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 behaviorof 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 infraredsingularities. When a proper regularization (resummation of a thermal mass, forinstance) is taken into account, this amplitude vanishes as expected. We alsostudy the amplitude $\pi^o\sigma\to\gamma$ and show that it does not vanish inthe same limit.