  We present a study of the mechanism of anomaly cancellation using only
transverse invariant amplitudes on anomaly diagrams at higher perturbative
orders. The method is the realization of the Green-Schwarz (GS) mechanism at
field theory level, which restores the Ward identities by a subtraction of the
anomaly pole. Some of the properties of the GS vertex are analyzed both in the
context of unitarity and of the organization of the related perturbative
expansion. We investigate the role played by the GS and the Wess-Zumino
vertices in the anomalous magnetic moment of the muon and in the hyperfine
splitting of muonium, which are processes that can be accompanied by the
exchange of a virtual anomalous extra Z-prime and an axion-like particle.
