Abstract
The free energy density of $\mathcal{N}=4$ supersymmetric Yang-Mills theory in four space-time dimensions is derived through second order in the 't Hooft coupling $\ensuremath{\lambda}$ at finite temperature using effective-field theory methods. The contributions to the free energy density at this order come from the hard scale $T$ and the soft scale $\sqrt{\ensuremath{\lambda}}T$. 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 $\sqrt{\ensuremath{\lambda}}T$ are taken into account by perturbative calculations in the effective theory.
Highlights
The thermodynamics of N 1⁄4 4 supersymmetric Yang-Mills theory in four dimensions (SYM4;4) is of great interest since, at finite-temperature, the weak-coupling limit of this theory has many similarities with quantum chromodynamics (QCD)
3 4 þ where λ 1⁄4 Ncg2 is the ‘t Hooft coupling, S is the entropy density, and Sideal 1⁄4 2dAπ2T3=3 is the corresponding noninteracting entropy density (Stefan-Boltzmann limit) with dA 1⁄4 N2c − 1 being the dimension of the adjoint representation
In the case of the high-temperature dimensional reduction, one obtains a three-dimensional effective field theory (EFT) that can be written in terms of dimensionally reduced fields
Summary
Mills theory in four dimensions (SYM4;4) is of great interest since, at finite-temperature, the weak-coupling limit of this theory has many similarities with quantum chromodynamics (QCD). [3] we will compute the SYM4;4 thermodynamic functions to Oðλ2Þ This will serve as a check on the calculation performed in Ref. In the case of the high-temperature dimensional reduction, one obtains a three-dimensional EFT that can be written in terms of dimensionally reduced fields. It is important to note that in supersymmetric theories, one must take some care with the dimensionality of the vector and spinor spaces describing the fields to ensure that supersymmetry is preserved by the regularization scheme used to evaluate Feynman diagrams For this purpose, we make use of the regularization by dimensional reduction (RDR) scheme [9,10,11,12,13,14].
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