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

Abstract

A recent analysis by T\"ornqvist and Roos suggests that the \ensuremath{\sigma} meson has a mass of 860 MeV with a width of 880 MeV. In this work we calculate the properties of the \ensuremath{\sigma} meson using a generalized Nambu--Jona-Lasinio model that includes a model of confinement. We describe, in some detail, how the \ensuremath{\sigma} 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 \ensuremath{\sigma} meson by considering $t$-channel scalar-isoscalar exchange between two quarks. The resulting quark-quark $T$ matrix ${t}_{\mathrm{qq}}{(q}^{2})$ has $\mathrm{Re}{t}_{\mathrm{qq}}{(q}^{2})=0$ for ${q}^{2}=(0.823\mathrm{GeV}{)}^{2}.$ Thus, we have ${m}_{\ensuremath{\sigma}}=0.823\mathrm{GeV}.$ However, the coupling of the \ensuremath{\sigma} 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.