Scale without Conformal Invariance: Theoretical Foundations

TL;DR

The paper argues that scale invariance does not necessarily imply conformal invariance in four-dimensional quantum field theories, by constructing and analyzing scale-invariant trajectories with a nontrivial virial current. It develops the CCJ improved energy–momentum tensor framework to derive precise constraints linking beta functions, anomalous dimensions, and the virial current, showing that dilatation generators can generate dilatations along these trajectories via generalized dimensions. A central result is that scale-invariant trajectories correspond to recurrent RG behavior (limit cycles or ergodicity), implying RG flows are not gradient flows and challenging the strongest form of the a-theorem. The authors present a systematic perturbative approach to search for scale-invariant trajectories and discuss scheme-dependence issues, emphasizing that three-loop data may be required to decisively establish such trajectories and their properties.

Abstract

We present the theoretical underpinnings of scale without conformal invariance in quantum field theory. In light of our results the gradient-flow interpretation of renormalization-group (RG) flow is challenged, due to deep connections between scale-invariant theories and recurrent behaviors in the RG. We show that, on scale-invariant trajectories, there is a redefinition of the dilatation current that leads to generators of dilatations that generate dilatations. Finally, we develop a systematic algorithm for the search of scale-invariant trajectories in perturbation theory.