Hard thermal loops and the entropy of supersymmetric Yang-Mills theories

TL;DR

This work extends a UV-finite, gauge-invariant hard-thermal-loop (HTL) resummation framework to compute the entropy of hot gauge theories, applying it first to QCD and then to N=4 supersymmetric Yang-Mills (SYM) theory. By expressing the entropy as a functional of dressed propagators and self-energies and truncating at two loops with ${\mathcal S}'=0$, the authors extract leading hard-mode and plasmon contributions through asymptotic masses $m_{\infty}$ and a quadratic gap equation, yielding a quasiparticle interpretation of thermal excitations. In QCD, the HTL-resummed (NLA) entropy agrees well with lattice results for temperatures $T \gtrsim 3 T_c$, supporting a picture of weakly interacting HTL quasiparticles dominating the entropy in this regime. For N=4 SYM, a unique $R_{[4,4]}$ Padé approximant interpolating between weak- and strong-coupling results is used, and the HTL-NLA entropy matches the Padé up to $\lambda \sim 3$ (with controlled $c_\Lambda$), suggesting that the entropy remains governed by HTL quasiparticles until strong-coupling corrections become significant once the entropy falls below roughly $0.85\mathcal S_0$. These findings imply a broad regime where QCD and SYM thermodynamics can be captured by effectively weakly coupled quasiparticle dynamics, with truly strong-coupling effects emerging only at lower entropies or higher couplings.

Abstract

We apply the previously proposed scheme of approximately self-consistent hard-thermal-loop resummations in the entropy of high-temperature QCD to N=4 supersymmetric Yang-Mills (SYM) theories and compare with a (uniquely determined) R[4,4] Padé approximant that interpolates accurately between the known perturbative result and the next-to-leading order strong-coupling result obtained from AdS/CFT correspondence. We find good agreement up to couplings where the entropy has dropped to about 85% of the Stefan-Boltzmann value. This is precisely the regime which in purely gluonic QCD corresponds to temperatures above 2.5 times the deconfinement temperature and for which this method of hard-thermal-loop resummation has given similar good agreement with lattice QCD results. This suggests that in this regime the entropy of both QCD and N=4 SYM is dominated by effectively weakly coupled hard-thermal-loop quasiparticle degrees of freedom. In N=4 SYM, strong-coupling contributions to the thermodynamic potential take over when the entropy drops below 85% of the Stefan-Boltzmann value.