Renormalization Group and the Anomalous Axial-Vector Ward-Takahashi Identities

Abstract

Un the basis of the anomalous Ward-Takahashi identities for the axial-vector current and homogeneous renormalization group equations in quantum electrodynamics anomalous dimensions of the axial-vector and pseudoscalar vertices are determined. As a by-product an elementary proof of the Adler-Bardeen theorem is obtained. § I. Introduction The failure of the naive Ward-Takahashi (W-T) identities for the axial-vector current and the appearance of the triangular anomalies are well-known subjects in spin or electrodynamics. P In the previous paper 2> (to be referred to as I here­ after) this problem was discussed on the basis of dispersion theory, and we have presented an unambiguous derivation of the anomalous axial-vector W-T identities. In connection with these identities the most remarkable feature consists in the absence of higher order radiative corrections to the coefficient of the anomalous term. The first observation of this feature was made by Adler and Bardeen 3> and this discovery is referred to as the Adler-Bardeen theorem. In the early stage of the investigation this statement was checked in perturbation theory. 3>. In the second stage proofs based on the Callan-Symanzik equations or the response equations have been devised, but these proofs exploit cutoff-dependent unrenor­ malized expressions.5l~n Then in order to avoid cutoff-dependent expressions a proof based on the normal-product method was presented.sJ,gJ The basic ingredients com­ mon to all these proofs are the W-T identities and the renormalization group (RG) equations. In the present paper we shall show that the use of the homogeneous RG equations!Ol is most suited for the present purpose. Namely, from the consistency between the axial-vector W-T identities and the homogeneous RG equations anom­ alous dimensions of the axial-vector and pseudoscalar vertices can be determined, and the Adler-Bardeen theorem follows immediately from these results. In § 2 the anomalous axial-vector W-T identities in the conventional form are introduced and are generalized. In § 3 the W-T identities in the generalized form are com­ bined with the homogeneous RG equations so that we can determine the anomalous dimensions of the axial-vector and pseudoscalar vertices. In § 4 an elementary proof of the Adler-Bardeen theorem is presented on the basis of the results obtained in the preceding section.