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

TL;DR

This paper computes the thermodynamic free energy of ${ m SYM}_4$ at finite temperature up to second order in the 't Hooft coupling, providing insights into the behavior of strongly coupled gauge theories relevant for holography.

Contribution

It presents a three-loop calculation of the free energy in ${ m SYM}_4$ including resummation and non-analytic terms, extending previous weak- and strong-coupling analyses.

Findings

The free energy is ultraviolet finite with infrared divergences canceled by ring diagrams.

Non-analytic terms at ${ m O}( ext{lambda}^{3/2})$ and ${ m O}( ext{lambda}^2 ext{log} ext{lambda})$ are identified.

The ${ m O}( ext{lambda}^2)$ weak-coupling result approximates the entropy density well for $ ext{lambda} extless 2$.

Abstract

We calculate the resummed perturbative free energy of ${\cal N}=4$ 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 ${\cal O}({\lambda}^{3/2}) $ and $ {\cal 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 ${\cal O}(\lambda^2)$. We compare our final result with prior results obtained in the weak- and strong-coupling limits and construct a generalized Pad\'{e} approximant that interpolates between the weak-coupling result and the large-$N_c$ strong-coupling result. Our results suggest that the ${\cal O}(\lambda^2)$ weak-coupling result for the scaled entropy density is a quantitatively reliable approximation to the scaled entropy density for $0 \leq \lambda \lesssim 2$.