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Theoretical Physics Institute

arXiv:hep-ph/9302298v1 24 Feb 1993Theoretical Physics InstituteUniversity of MinnesotaTPI-MINN-92/64-TNovember 1992On the Current Correlators in QCD at FiniteTemperatureV.L. Eletsky†,Theoretical Physics Institute, University of MinnesotaMinneapolis, MN 55455B.L.

IoffeInstitute of Theoretical and Experimental Physics, Moscow 117259, RussiaAbstractCurrent correlators in QCD at a finite temperature T are considered from the viewpointof operator product expansion. It is stressed that at low T the heat bath must be representedby hadronic, and not quark-gluon states.

A possibility to express the results in terms of T-dependent resonance masses is discussed. It is demonstrated that in order T 2 the masses donot move and the only phenomenon which occurs is a parity and isospin mixing.†Permanent address: Institute of Theoretical and Experimental Physics, Moscow 117259,Russia.

In the last years there has been an increasing interest in the study of the current corre-lators in QCD at finite temperatures. The hope is that investigating the same correlatorsboth at high temperatures where the state of quark-gluon plasma is expected, and at lowtemperatures where the hadronic phase persists, a clear signal for a phase transition couldbe found.

In this aspect of special interest is the temperature dependence of hadronic masseswhich manifest themselves through the behavior of correlators at large space-like distances,the appearance of poles in the correlation functions, etc. The calculations of the correlatorsare performed in the lattice simulations as well as by various analitical methods.

(For reviewssee, e.g., Refs. [1, 2]).Among the analitical methods one of the most popular is the extension of QCD sum ruleapproach to the case of finite temperature.

The idea is as follows. In the QCD sum rulemethod hadronic masses are obtained by investigation of current correlators and to a largeextent are determined by the values of vacuum condensates.

Therefore, if the temperaturedependence of the condensates were known, it would also allow to find the temperaturedependence of hadronic masses.Unfortunately, in carrying out this program certain wrong steps were taken and a mis-understanding exists in the literature. However, a clear understanding of the possibilitiesof finite temperature QCD sum rules, or more generally, the possibilities of the operatorproduct expansion in determination of current correlators at finite temperatures, is essen-tial especially in comparing the results obtained in this approach with the ones in latticecalculations.In this note we formulate (although partly it was done before in the literature) the basicpoints of the QCD sum rule method at finite temperature and principal results which areand could be obtained by this method.The object under consideration is a thermal average of a current correlator defined asC(q, T) = ⟨iZd4xeiqxT{j+(x), j(0)}⟩T =Pn⟨n|iR d4xeiqxT{j+(x), j(0)}e−H/T|n⟩Pn⟨n|e−H/T|n⟩(1)where j(x) is a colorless current which can have Lorentz and flavor indeces and can be aspinor, H is the QCD Hamiltonian and the sum is over all states of the spectrum.

It isassumed that q2 is space-like, q2 < 0, and |q2| is much larger than a characteristic hadronicscale, |q2| ≫R−2c , where Rc is the confinement radius, R−1c∼0.5 GeV .We consider the case of temperatures T below the phase transition temperature Tc. Inprinciple, the summation over n in eq.

(1) can be performed over any full set of states |n⟩in the Hilbert space. It is clear however that at T < Tc the suitable set of states is the set1

of hadronic states, but not the quark-gluon basis. Indeed, in this case the original particlesforming the heat bath, which is probed by external currents, are hadrons.

The summationover the quark-gluon basis of states would require to take into account the full range of theirinteraction. This point was first mentioned in Ref.([3]).

In the previous papers[4] devoted tothe extension of QCD sum rules to finite temperatures and even in some following papers[5]this was not understood and the summation over |n⟩at low T was performed in the quark-gluon basis without account of confinement.At low T ≪Tc the expansion in T/Tc can be performed. The main contribution comesfrom the pion states, |n⟩= |π⟩, |2π⟩, ....

In the chiral limit, when u and d quarks andpions are massless, the expansion parameter is T 2/f 2π where fπ = 1333 MeV is the piondecay constant: the one-pion contribution to Eq. (1) is proportional to T 2/f 2π, two-pioncontribution is of order T 4/f 2π, etc[3].

The contributions of massive hadronic states |n⟩aresuppressed by exp(−mn/T).At |q2| ≫R−2cthe operator product expansion (OPE) of T{j+(x), j(0)} on the lightcone can be performed so the coefficient functions are T-independent.Here generally isa difference in comparison with the case of T = 0, where expansion at small xµ (or nearthe tip of the light cone) takes place. This follows from the fact that at q0 >∼|q2/2mπ thematrix elements ⟨n|T{j+(x), j(0)}|n⟩are similar to the matrix elements of deep inelasticlepton-hadron scattering, where the process proseeds on the light cone, x2 ∼1/|q2|, but thelongitudinal distances, along the light cone are large and do not decrease with an increase of|q2|[6].

In the interesting special cases, when q0 = 0 or q = 0 and q0 pure imaginary, an OPEnear the tip of the light cone can be performed ans a few terms in this expansion must betaken into account. In this expansion operators with nonzero spin s appear, unlike the caseT = 0 where only spin zero operators contribute to the vacuum average.

This fact is now wellunderstood in the calculations done by QCD sum rule method at finite temperature[7, 8] aswell as at finite hadronic density[9]-[13]. At low temperatures where the pion states dominatein the chiral limit, the matrix elements of s ̸= 0 operators are proportional to T s+2.

It mustbe mentioned that even for s = 0 operators and their vacuum expectation values (v.e.v.) notall of the techniques which were successful at T = 0 can be applied at T ̸= 0.

For example,the factorization hypothesis which works well for v.e.v. of four-quark operators at T = 0cannot be directly applied at T ̸= 0, because pion intermediate states should be accountedfor[8, 14].In QCD sum rule method at T = 0 the v.e.v.

of a current correlator calculated by OPE,is on the other hand represented by the contributions of physical states using a dispersionrelation in q2 and in this way parameters of physical states (in particular, hadronic masses)2

were obtained. In this aspect the case of T ̸= 0 dramatically differs from the case of T = 0.The matrix elementsX′⟨n|iZd4xeiqxT{j+(x), j(0)}|n⟩e−En/T(2)where the sum is performed over all degrees of freedom of the state |n⟩, are functions oftwo Lorentz invariants, q2 and νn = pnq (or sn = (pn + q)2) where pn is the momentum ofthe state |n⟩.

Therefore, to consider analitical properties of the amplitudes in question andrepresent them in terms of physical states, it is necessary to take into account that they havediscontinuities in both q2 and sn (and also in the crossing channel, un = (pn −q)2). Thiswas not done in QCD sum rule calculations at finite temperature[7, 8] and density[10]-[12].

(In Refs. [9, 13] an attempt was made to partly account for these effects at finite density).It means that in writing down the dispersion relation in q2 it is necessary to specifyat which values of other invariants it takes place.For example, if ν = nq = q0 (n =(1, 0, 0, 0) is the time-like vector characterizing the heat bath) is fixed, then the contributionsof intermediate physical states in s- and u-channel must be taken into account.

In the caseof isospin 1 vector current jµ(x) it follows, e.g. that besides the ρ-meson pole πρ states arealso contributing.

It must be kept in mind that the representation of the physical spectrumas a lowest resonance pole plus continuum which is standard in the QCD sum rule methodat T = 0 is not suitable in the problem in question, because at least two poles and twodifferent continua in two (−q2 and s) channels should be taken into account. Further, it isevident that an effective mass of the lowest hadronic state to be obtained in this approach,depends on the relation between q0 and |q|.

E.g., it should be different for the cases q0 = 0,|q| ≫R−1cand imaginary q0, |q0| ≫R−1c , q = 0.For all these reasons the QCD sum rule approach does not generally look very promiss-ing for the problem of calculation of masses of lowest hadronic states at finite temperaturecontrary to the case of T = 0, where this method proved to be very effective. However,the calculation of q- and x-dependences (the latter was recently advocated in[2]) for variouscurrent correlators at finite temperature by the OPE on the light cone is still of a consider-able interest.

It would be very important, if lattice calculations of the same correlators atintermediate q2 and/or x2 could also be performed, because it would then give a possibilityto determine T-dependences of various condensates.As was shown in Ref. [3], the situation essentially simplifies if we confine ourselves tothe first order term in the expansion in T 2.

In this case apart from the vacuum state itis sufficient to take into account only the one-pion state in the sum over |n⟩in Eq.(1). Itscontribution can be calculated using PCAC and current algebra.

In Ref. [3] the correlators3

of isospin 1 vector and axial currents were considered and the following relations were foundCVµν(q, T)=(1 −ǫ)CVµν(q, 0) + ǫCAµν(q, 0)CAµν(q, T)=(1 −ǫ)CAµν(q, 0) + ǫCVµν(q, 0)(3)where CV,Aµν (q, T) are the correlators of V and A currents at finite temperature and CV,Aµν (q, 0)are the same correlators at T = 0. In the chiral limit, ǫ = (1/3)(T 2/f 2π).

If CV,Aµν (q, 0) arerepresented through dispersion relations by contributions of the physical states in V andA channels (ρ, a1, π, etc), then according to Eq. (3) the poles which are in the r.h.s.

ofEq. (3), i.e.

at T = 0, appear at the same positions in the l.h.s. Therefore, in order T 2 thepoles corresponding to ρ, a1 or π do not move1.

An important consequence of Eq. (3) is thatat T ̸= 0 in the vector (transverse) channel apart from the poles corresponding to vectorparticles, there arise poles corresponding to axial particles and visa verse, i.e.

a sort of paritymixing phenomenon occurs. The manifestation of this phenomenon is in complete accordwith the general considerations presented above: the appearence of an a1 pole in the vectorchannel corresponds to singularities in the s-channel.

In the same way a pion pole appearsin the longitudinal part of the vector channel.The statement that the poles do not move in order T 2 is very general: it is based only onPCAC and current algebra and can be immediately extended to any other current correlators. (The result that the nucleon pole does not move in order T 2 was obtained in the chiralperturbation theory in Ref.

[15] and by considering a current correlator in Ref.[16]). Theonly interesting physical phenomenon which occurs in this order is the parity mixing, i.e.the appearence of states with opposite parity in the given channel and, in some cases, alsoan isospin mixing.

The latter arises in baryon current correlators where, for example, in thecurrent with the quantum numbers of Λ there appears a Σ pole, and in the nucleon channelthere appear poles corresponding to baryon resonances with JP = 12± and T = 32,12.In the next order O(T 4) such a simple picture where the current correlator at finitetemperature is represented by the superposition of T = 0 correlators does not take place.Interpreted in terms of temperature dependent poles, it would mean that masses are shifted1In Ref. [3] it was not stressed that the poles do not move.

Instead, the l.h.s. of Eq.

(3) was representedby one effective pole (ρ in V -channel and a1 in A-channel). Such a representation is approximate, has nodeep sense and corresponds to the description of the current correlators at finite temperature by a one-polecontribution usually used in lattice calculations of hadron masses.

The fact that ρ and a −1 poles do notmove in order T 2 was overlooked in a recent paper[8]. For this reason the results obtained in Ref.

[8] are notreliable.4

in this order. But as explained above such an interpretation can be ambiguous.

We plan todiscuss this problem in a future publication.One of the authors (V.L.E.) would like to thank Joe Kapusta and Larry McLerran, forthe warm hospitality during his stay at the Nuclear Theory Group and Theoretical PhysicsInstitute, the University of Minnesota.

This work was supported in part by the Departmentof Energy under contract DE-FG02-87ER40328.5

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