Anomalies in finite amplitudes: Two-dimensional single axial-vector triangle

Abstract

An explicit and detailed investigation about the two-dimensional single axial-vector ($AVV$) triangle is performed. Such a perturbative amplitude is related to the anomalous axial-vector ($AV$) one through contractions with external momenta. Given this fact, before considering such a triangle we must give a clear point of view for the $AV$ amplitude. Such a point of view is constructed within the context of an alternative strategy to handle the divergences typical of perturbative solutions of quantum field theory. In this procedure all amplitudes in all theories, formulated in odd and even space-time dimensions, renormalizable or not, are treated in an absolutely identical way. The ambiguities are automatically eliminated and the symmetry relations preserved. The well-known divergent anomalous amplitudes are correctly described, in a predictive scenario. After performing, in a very detailed way, all the calculations involved we conclude that the same phenomenon occurring in the $AV$ amplitude is also present in the finite $AVV$ triangle. The conclusion gives support to the thesis that the phenomenon is present in all pseudoamplitudes in a chain where the divergent $AV$ one is only the most simple structure. The same must occur in all even space-time dimensions. In particular, the single and triple four-dimensional box amplitudes must exhibit anomalies too. A conclusive investigation is allowed due to the special features of the adopted procedure where regularization is completely avoided and an adequate systematization for the finite parts is introduced.