Scale Invariance + Unitarity => Conformal Invariance?

TL;DR

The paper investigates whether unitary scale-invariant quantum field theories in dimensions greater than two are necessarily conformally invariant. It revisits the D=2 proof that uses the stress-tensor two-point function and explains why that argument does not automatically extend to D≥3, highlighting the potential role of stress-tensor Ward identities. Through a thorough analysis of a multi-field scalar-fermion model with quartic and Yukawa interactions, it shows that at one-loop in the 4−ε expansion there are no scale-invariant but non-conformal fixed points, including a diagrammatic proof that potential Kμ candidates must vanish. The authors also discuss fake counterexamples that lack a stress tensor and emphasize that any genuine counterexample would require a different structural mechanism beyond the analyzed framework. Overall, the work narrows the possibility of counterexamples at weak coupling and sketches directions for a deeper, non-perturbative resolution with potential implications for phenomenology if scale vs. conformal invariance diverges in higher dimensions.

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.