Infrared problems with dimensional regularisation at finite temperature

Abstract

We address the question whether we have to perform the analytic continuation to 4 space-time dimensions before or after the perturbative expansion has been made. Using the simple model of large-N scalar field theory, we show how this affects quantities like the thermal mass and the pressure, why the two procedures give different results, and how this relates to the infrared behaviour of the theory. We conclude that the correct procedure is to take the limit d -> 4 before performing the perturbative expansion. Infrared divergences arise if one makes the perturbative expansion before going to the physical space-time dimension and they correspond to logarithms of the coupling in the exact (nonperturbative or resummed) expression. Finally, we outline how this may be relevant to the infrared problems encountered in thermal QCD.