Heavy Quarks in Quantum Chromodynamics

A pedagogical introduction to the heavy quark theory is given. It is explained that various expansions in the inverse heavy quark mass 1=mQ present a version of the Wilson operator product expansion in QCD. A systematic approach is developed and many practically interesting problems are considered. I show how the 1=mQ expansions can be built using the background eld technique and how they work in particular applications. Interplay between perturbative and nonperturbative aspects of the heavy quark theory is discussed. An extended version of the lectures given at Theoretical Advanced Study Institute QCD and Beyond, University of Colorado, Boulder, Colorado, June 1995. 1 Lecture 1. Heavy Quark Symmetry The statement that Quantum Chromodynamics (QCD) is the theory of hadrons has become common place. It is a very strange theory, since many questions concerning dynamics of the quarks and gluons at large distances { however simple they might seem { remain unanswered or, at best, understood only at a qualitative level. Progress in the direction of the quantitative description of the hadronic properties is slow { every step bringing us closer to such a description is painfully di cult. At the same time new results, even modest, have a special weight for obvious reasons { QCD, unlike many other trendy theories in the modern high energy physics, de nitely has a direct relation to Nature and will stay with us forever. Every hadron in a sense is built from quarks and/or gluons. I say \in a sense" because these are no ordinary building blocks. The number of degrees of freedom uctuates and is not xed; this we know for sure. At large distances we have to deal with a genuine strongly coupled eld theory, and, as usual, the strong coupling creates complicated structures which can not be treated by perturbative methods. Then we feel helpless and are ready to use every opportunity, no matter where it comes from, if only it gives the slightest hope of getting a solid quantitative approach based on QCD. QCD has two faces, two components { hard and soft. The hard component is the realm of perturbative QCD. Not much will be said in these lectures about this aspect. Instead, we will concentrate on the soft component. Many years ago, at the dawn of the QCD era, it was noted [1] that heavy quarks are, probably, the best probe of the soft component of the gluon elds out of all probes we have at our disposal. The developments we witnessed in recent years con rm this conclusion. The dynamics of soft degrees of freedom in QCD is the realm of non-perturbative phenomena. Having said this I hasten to add that there is an element of luck { transition from the perturbative regime to the non-perturbative one is very abrupt in QCD. In a sense the gauge coupling constant is abnormally small. I do not mean here the conventional logarithmic suppression of the running constant but, rather, the fact that b, the rst coe cient in the Gell-Mann-Low function, is numerically large. This fact allows us to forget, in the rst approximation, about perturbative e ects and focus on non-perturbative ones in a wide range of problems. It is more exact to say that we will concentrate on studying the soft degrees of freedom, but due to the fortunate circumstance of \abnormal" smallness of s( )= for as low normalization point as 1 GeV, all e ects due to the soft degrees of freedom are essentially non-perturbative. I will elucidate the precise meaning of this statement later. It would be great if we could just switch o { by adjusting some parameter { all hard processes in QCD without changing its soft component. Then we would be left with the con ning dynamics in a clean and uncontaminated form; formulation of the theory would be much easier. The only parameter which might do the job is b. If we could tend b ! 1 with QCD xed, the hard gluons would be suppressed