Three-loop hard-thermal-loop free energy for QED

Abstract

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, $e\ensuremath{\sim}2$. 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 $\ensuremath{\Phi}$-derivable approach.