We show that ultraviolet divergences found in fermionic Green's functions of massless $QED_2$ have an essentially non-perturbative nature. We investigate their origin both in gauge invariant formalism (the one where we introduce Wess-Zumino fields to restore quantum gauge invariance) and in gauge non-invariant formalism, mapping two different but equivalent mechanisms responsible for their appearance. We find the same results in both approaches, what contradicts a previous work of Jian-Ge, Qing-Hai and Yao-Yang, that found no divergences in the chiral Schwinger model considered in the gauge invariant formalism.

LaTeX2e, 21 pages, 12 figures, to appear in Journal of Physics G