Thermodynamics of N = 4 supersymmetric Yang-Mills theory using direct resummation and effective field theory methods

The thermodynamics of N = 4 supersymmetric Yang-Mills theory in four dimensions (SYM_4,4) is of great interest since, at finite-temperature, the weak-coupling limit of this theory has many similarities with quantum chromodynamics (QCD). Unlike QCD, however, in SYM_4,4 it is possible to make use of the AdS/CFT correspondence between gravity in anti-de Sitter space (AdS) and the large-N_c limit of conformal field theories (CFT) on the boundary of AdS to obtain results for SYM_4,4 thermodynamics in the strong coupling limit. The mathematical structure of SYM_4,4 is similar to that of QCD, the difference is mostly in the number of degrees of freedom and the representations of fields. There are four Majorana fermions and six scalars and all the fields are in the adjoint representation. In the last decade or so the thermodynamics of SYM_4,4 in a strong-coupling regime received a great deal of attention due to AdS/CFT where in SYM_4,4 is mapped to its gravity dual. In this limit, the thermodynamics has been computed to the order λ^−3/2, where λ = N_c g² is the ‘t Hooft coupling. In the opposite sector of weak coupling, prior to our work, the free energy density of SYM_4,4 was known to the order λ^3/2. In this regime, calculations are performed using perturbative field theory methods. This weak-coupling expansion of SYM_4,4 like QCD can pushed until λ^5/2, after which non-perturbative effects come into play. In this SYM_4,4 free energy density expansion interesting observations are made by constructing a generalized Padé which interpolates between strong and weak coupling expansion. The weak coupling expansion converges towards this Padé for λ ≲ 1 and the strong coupling for λ ≳ 10. The makes the weak and strong coupling expansion and their convergence in the intermediate region of 1 ≲ λ ≲ 10 of a great deal of interest.

On the weak-coupling side the free energy density calculations in SYM_4,4, like in QCD, are done and improved upon using various perturbative field theoretic techniques developed over the last decades. We used three different techniques with the aim of improving the convergence of the successive weak-coupling approximations in this SYM_4,4. Our first project involved the computation of the two-loop hard-thermal-loop (HTL) resummed thermodynamic potential for this SYM_4,4. Our final result is manifestly gauge-invariant and was renormalized using only simple vacuum energy, gluon mass, scalar mass, and quark mass counter terms. The HTL mass parameters m_D, M_D, and m_q are then determined self-consistently using a variational prescription which result in a set of coupled gap equations. Based on this, we obtain the two-loop HTL-resummed thermodynamic functions of N = 4 SYM. We compare our final result with known results obtained in the weak- and strong-coupling limits. We also compare to previously obtained approximately self-consistent HTL resummations and Padé approximants. We find that the two-loop HTL resummed results for the scaled entropy density is a quantitatively reliable approximation to the scaled entropy density for 0 ≤ λ ≲ 2 and is in agreement with previous approximately self-consistent HTL resummation results for λ ≲ 6.

Our next step is to extend the weak coupling calculations to higher orders in λ. To do the expansion to the order λ² we first used the direct resummation techniques. One caveat here is that supersymmetry is broken by conventional dimensional regularization, so we compute resummed perturbative free energy for SYM_4,4 in the regularization by dimensional reduction (RDR) scheme 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 SYM_4,4 ring diagrams. Non-analytic terms at O(λ^3/2) and O(λ² log λ) are generated by dressing the A₀ and scalar propagators. The gauge-field Debye mass m_D and the scalar thermal mass M_D are determined from their corresponding finite-temperature self-energies. Based on this, we obtain the three-loop thermodynamic functions of SYM_4,4 to 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-N_c strong-coupling result. Our results suggest that the O(λ²) weak- coupling result for the scaled entropy density is a quantitatively reliable approximation to the scaled entropy density for 0 ≤ λ ≲ 2.

Next, we revisit the resummed thermodynamic calculations in SYM_4,4 done previously to two-loop order within hard thermal loop perturbation theory (HTLpt) using the regularization by dimensional reduction (RDR) scheme. Earlier calculations were performed using the canonical dimensional regularization (DRG) scheme. Since this scheme breaks supersymmetry, the objective is to assess if and by how much the resummed perturbative results depend on the regularization scheme used. Comparing predictions for the scaled entropy obtained using the DRG and RDR schemes we find that for λ ≲ 6 they are numerically very similar. We then compare the results obtained in both schemes with the strict perturbative result, which is accurate up to order λ², and a generalized Padé approximant constructed from the known large-N_c weak- and strong-coupling expansions. Comparing the strict perturbative expansion of the two-loop HTLpt result with the perturbative expansion to order λ², we find that both the DRG and RDR HTLpt calculations result in the same scheme-independent predictions for the coefficients at order λ, λ^3/2, and λ² log λ, however, at order λ² there is a residual regularization scheme dependence.

Finally, we consider the resummed thermodynamics for N = 4 Super Yang-Mills theory using effective field theory (EFT) techniques. These techniques provide a computational simplification, and will allow us to extend the resummed perturbative thermodynamics beyond order λ². This also serves as a check of our previous results using direct resummation. The contributions to the free energy density at this order come from the hard scale T and the soft scale √λT . The effects of the scale T are encoded in the coefficients of an effective three-dimensional field theory that is obtained by dimensional reduction at finite temperature. The effects of the electric scale √λT are taken into account by perturbative calculations in the effective theory.