Calculation of the properties of the σ meson in a generalized Nambu–Jona-Lasinio model with Lorentz-vector confinement

A recent analysis by Törnqvist and Roos suggests that the σ meson has a mass of 860 MeV with a width of 880 MeV. In this work we calculate the properties of the σ meson using a generalized Nambu–Jona-Lasinio model that includes a model of confinement. We describe, in some detail, how the σ coupling to states in the two-pion continuum may be calculated, when using a Lorentz-vector confining interaction. As part of our work we provide a general procedure for calculating various loop diagrams in Minkowski momentum space for quarks in the presence of the confining interaction. We study the properties of the σ meson by considering $t$-channel scalar-isoscalar exchange between two quarks. The resulting quark-quark $T$ matrix $t_{\mathrm{qq}}{(q}^2)$ has $\mathrm{Re}{t}_{\mathrm{qq}}{(q}^2)=0\mathrm{}$ for $q^2=(0.823\mathrm{}\mathrm{GeV}{)}^2.$ Thus, we have $m_{\sigma }=0.823\mathrm{}\mathrm{GeV}.$ However, the coupling of the σ to the two-pion states is so large as to make $|t_{\mathrm{qq}}{(q}^2)|$ a rather smooth function over a broad range of $q^2.$ Therefore, we do not attempt to assign a width for the resonance.