Anomalies in finite amplitudes: Two-dimensional single and triple axial-vector triangles

Abstract

An explicit and detailed investigation about the two-dimensional (2D) single and triple axial-vector triangles is presented. Such amplitudes are related to the 2D axial-vector two-point function (AV) through contractions with the external momenta. Given this fact, before considering the triangles, we give a clear point of view for the AV anomalous amplitude. Such point of view is constructed within the context of an alternative strategy to handle the divergences typical of the perturbative solutions of quantum field theory. In the referred procedure all amplitudes in all theories, formulated in odd and even space–time dimensions, renormalizable or not, are treated on the same footing. After performing, in a very detailed way, all the calculations, we conclude that the same phenomenon occurring in the AV amplitude is present also in the finite single and triple axial-vector triangles. The conclusion gives support to the thesis that the phenomenon is present in pseudo-amplitudes belonging to a chain where the divergent AV one is only the simplest structure. It is expected that the same must occur in all even space–time dimensions. In particular, in four dimensions, the single and triple axial box amplitudes must exhibit anomalies too.