Pion observables and the notion of constituent quarks

The Dyson Schwinger equation for the quark self energy is employed to investigate the singularity structure of the quark propagator. An effective gluon two-point function, which is free of singularities and consistent with the known asymptotic form, is used to represent the quark-quark interaction. The parameters introduced by the gluon two-point function are constrained by pion observables through the equivalence of the Dyson Schwinger and pion Bethe-Salpeter equations for on-shell pions (a result of Goldstone's theorem). With the parameters fixed, the Dyson-Schwinger equation is used to investigate the singularity structure of the quark propagator through continuation into the complex plane. The quark appears as a broad resonance centered at the expected constituent quark mass. Comments are made on the relevance of this result to confinement and the relativistic investigation of meson spectroscopy