The Nambu–Jona-Lasinio model and its development

The historical development of the Nambu–Jona-Lasinio (NJL) model is briefly reviewed. The $\mathrm{SU}(2)\times\mathrm{SU}(2)$ and $\mathrm{U}(3)\times\mathrm{U}(3)$ local quark NJL models are considered. The mechanisms responsible for spontaneous breaking of chiral symmetry and vector dominance are exhibited. The local NJL model is adequate in describing the mass spectrum and the strong and electroweak decay modes of the four ground-state meson nonets: pseudoscalar, scalar, vector, and axial-vector. The applicability of the model to mesons in a hot dense medium is discussed. It is shown that solving problems related to the description of meson radial excitations and quark confinement requires the nonlocal extension of the NJL model. The primary emphasis of this review is on the methods that are used in various versions of the NJL model. The reader is referred to the cited works for what these models predict in low-energy hadron physics.