Quantum Chromodynamics becomes now – moving into the second decade of the twentyfirst century – a very extended theory with many branches. Therefore, it is impossible to expose in one book the whole QCD with all its branches and applications. On the other hand, it is a rapidly developing theory and for the authors of a book on QCD it is a serious danger to go too close to the border between that which is known with certainty and the domain of unknown. For these reasons, we decided to present in this book some aspects of QCD that are not very closely related to one another. In the selection of these aspects, surely, the individual preferences of authors also played a role.

What is not discussed in this book includes the theory of heavy quark systems – charm, bottom, top mesons and baryons; QCD at finite temperature and density; the problem of phase transitions in QCD; and lattice calculations. The QCD corrections to weak interactions, particularly to the production of W, Z, and Higgs bosons, are also beyond the scope of the book. We intentionally avoided various model approaches related to QCD, restricting ourself to exposition of the results following from the first principles of the theory. The phenomenology of hard processes and of the parton model was only lightly touched on in this book. We refer the reader to our previous book: B. L. Ioffe, V. A. Khoze and L. N. Lipatov, Hard Process, North Holland, 1984, where these topics were considered in detail.

Generalities

The phenomenon of anomaly plays an important role in quantum field theory: in many cases it determines whether or not a theory is self-consistent and can be realized in the physical world and, therefore, allows one to select the acceptable physical theories. In a given theory anomalies are often related to the appearance of new quantum numbers (topological quantum numbers; see Chapter 4), result in the emergence of mass scales, determine the spectrum of physical states. So, despite its name, anomalies are a normal and significant attribute of any quantum field theory.

The term “anomaly” has the following meaning: Let the classical action of the theory obey some symmetry, i.e. let it be invariant under certain transformations. If this symmetry is violated by quantum corrections, such a phenomenon is called an “anomaly.” (Reviews of anomalies are given in [1]–[4].) There are two types of anomalies – internal and external. In the first case, the gauge invariance of the classical Lagrangian is destroyed at the quantum level. The theory becomes nonrenormalizable and cannot be considered as a self-consistent theory. The standard method to solve this problem is a special choice of fields in the Lagrangian in such a way as to cancel all internal anomalies. (Such an approach is used in the standard model of electroweak interaction – the Glashow–Illiopoulos–Maiani mechanism.) An external anomaly corresponds to violation of symmetry of interaction with external sources, not related to gauge symmetry of the theory. Such anomalies arise in QCD and are considered below. There are two kinds of anomalies in QCD: axial (chiral) anomaly and scale anomaly.

Unlike QED, the vacuum state in QCD has nontrivial structure. In QCD vacuum there are nonperturbative fluctuations of gluon and quark fields. They are responsible for spontaneous violation of chiral symmetry and for the appearance of topological quantum numbers, which result in a complicated structure of an infinitely degenerate vacuum. The phenomenon of confinement is also attributed to these fluctuations.

Instantons were discovered in 1975 by Belavin, Polyakov, Schwarz, and Tyupkin [1]. They are the classical solutions for gluonic field in the vacuum, which indicate the nontrivial vacuum structure in QCD (papers on instantons are collected in [2]). In Euclidean gluodynamics (i.e. in QCD without quarks) at small g2 they realize the minimum of action. The instantons carry new quantum numbers – the topological (or winding) quantum numbers n. There is an infinite set of minima of the action, labelled by the integer n and, as a consequence, an infinite number of degenerate vacuum states. In Minkowski space-time instantons represent the tunneling trajectory in the space of fields for transitions from one vacuum state to another. Therefore, the genuine vacuum wave function is a linear super-position of the wave functions of vacua of different n characterized by a parameter θ. This is analogous to the Bloch wave function of electrons in crystals – the so-called θ vacuum. θ is the analog of the electron quasimomentum in a crystal. The existence of a θ vacuum at θ ≠ 0 implies violation of CP-invariance in strong interactions, which is not observed until now. This problem waits for its solution.

The general properties of QCD at low energies

The asymptotic freedom of QCD, i.e. the logarithmic decrease of the QCD coupling constantat large momentum transfers Q2 → ∞ (or, equivalently, the decrease of αs at small distances, αs (r) ∼ 1/ ln r) allows one to perform reliable theoretical calculations of hard processes, using perturbation theory. However, the same property of the theory implies an increase of the running coupling constant in QCD at small momentum transfer, i.e. at large distances. Furthermore, this increase is unlimited within the framework of perturbation theory. Physically such growth is natural and is even needed, because otherwise the theory would not be a theory of strong interactions.

QCD possesses two remarkable properties. The first is the property of confinement: quarks and gluons cannot leave the region of their strong interaction and cannot be observed as real physical objects. Physical objects, observed experimentally at large distances, are hadrons – mesons and baryons. The second important property of QCD is the spontaneous violation of chiral symmetry. The masses of light u, d, s quarks that enter the QCD Lagrangian, especially the masses of u and d quarks from which the usual (nonstrange) hadrons are built, are very small as compared to the characteristic QCD mass scale. In QCD, the quark interaction is due to the exchange of vector gluonic field. Thus, if light quark masses are neglected, the QCD Lagrangian (its light quark part) becomes chirally symmetric, i.e. not only vector, but also axial currents are conserved.