ultraviolet cut-off is removed $N\to\io$; on the other hand for any finite cut-off non vanishing corrections are expected.

It is indeed useful to compare the present result to the analogous computation for the Thirring model with non local interaction, that is neglecting the bosons; in such a case the Schwinger-Dyson equation is still given by but in the r.h.s. $\sum_{\a=\phi,\psi}
\hat\r_{\a,-\o,\a}(\pp)$ should be replaced by $\hat\r_{\psi,-\o}(\pp)$; by using the WI , (32) one would get an extra term in function of the chiral anomaly. As a result, one would find that the asymptotic behavior of $<\hat\Psi^-_{\psi,\o,\kk}\hat\Psi^+_{\psi,\o,\kk}>_{T,0,N}$ is [*different*]{} with respect to the non interacting case; for small $\kk$ $<\hat\Psi^-_{\psi,\o,\kk}
\hat\Psi^+_{\psi,\o,\kk}>_{T,0,N}$ would behave as $|\kk|^{-1+\h}$ with $\h>0$. In the present case, instead, the cancellation of the anomalies due to the supersymmetry has the effect that the asymptotic behavior of the two point function is equal to the free one, up to a small correction vanishing when the cut-off is removed.

The average of the product
--------------------------

Starting from for $E=0$, and using the notation (with $P(d\Psi)$ replaced by $P(d\Psi_a) P(d\Psi_b)$) we can write K\_[ł,0]{}()= &lt;\^-\_[a,ø,]{}\^+\_[a,ø,0]{}&gt;\_[T,0,N]{} &lt;\^-\_[b,ø,]{}\^+\_[b,ø,0]{}&gt;\_[T,0,N]{}+&lt;\^-\_[a,ø,]{}\^+\_[a,ø,0]{} \^-\_[b,ø,]{}\^+\_[b,ø,0]{}&gt;\_[T,0,N]{}\[ma1\] The computation of $<\psi^-_{a,\o,\xx}\psi^+_{a,\o,\yy}>_{T,0,N}$ can be done exactly as in the previous case. The 