We compute the weak-coupling free energy density of pure four-dimensional $\mathcal{N}=2$ supersymmetric Yang-Mills (SYM) theory through second order in the ’t Hooft coupling $\lambda$ at finite temperature using effective-field-theory (EFT) methods. The contribution from the hard scale $T$ is obtained from massless three-loop vacuum diagrams in the full theory and that from the electric scale $\sqrt{\lambda }T$ from a two-loop calculation in the dimensionally reduced three-dimensional effective theory. In contrast to $\mathcal{N}=4$ SYM, the theory is asymptotically free; the coupling-renormalization counterterm, together with the EFT unit-operator counterterm, cancels all $1/\epsilon$ poles. The remaining renormalization-scale dependence is fixed by the one-loop beta function. The expansion reproduces earlier results through order ${\lambda }^{3/2}$, and the order-${\lambda }^2$ term is new.

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• Effective field theory
• Finite temperature field theory
• Resummation methods
• Supersymmetric field theories