We consider a scalar Euclidean QFT with interaction given by a bounded, measurable function \(V\) such that \(V^{\pm }:=\lim _{w\rightarrow \pm \infty }V(w)\) exist. We find a field renormalization such that all the n-point connected Schwinger functions for \(n\ne 2\) exist non-perturbatively in the UV limit. They coincide with the tree-level one-particle irreducible Schwinger functions of the \(\textrm{erf}(\phi /\sqrt{2})\) interaction with a coupling constant \(\frac{1}{2} (V^+ - V^-)\) . By a slight modification of our construction we can change this coupling constant to \(\frac{1}{2} (V_+ - V_-)\) , where \(V_{\pm }:= \lim _{w\rightarrow 0^{\pm }} V(w)\) . Thereby, non-Gaussianity of these latter theories is governed by a discontinuity of \(V\) at zero. The open problem of controlling also the two-point function of these QFTs is discussed.
