Fixed and unfixed points: Infrared limits in optimized QCD perturbation theory

Abstract Perturbative QCD, when optimized by the principle of minimal sensitivity at fourth order, yields finite results for R e + e − ( Q ) down to Q = 0 . For two massless flavors ( n f = 2 ) this occurs because the couplant “freezes” at a fixed-point of the optimized β function. However, for larger n f ʼs, between 6.7 and 15.2, the infrared limit arises by a novel mechanism in which the evolution of the optimized β function with energy Q is crucial. The evolving β function develops a minimum that, as Q → 0 , just touches the axis at a p (the “pinch point”), while the infrared limit of the optimized couplant is at a larger value, a ⋆ (the “unfixed point”). This phenomenon results in R approaching its infrared limit not as a power law, but as R → R ⋆ − const / | ln Q | 2 . Implications for the phase structure of QCD as a function of n f are briefly considered.