Singlet-octet mixing of scalar mesons in a generalized Nambu–Jona-Lasinio model

We study the nonstrange scalar-isoscalar states of a generalized Nambu–Jona-Lasinio model that includes the ’t Hooft interaction and a model of confinement. We also include coupling of the $q\overline{q}$ states to the two-pion continuum in our analysis. (Our study is limited to energies $q^2<~1.8{\mathrm{GeV}}^2,$ since to go beyond $q^2=2.0\mathrm{}{\mathrm{GeV}}^2$ we need to change our method of calculation.) After introducing octet-singlet mixing we find two states, ${\sigma }_1$ and ${\sigma }_2,$ at energies of 1.00 and 1.28 GeV, respectively. The first of these is predominantly a SU(3) singlet, while the second state is rather strongly mixed. Of particular interest for nuclear physics is the behavior of the quark-quark $T$ matrix for small spacelike values of $q^2.$ Because of a strong $q^2$ dependence of the $T$ matrix at the opening of the two-pion continuum at $q^2{=4m}_{\pi }^2,$ the effective mass that parametrizes scalar-isoscalar exchange in the $T$ matrix is about 520 MeV. Thus, in a model of the nucleon-nucleon interaction, whose ingredients are a $T$ matrix for the interaction of off-mass-shell quarks and valence-quark nucleon form factors, we can provide a basis for the use of a low-mass effective σ meson in nuclear physics.