Dispersive approach to the axial anomaly and nonrenormalization theorem

Anomalous triangle graphs for the divergence of the axial-vector current are studied using the dispersive approach generalized for the case of higher orders of perturbation theory. The validity of this procedure is proved up to the two-loop level. By direct calculation in the framework of dispersive approach we have obtained that the two-loop axial-vector-vector (AVV) amplitude is equal to zero. According to the Vainshtein’s theorem, the transversal part of the anomalous triangle is not renormalized in the chiral limit. We generalize this theorem for the case of finite fermion mass in the triangle loop.

Figure 1

Effective $Z\gamma {\gamma }^{*}$ coupling contributing to $a_{\mu }^{EW}$.

Figure 2

Anomalous triangle diagram.

Figure 3

Two-loop triangle diagrams.