Renormalization of non-Abelian gauge theories in a background-field gauge. II. Gauge-invariant operators

Abstract

The Ward-Slavnov identities satisfied by the Green's functions with one insertion of a gauge-invariant operator are studied in the background-field gauge. As a consequence, the counterterms for a given gauge-invariant operator must satisfy a system of equations, whose general solution is found in the simplest cases of operators of low dimension ($d\ensuremath{\le}6$) or low twist ($\ensuremath{\tau}\ensuremath{\le}3$) and conjectured in the general case. It then follows that the renormalized Green's functions satisfy the same Ward identities as the bare, regularized ones. We deduce a definite prescription for the practical calculation of the anomalous dimensions of gauge-invariant operators which do not vanish in the classical limit: this prescription is formulated in the background gauge or in the usual Fermi-type gauge.