T-matrix of $Z^0$ Decay into Two Photons

Abstract

We first confirm that Nishijima's method of the $\pi^0 \rightarrow \gamma +\gamma $ calculation can precisely reproduce the observed life time of $\pi^0$ decay. Then, we calculate, for the first time, the T-matrix of the $Z^0 \rightarrow \gamma +\gamma $ process in which the vertex of the $\gamma^\mu \gamma_5$ is responsible for the decay of the weak vector boson. Even though the decay rate itself vanishes to zero due to the symmetry nature of two photons (Landau-Yang theorem), the T-matrix of the process has neither linear nor logarithmic divergences. Therefore, there is no room for the regularization of the triangle diagrams with the $\gamma^\mu\gamma^5$ vertex. Further, the T-matrices of all the triangle diagrams do not have any divergences at all, and therefore it is rigorously proved that the anomaly equation is spurious and appears only because of the improper regularization of unphysical amplitudes.