Non-Relativistic Scale Anomalies

Abstract: We extend the cohomological analysis in arXiv:1410.5831 of anisotropic Lifshitz scale anomalies. We consider non-relativistic theories with a dynamical critical exponent $z=2$ with or without non-relativistic boosts and a particle number symmetry. We distinguish between cases depending on whether the time direction does or does not induce a foliation structure. We analyse both $1+1$ and $2+1$ spacetime dimensions. In $1+1$ dimensions we find no scale anomalies with Galilean boost symmetries. The anomalies in $2+1$ dimensions with Galilean boosts and a foliation structure are all B-type and are identical to the Lifshitz case in the purely spatial sector. With Galilean boosts and without a foliation structure we find also an A-type scale anomaly. There is an infinite ladder of B-type anomalies in the absence of a foliation structure with or without Galilean boosts. We discuss the relation between the existence of a foliation structure and the causality of the field theory.