Constraints on Perturbative RG Flows in Six Dimensions

TL;DR

This paper investigates perturbative RG flows near six-dimensional CFTs, proving the $a$-theorem and scale implies conformal invariance, while identifying a new monotonic quantity differing from previous studies.

Contribution

It establishes the $a$-theorem in six dimensions using Weyl consistency conditions and clarifies the relationship between scale and conformal invariance in this context.

Findings

Proves the $a$-theorem perturbatively in six dimensions.

Shows scale invariance implies conformal invariance in this setting.

Identifies a monotonic quantity different from previous literature on $\, ext{phi}^3$ theory.

Abstract

When conformal field theories (CFTs) are perturbed by marginally relevant deformations, renormalization group (RG) flows ensue that can be studied with perturbative methods, at least as long as they remain close to the original CFT. In this work we study such RG flows in the vicinity of six-dimensional unitary CFTs. Neglecting effects of scalar operators of dimension two and four, we use Weyl consistency conditions to prove the $a$-theorem in perturbation theory, and establish that scale implies conformal invariance. We identify a quantity that monotonically decreases in the flow to the infrared due to unitarity, showing that it does not agree with the one studied recently in the literature on the six-dimensional $\phi^3$ theory.