Abstract
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 Padé 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.
Highlights
Where λ = g2Nc is the ’t Hooft coupling, which does not run and is independent of the temperature.1 This expansion is identical in form to the perturbative expansion of the QCD free energy [1,2,3,4,5]
Our final result is ultraviolet finite and all infrared divergences generated at three-loop level are canceled by summing over SYM4,4 ring diagrams
The leading term in this expression is the free energy of an ideal SYM4,4 plasma and the O(λ) correction can be obtained by computing two-loop Feynman diagrams
Summary
The evaluation of Feynman diagrams for theories that are obtained by dimensional reduction of SYM1,D can be carried out in two equivalent ways, one is working throughout in (D−2 )⊕(D−D+2 ) space for the theories we target, with being an infinitesimal quantity used for regularization and taking the dimension of the momentum-space to be d = D − 2 This prescription results in one having to introduce -scalars into the Lagrangian in order to preserve supersymmetry [24]. We will use this second RDR scheme since it is the most transparent and efficient for computations [14] With this in mind, up to thermal mass corrections, for any SYM theory that is characterized by (Dmax, d), massless (unresummed) contributions to the free energy can be calculated perturbatively by computing vacuum Feynman diagrams using N = 1 SYM in D = Dmax and restricting the momentum in loops to d dimensions to obtain the result in the target theory.
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