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

Abstract

By using a 2 → 3-body reaction as a basic process, a recurrence relation is derived for the phase-space integral $R_n$ of the reaction $p_a+p_b\to p_1+\cdots +p_n$. By iteration, $R_n$ can be written in the variables ${\hat{s}}_i={(p_1+\cdots +p_i)}^2$, $t_i={(p_a-p_1-\cdots -p_i)}^2$, and $s_i={(p_i+p_{i+1\mathrm{}})}^2$, $i=1,\cdots ,n-1\mathrm{}$. 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 ${\hat{s}}_i$ 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.