Three-loop Φ-derivable approximation in QED -: art. no. 025011

In this paper we examine Phi-derivable approximations in QED. General theorems tell us that the gauge dependence of the n-loop Phi-derivable approximation shows up at order g(2n) where g is the coupling constant. We consider the gauge dependence of the two-loop Phi-derivable approximation to the Debye mass and show that it is of order e(4) as expected. We solve the three-loop Phi-derivable approximation in QED by expanding sum-integrals in powers of e(2) and m/T, where m is the Debye mass which satisfies a variational gap equation. The results for the pressure and the Debye mass are accurate to order e(5).