Hardthermalloop QED thermodynamics
Abstract
The weakcoupling expansion for thermodynamic quantities in thermal field theories is poorly convergent unless the coupling constant is tiny. We discuss the calculation of the free energy for a hot gas of electrons and photons to threeloop order using hardthermalloop perturbation theory (HTLpt). We show that the hardthermalloop perturbation reorganization improves the convergence of the successive approximations to the QED free energy at large coupling, . The reorganization is gauge invariant by construction, and due to the cancellations among various contributions, we obtain a completely analytic result for the resummed thermodynamic potential at three loops.
Hardthermalloop QED thermodynamics
Nan Su Jens O. Andersen Michael Strickland 
1 (Frankfurt Institute for Advanced Studies, D60438 Frankfurt am Main, Germany)
2 (Department of Physics, Norwegian University of Science and Technology, N7491 Trondheim, Norway)
3 (Department of Physics, Gettysburg College, Gettysburg, PA 17325, USA)

Key words Resummation, HardThermalLoop, Thermal Field Theory, QED, Free Energy

PACS 11.15.Bt, 04.25.Nx, 11.10.Wx, 12.38.Mh
1 Introduction
The calculation of thermodynamic functions for finite temperature field theories has a long history. In the early 1990s the free energy was calculated to order for massless scalar theory^{[2, 3]}, QED^{[4, 3]} and QCD^{[3]} respectively. The corresponding calculations to order were soon obtained afterwards^{[5, 6]}^{[7, 8]}^{[9, 10]}. Recent results have extended the calculation of the QCD free energy by determining the coefficient of the contribution^{[11]}. For massless scalar theories the perturbative free energy is now known to order ^{[12]} and ^{[13]}.
Unfortunately, for all the abovementioned theories the resulting weakcoupling approximations, truncated orderbyorder in the coupling constant, are poorly convergent unless the coupling constant is tiny. In this proceedings we shall focus on the discussion of QED. Fig. 1 shows the successive perturbative approximations to the QED free energy. As can be seen from this figure, at couplings larger than the QED weakcoupling approximations exhibit poor convergence. To improve the bad convergence of perturbative expansions, several systematic resummation techniques have been introduced and they are summarized in references^{[14, 15, 16]}. In the following we will discuss recent advances in the application of hardthermalloop perturbation theory (HTLpt) ^{[17, 18, 19]}.
2 Threeloop hardthermalloop perturbation theory
Hardthermalloop perturbation theory is inspired by variational perturbation theory^{[20, 21, 22, 23]} and is a gaugeinvariant extension of screened perturbation theory^{[24, 25, 26, 27]}. The basic idea of the technique is to add and subtract an effective mass term from the bare Lagrangian and to associate the added piece with the free Lagrangian and the subtracted piece with the interactions. However, in gauge theories, one cannot simply add and subtract a local mass term since this would violate gauge invariance. Instead one adds and subtracts an HTL improvement term which modifies the propagators and vertices in such a way that the framework is manifestly gaugeinvariant.
HTLpt has recently been pushed to three loops or the nexttonexttoleading order (NNLO) and the details of the formalism and calculations are presented in Ref.^{[28]}. Here only a few selected results are listed.
With rescaled dimensionless parameters , , and , the renormalized NNLO thermodynamic potential reads
(1)  
There is also a corresponding nexttoleading order (NLO) thermodynamic potential that contains some numerical coefficients^{[28]}. We note that at NNLO all numerically determined coefficients drop out and we are left with a final result which is completely analytic.
The mass parameters and in HTLpt are in principle completely arbitrary. To complete a calculation, it is necessary to specify and as functions of and . In Ref.^{[28]} we considered two possible mass prescriptions: 1) the variational thermal masses obtained from the gap equations; 2) the perturbative Debye mass^{[29, 8]} and the perturbative fermion mass^{[30]}. The resulting predictions for the free energy are shown in Fig. 1. As can be seen from these figures both the variational and perturbative mass prescriptions seem to be consistent when going from NLO to NNLO. At the central value , both prescriptions are the same to an accuracy of 0.6% at . As a further check, we show a comparison of our NNLO HTLpt results with a threeloop calculation obtained previously using a truncated threeloop derivable approximation^{[31]} in Fig. 3. As can be seen from this figure, there is very good agreement between the NNLO derivable and HTLpt approaches even at large coupling.
3 Conclusions and outlook
In this proceedings we briefly discussed HTLpt, which is a gaugeinvariant reorganization of finite temperature perturbation theory. We presented results of a recent threeloop HTLpt calculation of the QED free energy^{[28]} and showed that the HTLpt reorganization improves the convergence of the successive approximations at large coupling, . We also compared the HTLpt threeloop result with a threeloop derivable approach^{[31]} and found agreement at the subpercentage level.
In closing, we mention that the corresponding threeloop calculation for puregauge QCD was just completed^{[32]} and the resulting thermodynamic quantities are consistent with lattice data^{[33]} down to temperatures which indicates that the lattice data at these temperatures are consistent with the picture of a plasma of weaklycoupled quasiparticles. Since HTLpt is formulated in Minkowski space, its application to realtime dynamics could be important to heavyion phenomenology.
N. S. was supported by the Frankfurt International Graduate School for Science. M. S. was supported in part by the Helmholtz International Center for FAIR Landesoffensive zur Entwicklung WissenschaftlichÖkonomischer Exzellenz program.
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