Bethe-salpeter equation with instantaneous harmonic oscillator exchange

The Bethe-Salpeter equation in the form due to Cung et al. (Ann. Phys. (N.Y.)98 (1976), 516) is investigated for the special case of instantaneous harmonic oseillator exchange, An exact reduction to a pair of coupled ordinary differential equations for the radial excitations of the ³(J ± 1)_J modes is achieved. The equations in the mass zero case are brought to a form which is quite close to Whittaker's equation. This similarity to Whittaker's equation is exploited in a computer study of the level structure as a function of the quark mass. This study covers the region from a highly relativistic spectrum depending only upon J to the nonrelativistic regime where the spectrum depends only upon L. An expression for the leptonic width of a ³S₁ state in terms of the Bethe-Salpeter wave function is derived and applied to the ψ-family. The effect of relativistic corrections is to reduce the predicted value of the leptonic width compared to the value calculated by assuming nonrelativistic kinematics. It is also shown that the relativistic treatment allows a ³D₁ state to couple directly to a virtual photon.