Historical and phenomenological aspects

The Schrödinger equation was invented at a time when electrons, protons and neutrons were considered to be the elementary particles. It was extremely successful in what is now called atomic and molecular physics, and it has been applied with great success to baryons and mesons, especially those made of heavy quark–antiquark pairs.

While before World War II approximation methods were developed in a heuristic way, it is only during the post-war period that rigorous results on the energy levels and the wave functions have been obtained and these approximation methods justified. Impressive global results, such as the proof of the ‘stability of matter’, were obtained as well as the properties of the two-body Hamiltonians including bounds on the number of bound states. The discovery of quarkonium led to a closer examination of the problem of the order of energy levels from a rigorous point of view, and a comparison of that order with what happens in cases of accidental degeneracy such as the Coulomb and harmonic oscillator potentials. Comparison of these cases also leads to interesting results on purely angular excitations of two-body systems.

Born's prediction turned out to be true, and will remain true for atomic and molecular physics, and — as we shall see — even for particle physics.

Until 1975 the Schrödinger equation had rather little to do with modern particle physics, with a few exceptions. After November 1974, when it was understood that the J/ψ was made of heavy quark–antiquark pairs, there was a renewed interest in potential models of hadrons, which continued with the discovery of the b quark in 1977. The parallel with positronium was obvious; this is the origin of the neologism “quarkonium”. However, in contrast to positronium, which is dominated by the Coulomb potential, the potential between quarks was not known and outside explicit numerical calculations with specific models, there was a definite need for new theoretical tools to study the energy levels, partial widths, radiative transitions, etc. for large classes of potentials. This led to the discovery of a large number of completely new rigorous results on the Schrödinger equation which are interesting not only for the qualitative understanding of quarkonium and more generally hadrons but also in themselves and which can be in turn applied to other fields such as atomic physics. All this material is scattered in various physics journals, except for the Physics Reports by Quigg and Rosner on the one hand and by the present authors on the other hand, which are partly obsolete, and the review by one of us (A.M.) in the proceedings of the 1986 Schladming “Internationale Universitätswochen für Kernphysik”, to which we will refer later. There was a clear need to collect the most important exact results and present them in an orderly way.

In recent years, the study of strong interaction physics within the framework of Quantum Chromodynamics (QCD) has largely been restricted to processes which involve a single hard scale (of the order of the centre-of-mass energy). There is a whole wealth of strong interaction physics which is ignored in such a study, including the connection between QCD and Regge theory, which was successfully used to describe certain aspects of the strong interactions before the advent of QCD.

The connection between QCD and Regge theory has attracted much attention in the theoretical community for many years now. Indeed the BFKL equation, which describes what we shall refer to as the perturbative Pomeron, has been known for nearly twenty years. Only recently with the arrival of the HERA and Tevatron colliders has it been possible to perform experiments in the kinematic regime relevant to the perturbative Pomeron. Structure functions at low values of Bjorken x and the observation of rapidity gaps are examples of phenomena which can be used to test the perturbative Pomeron.

The work of those many authors who have contributed to the understanding of the Pomeron in QCD is indeed very formidable. However, to our knowledge, no single self-contained compendium of such work exists. Furthermore many of the papers which have been published on this subject have not been written in a particularly pedagogical style and are therefore not easily understood by a pedestrian reader who wishes learn about the perturbative Pomeron.

Before the advent of the field theoretic approach (QCD), a good deal of progress had already been made in developing an understanding of the scattering of strongly interacting particles. This progress was founded on some very general properties of the scattering matrix. Regge theory provided a natural framework in which to discuss the scattering of particles at high centre-of-mass energies.

With the arrival of QCD much attention was diverted away from the ‘old fashioned’ approach to the strong interactions. Interest was re-ignited within the particle physics community with the arrival of colliders capable of delivering very large centre-of-mass energies (e.g. the HERA collider at DESY and the Tevatron collider at FNAL). For the first time physicists started to investigate in earnest the properties of QCD at high energies and compare them with the predictions of the Regge theory.

The high energy limit provides the arena in which the Regge properties of QCD can be studied. It is the meeting place of the ‘old’ particle physics with the ‘new’. Since by ‘old’ we mean over 30 years ago it is necessary to commence our study of high energy scattering in QCD with an introduction to (or recap of) Regge theory. This chapter will contain a ‘whistle-stop tour’ of Regge theory and Pomeron phenomenology. We keep this to the minimum which will be required in order to follow the subsequent chapters and refer the interested reader to the literature (e.g. Collins (1977)) for further details.

Following the success of the reggeization of various different elementary particles it was hoped that a particle could be identified with the quantum numbers of the Pomeron which would reggeize to give the Pomeron trajectory.

Unfortunately this turned out not to be possible. In particular, in QCD all the elementary particles carry colour so there is no basic QCD constituent with the quantum numbers of the Pomeron. In QCD the lowest order Feynman diagram that can simulate the exchange of a Pomeron is a two-gluon exchange diagram. This led Low (1975) to use two-gluon exchange as a model for the bare Pomeron. He made numerical estimates of the amplitude for the exchange of two gluons between two hadrons using the then fashionable bag model of hadrons. This was then developed by Nussinov (1975, 1976), who considered contributions from more than two exchanged gluons as well as uncrossed ladder corrections to the two-gluon exchange amplitude.

We have already implicitly used the Low–Nussinov model in Chapter 2 to construct the Pomeron in the scalar theory model considered in that chapter. Combining this with our experience in deriving the reggeized gluon we can see what the picture of the Pomeron is in leading logarithm perturbative QCD.

The imaginary part of the amplitude for Pomeron exchange is given in terms of the multi-Regge exchange amplitude for two incoming particles (quarks for simplicity) to scatter into two quarks plus n gluons.

As we pointed out in the preceding chapter, there are several important differences between the behaviour of the perturbative QCD Pomeron which is the solution of the BFKL equation and that of the ‘soft’ Pomeron predicted by Regge theory and identified in total hadronic cross-sections and differential cross-sections at small tranverse momenta. Although one might have hoped that a purely perturbative analysis of QCD would yield results which were in qualitative agreement with the behaviour of the ‘soft’ Pomeron, it is not surprising that the results are in fact very different. Perturbative QCD theory can only be applied reliably to Green functions in which all the momenta and their scalar products are sufficiently large. In the subsequent two chapters we shall be discussing experimental situations in which such criteria are obeyed. However, total hadronic cross-sections or differential cross-sections with low momentum transfer do not obey these criteria and we must therefore expect that non-perturbative features of QCD will play a crucial role in describing such phenomena. Unfortunately a complete analysis of the non-perturbative behaviour of QCD is outside our present grasp. Nevertheless, we can investigate the ‘meeting points’ of perturbative and non-perturbative QCD in order to obtain some idea of how non-perturbative effects are likely to affect the Pomeron and to what extent we may expect to be able to reproduce the behaviour of hadronic cross-sections in QCD.

In the preceding chapter, we focused on some interesting total cross-sections. That is, we were concerned with the behaviour of the (imaginary part of the) scattering amplitudes in the forward direction (i.e. t = 0). It is now time to turn our attention to processes which involve the square of the scattering amplitude. Since in the Regge limit the centre-of-mass energy is much larger than the momentum transferred from the incoming particles to any of the outgoing particles such processes must produce a rapidity gap (see Section 1.10) in the final state.

After a brief word regarding elastic scattering at t = 0 we continue by looking at processes at large t. Of course we will find a high energy behaviour which is driven by the leading eigenvalue of the BFKL kernel. In addition, we demonstrate that large t is a good way of keeping the dynamics perturbative (recall that the impact factors were the only way to ensure this in the t = 0 case) and that the dominant contributions are characterized by the physics of diffusion in the transverse plane. After demonstrating these important points, we go on to discuss the specific example of vector meson production in two-photon collisions, i.e. γγ → VV where V denotes a vector meson.

The second part of this chapter will be concerned with the physics of diffraction dissociation. In particular, we look in some detail at the particular process of photon dissociation in deep inelastic scattering.

A particle of mass M and spin J is said to ‘reggeize’ if the amplitude, A, for a process involving the exchange in the t-channel of the quantum numbers of that particle behaves asymptotically in s as

A ∝ sα(t)

where α(t) is the trajectory and α(M2) = J, so that the particle itself lies on the trajectory.

The idea that particles should reggeize has a long history. It was first proposed by Gell-Mann et al. (1962, 1964a,b) and by Polkinghorne (1964). Mandelstam (1965) gave general conditions for reggeization to occur and this was developed by several authors (Abers & Teplitz (1967), Abers et al. (1970), Dicus & Teplitz (1971), Grisaru, Schnitzer, & Tsao (1973)). Calculations in Quantum Electrodynamics (QED) were carried out by Frolov, Gribov & Lipatov (1970, 1971) and by Cheng & Wu (1965, 1969a–c, 1970a,b), who showed that the photon had a fixed cut singularity (as opposed to a Regge pole). On the other hand McCoy & Wu (1976a–f) established that the fermion does indeed reggeize in QED. This was extended to non-abelian gauge theories by Mason (1976a,b) and Sen (1983). The demonstration of reggeization of the gluon was first shown to two-loop order by Tyburski (1976), Frankfurt & Sherman (1976), and Lipatov (1976) and to three loops by Cheng & Lo (1976). The reggeization to all orders in perturbation theory has been established by several authors using somewhat different techniques.

The preceding chapters have established the theoretical framework which ought to describe the perturbative scattering of strongly interacting particles at high centre-of-mass energies (in the Regge region). In this chapter (and the next), we shall attempt to place this framework under the experimental spotlight. That is to say, we shall turn the theoretical calculations of the preceding chapters into physical cross-sections for processes which can be measured at present or future colliders.

To construct these cross-sections, we need to specify the impact factors which define the coupling of the Pomeron to the external particles. These impact factors are then convoluted with the universal BFKL amplitude, f(ω, k1, k2, q) (see Eq.(4.33)) in order to obtain the relevant elastic-scattering amplitude. Remember that we are using perturbation theory and so can take our result seriously only if we are sure that the typical transverse momenta are much larger than ΛQCD. AS we showed in Section 5.1, for t = 0 the largeness of the typical transverse momenta is assured provided we pick processes with impact factors which are peaked at large transverse momenta. Clearly, this is not the case for proton–proton scattering and that is why we were not surprised to find that our results were incompatible with the relatively modest rise of the p–p total cross-section with increasing s. Another way of keeping our integrals away from the infra-red region is to work at high-t but we defer this topic until the next chapter.