  We consider the detailed renormalization of two (1+1)-dimensional gauge
theories which are quantized without preserving gauge invariance: the chiral
and the "anomalous" Schwinger models. By regularizing the non-perturbative
divergences that appear in fermionic Green's functions of both models, we show
that the "tree level" photon propagator is ill-defined, thus forcing one to use
the complete photon propagator in the loop expansion of these functions. We
perform the renormalization of these divergences in both models to one loop
level, defining it in a consistent and semi-perturbative sense that we propose
in this paper.
