Constructive expansion for vector field theories I. Quartic models in low dimensions

Abstract

This paper is the first of a series aiming to use the loop vertex expansion (LVE) to recover or prove analyticity and Borel summability for generic vector models with bosonic or fermionic statistics in various dimensions. We consider both non-relativistic and relativistic bosons and fermions coupled with a constant quartic tensor in zero-, one-, and two-dimensional space by limiting our investigations to the super-renormalizable models. This offers a unified perspective on classical constructive results, highlighting the usefulness of the LVE as a modern tool to address these questions and to tackle more challenging models in higher dimensions. Finally, we investigate the large N and massless limits along with quenching for fermions in one dimension. In particular, this work establishes the Borel summability of the Sachdev–Ye–Kitaev model.