Comments on scale and conformal invariance in four dimensions

TL;DR

This work analyzes the question of whether four-dimensional unitary scale-invariant quantum field theories are necessarily conformal by inspecting the scale anomalies in correlators of the trace $T$ of the stress-energy tensor. Using a Weyl/Wess–Zumino framework with sources for the metric, virial current, and relevant scalars, the authors derive the most general scale-violating structures for 2-, 3-, and 4-point functions, including nontrivial semi-local contributions that can account for the observed anomalies. A central result is that the anomaly coefficient $e_{TT}$ governs the possibility of enhanced conformal invariance, with $e_{TT}=0$ implying $T=0$ and hence conformality; yet semi-local terms can reconcile certain 3- and 4-point anomalies without forcing $e_{TT}$ to vanish. Consequently, while the theory is conformal precisely when $e_{TT}=0$, the paper finds no counterexample to conformality but acknowledges that a complete proof requires further work, given the subtleties of semi-local contributions and large-momentum/OPE considerations in the four-dimensional setting.

Abstract

There has been recent interest in the question of whether four dimensional scale invariant unitary quantum field theories are actually conformally invariant. In this note we present a complete analysis of possible scale anomalies in correlation functions of the trace of the stress-energy tensor in such theories. We find that 2-, 3- and 4-point functions have a non-trivial anomaly while connected higher point functions are non-anomalous. We pay special attention to semi-local contributions to correlators (terms with support on a set containing both coincident and separated points) and show that the anomalies in 3- and 4-point functions can be accounted for by such contributions. We discuss the implications of the our results for the question of scale versus conformal invariance.