The $a$-theorem and the Asymptotics of 4D Quantum Field Theory

TL;DR

The paper extends the a-theorem framework to constrain 4D RG flows, showing that perturbative UV and IR asymptotics must be conformal by linking the dilaton-dilaton amplitude to the a-anomaly and employing dispersion relations. It develops a dilaton EFT, analyzes operators of dimension ≤2, and uses higher partial waves and Wess–Zumino consistency to prove that beta-function coefficients vanish in the UV/IR, forcing CFT behavior in perturbative limits. A nonperturbative argument further excludes scale without conformal invariance under a technical assumption, and gravitational anomalies do not invalidate the conclusions. These results substantially limit exotic RG flows and highlight a deep irreversibility toward conformal fixed points in 4D quantum field theories, with connections to holography and potential extensions to nonperturbative regimes.

Abstract

We study the possible IR and UV asymptotics of 4D Lorentz invariant unitary quantum field theory. Our main tool is a generalization of the Komargodski-Schwimmer proof for the $a$-theorem. We use this to rule out a large class of renormalization group flows that do not asymptote to conformal field theories in the UV and IR. We show that if the IR (UV) asymptotics is described by perturbation theory, all beta functions must vanish faster than $(1/|\lnμ|)^{1/2}$ as $μ\to 0$ ($μ\to \infty$). This implies that the only possible asymptotics within perturbation theory is conformal field theory. In particular, it rules out perturbative theories with scale but not conformal invariance, which are equivalent to theories with renormalization group pseudocycles. Our arguments hold even for theories with gravitational anomalies. We also give a non-perturbative argument that excludes theories with scale but not conformal invariance. This argument holds for theories in which the stress-energy tensor is sufficiently nontrivial in a technical sense that we make precise.

Figures (2)

• Figure 1: A priori possible IR behavior of renormalization group flows.
• Figure 2: The integration contour in the complex $s$ plane used to argue that $\al > 0$.