Construction and Borel summability of planar 4-dimensional euclidean field theory

Abstract

We use the methods of [1] to show that the planar part of the renormalized perturbation theory forϕ 4 4 -euclidean field theory is Borel-summable on the asymptotically free side of the theory. The Borel sum can therefore be taken as a rigorous definition of theN→∞ limit of a massiveN×N matrix model with a +trgϕ4 interaction, hence with “wrong sign” ofg. Our construction is relevant for a solution of the ultra-violet problem for planar QCD. We also propose a program for studying the structure of the “renormalons” singularities within the planar world.