The operator product expansion plus renormalization group result (22.2.34) tells us how QCD controls the Q2 variation of the moments of the deep inelastic structure functions. But it does not give us the actual value of the moments, since they depend upon unknown, non-calculable, hadronic matrix elements ON,j of certain operators. In Section 22.2.5 we saw that the moment equations can be replaced by an equation controlling the Q2 variation of the structure functions themselves, and this could be interpreted [see (22.2.53)] as a Q2 variation of the parton densities. Again the equation does not give us the actual value of the parton distribution—only their Q2 evolution is calculable. Thus the rôle of the unknown ON,j in the moment equation is taken by the unknown in the evolution equation.

It should be clear that all the difficulty stems from the hadrons. They are a non-perturbative manifestation of QCD and the problem is to derive some consequences of QCD without being able to handle the genuinely non-perturbative aspect. One is seeking a blend of the perturbative and the non-perturbative and the boundary between them is subtle. If individual hadrons are not involved, for example, in the totally inclusive reaction e+e− → hadrons, we can use purely perturbative QCD and end up with a genuine calculation of the cross-section to some order in αs, with no unknown constants or functions appearing. This can be seen in (22.1.22).

In the present chapter we develop a well defined calculational scheme for handling reactions involving individual hadrons—the QCD-improved parton model.

In the previous chapter we studied the general ideas of renormalization of a field theory, in particular, the freedom in the choice of a renormalization scheme and the consequences thereof as embodied in the renormalization group. For simplicity we talked mainly in terms of scalar φ4 theory. But we did illustrate the very important property of asymptotic freedom that emerges when these results are used in QCD.

In this chapter we show how to extend these ideas to the realistic case of gauge theories, and especially to QCD. We begin with a general outline of gauge theories and point out some of their subtleties, highlighting differences between QCD and QED. We then extend the renormalization group results of Chapter 20 to the case of QCD.

Introduction

In earlier chapters, and in those to follow, we constantly quote QCD corrections to naive quark–parton model estimates in various processes. It is felt at present that QCD is a serious candidate for the theory of strong interactions. QCD has many beautiful properties. It is a non-Abelian gauge theory describing the interaction of massless spin ½ objects, the ‘quarks’, which possess an internal degree of freedom called colour, and a set of massless gauge bosons (vector mesons), the ‘gluons’ which mediate the force between quarks in much the same way that photons do in QED. Loosely speaking, the quarks come in three colours and the gluons in eight.

As with all theories of strong interactions things look simple to begin with, but as experimental accuracy improves models are forced to become increasingly complicated, with the continuous addition of new features much, it must be admitted, like the Ptolemaic cycles and epi-cycles of old. Thus the behaviour of the structure functions in deep inelastic lepton scattering will lead us initially to a simple picture of a hadron composed of granular constituents, partons. Soon thereafter we shall discover the need for antipartons and the beautiful association between partons and quarks. But the gross failure of the ‘momentum sum rules’ will indicate that some constituents are still missing; presumably the ‘gluons’. Although the gluons do not interact directly with the virtual photon used in our picture of deep inelastic scattering they are supposed to mediate the strong interactions and to give rise to the QCD corrections to the quark–parton model. These will be discussed in Chapters 20 and 21.

Finally we warn the reader that to meet the phenomenal accuracy of recent experiments it is necessary to give careful attention to kinematic details which to some extent detract from the elegant simplicity of the original picture. These effects are treated in the appendix to this chapter (Section 16.9) in which the parton model is reformulated as an impulse approximation which allows better control of the kinematic factors. The chapter can be perfectly well understood without reading the appendix.

The introduction of partons

We begin with a qualitative discussion of the quark-parton model. A more careful and quant it ive description is gradually developed thereafter.

This short chapter contains some results of the theory of extensions by non-abelian groups. It is shown that the study of such extensions can be reduced to an abelian cohomology problem: a G-kernel is extendible iff some three-cocycle with values in the centre of K, Ck, is trivial, and the possible extensions of G by K are in one-to-one correspondence with the extensions of G by the abelian group CK, The obstruction to the extension is given by a three-cocycle.

As a physical example the eight ‘covering’ groups of the complete Lorentz group are found (although this problem does not necessarily require the use of the techniques explained here).

The chapter is intended to provide a more complete description of the problem of group extensions, but may be omitted in a first reading since it is not necessary for the rest of the book.

The information contained in the G-kernel (K,σ)

Let us go back to the general problem of extending an abstract group G (chapter 4) and consider the case where K is not abelian. The main difference from the abelian case of chapter 5 is that not every G-kernel (K,σ) is associated with one or more extensions G of G by K: not every G-kernel (K,σ) is extendible. In fact, one of the aims of this chapter is to show that the G-kernel (K,σ) determines an obstruction to the extension in the form of a certain three-cocycle; the G-kernel (K,σ) is extendible iff this cocycle is trivial.

Before attacking the problem of giving a necessary and sufficient condition for G to exist, let us further analyse the information contained in the data, the G-kernel (K,σ).

This chapter is devoted to studying some concepts that will be extensively used in the last chapters, namely the cohomology of Lie algebras with values in a vector space, the Whitehead lemmas and Lie algebra extensions (which are related to second cohomology groups). The same three different cases of extensions of chapter 5 as well as the ℱ(M)-valued version of cohomology will be considered. In fact, the relation between Lie group and Lie algebra cohomology will be explored here, first with the simple example of central extensions of groups and algebras (governed by twococycles), and then in the higher order case, providing explicit formulae for obtaining Lie algebra cocycles from Lie group ones and vice versa.

The Lie algebra cohomology à la Chevalley-Eilenberg, which uses invariant forms on a Lie group, is also presented in this chapter. This will turn out to be specially useful in the construction of physical actions (chapter 8), i.e. in the process of relating cohomology and mechanics. The BRST formulation of Lie algebra cohomology will also be discussed here due to its importance in gauge theories.

Cohomology of Lie algebras: general definitions

We now discuss the cohomology of Lie algebras with values on a vector space V following the same pattern as used in chapter 5 for the cohomology of groups with values in an abelian group. Let G be a Lie algebra, and let the vector space V over the field K be a ρ(G)-module.

This book has its origin in a graduate course delivered at DAMTP in the Lent term of 1989 by one of the authors (J. de A.) on group cohomology. We have aimed at a self-contained presentation. Some of the background sections, however, are provided as review or reference material and occasionally they may not be suitable for first study if the reader is completely unfamiliar with their contents. There are already several good textbooks on differential geometry for physicists and the authors felt that there was no point in reproducing here topics not relevant to the main subject that could be found without much effort in a single source. Thus, a modicum of knowledge of differential manifolds, Cartan calculus and of the theory of connections is assumed. From the physics side, a knowledge of quantum field theory is also required to read certain parts of the book. Nevertheless, almost all the necessary background material is discussed in it, and often repeated in different settings of increasing complexity so that the various concepts can be assimilated more easily.

A comment should be made concerning rigour. The book uses generously the mathematical ‘Theorem, Proposition, etc’ presentation style. We have found this convenient in order to stress certain key ideas or to organize and separate parts of the text that would otherwise look unreasonably long. Nevertheless, the book is primarily intended for physicists wishing to familiarize themselves with certain aspects of the theory of Lie groups, fibre bundles and a few of their physical applications, particularly to problems where cohomology and extension theory play an important rôle.