Reductions of the Phase-Space Integral in Terms of Simpler Processes

By using a 2 {→} 3-body reaction as a basic process, a recurrence relation is derived for the phase-space integral Rₙ of the reaction pₐ+pb→p₁+⋯+pₙ. By iteration, Rₙ can be written in the variables ^sᵢ=(p₁+⋯+pᵢ)², tᵢ=(pₐ-p₁-⋯-pᵢ)², and sᵢ=(pᵢ+pᵢ₊₁)², i=1,⋯,n-1. The kinematical boundaries are stated explicitly, and the relation of this form to the form obtained from 1 {→} 2- and 2 {→} 2-body basic processes is clarified. This expression can also be applied to the Monte Carlo generation of events and to the summing of iterative production models by integral-equation techniques. The transformation of ^sᵢ to Toller angles is carried out, but the resulting expression is seen to be impractical at finite energies. Useful geometrical interpretations of Gram determinants of four-momentum vectors are given.