Resummation in hot field theories
There has been significant progress in the understanding of finite-temperature field theory over the past decade. In this paper the progress in perturbative thermal field theory focusing on thermodynamic quantities is reviewed. There are discussed first the breakdown of naive perturbation theory (loop expansion) at finite temperature due to infrared divergences. Diagrams which are nominally of higher order in the coupling constant contribute to leading order in the interaction parts \(g\). Thus, dividing the euclidean Lagrangian of a scalar field in a free part proportional to \(g^0\) and an interaction part proportional to \(g^2\), and calculating radiative corrections in a loop expansion which is equivalent to a power series in \(g^2\) (that means in \(\phi^4\)-interaction), one finds that the perturbative expansion breaks down at finite temperature and the week coupling expansion becomes an expansion in \(g\) rather than in \(g^2\). Thus, a consistent perturbative expansion requires the resummation of an infinite subset of diagrams from all orders of perturbation theory. The Braaten-Pisarski resummation approach is such a consistent expansion, it can be used to systematically calculate various static and dynamical quantities as a weak-coupling expansion in powers of \(g\). However, it turns out that the weak coupling expansion for thermodynamic quantities is useless unless the coupling constant is very small. In the paper it is shown that the perturbative expansion of the free energy for both simple scalar field theories and QCD have a very small, if not vanishing, radius of convergence. In the case of QCD this makes the resulting expansion useless at phenomenologically relevant temperatures. Then it is shown that dimensional reduction (DR) and effective field theory methods can provide an efficacious way of organizing weak-coupling calculations. By combining them with nonperturbative methods such as lattice gauge theory, the long-standing problem of calculating the order \(\alpha^3\)-contribution to the QCD pressure can be solved (the coupling constant \(\alpha\) being proportional to the square of the elementary charge \(e\)). Another advantage of using DR and effective field theory methods is that the explicit separation of the hard and soft scales allows for a different treatment of the physics of these scales. In the paper it is shown that if the soft-sector is treated non-perturbatively, the successive approximations to the thermodynamic functions are much better behaved, giving reasonable constrained results for the pressure in scalar field theories to \(g\sim4\). It is shown that two reorganizations of the soft sector computation, the DR screened perturbation theory (SPT) and the DR \(\Phi\)-derivable approach, both give better convergence of the successive approximations to the pressure at larger coupling. While the results are impressive, the methods are limited because they can only be applied to the computation of static quantities such as the pressure. It would be preferable to have a reorganization of the soft-sector physics which could be applied to the calculation of real-time quantities as well. This desire has motivated the development of the \(\Phi\)-derivable approach and the SPT and hard-thermal-loop perturbation theory (HTLPT) reorganizations. The \(\Phi\)-derivable approach has only been taken to three-loops for a scalar field theory where it improves convergence and scale variation of the approximations to the pressure. It would be nice to systematically extend this approach also to three loops for QCD. It has been demonstrated that when applied to the calculation of the pressure in QCD the HTLPT reorganization yields impressive results when one considers the convergence of the successive approximations. Additionally, results are explicitly gauge-invariant within HTLPT as opposed to \(\Phi\)-derivable approximations where gauge invariance can only be guaranteed to a fixed order in the coupling constant. Despite these theoretical benefits, when compared to existing lattice data for the pressure in the temperature range \(T_c<T<5T_c\) (\(T_c\) is the critical temperature for the deconfinement phase transition), the NLO HTLPT prediction for the pressure seems to understimate the correction to ideal gas behavior. The failure of the method in this region could come from the fact that within HTLPT the requirement of gauge invariance causes a simultaneous modification of both soft and hard modes. Perhaps a method that makes an explicit separation between these two scales is more appropriate. Another possible explanation is that HTLPT, as implemented, does not properly take into account the proper scaling relations for quantities near the phase transition which is governed (approximately) by the center group of SU(3), Z(3). However, deciding which of these two possibilities is the culprit is not possible at this stage. Results from the approximately self-consistent \(\Phi\)-derivable approach of \textit{J.-P.\ Blaizot, E.\ Iancu} and \textit{A.\ Rebhan} [Phys.\ Rev.\ D (3) 68, 025011 (2003)] seem to point to the first possibility, but are by no means conclusive. The only thing that can resolve this is further study.
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