The dynamics of the massless relativistic string (which we will meet at very many different places in this book) is a delightful theoretical laboratory to study the properties of the theory of special relativity. To make the book self-contained and also to define our notation we will briefly review in this chapter some properties of special relativity, in particular with respect to its implications for high-energy particle kinematics.

We will also review some properties of electromagnetic fields with particular emphasis on the features we are going to make use of later in the book. We will end with a description of the interaction ability of an electrically charged particle.

This is the first but not the last example in this book of the law of the conservation of useful dynamics. This says that every new generation of theoretical physicists tends to reinvent, reuse (and usually also rename) the most useful results of earlier generations. One reason is evidently that there are few situations where it is possible to find a closed mathematical expression for the solution to a dynamical problem.

Here our basic aim is to describe the interactions between charged particles which are moving with very large velocities (as they do in high-energy physics). As a charged particle interacts via its field the question can be reformulated into finding a way to describe the field of a charged particle which is moving very fast.

Introduction

Bremsstrahlung emission is an inherent property of all gauge field theories. It can be understood even within classical mechanics, at least for the soft part of the spectrum. Suppose that we consider a charge surrounded by its Coulomb field, which necessarily is extended in space outside the charge. Then suppose that there is a sudden change in the state of motion of the charge itself. The result will be that the outlying field will need some time to readjust to the new situation.

Therefore there will be, as in all other situations of sudden change in physics, a brief interlude of compressions and extensions in the field before it comes back to a stable state. The ensuing radiation field, to be described below, is a bremsstrahlung field. Its properties depend upon the way in which the charge distribution is changed. For a single charge with a sudden momentum transfer, or for the situation when a charge and anticharge suddenly emerge, the bremsstrahlung is essentially of a dipole character. This approximation means that the current contains a direction, the dipole axis, but the size of the interaction region is neglected. We will consider a ‘classical’ current with these properties.

Some warning is needed against taking the classical picture too far. We have shown in Chapter 2 how the method of virtual quanta describes the Coulomb field of a fast-moving charge.

Introduction

In this chapter we consider the way in which gluons are introduced into the Lund string fragmentation model. They are treated as internal excitations on the massless relativistic string (the MRS) similar to a sudden ‘hammer hit’ on an ordinary classical string. Thus they will be initially well localised in space-time. But we will find that they quickly disperse their energy-momentum to the surrounding string. This property means that the gluon excitation disappears and reappears periodically as a localised energy-momentum-carrying entity during the string cycle.

We will start as usual with a classical mechanics scenario and study some simple modes of motion of the MRS in order to get acquainted with the notion of an internal excitation. We start with the mode which has acquired the poetical name of ‘the dance of the butterfly’. It certainly does exhibit the grace and the beauty that goes with this name. After a brief snapshot description of the appearance of this mode in space coordinates we proceed to a description in space-time. This will lead us to the general equations of motion for the MRS and to an understanding of the way the string is built up in terms of moving wave fronts.

After that we consider more complex modes, although there is no reason to go into too many details.

Introduction

In this chapter we will consider the decay properties of a cluster. We start to derive some results from the internal-part formulas, Eqs. (8.41) and (8.43).

1. I1 If we sum over all available states in the decay formulas of a cluster of squared mass s we obtain asymptotically, i.e. for large values of s, the behaviour ∼ sa. We will consider these state equations both for the case of a single species of flavor and meson and also for the case of many flavors and many hadrons in each flavor channel.

2. I2 At the same time we will derive the finite-energy version, fs, of the fragmentation function f in Eqs. (8.16), (8.17). We will show that fs tends rapidly towards f when s is larger than a few squared hadron masses (just as Hs → H according to the results of Chapter 9).

The method we will use is to derive a set of integral equations and then to solve them. In that way we will find that there are some necessary relationships between the parameters a, b and the normalisation parameters that constitute a set of eigenvalue equations for the integral equations.

The whole procedure is very similar to that for obtaining the unitarity conditions for the S-matrix in a quantum field theory. We will exploit these relationships by showing that the results obtained under I1 are just the same as are obtained for the multiperipheral ladder equations in a quantum field theory.

Introduction

In this chapter we will provide the parton model, the PM, with a QCD field theoretical structure according to the conventional method; for more details see e.g. In the next chapter we continue the discussion and present the Lund model version of the properties of deep inelastic scattering (DIS) events, both the treatment of the fragmentation and, in particular, the use of the newly developed linked dipole chain model, to provide the fragmenting string state.

The method of virtual quanta (MVQ) in Chapter 2 describes the electromagnetic field from a fast-moving charge in terms of the photon flux from the bremsstrahlung spectrum, and we will make use of this as an analogy. It is evident that Feynman picked up the basic features of the MVQ to make the PM into a description of the corresponding flux of the hadronic field quanta. In that way he made the PM into a useful tool to describe the cross sections for DIS events. Those we consider in this book are initiated by an electromagnetic probe, i.e. they correspond to inelastic electron-baryon (or muon-baryon) scatterings. But it is also possible to use the PM to describe e.g. inelastic neutrino-baryon scattering events as well as to consider the interactions between the partons themselves.

Feynman assumed that the partons can be treated as a stream of free elastic scatterers with respect to the probe.

Introduction

In this chapter we start to consider the properties of the massless relativistic string (the MRS). We will begin with a simple situation in which the MRS plays the role of a constant force field, acting upon a ‘charge’ and an ‘anticharge’ placed at the endpoints of an open MRS. This means that the motion will be in one space dimension along the force direction. We will refer to it as the yoyo-mode for reasons that will become clear when it is exhibited.

In later chapters we will come back to more complex modes involving several dimensions. All these modes are used in the Lund model as semi-classical models for different high-energy interactions between hadrons. The yoyo-mode is used both to describe an e+e– annihilation event and as a simple model for stable hadrons. In the last section of this chapter we provide a possible dynamical analogy between the QCD vacuum and superconductivity as a justification for using string dynamics to describe hadronic states and interactions.

In the yoyo-mode the two charges at the endpoints of the string move like point particles, i.e. the momentum of the state is localised in these endpoint particles of the MRS force field. At any moment the total energy of the state can be decomposed into the energy in the force field, corresponding to a linearly rising potential, and the kinetic energies of the particles at the endpoints.

Introduction

In this and the next chapter we will consider some properties of quantum fields. The examples taken will be mostly scalar fields and only when necessary will we invoke the complexities stemming from the vector nature of the interactions in QED and QCD; there are many good text-books devoted to a detailed treatment of the subject.

We need only intuition and a set of understood formulas for the investigations contained in this book. We start with a discussion of the quantum mechanical harmonic oscillator coupled to an external force. There are several reasons to dwell on this particular system. Firstly its sine and cosine behaviour in time is matched by the corresponding harmonic behaviour of the plane wave solutions for the quanta in a field theory.

It was noted even in the first papers on quantum field theory that a free or weakly interacting quantum field is in a rather precise way a superposition of an infinite, although enumerable, set of harmonic oscillators, one for each degree of freedom.

A real interacting-field theory does not behave in this way with respect to its excitations. There is always, however, at the basis of any experiment in high-energy particle physics the idea of a three-act scenario in time.