Most zero-mass and high-energy cross sections are not directly calculable in perturbation theory, because of the presence of large logarithms of energy over mass (see the comments at the end of Chapter 13). Nevertheless, events with large momentum transfer are the result of violent short-distance collisions, which one can isolate quantitatively. The separation of calculable short-distance from incalculable long-distance effects is known as factorization. Deeply inelastic scattering cross sections illustrate this property. They may be used, in turn, to compute a wide class of other inclusive and semi-inclusive cross sections with large momentum transfers. The evolution in momentum transfer of deeply inelastic and related cross sections may also be determined by methods related to both the renormalization group and the parton model. The operator product expansion gives an alternate interpretation, which generalizes factorization beyond leading power behavior, at least in deeply inelastic scattering.

The examples of this chapter, in which the above results are derived and discussed, are drawn primarily from QCD, but the techniques of factorization and the operator product expansion are generally applicable in field theory. They transcend low-order calculations, by systematically organizing contributions from arbitrary orders of the perturbative expansion.

Deeply inelastic scattering

As an introduction, and to facilitate calculation, we first discuss tensor analysis and kinematics for the leptoproduction amplitude (Roy r1975, Close r1979), in which a lepton scatters from a hadron of momentum pμ to produce an arbitrary hadronic state with momentum pn.

In this appendix, we discuss three related topics: the Goldstone theorem, chiral symmetry breaking in the strong interactions, and the extraordinary application of the axial anomaly to the decay of the neutral pion. These topics draw heavily on nonperturbative reasoning, and therefore stand somewhat outside our main line of discussion. At the same time, they are so central to our understanding of the standard model as to bear at least brief description.

The Goldstone theorem

As defined in Section 5.4, a symmetry is spontaneously broken when it does not leave the vacuum state invariant. According to the Goldstone theorem, a spontaneously broken symmetry implies the existence of zero-mass scalar particles. The proof of the Goldstone theorem given by Goldstone, Salam & Weinberg (1962) is very accessible, and the reader is referred to their paper for further details. We give a variant of their argument here.

Consider a set of generators Qa, which commute with the Hamiltonian, [Qa, H] = 0, but for which Qa∣0〉 = ∣a〉 ≠ ∣0〉. That is, each Qa acting on the vacuum gives a new, nonzero state. Evidently the vacuum is not unique. We are familiar with this situation from the standard-model Higgs Lagrange density with a wrong-sign mass term, eq. (5.110).

Factorization enables us to derive inclusive high-energy cross sections directly from the perturbative expansion in terms of elementary fields. Yet, the ubiquity of bound states in hadronic physics demands that we further bridge the gap between them and elementary fields. Similarly, to describe bound states such as positronium in QED fully, it is necessary to include field-theoretic corrections.

Under some circumstances, bound state masses and matrix elements are well understood in terms of infinite sums of Feynman diagrams, or of solutions to integral equations based on perturbation theory. The perturbative expansion can suggest expressions for bound state matrix elements, even when the corresponding wave functions are unknown. Certain quantities, however, do not seem accessible to perturbation theory, even summed to all orders. Among these are the matrix elements that appear in the operator product expansion in QCD, and the masses of light hadrons.

The Bethe–Salpeter equation and wave functions

The derivations that led to the reduction formulas eqs. (2.97) and (7.88) are not limited to elementary particles. It is easy to check that any asymptotic single-particle state produces a pole in any Green function with fields of the appropriate quantum numbers. Let us see, however, how such a pole can be generated in the language of perturbation theory.

Renormalization makes it possible to compute perturbative corrections to lowest-order amplitudes and cross sections. The one-loop correction to the electron–photon vertex is an instructive example. Here we encounter a new kind of infinity, associated with very-long-wavelength photons. These ‘infrared’ divergences are well understood in quantum electrodynamics, and cancel in suitably defined cross sections. Yet another variety of on-shell infinity, the ‘collinear divergence’ arises in quantum chromodynamics (QCD), and in any other theory in which massless particles couple among themselves. Some of the resulting difficulties can be avoided by working with inclusive cross sections in the high-energy limit. The total and jet cross sections for e+e− annihilation into hadrons afford a wide range of experimental tests of QCD.

One-loop corrections in QED

Tensor structure and form factors

The fermion–photon vertex may describe the scattering of an electron or positron, or the annihilation or creation of a pair. We pick the scattering of an on-shell electron. The corresponding matrix element is where 〈(p2, σ2)(−)∣jμ(0)∣(p1, σ1)(−)〉, where jμ is the electromagnetic current. We shall use the notation ui for spinors u(pi, σi) below.

The search for the underlying structure of physical reality is as old as speculative thought. Our deepest experimental insights to date are expressed in the language of quantum field theory, in terms of particles that interact at points in space–time, subject to the constraints of special relativity. The theoretical developments that lead to this portrait are the subject of this book. Its aim is to provide a self-contained introduction to relativistic quantum field theory and its applications to high-energy scattering. Some of the methods described predate quantum theory, while others are quite recent. What makes them vital is not only their considerable success thus far, but also the very limitations of that success.

There is every reason to believe that quantum field theory is not a closed chapter. A great deal of freedom remains in the choice of particles and their interactions within the field-theoretic description of fundamental processes. The ‘standard model’, which describes elementary processes as they are known at this time, is a grab bag of matter and forces, in which breathtaking theoretical elegance coexists with seemingly senseless arbitrariness. Whatever the next step in our understanding of elementary processes, however, the elements of quantum field theory will remain relevant to their description.

Chapter 9 has left us in an awkward position with regard to quantum corrections. Ultraviolet divergent diagrams can be computed only by continuing to unphysical numbers of dimensions (or otherwise regularizing). In this chapter, we develop the process of renormalization, which will allow us to return many theories to physical dimensions.

It is possible to make the perturbation series of many theories finite, at the price of making certain parameters in their Lagrangians infinite. From one point of view, renormalization provides a positivistic morality play, in which, by renouncing our ability to calculate a few unobservable quantities (‘bare’ quantities below), we gain many predictions relating observable quantities. It may be unsettling, however, to deal with a Lagrange density that becomes infinite when the regularization is removed, even if it does give a finite perturbation series. Thus the perennial conjecture arises, that the quantum fields in nature are low-energy manifestations of an underlying finite theory. The most recent, and most promising, candidate is string theory (Green, Schwarz & Witten r1987). Here we shall simply examine renormalization as a self-consistent procedure, without asking for the ultimate origin of renormalizable theories.

We begin our discussion with the simplest case in which renormalization is necessary, ϕ3 theory in four dimensions (ϕ34).

In the preceding chapters, much of the basic accelerator theory has been introduced and the ideas were so fundamental that they were often equally applicable to transfer lines and linear accelerators, as well as to circular machines, the central theme of this book. The present chapter will depart from this pattern and look briefly at some special aspects of circular colliders. This review will be rather superficial, but its aim is to make the reader aware of the problems and to provide references rather than to lay out detailed derivations of formulae.

From the introductory chapter, it is evident that high-energy physics was profoundly influenced by the inventions that made colliding beams feasible experimental tools. This did not happen suddenly. The main ideas on how to accumulate sufficiently intense beams to achieve significant interaction rates originated in the MURA Group in 1956, but considerable scepticism on the part of the physics community had to be overcome through practical demonstrations before the consensus of opinion turned in favour of colliders.

For electrons, many facilities were built and operated, but the highlights of this development were SPEAR and DORIS. Following their outstanding physics discoveries in the 1970s, all later proposals for electron machines were for colliders. LEP at CERN is the most recent and will probably remain the largest circular collider to be built for electrons.