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

TL;DR

This paper investigates the asymptotic behavior of 4D quantum field theories, demonstrating that only conformal field theories can describe their UV and IR limits within perturbation theory, and rules out theories with scale invariance without conformality.

Contribution

It generalizes the $a$-theorem proof to constrain the asymptotics of 4D QFTs, ruling out non-conformal scale-invariant theories in UV and IR regimes.

Findings

Perturbative asymptotics must have vanishing beta functions faster than $(1/| ext{ln}\mu|)^{1/2}$.

Only conformal field theories are consistent as perturbative UV and IR limits.

Non-perturbative arguments exclude theories with scale but not conformal invariance.

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 $(1/|\ln\mu|)^{1/2}$ as $\mu \to 0$ ($\mu \to \infty$). 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.