Limit Cycles in Four Dimensions
  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. 
Jean-Fran\c{c}ois Fortin, Benjam\'in Grinstein, Andreas Stergiou
hep-th hep-ph