Improved Resummations for the Thermodynamics of the Quark-gluon Plasma∗

At leading order, perturbation theory in the deconfined phase of QCD gives a reasonable estimate of the interaction pressure for temperatures a few times the critical one. But as soon as the beautiful machinery of resummed thermal perturbation theory comes into its own, its poor convergence properties seem to forbid its exploitation except at ridiculously high temperatures (or densities). This breakdown becomes apparent already at order g, which is entirely produced by the collective phenomenon of Debye screening (somewhat misleadingly dubbed “plasmon effect”), and already occurs in the simplest models such as scalar φ theory for rather small coupling. At least in scalar theory, it has been shown that this impasse can be breached by Padé resummation and, more promisingly, by judiciously optimized perturbation theory such as “screened perturbation theory” (SPT). In SPT a coupling expansion is performed only with respect to couplings in explicit interactions, while any coupling constants buried in thermal (quasiparticle) masses are not expanded out, leading to nonpolynomial, i.e. nonperturbative, expressions in g. This has recently been adapted for QCD under the trademark “HTL perturbation theory”. There, in place of a simple mass term, the hard-thermal-loop (HTL) effective action is added, and subtracted again as a formally higher-order counterterm, from the ordinary action. This approach differs from standard (HTL-)resummed perturbation theory in that resummed quantities are not only used in the soft momentum regime, but throughout. However, there is a price to be paid. At any finite loop order, the UV structure of the theory is modified—new (eventually temperature-dependent) divergences occur and must be subtracted, introducing a new source of renormalization scheme dependence. An alternative approach for a more extensive resummation of the physics of HTL’s has been worked out by J.-P. Blaizot, E. Iancu, and myself, which is based on a self-consistent (“Φ-derivable”) two-loop approximation to