High Temperature Behaviour of the Qcd Coupling Constant

This straightforward extension of the T = 0 renormalizat ion group ( R G ) results is not completely satisfactory. In ref. [ 2 ], Matsumoto et al. generalize the RG approach at finite T i n the framework of the real t ime formalism [3]. In this framework, the ultraviolet divergences are the same as at T = 0 but in addit ion to the freedom of choosing the usual m o m e n t u m renormalizat ion point, one has to deal with the arbitrariness of the temperature at which the renormalizat ion parameters are determined. Within this finite temperature RG approach, the behaviour of the QCD coupling constant with respect to the temperature has been investigated by working out the solution of the one loop RG equations [4 -7] . In this paper, we reanalyze this problem in a specific renormalizat ion scheme and show that contr ibut ions have been overlooked which in fact dominate the high T expansion of the QCD coupling constant. This scheme generalizes at finite T the MOM renormalizat ion procedure [ 8 ]. Propagators and vertices are renormalized at spacelike momenta and at a given temperature. We work in pure QCD and derive the one loop running coupling constant in the Feynman gauge in two MOM renormalizat ion schemes attached to the three gluon vertex and the gluon-ghost vertex in the symmetric spacelike configuration for external lines p2 = q2 = r 2 = _ M 2 (fig. 1 ). We find that terms depending on the second and third power of the gluon (or ghost) Bose-Einstein statistical distr ibution are present ~ and dominate the asymptotic temperature behaviour o f g2 ( T) at fixed M: