Dynamical renormalization group approach to transport in ultrarelativistic plasmas: The electrical conductivity in high temperature QED

The dc electrical conductivity of an ultrarelativistic QED plasma is studied in real time by implementing the dynamical renormalization group. The conductivity is obtained from the real-time dependence of a dissipative kernel closely related to the retarded photon polarization. Pinch singularities in the imaginary part of the polarization are manifest as secular terms that grow in time in the perturbative expansion of this kernel. The leading secular terms are studied explicitly and it is shown that they are insensitive to the anomalous damping of hard fermions as a result of a cancellation between self-energy and vertex corrections. The resummation of the secular terms via the dynamical renormalization group leads directly to a renormalization group equation in real time, which is the Boltzmann equation for the (gauge invariant) fermion distribution function. A direct correspondence between the perturbative expansion and the linearized Boltzmann equation is established, allowing a direct identification of the self-energy and vertex contributions to the collision term. We obtain a Fokker-Planck equation in momentum space that describes the dynamics of the departure from equilibrium to leading logarithmic order in the coupling. This equation determines that the transport time scale is given by $t_{\mathrm{tr}}=24\mathrm{}\pi {/e}^{4}T\ln \mathrm{}(1/e).$ The solution of the Fokker-Planck equation approaches asymptotically the steady-state solution as $\sim e^{-t/\left(4.038\mathrm{}\dots t_{\mathrm{tr}}\right)}.$ The steady-state solution leads to the conductivity $\sigma =15.698\mathrm{}{T/e}^2\ln \mathrm{}(1/e)\mathrm{}$ to leading logarithmic order. We discuss the contributions beyond leading logarithms as well as beyond the Boltzmann equation. The dynamical renormalization group provides a link between linear response in quantum field theory and kinetic theory.