Saddle Points in the Auxiliary Field Method

TL;DR

This paper compares two saddle point calculation methods within the auxiliary field approach, revealing their differences and similarities in bosonic and fermionic integrals, especially under weak coupling conditions.

Contribution

It introduces and compares two saddle point calculation methods, providing a detailed analysis and formalism applicable to Gamma functions and Grassmann integrals.

Findings

Methods coincide in bosonic case

Method(II) deviates in fermionic weak coupling region

Formalism allows high-order calculations up to O(1/N^{14})

Abstract

Investigations are made on the saddle point calculations (SPC) under the auxiliary field method in path integrations. Two different ways of SPC are considered, Method(I) and Method(II), to be checked in an integral representation of the Gamma function, \Gamma (N), as a bosonic example and in a four-fermi type of Grassmann integral where one "fermion mass" \omega_0 differs from the other N-degenerate species. The recipe of Method(I) seems rather complicated than that of (II) superficially, but the case turns out to be opposite in the actual situation. A general formalism allows us to calculate for \Gamma (N) up to O(1/N^{14}). It is found that both happen to coincide in the bosonic case but in the fermionic case Method(II) shows a huge deviation in the weak coupling region where \omega_0 \ll 1.