Nambu--Jona-Lasinio Model Revisited with a Simple Regularization-Renormalization Method

TL;DR

This paper revisits the Nambu-Jona-Lasinio model using a simplified regularization-renormalization method, deriving constraints and calculating fermion and boson masses as collective modes.

Contribution

It introduces a straightforward regularization-renormalization approach to the NJL model, providing clear constraints and mass calculations for fermions and bosons.

Findings

Derived an unambiguous constraint for the model parameters.

Calculated fermion mass after vacuum phase transition.

Determined boson masses as collective fermion-antifermion modes.

Abstract

According to the pioneering model proposed by Nambu and Jona-Lasinio (NJL) \cite{2,3}, a massless fermion acquires its mass via a vacuum phase transition (VPT) process. Our discussion is considerably simplified because an effective Hamiltonian for VPT is proposed and a simple regularization-renormalization method (RRM) is adopted. An unambiguous constraint is found as $\frac{\pi^2}{G\Delta^2_1}=\frac{2}{3}$, where $G$ is the coupling constant first introduced in the NJL model while $\Delta_1$ the mass of fermion ($f$) created after the VPT. The masses of bosons with spin-parity $J^P=0^+, 0^-,1^+$ and $1^-$ as collective modes composed of fermion-antifermion pairs ($f\bar{f}$s) are also calculated by the method of random phase approximation (RPA).