Scale Invariance plus Unitarity Implies Conformal Invariance in Four Dimensions

Abstract

We give a non-perturbative proof that any 4D unitary and Lorentz-invariant quantum field theory with a conserved scale current is in fact conformally invariant. We show that any scale invariant theory (unitary or not) must have either a vanishing anomaly for global scale transformations or an operator of spin 2 and dimension 2. Neither of these possibilities is allowed for unitary theories, proving the result. This is also a strong constraint on non-unitary Euclidean theories with scale but not conformal invariance, suggesting the conjecture that all such theories are free field theories.

This paper has been withdrawn by the authors. Paper withdrawn. This paper claimed to give a proof that scale invariance implies conformal invariance using only the OPE and unitarity. This cannot be correct due to a counterexample. The mistake in the present paper is the use of the OPE in momentum space, which does not hold in the cases it was used in the argument. A detailed discussion of this point can be found in arXiv:1402.3208 and arXiv:1402.6322

References in corpus (3)

• Unparticle Physics
• Comments on Unparticles
• On Scale and Conformal Invariance in Four Dimensions

Cited by in corpus (8)

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• Scale Invariance, Conformality, and Generalized Free Fields
• Dilaton Effective Action with $\mathcal{N}=1$ Supersymmetry
• Massless Wigner particles in conformal field theory are free
• Scale- versus Conformal Invariance Revisited