Scale without Conformal Invariance: Theoretical Foundations

TL;DR

This paper explores the theoretical foundations of scale invariance without conformal invariance in quantum field theory, challenging existing interpretations of RG flow and providing an algorithm to identify such trajectories.

Contribution

It introduces a systematic method to find scale-invariant trajectories and redefines dilatation generators, advancing understanding of scale invariance without conformal symmetry.

Findings

Reveals deep connections between scale invariance and recurrent RG behaviors

Challenges the gradient-flow interpretation of RG flow

Provides an algorithm for locating scale-invariant trajectories in perturbation theory

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.