The Static Quark-Antiquark Potential in QCD to Three Loops

The static potential between an infinitely heavy quark and antiquark is derived in the framework of perturbative QCD to three loops by performing a full calculation of the two-loop diagrams and using the renormalization group. The contribution of massless fermions is included. The force law between infinitely heavy quarks has been investigated since more than 20 years because of its importance for a deeper understanding of the strong interactions. The static quark-antiquark potential is a very fundamental concept, constituting the non-abelian analog to the Coulomb potential of electrodynamics, and also enters as a vital ingredient in the description of non-relativistic bound states like quarkonia. It is widely believed to consist of two parts: a Coulombic term at short distances which can be derived from field theory by using perturbative QCD, and a long-ranged confining term whose derivation from first principles presumably requires much more advanced methods. Although an analysis based on perturbation theory The complete paper, including figures, is also available via anonymous ftp at ttpux2.physik.uni-karlsruhe.de (129.13.102.139) as /ttp96-37/ttp96-37.ps, or via www at http://www-ttp.physik.uni-karlsruhe.de/cgi-bin/preprints alone thus cannot give the complete potential, the result of such an effort would nevertheless be very useful. It could provide an improved input for QCD inspired potential models or even describe very heavy systems to a reasonable accuracy by itself. It could be compared with the potential obtained from numerical studies using lattice gauge theory and it might also give some hints on the nonperturbative regime. The first investigation of the static perturbative QCD potential has been performed in [1]. Although this work has been extended by several groups shortly after [2, 3, 4, 5], and some of these also studied aspects of the two[2] and even three-loop diagrams [4], there is still no full calculation of the two-loop diagrams available. The purpose of the present paper is to fill this gap and hence, by exploiting the renormalization group equation, to obtain the three-loop potential. Before turning to the actual analysis, let us first recall the calculational procedure employed. It seems appropriate to begin with the simplest case, the Abelian theory without massless fermions. The static potential in QED can be defined in a way which makes its gauge invariance manifest via the vacuum expectation value of a Wilson loop taken about a rectangle of width R and length T R: V (R) = − lim T→∞ 1 iT ln〈P exp ( ie ∮ dxμA μ ) 〉 (1) where P denotes the path ordering prescription. The functional integral can be calculated exactly, and one indeed finds the Coulomb potential plus an additional term which represents the self-energy of the sources [2]. To compare with the non-Abelian theory it is, however, useful to go through the perturbative analysis as well. The Feynman rules for the source are as follows: a source-photon vertex corresponds to a factor iev, with an additional minus sign for the “anti-source”, and the source “propagator” reads SF (x− x ) = −iΘ(x0 − x ′ 0)δ(x− x ) (2) in coordinate space or SF (p) = 1 vp+ iε (3) in momentum space. The four vector v is given by v = (1,0) and has only been introduced for notational reasons. The appearance of a propagator for