Infrared behavior of the static potential in perturbative QCD

The static singlet potential between a heavy quark and an antiquark is an object of considerable interest both theoretically and phenomenologically. Strictly speaking, the potential being a dynamical quantity, it can be defined only in a context where the full dynamics of the system is consistently taken into account, for instance, in the proper Schrödinger equation. Nevertheless, the static singlet potential is usually defined as the logarithm of the static Wilson loop divided by the interaction time in the limit of infinite interaction time. Moreover, the above definition, in QCD, suffers from infrared ~IR! divergences when computed at finite orders of perturbation theory @1#. This has to do with the non-Abelian nature of QCD, which allows massless gluons to self-interact at arbitrarily small energy scales. The problem was posed in the past @1# of whether the static singlet potential could be defined in some way at any order in perturbation theory. The leading IR singularities in the static Wilson loop were found at a relative order as 3 in @1#. These singularities can indeed be regulated upon resummation of a certain class of diagrams which give rise to a dynamical cutoff provided by the difference between the singlet and the octet potentials. However, the extraction of the static singlet quark-antiquark (Q-Q̄) potential from this resummation has been regarded as suspect by other authors @2#, who noticed that the difference between the singlet and the octet potential provides a dynamical scale which, at least in perturbation theory, is of the same order of the kinetic energy for quarks of large but finite mass. A similar situation occurs in QED for the Lamb shift, where one of the contributions has to be interpreted as a nonpotential correction to the hydrogen atom energy levels. The novel feature in QCD is that such an effect appears to be related to the definition of the static potential, while in QED it is of order 1/m. No quantitative procedure has been developed so far to deal with this problem in QCD.