Anomalous Scaling at Non-Thermal Fixed Points of the Sine-Gordon Model

We extend the theory of non-thermal fixed points to the case of anomalously slow universal scaling dynamics according to the sine-Gordon model. This entails the derivation of a kinetic equation for the momentum occupancy of the scalar field from a non-perturbative two-particle irreducible effective action, which re-sums a series of closed loop chains akin to a large-$N$ expansion at next-to-leading order. The resulting kinetic equation is analyzed for possible scaling solutions in space and time that are characterized by a set of universal scaling exponents and encode self-similar transport to low momenta. Assuming the momentum occupancy distribution to exhibit a scaling form we can determine the exponents by identifying the dominating contributions to the scattering integral and power counting. If the field exhibits strong variations across many wells of the cosine potential, the scattering integral is dominated by the scattering of many quasiparticles such that the momentum of each single participating mode is only weakly constrained. Remarkably, in this case, in contrast to wave turbulent cascades, which correspond to local transport in momentum space, our results suggest that kinetic scattering here is dominated by rather non-local processes corresponding to a spatial containment in position space. The corresponding universal correlation functions in momentum and position space corroborate this conclusion. Numerical simulations performed in accompanying work yield scaling properties close to the ones predicted here.