Determination of scalar meson masses in QCD-inspired quark models

We compare two $QCD$-inspired quark models with four-fermion interaction, without and with the remnant coupling to low-energy gluons, in the regime of dynamical chiral symmetry breaking (DCSB). The first one, the Nambu–Jona–Lasinio (NJL) model ensures the factorization of scalar and pseudoscalar meson poles in correlators, the well-known Nambu relation between the scalar meson mass and the dynamical quark mass, $m_\sigma=2m_{\mathrm{dyn}}$, and the residual chiral symmetry in coupling constants characteristic for the linear $\sigma$-model. The second one, the Gauged NJL model, happens to be qualitatively different from the NJL model, namely the Nambu relation is not valid and the factorization of light meson poles does not entail the residual chiral symmetry, i.e., it does not result in a linear $\sigma$-model. The more complicated DCSB pattern in the GNJL model is fully explained in terms of excited meson states with the same quantum numbers. The asymptotic restrictions on parameters of scalar and pseudoscalar meson states are derived from the requirement of chiral symmetry restoration at high energies.