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

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).