The Auxiliary Field Method in Quantum Mechanical Four-Fermi Models -- A Study Toward Chiral Condensation in QED

TL;DR

This paper evaluates the auxiliary field method (AFM) in quantum mechanical four-fermi models, demonstrating its increasing accuracy with higher order corrections and exploring its implications for chiral symmetry breaking in QED.

Contribution

It provides a detailed validation of AFM in simple models and discusses its potential application to chiral symmetry breaking in QED.

Findings

AFM accuracy improves with higher order terms in bosonic and four-fermi models.

Higher order corrections approach the exact ground state energy in 1D four-fermi models.

One model exhibits WKB-exactness with no higher order corrections.

Abstract

A study for checking validity of the auxiliary field method (AFM) is made in quantum mechanical four-fermi models which act as a prototype of models for chiral symmetry breaking in Quantum Electrodynamics. It has been shown that AFM, defined by an insertion of Gaussian identity to path integral formulas and by the loop expansion, becomes more accurate when taking higher order terms into account under the bosonic model with a quartic coupling in 0- and 1-dimensions as well as the model with a four-fermi interaction in 0-dimension. The case is also confirmed in terms of two models with the four-fermi interaction among $N$ species in 1-dimension (the quantum mechanical four-fermi models): higher order corrections lead us toward the exact energy of the ground state. It is found that the second model belongs to a WKB-exact class that has no higher order corrections other than the lowest correction. Discussions are also made for unreliability on the continuous time representation of path integration and for a new model of QED as a suitable probe for chiral symmetry breaking.