Limit Cycles in Four Dimensions

TL;DR

This paper demonstrates the existence of a limit cycle in a four-dimensional gauge theory, showing that beta functions can exhibit non-gradient flow behavior, proven through detailed three-loop perturbative calculations.

Contribution

It provides the first explicit example of a limit cycle in a four-dimensional gauge theory, challenging the assumption that beta functions always form gradient flows.

Findings

Existence of a limit cycle in a four-dimensional gauge theory.

Beta functions do not necessarily produce gradient flows.

Limit cycle confirmed through three-loop perturbative analysis.

Abstract

We present an example of a limit cycle, i.e., a recurrent flow-line of the beta-function vector field, in a unitary four-dimensional gauge theory. We thus prove that beta functions of four-dimensional gauge theories do not produce gradient flows. The limit cycle is established in perturbation theory with a three-loop calculation which we describe in detail.