Effective coupling constant of plasmons

We study an ultrarelativistic QED plasma in thermal equilibrium. Plasmons-photon collective excitations-are postulated to correspond not to poles of the retarded photon propagator but to poles of the propagator multiplied by the fine-structure constant. This product is an invariant of the renormalization group that is independent of an arbitrarily chosen renormalization scale. In addition, our proposal is physically motivated since one needs to scatter a charged particle off a plasma system to probe its spectrum of collective excitations. We present a detailed calculation of the QED running coupling constant at finite temperature using the Keldysh-Schwinger representation of the real-time formalism. We discuss the issue of how to choose the renormalization scale and show that the temperature is a natural choice which prevents the breakdown of perturbation theory through the generation of potentially large logarithmic terms. Our method could be applied to anisotropic systems where the choice of the renormalization scale is less clear, and could have important consequences for the study of collective modes.