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