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

We use the methods of 1] to show that the planar part of the renormalized perturbation theory for 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 +trg ⁴ 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.