Anomalous processes at high temperature and density in a 2-dimensional linear $\sigma$ model

We use the 2-dimensional $\sigma$ model as a toy model to study the behavior of anomalous amplitudes in the limit where the constituent quark mass is small. Symmetry arguments tell that the ${\pi }^o\to \gamma$ amplitude should vanish if $m\to 0$, but we show that this conclusion is spoiled by infrared singularities. When a proper regularization (resummation of a thermal mass, for instance) is taken into account, this amplitude vanishes as expected. We also study the amplitude ${\pi }^o\sigma \to \gamma$ and show that it does not vanish in the same limit.