Ward identities and some clues to the renormalization of gauge-invariant operators

Abstract

The problem of the renormalization of gauge-invariant operators in the non-Abelian Yang-Mills theory is tackled through the study of a specific example, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{F}}_{\ensuremath{\mu}\ensuremath{\nu}}^{}{}_{}{}^{2}$, for which the explicit solution can be derived from renormalization-group considerations. It is shown that the operator $\stackrel{\ensuremath{\rightarrow}}{\mathrm{F}}_{\ensuremath{\mu}\ensuremath{\nu}}^{}{}_{}{}^{2}$ mixes with non-gauge-invariant operators and that this mixing must be taken into account for the computation of the anomalous dimension of the renormalized gauge-invariant operator. The explicit solution is examined with the help of Ward identities derived from a new type of gauge transformations which appear very convenient from a technical point of view. The multiplicatively renormalizable gauge-invariant operator is shown to satisfy Ward identities and to possess an $\ensuremath{\alpha}$-independent anomalous dimension. As a by-product, we analyze the gauge dependence of the Callan-Symanzik function $\ensuremath{\beta}$.