The Standard Model of elementary particles and their interactions has two basic components: the spontaneously broken SU(2)xU(1) electroweak theory, and the unbroken SU(3) color gauge theory, known as quantum chromodynamics (QCD). If we date the birth of the theory of strong interactions to the discovery of the neutron, QCD has existed for about a third of the ensuing time, profoundly deepening and enlarging our view of the subject.
Perhaps it is worthwhile to recall the situation in strong-interaction studies at the time when QCD emerged. Into the mid-sixties, the picture of strong interactions centered on general principles of scattering amplitudes (analyticity, unitarity, crossing, etc.) that could be developed without information on elementary constituents. The idea was widely entertained that the strong interactions were not to be described by a renormalizable field theory of point particles, which had been so successful for quantum electrodynamics (Weinberg, 1977; Schweber, 1994). Whether one accepted this viewpoint or not [1], in the absence of a viable theory of strongly interacting elementary particles it was clearly necessary to rely on general properties of the scattering matrix. Perturbative field theory, if utilized at all, could be employed primarily to illustrate and explore the consequences of these properties (Eden, Landshoff, Olive, and Polkinghorne, 1966).
In this context, Regge theory (Regge, 1959; Chew and Frautschi, 1961; P.D.B. Collins, 1971) and its allies and generalizations, such as the dual model (Veneziano, 1968; Mandelstam, 1974) and Reggeon calculus (Gribov, 1968; Abarbanel, Bronzan, Sugar, and White, 1975; Baker and Ter-Martirosyan, 1976), which described particles primarily as analytic features of the S-matrix, flourished. A large body of experimental data, including near-forward elastic (Giacomelli, 1976), diffractive (Goulianos, 1983), and high-multiplicity inelastic scattering (Mueller, 1970, Frazer et al., 1972), are still best understood in this language. These developments also gave rise, of course, to string theory (Nambu, 1970, Goto, 1971, Green, Schwartz, and Witten, 1987). The weak and electromagnetic interactions of hadrons with leptons was, and still is, profitably described by current algebra (Gell-Mann and Lévy, 1960; Adler and Dashen, 1968), which provided elementary operators, the currents, even without elementary particles. The currents themselves are linked to strong dynamics by the partially conserved axial vector current hypothesis, which led to an effective field theory for pions (Weinberg, 1970) that remains today our fundamental picture of low-energy strong interactions (Weinberg, 1979, Leutwyler, 1992). Into this rich and complex set of investigations and viewpoints came partons and quarks.
The study of the strong interactions was transformed with the advent of accelerators in the multi-GeV energy range. The famous SLAC experiments of the nineteen sixties and seventies were the first to show the point-like substructure of hadrons (Bloom et al., 1969; Friedman and Kendall, 1972). The parton model (Feynman, 1969, 1972; Bjorken and Paschos, 1969) showed that elementary constituents, interacting weakly, could convincingly explain the central experimental results. In the same period, the quark model (Gell-Mann, 1964; Zweig, 1964; Kokkedee, 1969) rationalized hadron spectroscopy. Out of it grew the idea of color (Han and Nambu, 1965; Greenberg, 1964), a new quantum number postulated in the first instance to avoid the apparent paradox that the quark model seemed to require spin-1/2 quarks with bosonic statistics.
The idea of extending the global color model to a gauge theory (Fritzsch et al., 1973; Gross and Wilczek, 1973b; Weinberg, 1973) was in many ways a natural one [2] but the motivation for doing so was incalculably strengthened by the new-found ability to quantize gauge theories in a manner that was at once unitary and renormalizable [3] motivated in large part to describe electroweak interactions. Concurrently, the growth of the technology of the renormalization group and the operator product expansion (Wilson, 1969; Callan, 1970; Symanzik, 1970; Christ, Hasslacher, and Mueller, 1972; Frishman, 1974) made it clear that any field theory of the strong interactions would have to have an energy-dependent coupling strength, to harmonize the low-energy nature of the strong interactions, that give them their name, with their weakness at high energy (or short distances). The concept of asymptotic freedom (Gross and Wilczek, 1973a; Politzer, 1973), which is satisfied almost uniquely by quantum chromodynamics, brilliantly filled these demands.
As will appear in the following sections, asymptotic freedom is a perturbative concept. Yet, as searches for free quarks, let along gluons, continued to give null results, it became evident that the perturbation theory of quantum chromodynamics had to be approached somewhat differently than that of, say, quantum electrodynamics. The usual S matrix and cross sections for isolated quarks and gluons in QCD all vanish, completely replaced by bound-state dynamics. This is the hypothesis of "confinement". After some time it also became obvious that although asymptotic freedom is a perturbative prediction, confinement is not. The use of perturbation theory in quantum chromodynamics, that is, "perturbative QCD", or pQCD, therefore developed rather slowly and even haltingly, amid considerable scepticism. Nevertheless, many predictions of the theory, primarily but not exclusively associated with inclusive processes, do not depend upon its long-distance behavior. These short-distance predictions are realm of perturbative QCD.
Since QCD remains an "unsolved" theory, with no single approximation method applicable to all length scales, the justification for the use of perturbative QCD rests in large part directly on experiment. In this regard, many of us remember vividly the rapid transformation of quantum chromodynamics from a promising but controversial candidate theory to a full-fledged part of the Standard Model, taken perhaps too confidently for granted. In this transformation, the achievements of lattice-based numerical studies also played an important role (Wilson, 1974; Kogut and Susskind, 1975; Creutz, 1983).
Over time, it has become ever clearer that perturbative QCD naturally describes a large set of high energy, large-momentum-transfer cross sections. It is in this restrictive yet important area that its formalism has developed, and in which it has proved an invaluable tool in the study of the strong interactions. Beyond this, however, the very successes of a purely perturbative approach challenge us to bridge the gap between perturbative and nonperturbative aspects of the theory. Every experiment in strong interactions tests QCD from some fixed "short" distance to its very longest distance scales, over which the value of the strong coupling may change radically. From a mathematical point of view, as well, QCD has special features. As we shall outline below, many of the basic tests and predictions of the theory rely on arguments "to arbitrary order" in perturbation theory. Thus, the very role of perturbative expansions in four-dimensional quantum field theories is accessible in QCD as in no other component of the Standard Model. Perhaps the greatest legacy of QCD will be in the theoretical and experimental methods that must still be developed to meet its unique demands.
Our intention in this handbook is to review the basic ideas and methods of perturbative QCD, especially in those areas for which there is ample experimental verification. This work is meant to be a sourcebook on perturbative QCD, accessible and useful to experts and novices, experimentalists and theorists alike. In it, we have collected discussions of the basic ideas and applications of the theory. While we have no intention of replacing more scholarly presentations of field-theoretic techniques and experimental reviews, we have included in the next two sections and in the appendices considerable introductory material on the basic concepts of QCD, its perturbative treatment, and on the parton model, out of which it grew. In the fourth section, we summarize the basic theorems upon which the perturbative treatment rests. We hope that sophisticated readers will find useful the discussions, applications and experimental reviews of specific processes and techniques in the sections that follow. These are organized according to process, including electron-positron annihilation, deeply inelastic scattering and hadron-hadron cross sections, first those induced by electroweak interactions and then those induced by QCD itself. We conclude with a description of the "global" approach to nucleon parton distributions. For the simplest processes, we have exhibited theoretical predictions explicitly. Given the complexity of many recent results, this is not always possible, and we have relied in this case on references to the literature and, as is increasingly becoming relevant, to specialized computer programs.
This article is the product of the CTEQ collaboration [4] as a whole, consisting of both experimentalists and theorists, and we have not attempted to enforce on ourselves an artificial uniformity of presentation and style. We hope and believe, however, that readers will find below a coordinated and fundamentally unified text. We would also like to think of this as an evolving document, and in this initial version, concentrate on inclusive high-energy reactions, for which the most basic results and processes are treated in detail. Directions abound for expansion, particularly toward moderate energy, the perturbative-nonperturbative junction, and hadronic structure: elastic scattering [5], "small-x" evolution [6], pQCD in the Regge limit [7], Sudakov resummation techniques [8], asymptotic behavior [9], QCD coherence [10], QCD in nuclei [11], and transparency [12], on the (supporting) roles of pQCD in Monte Carlo simulations of event structure, QCD sum rules and heavy-quark effective theory, and much more.