Systematic $1/N$ corrections for bosonic and fermionic vector models without auxiliary fields

TL;DR

This paper develops a systematic method using bilocal fields to analyze large N vector models, providing exact effective actions, reproducing perturbative expansions, and confirming non-perturbative results including the S-matrix.

Contribution

It introduces a $1/N$ correction framework for bosonic and fermionic vector models without auxiliary fields, using bilocal variables and exact Jacobian calculations.

Findings

Exact effective action reproduces perturbative two and four point functions.

Stationary points yield large N gap equations.

Leading $1/N$ correction matches the exact S-matrix.

Abstract

In this paper, colorless bilocal fields are employed to study the large $N$ limit of both fermionic and bosonic vector models. The Jacobian associated with the change of variables from the original fields to the bilocals is computed exactly, thereby providing an exact effective action. This effective action is shown to reproduce the familiar perturbative expansion for the two and four point functions. In particular, in the case of fermionic vector models, the effective action correctly accounts for the Fermi statistics. The theory is also studied non-perturbatively. The stationary points of the effective action are shown to provide the usual large $N$ gap equations. The homogeneous equation associated with the quadratic (in the bilocals) action is simply the two particle Bethe Salpeter equation. Finally, the leading correction in $1\over N$ is shown to be in agreement with the exact $S$ matrix of the model.