DECONFINEMENT AND QUARK–GLUON PLASMA

Deconfinement and Quark–gluon Plasma

A.V. NEFEDIEV, YU.A. SIMONOV, M.A. TRUSOV
Abstract

The theory of confinement and deconfinement is discussed as based on the properties of the QCD vacuum. The latter are described by field correlators of colour-electric and colour-magnetic fields in the vacuum, which can be calculated analytically and on the lattice. As a result one obtains a self-consistent theory of the confined region in the plane with realistic hadron properties. At the boundary of the confining region, the colour-electric confining correlator vanishes, and the remaining correlators describe strong nonperturbative dynamics in the deconfined region with (weakly) bound states. Resulting equation of state for , , are in good agreement with lattice data. Phase transition occurs due to evaporation of a part of the colour-electric gluon condensate, and the resulting critical temperatures for different are in good correspondence with available data.

1 Introduction

The fundamental problems of confinement and deconfinement have recently become a hot topic because of a possible observation of deconfined phase — the Quark-Gluon Plasma (QGP) at RHIC . The theory of the QGP was mainly centered on perturbative ideas with an inclusion of some resummations and nonperturbative effects . However, the experimental data display a strong interacting QGP, with some properties more similar to a liquid, rather than to a weakly interacting plasma, with strong collective effects and large energy loss of fast constituents. These effects call for a nonperturbative treatment of QGP, and hence for a nonperturbative theory of confined and deconfined phases of QCD.

For the confined phase the corresponding methods have been suggested in and formulated as the Field Correlator Method (FCM) — see for a review. In the FCM at , the vacuum configurations support mostly two scalar quadratic in field correlators, and . The first correlator is purely nonperturbative and ensures confinement, while the second one contains both perturbative and nonperturbative parts. As a result one obtains linear confining potential from the plus perturbative and nonperturbative corrections coming from the . The spectra of all possible bound states (mesons, baryons, hybrids and glueballs) have been calculated in FCM, with current masses of quarks, the string tension , and the strong coupling constant used as input, in good agreement with the experiment (see for a review) and lattice data . One can conclude therefore that the FCM describes well the confining region at .

For several complications occur (see for the details). First of all, colour-electric and colour-magnetic field correlators are no longer identical to each other and one has to do with five correlators instead of two: , , , , and . Correspondingly, the colour-electric and colour-magnetic condensates are now different, as well as the string tensions, which are simply related to the correlators,

(1)

It is important to notice that only the electric tension is responsible for confinement, while all others ensure spin-orbit forces. Thus it was conjectured in that deconfinement occurs if the correlator vanishes at . Indeed, this conjecture was confirmed later on the lattice, while all other correlators were measured to stay intact . Finally, it was argued in (see also ) that, since the vacuum energy density is proportional to the gluonic condensate, one can derive the condition of vanishing of the directly from the second thermodynamic law. Consequently, general features of the phase transitions and the very value of the critical temperature can be calculated in terms of the difference of the gluonic condensates in the confining and deconfining phases .

We now come to the central point of the paper: namely what is the dynamics of the QGP and its Equation of State (EoS)? As will be shown below, the basic role in the EoS is played by two effects: the colour-electric (nonconfining) forces due to the correlator and colour-magnetic forces due to the correlator . The former one is dominant for and creates an effective mass (energy) of isolated quarks and gluons, thus leading to typical curves for the pressure similar to those observed in lattice calculations. Furthermore, the same generates potential which is able to bind quarks and gluons , and provides strong correlations in white systems. Colour-magnetic forces which stem from the also can bind quarks at large distances and provide a large ratio specific for liquids . Moreover, (and thence ) grow with the temperature, and finally become the basic interaction in the dimensionally reduced QCD, with the transition temperature around .

The paper is organised as follows. In Section 2 we introduce the FCM at . In particular, we give necessary essentials of the method, discuss colour-electric and colour-magnetic interactions, explain the QCD string approach to hadrons, and consider spin-dependent interactions. In Section 3 we generalise the FCM for nonzero temperatures: we introduce the winding measure of integration, define the Single Line Approximation (SLA), and derive the EoS of the QGP in the framework of the SLA. In Section 4 we discuss interactions between quarks and gluons above the and investigate bound states of quarks and gluons due to these interactions. In Section 5 we generalise the results of Section 3 for nonzero chemical potentials . In Section 6 we compare the results discussed in this review with other approaches found in the literature and discuss various solved and unsolved problems.

2 FCM at

2.1 Essentials of the method

To describe the dynamics of quarks and gluons in both confining and deconfined phases of QCD we start from the gauge-covariant Green’s function of a single quark (or gluon) in the field of other quarks and gluons plus vacuum fields and use the Fock–Feynman–Schwinger Representation (FFSR) (in Euclidean space-time) :

(2)

where is the kinetic energy term,

(3)

with being the pole mass of the quark. The parallel transporter along the trajectory of the quark propagating from point to point is given by

(4)

while

(5)

where

(6)

and are the usual Pauli matrices. The symbols and in (2.1) and (2.1) stand for the ordering operators for the matrices and , respectively, along the quark path.

Similarly, for one gluon Green’s function, one writes

(7)

and, proceeding in the same lines as for quarks, one arrives at the gluon Green’s function in the FFSR:

(8)

where

(9)

Here and below the tilde sign denotes quantities in the adjoint representation, for example, .

The single-quark(gluon) Green’s functions (2.1) and (2.1) can be used now as building blocks to write the Green’s function of a hadron. For example, for the quark–antiquark meson one has:

(10)

where is a Dirac matrix which provides the correct quantum numbers of the meson (=1, , , , ) and the trace is taken in both Dirac and colour indices. The next important step is building the physical Green’s function of the meson,

(11)

where the averaging over gluonic fields is done with the usual Euclidean weight containing all gauge-fixing and ghost terms. The exact form of this weight is inessential for what follows.

The actual average one needs to evaluate in order to proceed reads:

(12)

where the closed contour runs along the trajectories of the quark, , and that of the antiquark, . Since the orderings and in , and , are universal, then is just a Wegner–Wilson loop with the insertion of the operators and at proper places along the contour .

To proceed it is convenient to rewrite (2.1) in the form

(13)

where the non-Abelian Stokes theorem was used and the integral is taken over the surface bounded by the contour . Notice that the differential

(14)

contains not only the surface element but it also incorporates the spin variables. This is the most economical way to include into consideration spin-dependent interactions between quarks ( and are the proper-time variables for the quark and the antiquark, respectively). Furthermore, if a cluster expansion theorem is applied to the right-hand side of (2.1), then the result reads:

(15)

where and . Double brackets denote irreducible (connected) averages, and the reference point is arbitrary.

Equation (2.1) is exact and therefore its right-hand side does not depend on any particular choice of the surface . At this step one can make the first approximation by keeping only the lowest (quadratic, or Gaussian) field correlator , while the surface is chosen to be the minimal area surface. As it was argued in , using comparison with lattice data, this approximation (sometimes called the Gaussian approximation) has the accuracy of a few per cent.

As was mentioned before, the factors in the Green’s function (2.1) need a special treatment in the process of averaging over the gluonic fields — it is shown in that one can use a simple replacement,

(16)

Therefore, the first step is fulfilled, namely the derivation of the physical Green’s function of a quark–antiquark meson (one can proceed along the same lines for baryons and hadrons with an excited glue) in terms of the vacuum correlator . In the next chapter the structure of this Gaussian correlator is studied, in particular, its separation into colour-electric and colour-magnetic parts.

2.2 Colour-electric and colour-magnetic correlators

The Gaussian correlator of background gluonic fields can be parameterized via two scalar functions, and :

where . Notice that, in order to proceed from (2.1) to (2.2) one needs to show that, for the correlators in (2.1) taken at close points, aaaThe gluonic correlation length defines the distance at which the background gluonic fields are correlated or, more specifically, the correlator defined in (2.2) decreases in all directions of the Euclidean space, and the length governs this decrease. The correlation length extracted from the lattice data is quite small, fm ., the parallel transporters passing through the point can be reduced to the straight line between the points and . To this end, notice that, for a generic configuration with ( is the radius of the Wilson loop, of order of the hadron size), one can write .

For future convenience let us also write the correlator (2.2) in components:

(18)

For the sake of brevity we omit, in this chapter, the spin-dependent terms in the Wilson loop (2.1) and consider only nonperturbative contributions to the Gaussian correlator responsible for confinement and given by the correlator . This correlator enters the Wilson loop (2.1) multiplied by the surface elements and, in what follows, we distinguish between the colour-electric and colour-magnetic contributions in (2.2). The former are accompanied by the structure and enters (2.1) multiplied by electric correlator , whereas, for the latter, these are and , respectively. The electric and magnetic string tensions are defined then according to (1).

We now proceed from the proper-time variables to the laboratory times by performing the following change of variables:

(19)

and synchronise the quarks in the laboratory frame, putting . The new variables which appeared in (19) are known as the auxiliary (or einbein) fields . The physical meaning and the role played by the einbeins will be discussed in detail below. Now we can write the surface element through the string profile function as

(20)

where , , and we adopt the straight–line ansatz for the minimal string profile, so that the latter is defined by the trajectories of the quarks :

(21)

For further convenience let us introduce two vectors:

(22)

where is the angular rotation vector. This allows one to write the differentials in a compact form,

(23)

Presenting the averaged Wilson loop (2.1) (without spins) as

one can write for the electric and magnetic contributions separately:

(24)
(25)

The correlation functions decrease in all directions of the Euclidean space–time with the correlation length which is measured on the lattice to be rather small,  fm or even smaller,  fm . Therefore, only close points and are correlated, so that one can neglect the difference between and , and in (24), (25) and also write:

(26)

The induced metric tensor is , . Now, after an appropriate change of variables and introducing the string tensions, according to (1), one readily finds:

(27)
(28)

For the sum of and reproduces the well-known action of the Nambu–Goto string (see, for example, ),

(29)

Notice that, for , and , that is confinement is of a purely colour-electric nature while the colour-magnetic contribution is entirely due to the rotation of the system.

2.3 QCD string approach

With the form of the colour-electric and colour-magnetic spin-independent interactions (27) and (28) in hand we are in a position to build a quantum-mechanical model of hadron consisting of quarks (gluons) connected by straight-line Nambu-Goto strings. Below we consider, as a paradigmatic case, a quark–antiquark meson.

We start from the kinetic energies of the quarks,

(30)

where (19) was used, and path integrals in appear through the substitution . One can see therefore that the einbeins should be treated on equal footings with other coordinates. However, since time derivatives of the einbeins, , do not enter (30), they can be eliminated with the help of the well-known integral:

(31)

Then, with the help of (2.1), (27), (28), and (30), one can extract the Lagrangian of the quark–antiquark system in the form (notice that hereinafter in this chapter we work in Minkowski space-time):

(32)

In fact, the einbein form of the kinetic energies is much more convenient since it does not contain square roots (unbearable for path integrals) and one deals with formally nonrelativistic kinetic terms, while the entire set of relativistic corrections is summed by taking extrema in the einbeins. Furthermore, extra (continuous) einbeins, and , can be introduced in order to simplify the string terms in (32), through the substitutions:

(33)

Then, introducing the centre-of-mass position and the relative coordinate as

(34)

one can rewrite the Lagrangian in the form (the centre-of-mass motion is omitted for simplicity):

(35)

where , and we have defined the reduced masses for the angular and radial motion separately:

(36)
(37)

The original form of the Lagrangian is readily restored once extrema in all four einbeins, , are taken. Generally speaking, the einbein fields appear in the Lagrangian and, as was mentioned before, even in absence of the corresponding velocities, they can be considered as extra degrees of freedom introduced to the system. The einbeins can be touched upon when proceeding from the Lagrangian of the system to its Hamiltonian and thus they mix with the ordinary particles coordinates and momenta. Besides, in order to preserve the number of physical degrees of freedom, constrains are to be imposed on the system and then the formalism of constrained systems quantisation is operative (see, for example, for the open straight–line QCD string quantisation using this formalism). A nontrivial algebra of constraints and the process of disentangling the physical degrees of freedom and non-physical ones make the problem very complicated. In the meantime, a simpler approach to einbeins exists which amounts to considering all (or some) of them as variational parameters and thus to taking extrema in the einbeins either in the Hamiltonian or in its spectrum . Being an approximate approach this technique appears accurate enough (see, for example, ) providing a simple but powerful and intuitive method of investigation. The physical meaning of the variables () is the average kinetic energy of the -th particle in the given state, namely, (see the discussion in ). The continuous einbein variable has the meaning of the QCD string energy density .

Following the standard procedure, we build now the canonical momentum as

(38)

with its radial component and transverse component being

(39)

respectively. Thus we arrive at the spin-independent part of the Hamiltonian :

(40)

Notice that the kinetic part of the Hamiltonian (40) has a very clear structure: the radial motion of the quarks happens with the effective mass , whereas for the orbital motion the mass is somewhat different, containing the contribution of the inertia of the string. For , the field drops from the Hamiltonian and the standard expression for the string with quarks at the ends readily comes out from (40).

2.4 Spin-dependent interactions in hadrons

In heavy quarkonia (up to the order ) the standard Eichten–Feinberg decomposition is valid :

where each potential (0-4) contains both perturbative (P) and nonperturbative (NP) contributions: . The static interquark potential , together with the potentials and , satisfies the Gromes relation ,

(42)

Notice that this relation refers both to the perturbative and nonperturbative parts of the potentials .

We now return to the Green’s function of the quark–antiquark system (2.1) and restore spin–dependent terms in order to derive the generalisation of (2.4).

Let us quote here without derivation the full set of spin-dependent potentials, both perturbative and nonperturbative, obtained in the framework of FCM and with the string rotation taken into account (an interested reader can find the details of the derivation in ):

(43)

where the masses are replaced by and , which makes this result applicable also to light quarks. Notice that this result , is not due to the expansion, but is obtained with the only approximation made being the Gaussian approximation for field correlators (corrections may come from triple and quartic correlators). Accuracy of this approximation was checked both at and at to be of the order of one percent.

With this explicit form of the potentials and taking the limit of heavy quarks one can check the Gromes relation (42), which now reads:

(44)

At , and , so that the Gromes relation (42) is satisfied. In the next Section we shall discuss the FCM at nonzero temperatures, in particular at , where the latter equalities between colour-electric and colour-magnetic correlators do not hold, and the Gromes relation is therefore violated.

3 FCM at

3.1 The winding measure of integration

Now we turn to the case and use Matsubara technique for the path integrals in the FFSR, first introduced in ,