We calculate the free energy of a hot gas of electrons and photons to three loops using the hard-thermal-loop perturbation theory reorganization of finite-temperature perturbation theory. We calculate the free energy through three loops by expanding in a power series in $m_D/T$, $m_f/T$, and $e^2$, where $m_D$ and $m_f$ are thermal masses and $e$ is the coupling constant. We demonstrate that the hard-thermal-loop perturbation reorganization improves the convergence of the successive approximations to the QED free energy at large coupling, $e\sim 2$. The reorganization is gauge invariant by construction, and due to cancellation among various contributions, we obtain a completely analytic result for the resummed thermodynamic potential at three loops. Finally, we compare our result with similar calculations that use the $\Phi$-derivable approach.

• Received 7 July 2009

DOI:https://doi.org/10.1103/PhysRevD.80.085015