We have seen how convenient it is to regulate the UV divergences of perturbation theory by continuation in the dimension of space-time. To date, no-one has shown how to use the method in the complete theory. But in perturbation theory, as we will now demonstrate, it is consistent and well-defined. Now all results obtained by this method can be obtained by other, more physical methods (say, a lattice regulator). But frequently much more labor is involved. This is not a triviality, for in complicated situations, especially in gauge theories, it enables us to handle the technicalities of renormalization in a simple way.

The idea of dimensional continuation has been used for a long time in statistical mechanics (see, for example, Fisher & Gaunt (1964)). It became very prominent when Wilson & Fisher (1972) discovered the ε-expansion and applied it to field-theoretic methods in statistical mechanics (Wilson (1973), Mack (1972), and Wilson & Kogut (1974)). In the ε-expansion one works in 4 — ε spatial dimensions, and expands in powers of ε. At the same time, in a purely field-theoretic context, a need arose to find a way of regulating non-abelian gauge theories that preserved gauge invariance and Poincaré invariance. This led to dimensional regularization ('t Hooft & Veltman (1972a), Bollini & Giambiagi (1972), Cicuta & Montaldi (1972), and Ashmore (1972)).

In Chapter 2 we met a number of equations involving products of field operators at the same point. Examples are given by the equations of motion (2.1.10) and the Ward identities (2.7.6). These products we will call composite operators. When computed directly they have ultra-violet divergences: the product ø(x)ø(y) makes unambiguous sense if x is not equal to y, but if x equals y then we have ø(x)2, which diverges. Since the equations of motion and the Ward identities express fundamental properties of the theory, it is useful to construct finite, renormalized composite operators with which to express these same properties.

It could be argued that there is no need to have renormalized equations of motion. One could say that one only actually needs the equations of motion in the regulated theory, where they are finite. A situation of practical importance where we actually do need renormalized composite operators is the operator-product expansion, to be discussed in Chapter 10. This is used in a phenomenological situation such as deep-inelastic scattering (Chapter 14) where we wish to compute the behavior of a Green's function when some of its external momenta get large. Equivalently, we need to know how a product of operators, like ø(x)ø(y), behaves as x → y.

The structure of a quantum field theory often simplifies when one considers processes involving large momenta or short distances. These simplifications are important in improving one's ability to calculate predictions from the theory, and in essence form the subject of this book.

The first simplification to be considered involves the very existence of the theory. The problem is that there are usually ultra-violet divergences caused by large fluctuations of the field(s) on short distance scales. These manifest themselves in Feynman graphs as divergences when loop momenta go to infinity with the external momenta fixed. The simplification is that the divergences can be cancelled by renormalizations of the parameters of the action. Consequently our first task will be to treat the ultra-violet renormalizations. Renormalization is essential, for otherwise most field theories do not exist.

We will then expose the methods needed to handle high-energy/short-distance problems. The aim is to be able to make testable predictions from a strong interaction theory, or to improve the rate of convergence of the perturbation expansion in a weakly coupled theory. The simplifications generally take the form of a factorization of a cross-section or of an amplitude, each factor containing the dependence of the process on phenomena that happen on one particular distance scale.

A characteristic feature of relativistic quantum field theories is that symmetries of the classical theory are not always present after quantization. We do not mean here the spontaneous breaking that is characterized by a non-invariant vacuum and by the presence of the Goldstone bosons. Rather we mean a situation where there is no conserved current for the symmetry despite the absence of any terms in the action that appear to break the symmetry. Such breaking of a symmetry is called anomalous.

If the classical action is invariant, then a naive application of Noether's theorem gives us a conserved current. That is, there is no anomalous symmetry breaking. What prevents the argument from being correct is the presence of UV divergences. The current is a composite operator, i.e., a product of elementary fields at the same point, and to define it, some kind of regularization and renormalization is needed. The renormalization may invalidate the equations used to prove Noether's theorem.

For simplicity, we will consider only global symmetries, as opposed to local, or gauge, symmetries. The simplest cases of global symmetries were considered in Chapter 9. These could be treated by using an ultra-violet regulator that preserved the symmetry. The proof of Noether's theorem can then be made in the cut-off theory.

In this chapter we come to the general theory of renormalization. The basic difficulty is that a graph may not only possess an overall divergence. It may have in addition many subdivergences which can be nested or can overlap in very complicated ways. Most of our effort must go to disentangling these complications.

We will begin by investigating some simple graphs. These will show us how to set up the formalism in the general case. The ultimate result is the forest formula of Zimmermann (1969). Contrary to its reputation, this is not an esoteric procedure, designed for pedantically rigorous treatments. Rather, the forest formula is merely a general way of writing down what is in fact the natural and obvious way of extracting the divergences from any integral. Its power is demonstrated by the ease of treating overlapping divergences, the handling of which is normally considered the bête noire of renormalization theory.

The forest formula is applied to individual Feynman graphs. It extracts the finite part of a graph by subtracting its overall divergence and its subdivergences. We will, of course, need to show that the subtractions can be implemented as actual counterterms in the Lagrangian. We will also show that the counterterms are local, i.e., polynomial in momentum.

A common situation in physics is that in investigating phenomena on a certain distance scale, one sees no hint of those phenomena that happen at much shorter distance scales. In a classical situation this observation seems evident. For example, one can treat fluid dynamics without any knowledge of the atomic physics that generates the actual properties of the fluids. However, in a quantum field theory this decoupling of short-distance phenomena from long-distance phenomena is not self-evident at all.

Consider an e+ — e- annihilation experiment at a center-of-mass energy well below 10GeV, the threshold for making hadrons containing the b-quark. There is, for practical (or experimental) purposes, no trace of the existence of this quark. However, the quark is present in Feynman graphs as a virtual particle, and can have an apparently significant effect on cross-sections. Our task in this chapter is therefore to prove what is known as the decoupling theorem. This states that a Feynman graph containing a propagator for a field whose mass is much greater than the external momenta of the graph is in fact suppressed by a power of the heavy mass. The physics at low energy is described by an effective low-energy theory that is obtained by deleting all heavy fields from the original theory.

It is important to show that renormalization of a gauge theory can be accomplished without violating its gauge invariance. Gauge invariance is physically important; among other things it is used (via the Ward identities) to show that the unphysical states decouple ('t Hooft (1971a)).

In Chapter 9 we considered the case that the basic Lagrangian of a theory is invariant under a global symmetry, as opposed to a gauge symmetry, such as we will be investigating in this chapter. We showed that the counterterm Lagrangian is also invariant under the symmetry. Suppose now that the basic Lagrangian is invariant under a gauge symmetry. One might suppose that the counterterms are also invariant under the symmetry, just as for a global symmetry. This is not true, however, since the introduction of gauge fixing (as explained in Sections 2.12 and 2.13) destroys manifest gauge invariance of the Lagrangian. One might instead point out that the theory with gauge fixing is BRS invariant and deduce that the counterterms are BRS invariant. This deduction is false. To see this, we recall that an ordinary internal symmetry relates Green's functions with certain external fields to other Green's functions differing only by change of symmetry labels. However, BRS symmetry relates a field to a composite field (2.13.1).

In the previous chapters we set up renormalization theory in momentum space. In this chapter, we will give a treatment in coordinate space. Now, the utility of a momentum-space description, such as we gave in the earlier chapters, comes from the translation in variance of a problem. However, the momentum-space formulation rather obscures the fact that UV divergences arise from purely short-distance phenomena. Thus a coordinate-space treatment is useful from a fundamental point of view. There are also a number of situations, essentially external field problems, where a coordinate-space treatment is the most appropriate from a more practical point of view. A particular advantage is that the coordinate-space method makes it easy to see that the counterterms are the same as with no external field.

An important case, which we will treat in detail in this chapter, is that of thermal field theory at temperature T (Fetter & Walecka (1971), Bernard (1974), and references therein). There one works with imaginary time using periodic boundary conditions (period 1/T).

It is first necessary to work out the short-distance singularities of the free propagator. This we will do in Section 11.1. A number of forms of the propagator will be given, whose usefulness will become apparent when we treat some examples in Section 11.2.

As we saw in Chapter 3, the renormalization procedure has considerable arbitrariness: the counterterm for a graph must cancel its divergence but may contain any amount of finite part. A rule for choosing the value of the counterterm we called a renormalization prescription. In one-loop order it was clear from the examples that a change in renormalization prescription can be cancelled by a change in the finite, renormalized couplings corresponding to each divergence. Thus a change in renormalization prescription does not change the theory but only the parametrization by renormalized coupling and mass. What is not so easy is to see that this property is true to all orders. This we will show in Section 7.1. The invariance of the theory under such transformations is called renormalization-group (RG) invariance.

A particularly useful type of change of renormalization prescription is to change the renormalization mass μ. Infinitesimal changes are conveniently described by a differential equation, called the renormalization-group equation, which is derived in Section 7.3. This leads to the concept of the effective momentum-dependent coupling. This concept is very useful in calculations of high-energy behavior, as explained in Section 7.4. The coefficients in the renormalization-group equation are called the renormalization-group coefficients and are important properties of a theory.

Most of the work in this book will be strictly perturbative. However it is important not to consider perturbation theory as the be-all and end-all of field theory. Rather, it must be looked on only as a systematic method of approximating a complete quantum field theory, with the errors under control. So in this chapter we will review the foundations of quantum field theory starting from the functional integral.

The purpose of this review is partly to set out the results on which the rest of the book is based. It will also introduce our notation. We will also list a number of standard field theories which will be used throughout the book. Some examples are physical theories of the real world; others are simpler theories whose only purpose will be to illustrate methods in the absence of complications.

The use of functional integration is not absolutely essential. Its use is to provide a systematic basis for the rest of our work: the functional integral gives an explicit solution of any given field theory. Our task will be to investigate a certain class of properties of the solution.

For more details the reader should consult a standard textbook on field theory.

In this chapter I shall tie up a few loose ends and then have a look at the broader implications of all the detailed and complicated work I have described.

First I shall deal with quark confinement. The enormous accelerators at CERN and Fermilabs have not produced quarks in any considerable numbers; in fact most people doubt if any have been produced. So quarks are at least very difficult to extract from nucleons. They are bound (or confined) very strongly in them. Some eminent theoreticians think they are absolutely confined, others are not so sure. But, at least, they are confined to a very considerable extent. Yet when we examine the quarks inside a proton or neutron they appear to be moving around freely – ‘asymptotic freedom’, it is called. How can this be? The answer turned out to be easy, but surprising.

The first force field physicists learned about was gravity. Now the force between two gravitating objects decreases as we move them apart. It is a rapid decrease, getting smaller as the square of the distance between them. If we double the distance, the force is less by a factor of four. When the electrical force was investigated it was found to obey a similar law – the force between two electrical charges varies inversely as the square of the distance between them. There is a difference between the two fields: gravity is always attractive, while the electrical force can be either attractive or repulsive, depending on whether we are dealing with unlike or like charges. But whether it is attractive or repulsive it is always ‘inverse square’.