\cal D}_{\perp}^{E}$ across the phase transition, as expected.

The analogous behaviour for ${\cal D}_{\parallel}^{B}$ and ${\cal D}_{\perp}^{B}$ is shown in Figs. 3 and 4. No drop is visible across the transition for the magnetic correlations.

Finally, we have also measured the mixed electric–magnetic correlator of Eq. (2.9). When computed at equal times ($u_{4}=0$), this correlator turns out to be zero both at zero temperature and at finite temperature. This is because the function $D_{1}^{BE}$ at the right–hand side of Eq. (2.9) depends on $u_{4}^{2}$, as a consequence of the invariance of the theory under time–reversal. We have verified this directly on the lattice. Then, we have measured the same correlator (2.9) for $u_{4}=1$ lattice spacing and $|\vec{u}|=3,4,5,6,7,8$ lattice spacings. The results of these measurements are shown in Fig. 5.

Five tables with the values of the various correlators together with the errors (including an estimate of the systematic uncertainties) are included for the convenience of the interested reader.

Our results can be summarized as follows:

(i) In the confined phase ($T<T_{c}$), until very near to the temperature of deconfinement, the correlators, both the electric–electric type (2.7) and the magnetic–magnetic type (2.8), are nearly equal to the correlators at zero temperature [4]: in other words, $D^{E}\simeq D^{B}\simeq D$ and $D_{1}^{E}\simeq D_{1}^{B}\simeq D_{1}$ for $T<T_{c}$.

(ii) Immediately above $T_{c}$, the electric–electric correlators (2.7) have a clear drop, while the magnetic–magnetic correlators (2.8) stay almost unchanged, or show a slight increase.

Acknowledgements

This work was done using the CRAY T3D of the CINECA Inter University Computing Centre (Bologna, Italy). We would like to thank the CINECA for having put the CRAY T3D at our disposal and for the kind and highly qualified technical assistance.

Stimulating discussions with Yuri Simonov are warmly acknowledged.