Kinematics of many-particle processes in invariant variables: Physical regions, phase space and all that

The kinematics of processes involving N spinless particles, in terms of invariant variables, are studied by means of Gram determinants. It is proposed that the Gram determinant method may provide a convenient unified treatment for general kinematical purposes. Using this approach, we first obtain the necessary and sufficient conditions for describing the physical regions; their solutions produce a systematic, practical scheme for constructing the physical regions explicitly. Secondly, a simple way of obtaining relations between geometrical and invariant quantities is described; some such relations are exhibited. The application of the Gram-determinant method is illustrated mainly with a discussion on the Toller angle dependence of the reggeon-reggeon-particle vertex function; in particular, we find a certain form of such dependence, in the asymptotic Regge limit, that arises naturally from the internal consistency of multi-Reggeism. Finally, the phase-space factor for many-particle productions is discussed.