Critical dynamics of a scalar field near four spatial dimensions

A bstract The critical dynamics of a non-conserved order parameter is generally expected to become overdamped at long distances, even when propagating modes occur at microscopic or intermediate scales. We investigate the critical dynamics of a scalar field theory in thermal equilibrium which, in addition to local friction and noise, also contains a time-dependent second-order kinetic term. We show how to build a supersymmetric field-theory formulation. Using a two-loop expansion about four spatial dimensions, we show that the propagating and strictly overdamped limits share the same static Gaussian and Wilson-Fisher fixed points but realize distinct dynamical scaling regimes. The overdamped limit reproduces Model A. On the surface where local friction and noise vanish, the theory instead supports an interacting propagating fixed point whose dynamic exponent receives corrections at two loops. We demonstrate that coarse-graining does not generate a local dissipative operator on this surface, which therefore remains invariant under the RG flow. Local dissipation is nevertheless relevant at the propagating fixed point: an arbitrarily small equilibrium friction-noise perturbation drives the flow away from propagating scaling. Propagating critical dynamics thus defines a consistent but fine-tuned regime that is unstable to local equilibrium dissipation.