Dynamical Renormalization Group Approach to Quantum Kinetics in Scalar and Gauge Theories

We derive quantum kinetic equations from a quantum field theory implementing a diagrammatic perturbative expansion improved by a resummation via the dynamical renormalization group. The method begins by obtaining the equation of motion of the distribution function in perturbation theory. The solution of this equation of motion reveals secular terms that grow in time, the dynamical renormalization group resums these secular terms in real time and leads directly to the quantum kinetic equation. This method allows to include consistently medium effects via resummations akin to hard thermal loops but away from equilibrium. A close relationship between this approach and the renormalization group in Euclidean field theory is established. In particular, coarse graining, stationary solutions, relaxation time approximation and relaxation rates have a natural parallel as irrelevant operators, fixed points, linearization and stability exponents in the Euclidean renormalization group, respectively. We used this method to study the relaxation in a cool gas of pions and sigma mesons in the O(4) chiral linear sigma model. We obtain in relaxation time approximation the pion and sigma meson relaxation rates. We also find that in large momentum limit emission and absorption of massless pions result in threshold infrared divergence in sigma meson relaxation rate and lead to a crossover behavior in relaxation. We then study the relaxation of charged quasiparticles in scalar electrodynamics (SQED). We begin with a gauge invariant description of the distribution function and implement the hard thermal loop resummation for longitudinal and transverse photons as well as for the scalars. While longitudinal, Debye screened photons lead to purely exponential relaxation, transverse photons, only dynamically screened by Landau damping lead to anomalous (non-exponential) relaxation, thus leading to a crossover between two different relaxational regimes. We emphasize that infrared divergent damping rates are indicative of non-exponential relaxation and the dynamical renormalization group reveals the correct relaxation directly in real time. Furthermore the relaxational time scales for charged quasiparticles are similar to those found in QCD in a self-consistent HTL resummation. Finally we also show that this method provides a natural framework to interpret and resolve the issue of pinch singularities out of equilibrium and establish a direct correspondence between pinch singularities and secular terms in time-dependent perturbation theory. We argue that this method is particularly well suited to study quantum kinetics and transport in gauge theories.