Local operator products in gauge theories. I

Abstract

In this and a subsequent paper, we discuss the renormalization and gauge invariance of products of local gauge-invariant operators and their operator product expansions in gauge theories in detail. In this paper, we study in detail the following problem which when solved leads to the complete understanding of the operator product expansion of a product of two gauge-invariant operators. The problem is to consider the complete set of color singlet local functionals of gauge, matter, and ghost fields (F[A,c,c¯,η0]) in a non-Abelian unbroken gauge theory of strong interactions containing fermions and find what subset of these local operators have the property that all of their “physical matrix elements” are η-independent to all orders in the perturbation theory. (Note: All the phrases in quotes and the above statement are to be defined in the text by certain limiting procedures where-upon they become precise mathematical statements.) Here, η is a gauge parameter. We believe that the discussion given here can be extended to spontaneously broken gauge theories in a straightforward manner, where of course, the physical matrix elements exist in perturbation theory. The conclusion is that the only nontrivial operators that have this property to all orders are all the gauge-invariant operators and certain gauge variant operators which are a subset of the renormalization counterterms a gauge-invariant operator at zero momentum needs. However, these gauge-variant operators are not new physical entities in the sense that their “renormalized physical matrix elements” are not independent to the “S-matrix elements.” These results will be useful in understanding which operators appear in the operator product expansion of, say two currents, letting one use these methods with full understanding in gauge theories. As a by-product, certain technical discussions in this paper can be used to considerably shorten the proof of a theorem in the renormalization of gauge-invariant operators previously given.