A Convergent Reformulation of QCD Perturbation Theory

We propose a generalization of Grunberg’s method of effective charges in which, starting with the effective charge for some dimensionless QCD observable dependent on the single energy scale $Q,R(Q)$, we introduce an infinite set of auxiliary effective charges, each one describing the sub-asymptotic Q-evolution of the immediately preceding effective charge. The corresponding infinite set of coupled integrated effective charge beta-function equations may be truncated. The resulting approximations for $R(Q)$ are the convergents of a continued function. They are manifestly RS-invariant and converge to a limit equal to the Borel sum of the standard asymptotic perturbation series in ${\alpha }_s({\mu }^2)$, with remaining ambiguities due to infra-red renormalons. There are close connections with Pad{'e} approximation.