Noninstantaneous approximation to the Bethe-Salpeter equation

Abstract

A new relativistic three-dimensional two-body equation is presented. If the negative energy poles of the two-body propagator are neglected, this equation is exactly equivalent to the Bethe-Salpeter equation. In the derivation of this equation from the Bethe-Salpeter equation, retardation effects in the two-body interaction are treated carefully by taking fully into account contributions of the exchanged-meson singularities. The results are presented for the two cases: (a) the driving term in the Bethe-Salpeter equation is given by the relativistic one-meson exchange diagram (the ladder approximation) and (b) the driving term also includes relativistic crossed diagrams. Using these results it is shown that the Bethe-Salpeter equation in the ladder approximation goes over to the Lippmann-Schwinger equation in the nonrelativistic limit. But unlike the Blankenbecler-Sugar and Gross equations, the two-body potential is not the nonrelativistic Yukawa potential. It is given by an infinite sum of terms corresponding to instantaneous multimeson exchange contributions in the nonrelativistic two-body interaction. A method is presented which in the low energy limit greatly simplifies the calculation with crossed diagrams. The theory is applied to the problem of two scalar nucleons exchanging scalar mesons, and comparisons are made with other relativistic equations. Also a simple model calculation is performed to demonstrate the sensitivity of phenomenological meson masses and meson-nucleon coupling constants to the theoretical reduction scheme chosen.NUCLEAR STRUCTURE Three-dimensional reduction of the Bethe-Salpeter equation. Ladder approximation. Crossed diagrams. Nonrelativistic limit. Instantaneous multimeson exchange.