Temperature Dependence of Gluon and Quark Condensates as from Linear Confinement

Abstract

The gluon and quark condensates and their temperature dependence are investigated within QCD premises. The input for the former is a gauge invariant $gg$ kernel made up of the direct (D), exchange (X) and contact(C) QCD interactions in the lowest order, but with the perturbative propagator $k^{-2}$ replaced by a `non-perturbative $k^{-4}$ form obtained via two differentiations: $ μ^2 \partial_m^2 (m^2+k^2)^{-1}$, ($μ$ a scale parameter), and then setting $m=0$, to simulate linear confinement. Similarly for the input $q{\bar q}$ kernel the gluon propagator is replaced by the above $k^{-4}$ form. With these `linear&#39; simulations, the respective condensates are obtained by `looping&#39; up the gluon and quark lines in the standard manner. Using Dimensional regularization (DR), the necessary integrals yield the condensates plus temperature corrections, with a common scale parameter $μ$ for both. For gluons the exact result is $$ <GG> = {36μ^4}π^{-3}α_s(μ^2)[2-γ- 4π^2 T^2/(3μ^2)]$$. Evaluation of the quark condensate is preceded by an approximate solution of the SDE for the mass function $m(p)$, giving a recursive formula, with convergence achieved at the third iteration. Setting the scale parameter $μ$ equal to the universal Regge slope $1 GeV^2$, the gluon and quark condensates at T=0 are found to be $0.586 Gev^4$ and $(240-260 MeV)^3$ respectively, in fair accord with QCD sum rule values. Next, the temperature corrections (of order $-T^2$ for both condensates) is determined via finite-temperature field theory a la Matsubara. Keywords: Gluon Condensate, mass tensor, gauge invariance, linear confinement, finite-temperature, contour-closing. PACS: 11.15.Tk ; 12.38.Lg ; 13.20.Cz