We calculate the resummed perturbative free energy of $\mathcal{N}=4$ supersymmetric Yang-Mills in four spacetime dimensions (${\mathrm{SYM}}_{44}$) 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 ${\mathrm{SYM}}_{44}$ 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 Padé 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 ${\mathrm{SYM}}_{44}$ has better convergence properties.

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• Quantum field theory
• Supersymmetric field theories