The Gauge-Independent QCD Effective Charge

Abstract

It is shown how the QED concept of a gauge-, scale- and scheme-independent one-loop effective charge can be extended directly at the diagrammatic level to QCD, thus justifying explicitly the ``naive non-abelianization'' prescription used in renormalon calculus. It is first argued that, for on-shell external fields and at the strictly one-loop level, the required gluon self-energy-like function is precisely that obtained from S-matrix elements via the pinch technique. The generalization of the pinch technique to explicitly off-shell processes is then introduced. It is shown how, as a result of a fundamental cancellation among conventional perturbation theory diagrams, encoded in the QCD Ward identities, the pinch technique one-loop gluon self-energy i\hat{\Pi}_{\mu\nu}^{ab}(q) remains gauge-independent and universal regardless of the fact that the ``external'' fields in the given process are off-shell. This demonstration involves a simple technique enabling the isolation, in an arbitrary gauge, of i\hat{\Pi}_{\mu\nu}^{ab}(q) from subclasses of up to several hundred diagrams at once. Furthermore, it is shown how this one-loop cancellation mechanism iterates for the subclasses of n-loop diagrams containing implicitly the Dyson chains of n one-loop self-energies i\hat{\Pi}_{\mu\nu}^{ab}(q). The gauge cancellation required for the Dyson summation of i\hat{\Pi}_{\mu\nu}^{ab}(q) is thus demonstrated explicitly in a general class of ghost-free gauges for all orders n.