Ladder Amplitude and Its Cluster Expansion - A New Approach to Summing Diagrams

We use the $t$-channel ladder diagrams in a ${\phi }^3$ theory up to tenth order in coupling constants as an example to study the possible connection between fragmentation and pionization and to suggest a new approach to summing diagrams. We demonstrate that, at infinite energies, the nonleading logarithms are associated with the degrees of freedom in the longitudinal phase space of subsystems of two or more particles with small invariant mass, called the clusters, just as the leading logarithsm are associated with the degrees of freedom in the longitudinal phase space of uncorrelated single particles. The sum of nonleading logarithms associated with each cluster also exponentiates to a power of $s$. The resultant $s$ dependence of the full amplitude is the product of power dependences for individual clusters. In terms of fragmentation and pionization, we find that the ladder amplitude factors into three parts, corresponding to fragmentation of the target, pionization, and fragmentation of the projectile. Including fragmentation events only modifies the scattering amplitude by an $s$-independent factor; all the $s$ dependence is contained in the pionization part.