Temperature dependence of electric and magnetic gluon condensates.

The contribution of Lorentz non-scalar operators to finite temperature correlation functions is discussed. Using the local duality approach for the one-pion matrix element of a product of two vector currents, the temperature dependence of the average gluonic stress tensor is estimated in the chiral limit to be 〈E2 + B〉T = π 2 10 bT . At a normalization point μ = 0.5 GeV we obtain b ≈ 1.1. Together with the known temperature dependence of the Lorentz scalar gluon condensate we are able to infer 〈E〉T and 〈B〉T separately in the low-temperature hadronic phase. Permanent address: Institute of Theoretical and Experimental Physics, Moscow 117259, Russia. Correlators of currents with the quantum numbers of hadrons are known to be useful to obtain information about the masses and couplings of hadrons; they are employed in the QCD sum rule approach and in lattice calculations. In both approaches the correlators are considered at large Euclidean distances or imaginary times where the dominant contribution comes from the lowest state with the corresponding quantum numbers. QCD sum rules give predictions also for form factors and structure functions of hadrons. (For a recent review of applications of QCD correlation functions see ref. [1].) In recent years there has been increasing interest in finite temperature QCD and hadronic physics due to the expectation that at high enough temperatures the QCD vacuum, specified by nonperturbative condensates of quark and gluon fields, will “melt” and undergo a transition to a quark-gluon plasma. Melting is usually understood in the sense that chiral symmetry restoration and deconfinement take place. The former means that with increasing temperature quark condensates evaporate, while the latter means that hadrons do not represent stable degrees of freedom. It was shown by Leutwyler and his collaborators [2] using the chiral Lagrangian approach that the quark condensate indeed decreases with rising temperature. From the usual QCD sum rules at T = 0 it is well known that the properties of hadrons are, to a large extent, determined by nonperturbative quark and gluon condensates [3]. Naturally, a large number of papers were devoted to the generalization of QCD sum rules to finite temperature in attempts to relate the temperature dependence of the hadronic spectrum to the temperature dependence of the condensates (see, e.g. [4, 5, 6]). In this case the vacuum average of the product of currents becomes the Gibbs average over the thermal ensemble. To calculate the Gibbs average one must choose a basis for the states. As argued in refs. [5, 7] at temperatures which are much less than the energy scale of confinement the appropriate basis is that of hadronic states, rather than the quark-gluon basis used in early papers on the subject (see, e.g. ref. [4]). Using this basis it was also shown [5] that at low T the thermal correlators are expressed as a mixture of zero-temperature correlators with different parity. It is also clear that if the operator product expansion (OPE) is applied to a thermal correlator then the temperature dependence appears only in the matrix elements of the operators (condensates), the coefficient functions being obtained through a perturbative calculation at T = 0. QCD sum rules at low temperature were recently reexamined along these lines in ref. [8]. At high temperatures, corresponding to the quark-gluon plasma, the calculation of thermal correlators should be performed in a basis consisting of quark and gluon states. In this case the perturbative temperature-dependent parts of the condensates due to quarks and gluons from the thermal ensemble may be included in the coefficient We thank T. Hatsuda for drawing our attention to this paper.