$\gamma +\gamma \to 3{\pi }^0$ and ${\pi }^{+}+{\pi }^{-}+{\pi }^0$ in a Linear $\sigma$ Model

We present the low-energy limit of the one-loop contribution to $\gamma +\gamma \to 3{\pi }^0$ and ${\pi }^{+}+{\pi }^{-}+{\pi }^0$ in a linear $\sigma$ model. In this model calculation we have the Adler-Schwinger anomaly and we do not have to set all pion momenta equal to zero. Our result maintains gauge invariance and both amplitudes vanish in contradiction to the previous result of Aviv, Hari Dass, and Sawyer.