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.