10 - The internal-part fragmentation formulas and their relations to the unitarity equations of a field theory; Regge theory

Summary

Introduction

In this chapter we will consider the decay properties of a cluster. We start to derive some results from the internal-part formulas, Eqs. (8.41) and (8.43).

1. I1 If we sum over all available states in the decay formulas of a cluster of squared mass s we obtain asymptotically, i.e. for large values of s, the behaviour ∼ s^a. We will consider these state equations both for the case of a single species of flavor and meson and also for the case of many flavors and many hadrons in each flavor channel.

2. I2 At the same time we will derive the finite-energy version, f_s, of the fragmentation function f in Eqs. (8.16), (8.17). We will show that f_s tends rapidly towards f when s is larger than a few squared hadron masses (just as H_s → H according to the results of Chapter 9).

The method we will use is to derive a set of integral equations and then to solve them. In that way we will find that there are some necessary relationships between the parameters a, b and the normalisation parameters that constitute a set of eigenvalue equations for the integral equations.

The whole procedure is very similar to that for obtaining the unitarity conditions for the S-matrix in a quantum field theory. We will exploit these relationships by showing that the results obtained under I1 are just the same as are obtained for the multiperipheral ladder equations in a quantum field theory.