On the elimination of scale ambiguities in perturbative quantum chromodynamics

We present a new method for resolving the scheme-scale ambiguity that has plagued perturbative analyses in quantum chromodynamics (QCD) and other gauge theories. For Abelian theories the method reduces to the standard criterion that only vacuum-polarization insertions contribute to the effective coupling constant. Given a scheme, our procedure automatically determines the coupling-constant scale appropriate to a particular process. This leads to a new criterion for the convergence of perturbative expansions in QCD. We examine a number of well known reactions in QCD, and find that perturbation theory converges well for all processes other than the gluonic width of the $\ensuremath{\Upsilon}$. Our analysis calls into question recent determinations of the QCD coupling constant based upon $\ensuremath{\Upsilon}$ decay.