On Scale and Conformal Invariance in Four Dimensions

TL;DR

This paper investigates the relationship between scale and conformal invariance in four-dimensional quantum field theories, proposing that under unitarity, scale invariance implies conformality, with some potential exceptions.

Contribution

It provides a new argument suggesting that scale invariance combined with unitarity in four dimensions generally leads to conformal invariance, linking these symmetries.

Findings

Many matrix elements vanish under scale invariance in unitary theories

Vanishing matrix elements serve as necessary conditions for conformality

Discussion of potential exceptions to the scale-conformal link

Abstract

We study the implications of scale invariance in four-dimensional quantum field theories. Imposing unitarity, we find that infinitely many matrix elements vanish in a suitable kinematical configuration. This vanishing is a nontrivial necessary condition for conformality. We provide an argument why this is expected to be a sufficient condition as well, thereby linking scale and conformal invariance in unitary theories. We also discuss possible exceptions to our argument.