N = 4 supersymmetric Yang-Mills thermodynamics to order λ 5 / 2

We calculate the resummed perturbative free energy of N = 4 supersymmetric Yang-Mills in four spacetime dimensions ( SYM 44 ) to order λ 5 / 2 in the ‘t Hooft coupling at finite temperature and zero chemical potential. All infrared divergences cancel when we include contributions from SYM 44 ring diagrams and the final result is both ultraviolet and infrared finite. Our result has special significance since order λ 5 / 2 is the highest order calculation that can be done with perturbation theory, because there are nonperturbative effects associated with the magnetic mass scale that come into play at order λ 3 . We compare results obtained with regularization by dimensional reduction (RDR), which preserves supersymmetry, and canonical dimensional regularization (DR). We also compare with a generalized Padé approximant constructed by matching the weak coupling result at order λ 2 and the large N c strong coupling result at order λ − 3 / 2 . Finally we make a comparison between our result and the QCD free energy and show that SYM 44 has better convergence properties.