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

Abstract

We calculate the resummed perturbative free energy of $\mathcal{N}=4$ supersymmetric Yang-Mills in four spacetime dimensions (${\text{SYM}}_{4,4}$) through second order in the 't Hooft coupling $\lambda$ 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 ${\text{SYM}}_{4,4}$ ring diagrams. Non-analytic terms at $\mathcal{O}({\lambda }^{3/2})$ and $\mathcal{O}({\lambda }^2\log \lambda )$ are generated by dressing the $A_0$ and scalar propagators. The gauge-field Debye mass $m_D$ and the scalar thermal mass $M$ are determined from their corresponding finite-temperature self-energies. Based on this, we obtain the three-loop thermodynamic functions of ${\text{SYM}}_{4,4}$ to $\mathcal{O}({\lambda }^2)$. We compare our final result with prior results obtained in the weak- and strong-coupling limits and construct a generalized Padé approximant that interpolates between the weak-coupling result and the large-$N_c$ strong-coupling result. Our results suggest that the $\mathcal{O}({\lambda }^2)$ weak-coupling result for the scaled entropy density is a quantitatively reliable approximation to the scaled entropy density for $0\le \lambda \lesssim 2$.

arXiv comment28 pages, 8 figures; v3: fix to soft contribution and final result, misc typos fixed, conclusions unchanged

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• Chromoelectric field correlator for quarkonium transport in the strongly coupled $\mathcal{N}=4$ Yang-Mills plasma from AdS/CFT
• $\mathcal{N}=4$ supersymmetric Yang-Mills thermodynamics from effective field theory
• Constrained Padé Ensembles for Thermal N=4 SYM: Quantified Uncertainties and Next-Order Predictions
• Scheme dependence of two-loop HTLpt-resummed ${\text{SYM}}_{4,4}$ thermodynamics