Cluster-Decomposition Properties ofϕ3-Perturbation-Theory Amplitudes at High Energy. II

In a previous paper, to which we shall refer as I, we demonstrated that a cluster-decomposition technique similar to that used in statistical mechanics can be applied to the study of high-energy scattering processes. Specifically, we examined in detail the ladder diagrams in a ${\varphi }^3$ field theory. In the analysis of I we focused attention on the cluster-decomposition properties of the differential multiparticle-production cross sections expressed as functions of the "momentum-transfer variables," $k_i$; in terms of ladder diagrams these variables correspond to the momenta carried by the sides of the ladders. Although the $k$-variable cluster decomposition arises naturally in theoretical analysis, as we emphasized in I, a phenomenologically potentially more useful approach would apply a cluster decomposition in terms of the actual final-state particle momenta, $q_i$; these variables, of course, correspond to the momenta carried by the rungs of the ladder diagrams in the simple model. In the present paper, we investigate the validity of such a $q$-variable cluster decomposition in ${\varphi }^3$ field theories. We explore the relationship between the $k$- and $q$-variable approaches and discuss and clarify a number of subtleties involved in the introduction of $q$-variable clusters. A feature that distinguishes the $q$-variable analysis from the earlier $k$-variable analysis is that a complete cluster decomposition - that is, a decomposition in terms of all components of the momenta $q_i$ - of the differential exclusive-production cross sections is not possible even in the simple model in which these cross sections are calculated from ladder diagrams in a ${\varphi }^3$ field theory in three space and one time dimensions. We are thus led to consider the $q$-variable cluster decomposition of the partially differential cross sections obtained by integrating over the transverse-momentum components, $q_i^{\perp }$. The resulting cluster decomposition, essentially in terms of the rapidities corresponding to the longitudinal components of the $q_i$, provides a direct and intuitive framework for theoretical calculations of inclusive multiparticle spectra and avoids the ambiguities of "particle ordering" which would have hindered application of the original $k$-variable clusters to phenomenological analysis. We illustrate the utility of the cluster approach in two brief model calculations of inclusive particle spectra.