Effective potential for composite operators and for an auxiliary scalar field in a Nambu-Jona-Lasinio model

TL;DR

This paper derives and compares effective potentials for composite operators and an auxiliary scalar field in the NJL model, demonstrating their equivalence in describing symmetry breaking and restoration at zero and finite temperature.

Contribution

It shows the equivalence of two different effective potentials in the NJL model and their relation through the Schwinger-Dyson equation, extending understanding of symmetry phenomena.

Findings

Effective potentials are equivalent when the cutoff is large enough.

Both potentials can predict symmetry breaking and restoration.

The potentials are related through the SD equation of the dynamical mass.

Abstract

We derive the effective potentials for composite operators in a Nambu-Jona-Lasinio (NJL) model at zero and finite temperature and show that in each case they are equivalent to the corresponding effective potentials based on an auxiliary scalar field. The both effective potentials could lead to the same possible spontaneous breaking and restoration of symmetries including chiral symmetry if the momentum cutoff in the loop integrals is large enough, and can be transformed to each other when the Schwinger-Dyson (SD) equation of the dynamical fermion mass from the fermion-antifermion vacuum (or thermal) condensates is used. The results also generally indicate that two effective potentials with the same single order parameter but rather different mathematical expressions can still be considered physically equivalent if the SD equation corresponding to the extreme value conditions of the two potentials have the same form.