Perturbative search for dead-end CFTs

TL;DR

This paper investigates the existence of dead-end conformal field theories (CFTs) without relevant scalar deformations in various dimensions, finding potential candidates in four-dimensional chiral gauge theories that challenge previous assumptions.

Contribution

It identifies and analyzes potential dead-end CFT candidates in three and four dimensions using perturbative beta functions, suggesting their existence in four dimensions.

Findings

Pure Abelian gauge theories in 3D are candidates.

Infinite non-trivial candidates in 4D based on chiral gauge theories.

Scaling dimension gaps can be as small as 10^{-5} and are perturbatively stable.

Abstract

To explore the possibility of self-organized criticality, we look for CFTs without any relevant scalar deformations (a.k.a dead-end CFTs) within power-counting renormalizable quantum field theories with a weakly coupled Lagrangian description. In three dimensions, the only candidates are pure (Abelian) gauge theories, which may be further deformed by Chern-Simons terms. In four dimensions, we show that there are infinitely many non-trivial candidates based on chiral gauge theories. Using the three-loop beta functions, we compute the gap of scaling dimensions above the marginal value, and it can be as small as $\mathcal{O}(10^{-5})$ and robust against the perturbative corrections. These classes of candidates are very weakly coupled and our perturbative conclusion seems difficult to refute. Thus, the hypothesis that non-trivial dead-end CFTs do not exist is likely to be false in four dimensions.