Approximately self-consistent resummations for the thermodynamics of the quark-gluon plasma. I. Entropy and density

We propose a gauge-invariant and manifestly UV finite resummation of the physics of hard thermal/dense loops (HTL/HDL) in the thermodynamics of the quark-gluon plasma. The starting point is a simple, effectively one-loop expression for the entropy or the quark density which is derived from the fully self-consistent two-loop skeleton approximation to the free energy, but subject to further approximations, whose quality is tested in a scalar toy model. In contrast to the direct HTL/HDL-resummation of the one-loop free energy, in our approach both the leading-order (LO) and the next-to-leading order (NLO) effects of interactions are correctly reproduced and arise from kinematical regimes where the HTL/HDL are justifiable approximations. The LO effects are entirely due to the (asymptotic) thermal masses of the hard particles. The NLO ones receive contributions both from soft excitations, as described by the HTL/HDL propagators, and from corrections to the dispersion relation of the hard excitations, as given by HTL/HDL perturbation theory. The numerical evaluations of our final expressions show very good agreement with lattice data for zero-density QCD, for temperatures above twice the transition temperature.