Application of a generalized Nambu-Jona-Lasinio model to the calculation of the properties of scalar mesons and nuclear matter

In this work we demonstrate the range of application of our generalized Nambu\char21{}Jona-Lasinio (NJL) model. We discuss the scalar-isoscalar mesons that are thought to have a $q\overline{q}$ structure and also consider some properties of nuclear matter. The spectrum of scalar-isoscalar mesons provides some insight into the sigma models that may be used to describe the dynamics of pseudoscalar and scalar mesons. Some authors have suggested that the lowest $q\overline{q} {0}^{++}$ state is the ${f}_{0}(1370).$ [The ${f}_{0}(400\char21{}1200)$ that now appears in the data tables may be generated dynamically in \ensuremath{\pi}-\ensuremath{\pi} scattering and is not a ``preexisting'' state of the spectrum.] The ${f}_{0}(980)$ and ${a}_{0}(980)$ are often designated as $K\ensuremath{-}\overline{K}$ molecules. However, recent work does not support that assignment. Since there is no low-lying scalar state, the model of choice in describing pseudoscalar meson dynamics is the nonlinear sigma model. However, there are a number of applications of the Nambu\char21{}Jona-Lasinio model in the study of nuclear matter properties that can be described in terms of the linear sigma model. In the linear sigma model there is a low-mass \ensuremath{\sigma} meson, whose effects would be generated dynamically in the nonlinear model. While it is not necessary to ever introduce a low-mass \ensuremath{\sigma} meson, we describe some calculations which may be interpreted in terms of such a meson. That only represents a formal device, and no low-mass \ensuremath{\sigma} is to be found in experiment. However, the use of the low-mass \ensuremath{\sigma} as an order parameter for partial restoration of chiral symmetry at finite density is quite consistent with the picture obtained in the application of QCD sum rules to the calculation of the nucleon self-energy in nuclear matter. We discuss such sum rules to show that the linear sigma model has some limited application if we consider only spacelike values of the \ensuremath{\sigma} meson momentum. We also use our generalized NJL model to discuss the spectrum of scalar-isoscalar states under the assumption that the ${f}_{0}(1370)$ is the $1{}^{3}{P}_{0}$ $n\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ state and the ${f}_{0}(1710)$ is the $1{}^{3}{P}_{0}$ $s\overline{s}$ state before consideration of quarkonium-glueball mixing. That analysis allows us to provide a microscopic model of the two $q\overline{q}$ states used by Close and Kirk in their recent description of quarkonium-glueball mixing in which they obtain wave functions of the ${f}_{0}(1370),$ ${f}_{0}(1500),$ and ${f}_{0}(1710)$ resonances. However, we also present an argument to show that the configuration assignments implied in the work of Close and Kirk are probably incorrect and that the standard scheme for the discussion of quarkonium-glueball mixing needs modification.