Accurate solution of Bethe-Salpeter equations for tightly bound fermion-antifermion systems

Summary  The Bethe-Salpeter equation for a fermion-antifermion system for the dynamical generation of a bound state via the exchange of a pseudoscalar particle leads to a set of four coupled integral equations for the stateJ P =0− and a system of eight for 1−. It is argued that physically sensible solutions exist only after the introduction of a cut-off. These equations are solved with high accuracy by expanding the invariant amplitudes in terms of hyperspherical harmonics and then retaining the lowestn contribution only. It is found that for weakly bound systems the 1− state is always lighter than the 0− while for strong couplings the situation is reversed. Considering this solution as a relativistic quark model to form mesons as quark-antiquark bound states, the ratio 4∶1 between the masses of the vector and the pseudoscalar mesons occurs for quark masses of about two nucleon masses. For the case of scalar particles we obtain an algebraic perturbation formula which is very accurate in the limit of small exchange masses.