Anomalous Kinetics of Hard Charged Particles: Dynamical Renormalization Group Resummation

We study the kinetics of the distribution function for charged particles of hard momentum in scalar QED. The goal is to understand the effects of infrared divergences associated with the exchange of quasistatic magnetic photons in the relaxation of the distribution function. We begin by obtaining a kinetic transport equation for the distribution function for hard charged scalars in a perturbative expansion that includes hard thermal loop resummation. Solving this transport equation, the infrared divergences arising from absorption and emission of soft quasi-static magnetic photons are manifest in logarithmic secular terms. We then implement the dynamical renormalization group resummation of these secular terms in the relaxation time approximation. The distribution function (in the linearized regime) is found to approach equilibrium as $\delta n_k(t) =\delta n_k(t_o) e^{-2\alpha T (t-t_o) \ln[(t-t_o)\bar{\mu}]}$, with $\bar{\mu}\approx \omega_p$ the plasma frequency and $\alpha =e^2/4\pi$. This anomalous relaxation is recognized to be the square of the relaxation of the single particle propagator, providing a generalization of the usual relation between the damping and the interaction rate. The renormalization group approach to kinetics reveals clearly the time scale $t_{rel} \approx (\alpha T \ln[1/\alpha])^{-1}$ arising from infrared physics and hinges upon the separation of scales $t_{rel} >>\omega_p^{-1}$.