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

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] Pade 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