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 R(e+e-)(Q) down to Q=0. For two massless flavours (n_f=2) this occurs because the couplant "freezes" at a fixed point of the optimized beta function. However, for larger n_f's, between 6.7 and 15.2, the infrared limit arises by a novel mechanism in which the evolution of the optimized beta function with energy Q is crucial. The evolving beta 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^star (the "unfixed point"). This phenomenon results in R approaching its infrared limit not as a power law, but as R -> R^star-const./|ln Q|^2. Implications for the phase structure of QCD as a function of n_f are briefly considered.