The Renormalization Group Improvement of the QCD Static Potentials

We resum the leading ultrasoft logs of the singlet and octet static QCD potentials within potential NRQCD. We then obtain the complete three-loop renormalization group improvement of the singlet QCD static potential. The discrepancies between the perturbative evaluation and the lattice results at short distances are slightly reduced. Preprint submitted to Elsevier Preprint 2 March 2008 Recently, there has been quite a lot of effort in identifying the large logs that arise from the different scales involved in heavy quark systems near threshold [1–7]. The proper renormalization group (RG) resummation of these logs is a non-trivial issue. It has recently been addressed within the so-called vNRQCD approach [8] for the O(1/m) potentials [9], and for the O(1/m) potential and the production current [10]. Since the proper resummation of the logs may be important for the physics of the top quark production near threshold, as well as for heavy quarkonium systems, and the existing calculations (see [2]) are done in a formalism closer to potential NRQCD (pNRQCD) [11,4], it would be desirable to know how to RG-improve within the latter formalism 1 . In this note we provide the first step towards this goal by showing how the static potentials of QCD can be RG-improved within that framework. Moreover, the RG-improved static potentials obtained in this paper represent a new result by themselves. They provide the complete three-loop RG evolution of the static potentials within an expansion in αs (in order to be so, besides our calculation, one needs to know the static potentials at two loops [12,13]). Our results may also be relevant in order to understand the discrepancies, at relatively short distances, between the perturbative evaluation and the lattice results. Since we are only interested in the static potentials, we only need to consider the static limit of NRQCD [14] and pNRQCD, i.e. we only need to work at leading order in 1/m. The matching between NRQCD and pNRQCD in the static limit, in the situation where ΛQCD ≪ 1/r (the limit we will consider in this paper), has been worked out in detail in ref. [4]. The pNRQCD lagrangian at leading order in 1/m and next-to-leading order in the multipole expansion reads