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Effective field theory treatment of ${\cal N}=4$ supersymmetric Yang-Mills thermodynamics

At finite temperature the free energy density of ${\cal N}=4$ supersymmetric Yang-Mills can be calculated using resummed perturbation theory through the order $λ^{5/2}$. Effective field theory methods provide a useful alternative approach to streamline these calculations. In this proceedings contribution, I review recent work with my collaborators where we used effective field theory methods to calculate the free energy density of ${\cal N}=4$ supersymmetric Yang-Mills in four spacetime dimensions through second order in the 't Hooft coupling $λ$. At this order the contributions to the free energy density come from the hard scale $T$ and the soft scale $\sqrtλT$. The contribution from the scale $T$ enters through the coefficients in the effective Lagrangian obtained by dimensional reduction and the effects of the scale $gT$ can be calculated using perturbative methods in the effective theory.