Large N

TL;DR

This paper surveys the 1/N expansion in large-N quantum field theories, illustrating how planarity organizes Feynman diagrams and how the expansion can be formulated for gauge theories, lattice theories, and zero-dimensional counting. The large-N sigma model example shows the 1/N counting via an auxiliary field and yields an exact dressed propagator for the auxiliary field, highlighting how higher-point effective vertices are suppressed by powers of $1/\sqrt{N}$ when $\tilde{g}^2 = g^2 N$ is held fixed. The zero-dimensional matrix-model counting provides exact generating functions for planar diagrams, revealing the structure of planar amplitudes and their singularities, including renormalons and a Landau ghost in nonrenormalizable cases. The work also discusses planar renormalization and proposes a scheme to construct perturbative planar amplitudes that avoids ultraviolet divergences by using primary vertex functions and momentum-difference relations.

Abstract

In the first part of this lecture, the 1/N expansion technique is illustrated for the case of the large-N sigma model. In large-N gauge theories, the 1/N expansion is tantamount to sorting the Feynman diagrams according to their degree of planarity, that is, the minimal genus of the plane onto which the diagram can be mapped without any crossings. This holds both for the usual perturbative expansion with respect to powers of {tilde g}^2=g^2 N, as well as for the expansion of lattice theories in positive powers of 1/{tilde g}^2. If there were no renormalization effects, the tilde g expansion would have a finite radius of convergence. The zero-dimensional theory can be used for counting planar diagrams. It can be solved explicitly, so that the generating function for the number of diagrams with given 3-vertices and 4-vertices, can be derived exactly. This can be done for various kinds of Feynman diagrams. We end with some remarks about planar renormalization.