All-orders infrared freezing of observables in perturbative QCD

Abstract

We consider a Borel sum definition of all-orders perturbation theory for Minkowskian QCD observables such as the ${R}_{{e}^{+}{e}^{\ensuremath{-}}}$ ratio, and show that both this perturbative component and the additional nonperturbative all-orders operator product expansion (OPE) component can remain separately well-defined for all values of energy $\sqrt{s},$ with the perturbative component dominating as $\stackrel{\ensuremath{\rightarrow}}{s}\ensuremath{\infty},$ and with both components contributing as $\stackrel{\ensuremath{\rightarrow}}{s}0.$ In the infrared $\stackrel{\ensuremath{\rightarrow}}{s}0$ limit the perturbative correction to the parton model result for ${R}_{{e}^{+}{e}^{\ensuremath{-}}}$ has an all-orders perturbation theory component which smoothly freezes to the value $\mathcal{R}(0)=2/b,$ where $b=(33\ensuremath{-}{2N}_{f})/6$ is the first QCD beta-function coefficient, with ${N}_{f}$ flavors of massless quark. For freezing one requires ${N}_{f}<9.$ The freezing behavior is manifested by the ``contour-improved'' or ``analytic perturbation theory'' (APT), in which an infinite subset of analytical continuation terms are resummed to all-orders. We show that for the Euclidean Adler-$D$ function, ${D(Q}^{2}),$ the perturbative component remains defined into the infrared if all the renormalon singularities are taken into account, but no analogue of the APT reorganization of perturbation theory is possible. We perform phenomenological comparisons of suitably smeared low-energy data for the ${R}_{{e}^{+}{e}^{\ensuremath{-}}}$ ratio, with the perturbative freezing predictions, and find good agreement.