Strong interaction and bound states in the deconfinement phase of QCD

Recent striking lattice results on strong interaction and bound states above T_c can be explained by the nonperturbative Q\bar Q potential, predicted more than a decade ago in the framework of the field correlator method. Explicit expressions and quantitative estimates are given for Polyakov loop correlators in comparison with lattice data. New theoretical predictions for glueballs and baryons above T_c are given.


Introduction
There is a growing understanding nowadays that nonperturbative dynamics plays important role in the deconfinement phase, for reviews and references see [1].
An additional part of this understanding, not contained in [1], is the realization of the fact, that at T c < T < 2T c , the colormagnetic fields are as strong as they are in the confinement phase, (where colormagnetic and colorelectric fields are of the same order) and become even stronger above 2T c . More than a decade ago the author has argued [2,3] that the deconfinement phase transition is the transition from the color-electric confinement phase to the colormagnetic phase, confining in 3d. This observation was supported theoretically by the calculation of T c [2,3] and on the lattice by the calculation of the spacial string tension at T > T c [4].
In 1991 the author has found [5] that colorelectric fields also survive the deconfinement transition in the form of potential V 1 (r), which can support QQ bound states in some temperature interval T c < T < T D , while quarks acquire self-energy parts equal to 1 2 V 1 (∞). As will be shown below this predicted picture is fully supported by recent numerous lattice experiments [6]- [13] (for a review see [10]), where QQ bound states have been discovered. At the same time the evidence for V 1 (r) and selfenergies 1 2 V 1 (∞) has been also obtained on the lattice in the form of Polyakov loop averages and of free and internal energies above T c [10,11,12]. The light quark (m q ∼ m s )qq bound states have also been observed in [13].
The theory used in [5] and below is based on the powerful Method of Field Correlators MFC [14], (for a review see [15]), where the basic dynamic ingredients are the field correlators trF µ 1 ν 1 (x 1 )...F µnνn (x n ) 1 . It was shown later [16] that the lowest quadratic (so-called Gaussian) correlator explains more than 90% of all dynamics and it will be considered in what follows. The quadratric correlator consists of two scalar form-factors, D and D 1 : (1) which produce the following static potential [17] between heavy quarks at zero temperature (obtained from the Wilson loop r × t with t → ∞) One can notice that linear confinement part of potential, V D = σR, is due to correlator D(x), σ = 1 2 D(x)d 2 x. At T > 0 one should distinguish between electric and magnetic correlators, D E (x), D E 1 (x), and D H (x), D H 1 (x) and correspondingly between σ (E) and σ (H) . It was argued in [2,3] that the principle of minimality of free energy requires D E and electric confinement , σ (E) , to vanish, while colormagnetic correlators, D H (x), D H 1 (x) should stay roughly unchanged at least up to 2T c . Several years later in detailed studies on the lattice in [18] these statements have been confirmed, and indeed magnetic correlators do not change at 1.5T c > T > T c while D E (x) vanishes in vicinity of T c .
Not much was said about the second electric correlator D E 1 (x), since in the parametrization of [18] it was found to be smaller than D E (x) and hence not so important at T < T c .
Meanwhile a lot of information was being accumulated on the lattice. First of all, the Polyakov loop averages already imply the presence of strong electric fields above T c , and the main point is that those can not be reduced to the perturbative electric and magnetic, (see the analysis in [11,12]).
Recently a detailed analysis of Polyakov loop correlators was done by the Bielefeld group [9]- [12] and the singlet free energy F 1 (r, T ) and internal energy U 1 (r, T ) were calculated at T < T c and at T > T c . In the latter case F 1 (r, T ) was found to saturate at large r at the values of the order of several hundred MeV (e.g. for T = 1.2T c the value of F 1 (∞, T ) found in [12] is around 0.7 √ σ, while the internal energy is around 3T c ) and this fact cannot be explained by perturbative contributions alone -we consider it as the most striking revelation of nonperturbative electric fields above T c . At the same time several groups have calculated the so-called spectral function of heavy [6]- [10] and light quarkonia [13] at T > T c . In all cases sharp peaks have been observed, corresponding to the ground state levels of cc at L = 0, 1 and of light (m q ≈ m s ) quarkonia in V, A, S, P S channels. In both heavy and light cases the peaks are possibly displaced as compared to T = 0 positions and apparently almost degenerate in different nn channels.
All these facts cannot be explained in the framework of the commonly accepted perturbative quark-gluon plasma and call for a new understanding of the nonperturbative physics at T > T c . In what follows we shall argue following [5] that at T > T c not only nonperturbative magnetic fields, but also strong nonperturbative electric fields are present, which can be calculated in MFC and explain the observed data.
2 Dynamics of Polyakov loops and the correlator D 1 In this section we consider the Polyakov loop and apply to it the nonabelian Stokes theorem and Gaussian approximation, taking first the loop as a circle on the plane and making limiting process with the cone surface S inside loop and finally transforming cone into the cylinder by tending the vertex of the cone to infinity 2 . In doing so we are writing the nonabelian Stokes theorem and cluster expansion for the surface S which is transformed from the cone to the (half) cylinder surface. As a result one has for the Polyakov loop average In obtaining (3) we have omitted the contribution of D(x) in D µν,ρλ , since this would cause vanishing of L in the limiting process described above due to the infinite cone surface S. This exactly corresponds to vanishing of L in the confinement region, observed on the lattice. Therefore the result (3) refers to the deconfinement phase, T > T c .
As it is known from lattice [18] and analytic calculations, D 1 (x) [19] exponentially falls off at large x as exp(−M 1 x), with M 1 > ∼ 1 GeV and for T ≪ M 1 one can approximate (3) as follows 3 (4) We turn now to the correlator of Polyakov loops following notations from [11]. Using the same limiting procedure as for one Polyakov loop, one can apply it to the correlator t rL xt rL + y ≡ P (x − y),tr = 1 Nc tr, representing the loops L x and L y as two concentric loops on the cylinder separated by the distance |x − y| along its axis, the cylinder obtained in the limiting procedure from the cone with the vertex tending to infinity. One can apply in this situation the same formalism as was used in [20] for the case of the vacuum average of two Wilson loops. For opposite orientation of loops using Eqs. (21)(22)(23)(24)(25)(26)(27)(28) from [20] one arrives at the familiar form found in [21] P 2 In doing so one is changing topology of the surface and as a result loses the Z(N ) subgroup of SU (N ). This however does not influence our results as long as one is remaining in the j = 0 sector of Z(N ) broken vacua (see last ref. of [1] for more discussion of Z(N )) 3 The correlators D, D 1 in (3) in principle should be taken in the periodic form, as was suggested in [22]. However for T ≤ 2T c this modification brings additional terms of the order of exp(−M 1 /T ) which are neglected below.
In the Appendix two different ways of derivation of Eq.(5) are given, with the resultF Here is the vacuum average of the adjoint Polyakov loop, which vanishes in the leading approximation in the confinement phase, as it is explained in the Appendix, and nonzero when gluon loops are taken into account, in which case 9 4 The suppression of exp(−F 8 /T ) in our approach in the confinement phase has thus the same origin as the strong damping of the adjoint Polyakov loop in that phase [23] and the persistence of the Casimir scaling for adjoint static potential in the interval 0 ≤ r < 1.2 fm (see [24] for discussion and references).
It is clear that in the deconfinement phase with D ≡ 0, V D ≡ 0 one has only V 1 (r) in bothF 1 andF 8 , and all these quantities are finite (after the renormalization of the perturbative divergencies specific for the fixed contours, which are discussed in section 3). Thus in the deconfined phase one can write where V 1 (r) and V 1 (∞) are renormalized. It is clear from (9) that at small r one has lim r→0 (Fq q (r) − V 1 (r)) → T lnN 2 c as was noticed and measured in [11].
At this point one should stress the difference between the genuine free energy F i (r, T ), i = 1.8, which is measured with some accuracy on the lattice, and the calculated aboveF i (r, T ). It is clear thatF i do not contain the contribution due to excitation of QQ and gluon degrees of freedom existing at finite T . The latter is contained in the free energy F 1 (r, T ) and in the internal energy, which we denote U i (r, T ) = F i + ST to distinguish from our V i (r, T ), since they are not equal.
In general for nonzero temperature and comparing to the lattice data on heavy-quark potential one should have in mind, that temperature effects might be of two kinds. First, the intrinsic temperature dependence due to changing of the vacuum structure and the vacuum correlators and hence of our potentialsF (r, T ). Second, the physical quantities like F i (r, T ), U i (r, T ) are thermal averages over all excited states, e.g.
One can associateF 1 (r, T ) = E 0 (r, T ), while the structure of excited spectrum can be traced in the temperature dependence of F 1 and U 1 . E.g. assuming in the confinement phase the string-like spectrum and multiplicity for the multihybrid spectrum with two static quarks, E n = σr + πn/r, n = 1, 2, .. and multiplicity ρ(m) = exp(m/m 0 )θ(m − m 1 ), m = πn/r, one arrives at The increase of U 1 (r, T ) below T c in the quenched case was indeed observed in lattice calculations (see Fig.3 of [12]). Above T c one can see in lattice data [10] the striking drop of entropy S 1 (∞, T ) and U 1 (∞, T ) in the region T c ≤ T ≤ 1.2T c which can be possibly explained again by the multihybrid states occurring due to the potential V 1 (r, T ) =F 1 (r, T ) connecting quarks and gluons, and assuming that the magnitude of V 1 (r, T ) decreases with temperature passing at T ≈ 1.05T c the critical value enabling to bind those states of high multiplicity. In this way one assumes that both below and above T c in quenched and unquenched cases the dominant (in entropy) configuration is the gluon chain connecting Q and Q with gluons bound together by confining string (below T c ) and potential V 1 (above T c ). Thus the comparison to the lattice data on F 1 , U 1 needs the exact knowledge of the spectrum. In what follows we shall associate our F 1 (r, τ ) with the free energy F 1 (r, T ), since its temperature dependence is not so steep as that of U 1 (r, T ) in this region and this discussion will be of qualitative character, leading detailed discussion of the spectrum to future publications.
3 Properties of D 1 (x) and F 1 (r, T ) The correlator D 1 (x) was measured on the lattice [18] both below and above T c , and decays exponentially with M 1 ≈ 1 ÷ 1.5 GeV (in the quenched case). At the same time D 1 (x) can be connected to the gluelump Green's function, and the corresponding M 1 for the electric correlator D E 1 (x) is M 1 ≈ 1.5 GeV at zero T [26]. Moreover in a recent paper [19] D E 1 (x) was found analytically for T ≤ T c , and can be represented symbolically as a sum, with perturbative part acting at small x, and the nonperturbative part having the asymptotic form As will be argued below, the form of D 1 (x) (14) does not change for T > T c , however the mass M 1 and A 1 may be there different.
Using the asymptotics (15) in the whole x region for a qualitative estimate, one has Finally the Polyakov loop exponent is One can see from (17) that V , contains also perturbative contribution at small r, which to the order O(α s ) is (r) ∼ r 2 at small r, one can renormalize matching V 1 (r, T ) with the Coulomb interaction at small r, as it was done in [9,11] for F i (r, T ).
As a result in the renormalized V (∞) can be put equal to zero, and we shall use it in what follows.
At this point we are able to compare V 1 (r, T ) with the lattice data for F 1 (r, T ) at T ≥ T c . In Fig.1 we compare the lattice data for F 1 (r, T ) taken from [12] for T = 1.05T c , 1.2T c and 1.5T c with the potential V 1 (r, T ) in the form (16) parametrizing M 1 and a(T ) ≡ A 1 M 1 in it as and find that M 1 = 0.69 GeV and a 0 = a(conf ) = 2C 2 (f )α s σ adj ∼ = 0.432 GeV 2 , c = 0.36 provides a good agreement with the data points at 1.5T c ≥ T ≥ T c , while a(T ) in (19) smoothly matches at T = T c the amplitude of the gluelump Green's function [19]. One can see that the behaviour of the total V 1 (r, T ) = V C 1 (r, T )+V (∞) is qualitatively very similar to the behaviour of F 1 (r, T ) as a function of r. We also compare in Fig.2 our results with lattice data [11] for the Polyakov loop (17) and find reasonable agreement. It is clear that both Fig.1 and Fig.2 are qualitative illustrations, and for quantitative comparison one needs knowledge of excitation spectrum and analytic or lattice predictions for a(T ), M 1 (T ) which will be given elsewhere [25], [37].

Bound states of qq in the deconfinement region
In the recent lattice studies sharp peaks have been found in the spectral function of cc system [6]- [10], which can be associated with the quark-antiquark bound states surviving at T ≥ T c . To understand qualitatively whether the interaction V 1 (r, T ) can support bound states, one can use the Bargmann condition [27] for monotonic attractive potentials 2m (r, T ) can support bound states in some interval of temperature T c < T < T D where T D ∼ 1.5 ÷ 2T c and exact value depends on terms of the order O(e −M 1 /T ) and therefore is beyond the scope of the present paper. This conclusion roughly agrees with lattice calculations in [6]- [9] and with the calculation done in [28] and [29].
One can consider gluons and the gg system in the same way as it was done for the cc system. To this end one should first multiply V 1 (r, T ) found earlier in (16)(17)(18) by C 2 (adj) C 2 (f ) = 9 4 , thus defining V (adj) 1 (r, T ) = 9 4 V 1 (r, T ). This potential can be considered as the interaction kernel in the Hamiltonian of the gg system as it was done in [30]. This Hamiltonian with the introduction of the einbein gluon mass µ g ∼ = √ p 2 has the form of the Schroedinger equation and the condition (20) can be applied. Since µ g ≈ 0.6 GeV [30], an additional (with respect to quarks) factor for the nonperturbative part in (20) is 9 4 µg mc ≈ 0.96. Hence two-gluon glueballs should be formed in approximately the same temperature interval as charmonium states.In lattice calculations [31] scalar glueball was studied below and around T c , and its width is increasing with T . Now we come to the baryon case and using the general formalism [32] to represent the triple Wilson loop of trajectories of 3 quarks and of the string junction in terms of field correlators D and D 1 . From [32] one has for 3 static quarks at distances r 1 , r 2 , r 3 from the string junction where V 1 (r, T ) is given in (16), (18). In the deconfinement phase when r i = R, i = 1, 2, 3, one obtains V 3q = 3 2 V 1 ( √ 3R). For the perturbative part one has from (21) V For the nonperturbative part from Eq.(21) it follows that V (∞). One can check that this prediction and the general form of V 3q = 3 2 V 1 ( √ 3R) as function of R is supported by the recent measurement of singlet free energy of the 3Q system in [33] at T > T c . Thus it is of interest to measure the spectral functions of baryons at T > T c in the same way as it was done for mesons.

Summary and conclusions
Citing the 1991 paper [5] when the magnitude of D 1 was not exactly known "...Using an exponential parametrization for D 1 , we can find D E 1 with parameter values which satisfy the condition for the appearance of levels. In this case ε(r) (our V (np) 1 (r, T )) is a well with a behaviour ε(r) ∼ r 2 as r → 0 and ε(r) → const > 0 as r → ∞. The quark and antiquark are thus bound but there exists a threshold ε(∞) above which quarks fly apart, each acquiring a nonperturbative mass increment δm = 1 2 ε(∞)...". In the present paper this picture was further substantiated and quantified using lattice and analytic knowledge on D 1 . Comparing to recent lattice data in [6]- [13] it was shown that this picture is qualitatively supported by data, and new proposals have been done for searching the glueball and baryon systems at T > T c .
The results obtained on the lattice [6]- [13] and in the present approach establish a new picture of the QCD thermodynamics at 1.5T c > T ≥ T c widely discussed in [28]. As a new feature compared to the works [28] the main emphasis in this paper is done on the selfenergies ( 1 2 V 1 (∞) ≡ ε q for quarks and 9 8 V 1 (∞) ≡ ε g for gluons) which are large (V 1 (∞, T c ) > F 1 (∞, T c ) ∼ = 600 MeV for n f = 2 [34]) and cancel each other at small distances for white bound states, like qq, (gg) 1 , (qgq) 1 , (qg...gq) 1 etc. In contrast to that colored states are higher in potential and mass by several units of ε q and ε g and are suppressed by the corresponding Boltzmann factors. As a result in this region white bound states of quarks and gluons are energetically preferable, while individual quarks and gluons acquire selfenergies, so that the thermodynamics of the system resembles that of the neutral gas, and for higher temperature T > 1.5 ÷ 2T c a smooth transitoon to the "ionised" plasma of colored quarks and qluons possibly occurs.
This new state of the quark-gluon matter should be taken into account when considering ion-ion collisions. For more discussion of the thermodynamics above T c see [28] and refs. therein.
It was noted before [35] that the behaviour of the free and internal energies above T c , with a bump around T ∼ 1.1 ÷ 1.2T c in ε−3P T can be explained if gluons are supplied with the nonperturbative mass term of the order of 0.6 GeV, while for higher T this mass is less important. This can be easily understood now taking into account the value of 1 2 V 1 (∞, T ) and its decreasing with growing T . In this way the nonperturbative dynamics in the form of correlator D 1 can explain the observed dynamics of the deconfined QCD.
A more detailed analysis of bound states requires explicit calculation of QQ and 3Q bound states taking into account spin splitting in the mass of P -wave charmonia and quasi-degeneration of spectra of light qq V, A, S, P S states observed in [13]. Here spin-dependent forces are different from the confining case, since only the correlator D 1 contributes, and one can list the corresponding terms in [17,36]. It is interesting to note, that due to the vector character of V 1 (r, T ), not violating chiral symmetry, bound states of massless quarks should exhibit parity doubling. All this analysis is now in progress [37]. The

Appendix 1 Derivation of the Polyakov loop correlator
We give here two different derivations ofF i (r, T ), i = 1, 8. The first is based on the correlator of two concentric Wilson loops, derived in [20], in which caseF 1,8 are expressed in terms of surface integrals of field correlators D µν,ρλ (u, v) In this way one obtains for two oppositely directed Polyakov loops from Eqs. (29), (23), (20) of [20] F 1 (r, T )/T = 1 2 I(S 12 , S 12 ) (A1.2) Here S 1 is the surface on the cylinder with circumference 1/T extending from the loop 1 at coordinate x in the direction y to infinity, the surface S 2 is also infinite surface from the loop 2 at coordinate y in the same direction (the answer does not depend on the choice of this direction).
The surface S 12 lies on the cylinder between the loops 1 and 2. Note that surface orientation in (A1.1) is fixed to be same. Calculation ofF 1 according to (A1.2) reduces to that of the Wilson loop and yields where V 1 (r, T ) is given in (4) and V D is Calculation ofF 8 is more subtle. To this end one can use connection of 4π 2 x 2 and inserting this into (4), one has (A1.6) One can see that V 1 (∞, T ) ≡ V Q + VQ is the sum of equal selfenergy parts of Q andQ, while v ex (r, T ) describes interaction due to one gluelump exchange between Q andQ.
Note that V 1 (0, T ) = V 1 (∞, T ) + v ex (0, T ) = 0 and v ex (∞, T ) = 0. Therefore v ex appears only in I(S 1 , S 2 ) in (A1.3) and one should restore there the original (opposite) orientation of loops L x and L + y to get the correct sign of v ex (the same sign and factor appears in the second derivation below).
From (A1.1) one obtains for I(S i , S i ) As a result one can write using (A1.3) for N c = 3 In (A1.8) the value of r * is infinitely large, when one neglects the valence gluon loops, as it is done everywhere above. In this case V D (r * , T ) → ∞ and the term exp(−F 8 /T ) vanishes in the confinement region. This is in line with the strong damping of the adjoint Polyakov loop in this region observed on the lattice [23], and with the persistence of Casimir scaling for adjoint static potential for 0 ≤ r ≤ 1.2 fm found on the lattice (see discussion and refs.in [24]. The correction due to the gluon determinant, producing additional gluon loops becomes important for r ≥ 1.2 fm (see discussion in the second ref. in [24]) and makes finite the value of V D (r * , T ) ≈ σr * , r * ≈ 1.2 fm/2 =0.6 fm. In (A1.9) the effects of loop-loop interaction and of the total (adjoint) loop are separated. Physically the result (A1.9) can be easily understood: in absence of the internal interaction one has do with the adjoint Polyakov loop, which strongly changes around T c , namely L adj (T < T c ) is much smaller than L adj (T > T c ).
Alternatively one can use the technic exploited in [39] to separate the contributions of perturbative exchanges from the nonperturbative confining terms.
For the first ones one commutes as in [39] the color generators t c of exchanged gluon (gluelump) with t a (A1.11) acording to the equality t c t a t c = −t a /2N c which finally gives in (A1.11) the adjoint Coulomb interaction αs 6r → − 1 8 V 1 (r), while the selfenergy parts and confining terms arise from sequences t c t c ′ t a → δ cc ′ t a and do not change sign. In this way one arrives at the same answer as given in (A1.9).