N $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills thermodynamics to order λ2

We calculate the resummed perturbative free energy of $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills in four spacetime dimensions (SYM4,4) through second order in the ’t Hooft coupling λ at finite temperature and zero chemical potential. Our final result is ultraviolet finite and all infrared divergences generated at three-loop level are canceled by summing over SYM4,4 ring diagrams. Non-analytic terms at $$ \mathcal{O} $$ (λ3/2) and $$ \mathcal{O} $$ (λ2 log λ) are generated by dressing the A0 and scalar propagators. The gauge-field Debye mass mD and the scalar thermal mass MD are determined from their corresponding finite-temperature self-energies. Based on this, we obtain the three-loop thermodynamic functions of SYM4,4 to $$ \mathcal{O} $$ (λ2). We compare our final result with prior results obtained in the weak- and strong-coupling limits and construct a generalized Pade approximant that interpolates between the weak-coupling result and the large-Nc strong-coupling result. Our results suggest that the $$ \mathcal{O} $$ (λ2) weak-coupling result for the scaled entropy density is a quantitatively reliable approximation to the scaled entropy density for 0 ≤ λ ≲ 2.