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

We apply the previously proposed scheme of approximately self-consistenthard-thermal-loop resummations in the entropy of high-temperature QCD to N=4supersymmetric Yang-Mills (SYM) theories and compare with a (uniquelydetermined) R[4,4] Pad\'e approximant that interpolates accurately between theknown perturbative result and the next-to-leading order strong-coupling resultobtained from AdS/CFT correspondence. We find good agreement up to couplingswhere the entropy has dropped to about 85% of the Stefan-Boltzmann value. Thisis precisely the regime which in purely gluonic QCD corresponds to temperaturesabove 2.5 times the deconfinement temperature and for which this method ofhard-thermal-loop resummation has given similar good agreement with lattice QCDresults. This suggests that in this regime the entropy of both QCD and N=4 SYMis dominated by effectively weakly coupled hard-thermal-loop quasiparticledegrees of freedom. In N=4 SYM, strong-coupling contributions to thethermodynamic potential take over when the entropy drops below 85% of theStefan-Boltzmann value.Comment: 14 pages, 2 figures, JHEP3. v2: revised and expanded, with unchanged HTL results but corrected NLO strong-coupling result from AdS/CFT (which is incorrectly reproduced in almost all previous papers comparing weak and strong coupling results of N=4 SYM) and novel (unique) Pade approximant interpolating between weak and strong coupling result