${\cal N}=4$ supersymmetric Yang-Mills thermodynamics to order $\lambda^{5/2}$

We calculate the resummed perturbative free energy of ${\cal N} = 4$ supersymmetric Yang-Mills in four spacetime dimensions (SYM$_{44}$) to order $\lambda^{5/2}$ 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 $\lambda^{5/2}$ 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 $\lambda^3$. We compare results obtained with regularization by dimensional reduction (RDR), which preserves supersymmetry, and canonical dimensional regularization (DR). We also compare with a generalized Pad\'e approximant constructed by matching the weak coupling result at order $\lambda^2$ and the large $N_c$ strong coupling result at order $\lambda^{-3/2}$. Finally we make a comparison between our result and the QCD free energy and show that SYM$_{44}$ has better convergence properties.