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

We use methods to show that the planar part of the renormalized perturbation theory for phi44-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 the N->infinite limit of a massive NxN matrix model with a+trgphi4 interaction, hence with ''wrong sign'' of g. Our construction is essentially equivalent to 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. (orig.)