Introduction

In the previous chapter we used some qualitative arguments to estimate the basic nature of synchrotron radiation. The results of this exercise are very useful for understanding the underlying physics, estimating the quantities involved, and judging the validity of certain approximations we will make. Now synchrotron radiation is treated in a quantitative manner. We will distinguish between the time t at which the radiation is observed and t′ when it was created by the moving charge at a distance r. Since the relation between the two is in general rather complicated, some of the derivations are lengthy. As final results we obtain expressions for the radiation field and the emitted power, which will be applied to calculate synchrotron and undulator radiation in the next two parts. Treatments of synchrotron radiation can be found in many books, journal publications, articles, proceedings of conferences and workshops, and laboratory reports. The first book on the topic of synchrotron radiation [3] was published in 1912. Complete coverage of the topic is presented in [4–8], some of which give also a quantum-mechanical treatment. Many books on electrodynamics treat the radiation from relativistic particles and cover also theoretical aspects of synchrotron radiation [9–13].

Introduction

An electron-storage ring consists of an assembly of deflecting bending magnets and focusing quadrupole magnets, called a lattice (Fig. 13.1). The bending magnets determine a closed equilibrium orbit for a particle having the nominal values for energy, initial position, and angle. Particles with small deviations from these nominal parameters are kept in the vicinity of the nominal orbit by the focusing quadrupoles. In addition a storage ring has magnet-free straight sections. These contain an acceleration system, which is usually called the radio-frequency system or just the RF system. It is used to replace the energy lost by the particles due to the emission of synchrotron radiation or to accelerate the electron beam to higher energy.

The straight sections also house insertion devices (undulator and wiggler magnets) and other equipment such as instruments necessary for the operation of the storage ring. The particles are made to circulate inside a vacuum chamber with a very low pressure of a few nTorr (0.13 μPa) in order to minimize the interaction between the particles and the residual gas.

We restrict ourselves to a plane storage ring and describe the beam dynamics in a coordinate system that follows the nominal equilibrium orbit as shown in Fig. 13.2. The longitudinal coordinate s is the path-length along this nominal equilibrium orbit.

Introduction

The renormalon problem is related to the well-known fact [372,375] (for more complete reviews, see for example [162,154]) that the QCD series is unfortunately divergent (no finite radius of convergence) like n!, which is the number of diagrams of nth order. Indeed, a given observable can be expressed as a power series of the coupling g as:

where the series are divergent:

and where the nth order grows like n!, such that it is not practicable to have a quantitative meaning of Eq. (29.1). For the approximation to be meaningful, the approximation should asymptotically approach the exact result in the complex g-plane, such that:

where the truncation error at order N should be bounded to the order gN+1. If fn behaves like in Eq. (29.2), KN usually behaves as aN N!Nb. The truncation error behaves similarly as the terms of the series. It first decreases until:

beyond which the approximation to F does not improve through the inclusion of higherorder terms. For N0 ≫ 1, the approximation is good up to terms of the order:

Provided fn ∼ Kn, the best approximation is reached when the series is truncated at its minimal term and the truncation error is given by the minimal term of the series.

In the previous parts of this book, we have studied different methods based on power corrections, namely renormalons, instantons and the SVZ-expansion in terms of condensates and eventual quadratic corrections, which are one important aspect of non-perturbative QCD. In this part, we shall shortly discuss the other most popular non-perturbative methods used in QCD for studying the low-energy properties of the hadrons and of QCD. These are:

• Lattice gauge theory

• Chiral perturbation theory (ChPT)

• Models of the QCD effective action

• Heavy quark effective theory (HQET)

• Potential approaches for quarkonia.

The method and phenomenology of QCD Spectral Sum Rules (QSSR) which will be discussed in more details will be devoted to a new part. Here, we do not aim to present a complete review and references but we shall limit ourselves to the outline of the general features of the different approaches. A more extensive list of non-perturbative methods, which will not be discussed here, such as the:

• Skyrme model

• Bag models

• Discretized light-cone quantization

can be found in [3]. We shall shortly discuss in the chapter dedicated to the heavy quarks, the two approaches:

• Potential and quark models

• Stochastic vacuum model

• Non-relativistic effective theories

which are discussed in details in some other reviews (see e.g. [51]).We shall complete this part by a short discussion on monopole and confinement.