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We study thermally activated, low temperature equilibrium dynamics of elasticsystems pinned by disorder using one loop functional renormalization group(FRG). Through a series of increasingly complete approximations, we investigatehow the field theory reveals the glassy nature of the dynamics, in particulardivergent barriers and barrier distributions controling the spectrum ofrelaxation times. A naive single relaxation time approximation for eachwavevector is found to be unsatisfactory. A second approximation based on arandom friction model, yields a size (L) dependent log-normal distribution ofrelaxation times (mean barriers ~L^\theta and variance ~ L^{\theta/2}) and aprocedure to estimate dynamical scaling functions. Finally, we study the fullstructure of the running dynamical effective action within the field theory. Wefind that relaxation time distributions are non-trivial (broad but notlog-normal) and encoded in a closed hierarchy of FRG equations. A thermalboundary layer ansatz (TBLA) appears as a consistent solution. It extends theone discovered in the statics which was shown to embody droplet thermalfluctuations. Although perturbative control remains a challenge, the structureof the dynamical TBLA which encodes barrier distributions opens the way fordeeper understanding of the field theory approach to glasses

Broad relaxation spectrum and the field theory of glassy dynamics for pinned elastic systems