Scale without Conformal Invariance at Three Loops

TL;DR

This paper demonstrates through a three-loop calculation that scale invariance can exist without conformal invariance in certain quantum field theories, analyzing scheme effects and stability of scale-invariant trajectories.

Contribution

It provides the first three-loop evidence of scale without conformal invariance in dimensional regularization and explores the stability and scheme dependence of such trajectories.

Findings

Existence of scale without conformal invariance at three loops.

Scheme dependence affects the properties of scale-invariant trajectories.

Stability analysis reveals both attractive and repulsive directions in specific models.

Abstract

We carry out a three-loop computation that establishes the existence of scale without conformal invariance in dimensional regularization with the MS scheme in d=4-epsilon spacetime dimensions. We also comment on the effects of scheme changes in theories with many couplings, as well as in theories that live on non-conformal scale-invariant renormalization group trajectories. Stability properties of such trajectories are analyzed, revealing both attractive and repulsive directions in a specific example. We explain how our results are in accord with those of Jack & Osborn on a c-theorem in d=4 (and d=4-epsilon) dimensions. Finally, we point out that limit cycles with turning points are unlike limit cycles with continuous scale invariance.