Scale Invariance, Conformality, and Generalized Free Fields

TL;DR

The paper investigates whether 4D Lorentz-invariant unitary QFTs can be scale invariant without being conformal, addressing a potential loophole where the trace $T$ is a generalized free field. It shows that a nonzero global scale anomaly with coefficient $e$ forces nontrivial structure in correlators that cannot be accommodated by a generalized free field unless a dimension-2 scalar operator exists to mix with $T$. When such an operator is present, an improvement can render an improved $T$ non-generalized-free, but a complete elimination of generalized free-field possibilities remains open. Additionally, the work clarifies that large-momentum behavior does not generally follow from the leading terms of the position-space OPE, refuting a previous argument (FLP) and highlighting the subtle role of contact terms and operator dimensions in momentum space. These results strengthen the argument that scale invariance in 4D unitary theories typically implies conformal invariance, while outlining precise caveats and directions for future research in other dimensions and holographic contexts.

Abstract

This paper addresses the question of whether there are 4D Lorentz invariant unitary quantum field theories with scale invariance but not conformal invariance. An important loophole in the arguments of Luty-Polchinski-Rattazzi and Dymarsky-Komargodski-Schwimmer-Theisen is that trace of the energy-momentum tensor $T$ could be a generalized free field. In this paper we rule out this possibility. The key ingredient is the observation that a unitary theory with scale but not conformal invariance necessarily has a non-vanishing anomaly for global scale transformations. We show that this anomaly cannot be reproduced if $T$ is a generalized free field unless the theory also contains a dimension-2 scalar operator. In the special case where such an operator is present it can be used to redefine ("improve") the energy-momentum tensor, and we show that there is at least one energy-momentum tensor that is not a generalized free field. In addition, we emphasize that, in general, large momentum limits of correlation functions cannot be understood from the leading terms of the coordinate space OPE. This invalidates a recent argument by Farnsworth-Luty-Prilepina (FLP). Despite the invalidity of the general argument of FLP, some of the techniques turn out to be useful in the present context.