Borel summability of $\phi^{4}_{4}$ planar theory via multiscale analysis

We review the issue of Borel summability in the framework of multiscale analysis and renormalization group, by discussing a proof of Borel summability of the $\phi^{4}_4$ massive euclidean planar theory; this result is not new, since it was obtained by Rivasseau and 't Hooft. However, the techniques that we use have already been proved effective in the analysis of various models of consended matter and field theory; therefore, we take the $\phi^{4}_4$ planar theory as a toy model for future applications.