The two-body problem in the extreme relativistic limit

Abstract

In view of the composite model for elementary particles the two-body problem is studied in the extreme relativistic limit. Spinless particle-antiparticle bound states are investigated for a single massless self-interacting Weyl field. High momenta are suppressed in a Lorentz invariant manner by introducing a heavy auxiliary boson. A Bethe-Salpeter equation is set up in the ladder approximation. The consequences of the continuity equation are studied in the same approximation and are shown to be fulfilled automatically only for scalar bound states. A new method is presented for solving the Bethe-Salpeter equation. A spectral ansatz is made for the wave function which leads to a one-dimensional integral equation for the weight function when the rest mass of the bound state is neglected. This integral equation is solved in the Fredholm approximation for a somewhat modified cut-off prescription in the scalar case. With the same restriction, the eigenvalue is found also in the pseudoscalar case. It is shown that the method of solution can be extended to any order in the rest mass of the bound state.