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

TL;DR

This work develops an exact effective action for bosonic and fermionic U(N) vector models in the large $N$ limit by rewriting the theory in terms of colorless unequal-time bilocal fields. The exact Jacobian for the bilocal change of variables is computed, and the resulting $S_{eff}$ reproduces the perturbative two- and four-point functions, while naturally incorporating Fermi statistics for fermions. Nonperturbatively, the stationary points yield the standard large $N$ gap equations, and the homogeneous quadratic sector reduces to a two-particle Bethe–Salpeter equation; the leading $1/N$ corrections are in agreement with the exact $S$-matrix. The framework thus provides a nonlocal bosonization valid in any dimension and a systematic route to compute subleading $1/N$ corrections for both bosonic and fermionic vector models.

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.