On Limit Cycles in Supersymmetric Theories

TL;DR

This paper proves that in unitary N=1 supersymmetric four-dimensional theories, the beta-function flow cannot form limit cycles, implying such theories are either superconformal or not scale-invariant.

Contribution

It demonstrates, to all orders in perturbation theory, that limit cycles do not occur in these supersymmetric theories, clarifying their scale and conformal invariance properties.

Findings

Limit cycles are absent in unitary N=1 supersymmetric theories.

Such theories cannot be scale-invariant without being superconformal.

Beta-function vector fields do not admit limit cycles in these theories.

Abstract

Contrary to popular belief conformality does not require zero beta functions. This follows from the work of Jack and Osborn, and examples in non-supersymmetric theories were recently found by some of us. In this note we show that such examples are absent in unitary N=1 supersymmetric four-dimensional field theories. More specifically, we show to all orders in perturbation theory that the beta-function vector field of such theories does not admit limit cycles. A corollary of our result is that unitary N=1 supersymmetric four-dimensional theories cannot be superscale-invariant without being superconformal.