Exact Schwinger functions for a class of bounded interactions in $d\ge 2$

Abstract

We consider a scalar Euclidean QFT with interaction given by a bounded, measurable function $V$ such that $V^{\pm }:={\lim }_{w\to \pm \infty }V(w)$ 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 $\mathrm{erf}(\varphi /\sqrt{2})$ 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 $V_{\pm }:={\lim }_{w\to 0^{\pm }}V(w)$. 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.

14 pages. To appear in Commun. Math. Phys