# 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.

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

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Source: https://tomesphere.com/paper/1812.11756