Static Bethe-Salpeter approach to the harmonic oscillator

An investigation of relativistic bound states in the framework of the covariant Bethe-Salpeter equation is conducted. For this end, we derive from the Bethe-Salpeter equation for the qq system, via charge conjugation, the corresponding bound state equation for the quark-pair, and formulate, four our static harmonic kernel, the coupled differential equations for the static Bethe-Salpeter amplitude, which is subsequently solved numerically for a two-body force consisting of a scalar and a fourth component vector interaction. We find that the energy spectrum for the ground and the radially excited state of massless quarks shows striking similarities to the corresponding energy spectrum of two nonrelativistic constituent quarks with masses around 300 MeV and a harmonic strength around 30 MeV/fm2, parameter set characteristic for non realistic quark models. (author)