Our journey to the very small has now brought us beyond the atoms, which are bulky and fragile objects compared with what we shall be occupied with next: the atomic nucleus and whatever is inside. The electrons, now seen ‘at a great distance’ circling around the nucleus, are themselves also small and extremely robust. I now invite you to have a look inside the nucleus, through the eyes of the scientists before 1970. I consider the years around 1970 as a crucial period, but I am now also choosing the year 1970 bcause this was the time I became acquainted with elementary particle physics myself, as a young graduate student at the State University of Utrecht in the Netherlands.

All the physics that I mentioned earlier (and of course a great deal more) was basic stuff for students of theoretical physics. Also, a lot was known about the structure of the atomic nucleus. The nucleus is built from two species of building blocks: protons and neutrons. The proton (Greek np{πρῶτoς = first) owes its name to the fact that the simplest atomic nucleus, that of hydrogen, consists of just one proton. It carries one positive unit of charge. The neutron resembles the proton as if it were its twin brother: its mass is practically the same, its spin is the same, but the electric charge is absent in a neutron; it is neutral.

And now I can report that implicating black holes in our considerations concerning the ultimate laws of physics means progress. I have now concentrated most of my own research towards this subject, asking myself this: suppose you have a tiny black hole and that you insist that it obey both the laws of quantum mechanics and those of gravity. Then how should one describe its behavior?

The shrewdness of this question is that I am assuming that an entire black hole should behave exactly as if it were an atom or molecule obeying the laws of quantum mechanics. Not everybody agrees with me here – not by any means. Some say that black holes are something altogether different. But what is so different about them? Black holes emit particles just like quite a few species of radioactive atoms do. Then why should they not do this by following the same rules? To put it more strongly, I happen to suspect that they absolutely have to obey such laws if you believe in any kind of ‘law and order’ at the Planck length.

One result of my calculations was a total surprise for me. I hit upon practically the same mathematical expressions as the ones in string theory! The formulae for capture and emission of particles by a black hole look exactly like Veneziano's formula. This is odd, since there is no question of strings.

So I liked the Higgs mechanism. But is it correct? Does it yield a healthy theory? What is and what is not allowed in describing elementary particles, and why?

Various universities in different countries organize courses at an international level, called ‘summer schools’, where in prestigious, yet quiet, recreational resorts experts and students convene to give lectures, and listen to them, and most of all to discuss their research subjects. The most prestigious summer school in my subject was in Les Houches, a winter sport resort near Chamonix, on the slopes of Mont Blanc. Presumably because I was too late in registering there, I was not admitted. But I was admitted at my second choice: Cargése. I went there in 1970.

The Institut des Sciences near Cargése on the island of Corsica was established by the French physicist Maurice Lévy. On a beautiful piece of land with a little beach, a small building had been placed where, from 1960 onwards, summer schools and conferences had been held. The 1970 summer school would be about the strong interactions. Together with Gell-Mann, Levy had constructed a model for the strong interaction. Although it was not expected to represent the complete truth, it had the nice feature that all symmetries of the strong interactions were reproduced in a very interesting way. But, because the forces are so strong, the particles will not even approximately move in straight lines, and that made the usual approximation scheme, called ‘perturbation expansion’, unreliable.

There was a way out! However, it originates in quite a different branch of theoretical physics, to wit the physics of metals at very low temperatures. At very low temperatures, ‘quantum phenomena’ can lead to quite remarkable effects. To describe these effects one uses quantized field theories, exactly the same as the ones employed in elementary particle physics. The physics of elementary particles has nothing to do with the physics of low temperatures, but the mathematics is very very similar!

The ‘field’ in some material that becomes important at low temperatures could be the field describing how atoms there oscillate about their equilibrium points, or else the field that describes electrons in this material. At very low temperatures we have to deal with the ‘quanta’ of these fields. For instance, the ‘phonon’ is the quantum of sound. Its behavior resembles that of the photon, the quantum of light, except that the numbers are very different: phonons travel with the speed of sound, hundreds or perhaps thousands of meters per second; the photon travels with light velocity, which is 300000kilometers/second, nearly one million times faster! The elementary particles we are interested in usually have velocities close to that of light.

One of the more spectacular ‘quantum phenomena’ that can take place in strongly cooled metals is called superconductivity, which is the fact that the resistance of such a material to electrical currents vanishes completely.

Quantum Chromodynamics (QCD) is the theory of the strong interaction, one of the four fundamental forces in nature. It describes the interactions between quarks and gluons, and in particular how they bind together to form the class of particles called hadrons, of which the proton and neutron are familiar examples. QCD emerged as a mathematically consistent theory in the 1970s, and nowadays is regarded as one of the cornerstones of the ‘Standard Model’ of the elementary particles and their interactions.

Collider physics has grown from a small part of the subject in the 1960s to the position it now holds as the principal method of high-energy physics research. Particle colliders provide the highest available centre-of-mass energies and permit us to probe the structure of matter at the shortest distances. On these scales, all of the known fundamental interactions except gravity can be studied in unprecedented detail. In this book, in order to limit the scope, and not stray too far from our field of expertise, we concentrate mainly on the strong interaction aspects of collider physics, which are described by QCD.

One of the triumphs of modern particle physics has been the extent to which QCD has successfully accounted for the strong interaction processes observed at colliders, most notably the dramatic phenomena of hadronic jet and heavy quark production. Our aim here is to show what has been achieved in this area, while indicating the directions that need further development. Increasingly the study of QCD also has an ‘engineering’ aspect. In order to exploit the potential of high-energy colliders to study other physics, especially at machines with circulating hadron beams, one needs a quantitative understanding of QCD.

Many of the basic ideas and properties of perturbative QCD can be illustrated by considering the production of hadrons in electron-positron annihilation. In this chapter we will discuss a variety of ways in which the theory can be tested. We begin by discussing the total cross section for e+e → hadrons: we show how the order αs2 corrections are calculated, and how renormalization scheme dependence enters at order. Because of its inclusive nature, this is one of the few quantities for which the first three terms in the QCD perturbation series are known, and for which non-perturbative corrections are expected to be small. As a result it provides potentially one of the most precise measurements of the strong coupling.

Perturbative QCD also predicts a rich ‘jet’ structure for the final-state hadrons in electron-positron annihilation. We discuss the various ways in which jet cross sections can be defined, how such cross sections are calculated in perturbation theory, and how the predictions compare with experiment. Jet cross sections can also be used to measure αs, and to test the basic interactions in the theory, in particular the triple-gluon vertex.

The total hadronic cross section

The production of a muon pair in electron-positron annihilation, e+e− → μ+μ−, is one of the fundamental processes of QED. The same annihilation process can also produce hadrons in the final state. As we have seen in the previous chapter, the formation of these hadrons is not governed by perturbation theory. Why then would one expect a perturbative approach to give an accurate description of the hadronic cross section? The answer can be understood by visualizing the event in space-time.

One of the motivations for collider experiments is to discover new heavy objects. It is therefore important to test our understanding of such processes by predicting the production rates for the known heavy objects, such as the top, bottom and charm quarks, and to compare the predictions with experimental measurements.

Aspects of the production of heavy quarks in e+e− annihilation have already been considered in Chapter 3. We shall concentrate here on the hadronic production of heavy quarks. An idea of the relative cross sections and signal-to-background ratios for c, b and t quark production at various present and future accelerators can be obtained from Table 10.1. As we shall see, the description of hadronic production is possible in perturbative QCD when the mass of the produced quark is large compared to 1 GeV, the typical scale of the strong interactions. Because the cross sections are generally large, hadronic interactions offer the potential to produce the large number of heavy quarks necessary to study their decays in detail. However, at present, much of the information on states containing c and b quarks comes from e+e− annihilation, because of the better signal-tobackground ratios at e+e− colliders.

As an illustration of the power of hadron colliders we consider the case of b production. For example, at the Fermilab Tevatron running at a luminosity of 2 × 1031 cm−2s-1, b quarks are produced at 400 Hz. This should be contrasted with an e+e− machine such as CESR operating at a luminosity of 2×1032 cm−2s−1, where the rate of 6 production is ∼ 0.1 Hz1.

We illustrated in Chapter 3 the use of perturbation theory to calculate ‘infra-red safe’ quantities such as the e+e− total cross section and thrust distribution. There are in addition predictions that can be made concerning some quantities that are ‘infra-red sensitive’, i.e. that have infra-red and collinear singularities in perturbation theory. Such quantities can still be handled provided the singularities can be collected into an overall factor which describes the sensitivity of the quantity to long-distance physics. The divergence of this factor corresponds to the fact that longdistance phenomena are not reliably predicted by perturbation theory. For example, quarks and gluons are free to propagate over macroscopic distances in perturbation theory, whereas in reality they are confined inside hadrons. Therefore the divergent factor should be replaced by a finite factor determined experimentally by measurements at a relatively short distance scale. Once this is done, perturbation theory can be used to predict the scale dependence of the quantity.

The best known example of a perturbative prediction of a factorizable infra-red-sensitive quantity is the phenomenon of scaling violation, discussed in Chapter 4. There we studied the parton distributions inside a hadron, probed by deep inelastic lepton scattering. Here we consider the related phenomenon for the fragmentation of a jet, produced for example in e+e− annihilation, into hadrons of a given type h.

Jet fragmentation

We start by defining fragmentation functions, which are dimensionless functions that describe single particle distributions in the final state.