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

TL;DR

This paper explores the quantum behavior of a $(2+1)$-dimensional Lifshitz scalar field, revealing an exactly marginal deformation and the breaking of classical scale invariance at the quantum level, with implications for bound state formation.

Contribution

It identifies an exactly marginal deformation in a charged Lifshitz scalar theory and analyzes how other scale-invariant interactions break scale symmetry quantum mechanically.

Findings

Existence of an exactly marginal deformation with derivative coupling.

Quantum breaking of classical scale invariance for certain interactions.

Conjecture of bound states arising from broken scale invariance.

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 $z=2N$, $n \in \mathbb{N}$.