Effective Field Theories in the Large $N$ Limit

TL;DR

This paper explores exact solutions of certain large N effective field theories, revealing how infinite couplings cancel infinities and providing qualitative scattering predictions, especially near phase transitions.

Contribution

It demonstrates that various four-dimensional effective field theories have exact solutions in the large N limit, despite non-renormalizability, by leveraging infinite coupling types.

Findings

Exact solutions exist for large N versions of certain models.

Infinite couplings cancel infinities in non-renormalizable theories.

Near phase transitions, the large N limit simplifies diagram calculations.

Abstract

Various effective field theories in four dimensions are shown to have exact non-trivial solutions in the limit as the number $N$ of fields of some type becomes large. These include extended versions of the U(N) Gross-Neveu model, the non-linear O(N) $\sigma$-model, and the $CP^{N-1}$ model. Although these models are not renormalizable in the usual sense, the infinite number of coupling types allows a complete cancellation of infinities. These models provide qualitative predictions of the form of scattering amplitudes for arbitrary momenta, but because of the infinite number of free parameters, it is possible to derive quantitative predictions only in the limit of small momenta. For small momenta the large-$N$ limit provides only a modest simplification, removing at most a finite number of diagrams to each order in momenta, except near phase transitions, where it reduces the infinite number of diagrams that contribute for low momenta to a finite number.