Non-Relativistic Scale Anomalies

TL;DR

This paper extends the analysis of scale anomalies in non-relativistic Lifshitz theories with dynamical exponent z=2, exploring their structure in various dimensions and symmetry conditions, revealing new types of anomalies and their relation to causality.

Contribution

It provides a detailed cohomological classification of non-relativistic scale anomalies for z=2 theories, including cases with and without foliation and boost symmetries, across 1+1 and 2+1 dimensions.

Findings

No scale anomalies in 1+1 dimensions with Galilean boosts.

All 2+1 dimensional anomalies with foliation are B-type and match purely spatial Lifshitz anomalies.

Presence of an infinite ladder of B-type anomalies without foliation, with or without Galilean boosts.

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.