Coupled-channel dynamics in the Nambu–Jona-Lasinio model

We study the scalar-isoscalar sector of the Nambu–Jona-Lasinio (NJL) model and extend the model to include a description of the coupling of the quark-antiquark states to the two-pion continuum. The qq¯ interaction gives rise to a sigma meson, which takes on a width and energy shift that depends upon the strength of the coupling for q+q¯→π+π. (For weak channel coupling, the resonance is located at the mass of the sigma, ${\mathit{m}}_{\mathrm{\sigma }}$≃2${\mathit{m}}_{\mathit{q}}^{\mathrm{cons}}$, where ${\mathit{m}}_{\mathit{q}}^{\mathrm{cons}}$ is the constituent quark mass of the NJL model.) We consider two models for the qq¯→ππ coupling. In the first model, we find a low-energy resonance, with the resonance energy ${\mathit{E}}_{\mathit{R}}$≤2${\mathit{m}}_{\mathit{q}}^{\mathrm{cons}}$. We then see that the values, obtained from the analysis of experimental data, of the scalar-isoscalar phase shift describing ππ scattering ${\mathrm{\delta }}_0^0$, are not compatible with the existence of a low-mass sigma. In the second model, the resonance is pushed upward into the region of the two-quark continuum, ${\mathit{E}}_{\mathit{R}}$>2${\mathit{m}}_{\mathit{q}}^{\mathrm{cons}}$. This second model provides an example of a phenomenon where the behavior of the qq¯ T matrix is parametrized for ${\mathit{q}}^2$≤0 by a mass that is smaller than the physical mass that characterizes the pole in the T matrix. The behavior of the second model suggests how the absence of experimental evidence for a low-mass sigma may be reconciled with the importance of such a meson in nuclear structure studies.