Scale Invariance + Unitarity => Conformal Invariance?

TL;DR

This paper examines whether scale invariance necessarily leads to conformal invariance in higher-dimensional unitary field theories, analyzing existing proofs, potential counterexamples, and specific models.

Contribution

It clarifies why the 2D proof does not extend to higher dimensions and shows no counterexamples among certain scalar-fermion theories in 4-epsilon dimensions.

Findings

No counterexamples in multi-field scalar-fermion models in 4-epsilon dimensions

Explanation of limitations of the 2D proof in higher dimensions

Discussion of fake counterexamples without stress tensors

Abstract

We revisit the long-standing conjecture that in unitary field theories, scale invariance implies conformality. We explain why the Zamolodchikov-Polchinski proof in D=2 does not work in higher dimensions. We speculate which new ideas might be helpful in a future proof. We also search for possible counterexamples. We consider a general multi-field scalar-fermion theory with quartic and Yukawa interactions. We show that there are no counterexamples among fixed points of such models in 4-epsilon dimensions. We also discuss fake counterexamples, which exist among theories without a stress tensor.