Partial-Wave Analysis of Spinor Bethe-Salpeter Equations for Single-Particle Exchange

The Bethe-Salpeter equation for spin-$\frac{1}{2}$ particles is analyzed by the use of helicity amplitudes. Decomposition into partial-wave amplitudes reduces the equation into three subsets of eight coupled integral equations, one set for singlet states, one set for triplet states ($L=J$), and one set for the coupled triplet states ($L=J\pm 1$). Both positive- and negative-energy states are included. Matrix elements are written for interaction kernels that come from single-particle exchange. The cases treated are scalar, pseudoscalar, and photon exchanges. Detailed expressions are given in the general case when all particles are off the mass shell, both for the full Bethe-Salpeter equation and for all the partial-wave equations.