Effective two-body equations for the four-body problem with exact treatment of (2+2)-subsystem contributions

Abstract

Effective two-body equations for the four-body problem are derived within the general $N$-body theory of Alt, Grassberger, and Sandhas. In contrast to usual treatments, the final expressions do not require separable (2 + 2) subamplitudes but incorporate these exactly. All four-body amplitudes can be calculated from the solution of a single integral equation for the reaction $\mathrm{}(3+1)\mathrm{}\to \mathrm{}(3+1)\mathrm{}$. With single-term separable approximations for the twoparticle and the (3 + 1) subsystem amplitudes the driving terms of the final equations are seen to reduce to those of the field-theoretical model by Fonseca and Shanley. Since our results are based on an exact and complete $N$-body theory, the investigation of subsystem reaction mechanisms is facilitated. As a consequence, we are led to a three-particle propagator which has the right pole behavior and includes exchange effects.