The curious behaviour of the scale invariant $(2+1)$-dimensional Lifshitz scalar

Abstract: We demonstrate the existence of an exactly marginal deformation, with derivative coupling, about the free theory of a $(2+1)$-dimensional charged, Lifshitz scalar with dynamic critical exponent $z=4$ and particle-hole asymmetry. We show that the other classically scale invariant interactions (consistent with translational and rotational invariance) break the scale symmetry at the quantum level and find a trace identity for the stress-energy-momentum tensor complex. We conjecture the existence of bound states of $(N+1)$-particles, as a manifestation of broken scale invariance, when we turn on an attractive, classically scale invariant, polynomial interaction in charged, scalar Lifshitz field theories with dynamic critical exponent $z=2N$, $n \in \mathbb{N}$.