Perturbative Finite-Temperature Results and Padé Approximants

Padé approximants are used to improve the convergence behavior of perturbative results in massless scalar and gauge field theories at finite temperature. In recent years, computational methods have been developed to analytically tackle three-loop vacuum diagrams and higher-order contributions of diagrams with less loops in massless field theories at finite temperature [1]-[5]. Consequently, the free energy density F at zero chemical potential could be computed analytically at the g level in both massless gφ theory [3] (the pressure given there is the negative of the free energy density) and in massless gauge theories [5, 6]. In [5]-[7], specializations to QED can be found, where the result was known before in partially numerical form [2]. However, for interesting values of the coupling constant in non-Abelian gauge theories, the convergence behavior of the perturbative series is not convincing [5, 6]. In this brief report, we note that the use of Padé approximants drastically improves this behavior in both φ and gauge theories. For the use of Padé approximants and other resummation techniques in other contexts in perturbative field theory and statistical physics, see e.g. [8] and references therein. Let us first review those features of the results of [3, 5, 6] which are essential for our analysis here. The perturbative series for the free energy density in both scalar and gauge theories has the structure (see the appendix for details) F = T [c0 + c2g 2 + c3g 3 + (c4a ln g + c4b)g 4 + (c5a ln g + c5b)g 5 +O(g ln g)] , (1) where T is the temperature and c0, c2, c3, c4a, c5a are constants, while c4b and c5b have a logarithmic dependence on ln(μ̄/T ), where μ̄ is the renormalization scale in the modified minimal subtraction scheme, MS. In φ theory, c4a = 0, while in gauge theories c5a = 0. 1 As in [5], we use the renormalization group to make g running, 1 g2(μ̄) ≈ 1 g T − β0 ln μ̄ T + β1 β0 ln (