Introduction

Though many-body final states provide the bulk of the high energy scattering cross-section, individual final states are hard to analyse. They are hard to extract experimentally because it is essential to test (using energy, momentum, and quantum-number arguments) that the final-state particles observed in the detecting apparatus were the only particles produced, and to exclude all the many other different types of events which could have occurred. In particular the production of neutral particles is especially hard to detect. And, as we have found in the previous chapter, final states are also hard to analyse theoretically both because the number of independent variables increases rapidly with the number of particles, and because only a fraction of the events occur in regions of phase space which are easy to parametrize, such as the low sub-energy resonance region, or the high sub-energy Regge region.

Because of these problems it has been found more useful to concentrate attention on so-called ‘inclusive processes’, that is, processes in which a given particle or set of particles is found to occur in the final state, but no questions are asked about all the other particles which may also be present in this final state.

Introduction

One of the most important conclusions of chapters 2 and 4 was that whenever a Regge trajectory, α(t), passes through a right-signature integral value of J - v a t-plane pole will occur in the scattering amplitude because of the vanishing of the factor sin[π(α(t) + λ′)] in (4.6.2). And, as we found in section 1.5, such poles correspond to physical particles; to a particle which is stable against strong-interaction decays if the pole occurs below the t-channel threshold, or to a resonance which can decay into other lighter hadrons if it occurs above threshold. If a given trajectory passes through several such integers it will contain several particles of increasing spin, and so it is possible to classify the observed particles and resonances into families, each family lying on a given Regge trajectory. Some examples are given in figs. 5.5 and 5.6 below.

This chapter is mainly devoted to presenting the evidence for this Regge classification, but as there will be a different trajectory for each different set of internal quantum numbers such as B, I, S, etc. it will be useful for us first to examine briefly the way in which the particles have been classified according to their internal quantum numbers using SU(3) symmetry and the quark model. Readers requiring a more complete discussion than we have space for here will find the books by Carruthers (1966), Gourdin (1967), and Kokkedee (1969) very helpful.

Introduction

In section 4.8 we demonstrated that the occurrence of Gribov–Pomeranchuk fixed poles at wrong-signature nonsense points, generated by the third double spectral function, ρsu, requires that there be cuts in the t-channel angular-momentum plane. Otherwise it is impossible to satisfy t-channel unitarity. We have also found in section 6.8 that, despite the many successes of Regge pole phenomenology, there are some features of the data that poles alone cannot explain. These are mainly failures of factorization, and it seems natural to try an invoke Regge cuts, which correspond to the exchange of two or more Reggeons and so are not expected to factorize, to make good these defects.

Unfortunately we still have a much less complete understanding of the properties of Regge cuts than of the properties of poles. On the phenomenological side, this is mainly because it is difficult to be sure whether cuts or poles are responsible for what is observed, since the main tests, logs behaviour (see (8.5.12) below) and lack of factorization, are hard to apply. Though cuts do not have to factorize some models suggest that they do, at least approximately. We shall review some of these problems in section 8.7.

Also the various theoretical models which have been used to gain insight into the behaviour of Regge poles (discussed in chapter 3) are harder to apply to cuts. For example in potential scattering, which has only elastic unitarity and no third double spectral function, there are no Regge cuts if the potentials are well behaved.

Introduction

Having identified, in the previous chapter, some of the leading Regge trajectories from the resonance spectrum, we next want to look more closely at the other main aspect of Regge theory, the way in which Regge poles in the crossed t channel control the high energy behaviour of scattering amplitudes in the direct s channel.

For spinless-particle scattering this presents few problems; we would simply use the expression (2.8.10) in the region where t is small and negative, and s is large. However, for real experiments with spinning particles it is a bit more difficult because, as we shall find in the next section, the t-channel helicity amplitudes contain various kinematical factors, and are subject to various constraints, which must also be incorporated in the Regge residues. Also we shall need to look closely at the behaviour of the residue function when a trajectory passes through the nonsense points discussed in section 4.5. Only when we have clarified these kinematical requirements can we write down correct expressions for the Regge pole contribution to a scattering amplitude based on (4.6.15).

In exploring these kinematical problems we shall discover that some of the difficulties at t = 0 may imply the occurrence of additional trajectories called ‘daughters’ and ‘conspirators’, and we shall briefly review the application of group theoretical techniques to such problems. Also we examine the way in which the internal SU(2) and SU(3) symmetries constrain Regge pole exchange models.

Introduction

So far in this book we have been solely concerned with hadronic interactions, which are the principal field in which Regge theory has been used. We have ignored electromagnetic effects in assuming that isospin is an exact symmetry of the scattering processes, and have not needed to mention the weak-interaction properties of the particles such as β-decay, etc. But of course any discussion of the electromagnetic or weak interactions of hadrons necessarily involves consideration of their hadronic properties too, because it is the strong interaction which is mainly responsible for the composite structure of the hadrons. Regge theory has played a small but not insignificant role in the development of theories of these weaker interactions, and clearly if there is to be any chance of unifying all the interactions they must be reconciled with Regge theory. In this chapter we shall look rather briefly at the problems which may arise in so doing.

Basically there are two such problems. First, weak interactions (and from now on we shall usually use the word ‘weak’ to refer to both electromagnetism and the weak interaction) are generally formulated in terms of a Lagrangian field theory for the interaction of a basic set of elementary particles. These are the leptons, l (i.e. electron e, muon μ, and neutrinos ve, vμ), photon γ, vector boson W, etc., and elementary hadrons (which at least initially do not lie on Regge trajectories but occur as Kronecker δαJ terms in the J plane).

In 1959 Regge showed that, when discussing solutions of the Schroedinger equation for non-relativistic potential scattering, it is useful to regard the angular momentum, l, as a complex variable. He proved that for a wide class of potentials the only singularities of the scattering amplitude in the complex l plane were poles, now called ‘Regge poles’. If these poles occur for positive integer values of l they correspond to bound states or resonances, and they are also important for determining certain technical aspects of the dispersion properties of the amplitudes. But it soon became clear that his methods might also be applicable in high energy elementary particle physics, and it is in fact here that the theory of the complex angular momentum plane, usually called ‘Regge theory’ for short, is now most fruitfully employed.

Apart from the leptons (electron, muon and neutrinos) and the photon, all the very large number of elementary particles which have been found, baryons and mesons, enjoy the strong interaction (i.e. the nuclear force which inter alia binds nucleons into nuclei) as well as the less forceful electromagnetic, weak and gravitational interactions. Such particles are called ‘hadrons’, from the Greek ὰδρός meaning large. Some are stable, but most are highly unstable and decay rapidly into other hadrons and leptons.

Introduction

In chapter 3 we showed how Regge trajectories could be generated by the imposition of unitarity on the basic exchange force, whether that force was a non-relativistic potential, a single-particle-exchange Feynman diagram in a field theory, or even a single Reggeon-exchange force in a bootstrap model. But the various bootstrap methods which we reviewed in section 3.5 all suffered from the very serious defect that they were limited to two-body unitarity in one channel or another. In chapters 9 and 10 we have found that Regge theory can also predict successfully the sort of behaviour to be expected in many-particle scattering amplitudes, so it is now possible to return to some of the most fundamental questions of Regge theory, such as how the Regge singularities are self-consistent under unitarity, and whether the bootstrap idea introduced in section 2.8 can be correct.

For this purpose we need models for many-particle production processes, and in the next two sections we examine two such models. One, the diffraction model, though inadequate by itself, does describe Pomeron-exchange effects and the fragmentation region, while the other, the multi-peripheral model, though applicable only in certain regions of phase space, allows one to approximate the effect of multi-Reggeon exchange. The so-called ‘two-component model’ which incorporates both these contributions seems to account quite well for the basic structure of many-particle cross-sections, if not all the details.

Introduction

In our discussion of S-matrix theory in chapter 1, and in the development of Regge theory in chapter 2, we have for simplicity ignored the possibility that the external particles entering or leaving a given process may have intrinsic spin. Only the internal Reggeons have been permitted non-zero angular momentum. Since most hadronic scattering experiments use the spin = ½ nucleon as the target, with beams of spin = 0 (π or K), spin = ½ (p, n, p, ∧ etc.) or spin = 1(γ), and since the particles produced in the final state may have any integer or halfinteger spin, it is essential to rectify this deficiency before we can confront the predictions of Regge theory with the real world.

There are three important points to bear in mind while doing this. First, an experiment may include in the initial state particles whose spin orientations have been predetermined (polarization experiments), or may involve detection of the spin direction of some of the final-state particles, by secondary scattering or by observing their subsequent decay. So there are further experimental observables (in addition to σtot and dσ/dt) which show how the scattering probability depends on these spin directions. Secondly, the dependence of the scattering process on the spin vectors means that the Lorentz invariance and crossing properties of the scattering amplitudes will generally be more complicated than those for spinless particles.

Introduction

So far we have limited our attention to four-particle scattering amplitudes (i.e. to processes of the form l + 2 → 3 + 4). These have the advantage of being kinematically rather similar to the potential-scattering amplitudes, for which the basic ideas of Regge theory were originally developed. In particular they depend on only two independent variables, s and t, and so it is a fairly straightforward matter to make analytic continuations in J and t. Also there is a wealth of two-body-final-state data with which to compare the predictions of the theory.

Though the initial state of any physical scattering process will always in practice be a two-particle state (counting bound states such as deuterons as single particles), except at very low energies particle production is always likely to occur. And as the energy increases twobody and quasi-two-body final states make up a diminishing fraction of all the events. So it is very desirable to be able to extend our understanding of Regge theory so as to obtain predictions for many-body final states. Theoretically, this is even more necessary, since models like fig. 3.3 for Regge poles or fig. 8.6 for Regge cuts demonstrate how even in 2 → 2 amplitudes Regge theory makes essential use of manybody unitarity. So if we are to have any hope of making Regge theory self-consistent (in the bootstrap sense, for example) we must be able to describe such intermediate states in terms of Regge singularities.