Scale without Conformal Invariance: An Example

TL;DR

This paper provides an explicit example of a four-dimensional scale-invariant but non-conformally invariant model, and proves conditions under which scale invariance implies conformal invariance in certain quantum field theories.

Contribution

It presents a concrete model demonstrating scale without conformal invariance and establishes theoretical proofs linking scale and conformal invariance in specific models.

Findings

Constructed a unitary, finite correlator model with limit cycle RG trajectory.

Proved scale invariance implies conformal invariance to second order in D=4-epsilon.

Showed scale implies conformal invariance to all orders for models with one scalar and Weyl spinors.

Abstract

We give an explicit example of a model in D=4-epsilon space-time dimensions that is scale but not conformally invariant, is unitary, and has finite correlators. The invariance is associated with a limit cycle renormalization group (RG) trajectory. We also prove, to second order in the loop expansion, in D=4-epsilon, that scale implies conformal invariance for models of any number of real scalars. For models with one real scalar and any number of Weyl spinors we show that scale implies conformal invariance to all orders in perturbation theory.