"Exact" relativistic theory of two-body bound-state wave functions

Abstract

An exact three-dimensional reduction of the Bethe-Salpeter equation is presented for the case of two spin-(1/2) fermions exchanging scalar and pseudoscalar bosons. The resulting three-dimensional integral equation involves only a truncated basis, and once it is solved to get an auxiliary scattering amplitude T^~, the fully off-shell Bethe-Salpeter amplitude T is given essentially as an integral over T^~, thereby considerably simplifying its calculation. Similar remarks apply to bound-state equations as well. Partial-wave decomposition of the latter equations is carried out leading to Schrodinger-type equations for two-body relativistic wave functions. By means of a simple order-of-magnitude estimate we show that in the deuteron wave functions the admixture of virtual states with both nucleons in negative-energy states is more likely than that of the states with one nucleon in a positive-energy and the other in a negative-energy state, provided the coupling is pseudoscalar-pseudoscalar. This effect has not received sufficient attention in the literature. Another interesting result is that if the effective interaction in our equations is calculated to the lowest order in the coupling constant, only six out of the eight components of the deuteron wave functions are nonzero, the other two appear only in the higher order.