  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.
