Particle multiplicity distribution a la invariant imbedding and natural scaling

We study the multiplicity distribution of particles produced in hard collisions on the basis of a gluon quark cascade developing through quark material (background). The process of hard collision [1] is assumed to proceed through two stages, in the first stage the collision is assumed to result in a certain number of quarks and gluons; each of these quarks and gluons so produced generate a cascade through the process of quark bremmstrahlung and gluon bremmstrahlung (single as well as double) and gluon annihilation into quark antiquark pairs, the cross sections for these processes being constants. The particle multiplicity, in the usual models in vogue is identified with the resulting gluon multiplicity. The technique of invariant-imbedding is used on the QCD parameter-space and this leads to a chain of equations (not necessarily linear in all cases) for the various probability generating functions of the multiplicity distribution. We study several models including a class of models in which 1. (i) the gluon evolution is dominant so that the contribution from cascades generated by quark antiquark pairs is negligible, and 2. (ii) the quark antiquark evolution is treated as a background effect. In the case of models of the type (i), it is shown by an appropriate use of the limiting procedure of Bellman and Harris [2] that the multiplicity obeys a gamma distribution with a scaling property characteristic of cascade processes. In the case of (ii), we show that the limit distribution by itself produces a natural scaling as the QCD parameter t becomes large. More general models are also dealt with and it is shown that the moments do possess the appropriate scaling property. In the end a short discussion is provided as KNO scaling and its possible violation. © 1991.