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

Perturbative QCD, when optimized by the principle of minimal sensitivity at fourth order, yields finite results for Re+e−(Q) down to Q=0. For two massless flavors (nf=2) this occurs because the couplant “freezes” at a fixed-point of the optimized β function. However, for larger nfʼs, 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 ap (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/|lnQ|2. Implications for the phase structure of QCD as a function of nf are briefly considered.