the coefficient of $I(0)$ in (\[eqn:final\]) would read \_[[M]{}[R]{}]{}\^[[P]{}[S]{}]{}(p\_1,p\_2)[S\^[(0)]{}]{}\_[[S]{}[N]{}]{}\^[[Q]{}[R]{}]{}(p\_2,p\_2) , where, as before, for theories including fermionic fields there is an additional sign given by $(-1)^{[{Q}][{R}]+[{R}][{S}]}$. Therefore, consistency between the two expressions requires the following condition on the tree-level S-matrix \[consistency\] [S\^[(0)]{}]{}\_[[M]{}[R]{}]{}\^[[S]{}[P]{}]{}(p\_1,p\_1)[S\^[(0)]{}]{}\^[[R]{}[Q]{}]{}\_[[S]{}[N]{}]{}(p\_1,p\_2)=[S\^[(0)]{}]{}\^[[P]{}[S]{}]{}\_[[M]{}[R]{}]{}(p\_1,p\_2)[S\^[(0)]{}]{}\^[[Q]{}[R]{}]{}\_[[S]{}[N]{}]{}(p\_2,p\_2) . In the case where there are also fermionic fields the consistency condition is generalized to \[consistency\_ferm\] (-1)\^[\[[P]{}\]\[[S]{}\]+\[[R]{}\]\[[S]{}\]]{}[S\^[(0)]{}]{}\_[[M]{}[R]{}]{}\^[[S]{}[P]{}]{}(p\_1,p\_1)[S\^[(0)]{}]{}\^[[R]{}[Q]{}]{}\_[[S]{}[N]{}]{}(p\_1,p\_2)= (-1)\^[\[[Q]{}\]\[[R]{}\]+\[[R]{}\]\[[S]{}\]]{} [S\^[(0)]{}]{}\^[[P]{}[S]{}]{}\_[[M]{}[R]{}]{}(p\_1,p\_2)[S\^[(0)]{}]{}\^[[Q]{}[R]{}]{}\_[[S]{}[N]{}]{}(p\_2,p\_2) . If / is satisfied, then the formula (\[eqn:final\])/ is free from ambiguities. We have checked this for the tree-level S-matrices of all the field theory models treated below.

Finally, let us comment on the issue of renormalization. It is clear that the results, and , following from our procedure are finite quantities as they only involve the scalar bubble integral in two dimensions. This follows from the discreteness of the set of zeroes of the arguments of the $\delta$-functions in -, which allows one to pull out the tree-level amplitudes from the loop integral and was used to derive . Consequently, no additional regularization is required and the result can be compared directly with the $2 \to 2$ particle S-matrix (following from the finite or renormalized four-point amplitude) found using standard perturbation theory.

Of course, this need 