pNRQCD: review of selected results
Non-relativistic bound-state systems are characterized by, at least, three widely separated scales: the mass m of the particle, the (soft) scale associated to its relative momentum ∼ mv, v ≪ 1, and the (ultrasoft) scale associated to its kinetic energy ∼ mv. In QED and in the perturbative regime of QCD the velocity v of the particle in the bound state may be identified with the coupling constant. Moreover, the inverse of the size of the system is also of order mv and the binding energy of order mv. Indeed, a systematic treatment of nonrelativistic bound-state systems in the framework of effective field theories (EFT), which takes full advantage of the above energy scale hierarchy, was initiated in QED 1 and in more recent years remarkable progress has been achieved in the analysis of tt̄ threshold production . For systems made of b and c quarks (I will denote them generically as heavy quarkonia: ψ, Υ, Bc, ...) non-perturbative contributions may be relevant. By comparing the energy level spacings of these systems (see Fig. 1) with the heavy-quark masses (e.g. mb ≃ 5 GeV and mc ≃ 1.6 GeV) we can still argue that the data are consistent with a kinetic energy of the bound quark much smaller than the heavy-quark mass and, therefore, with a non-relativistic (NR) description of the heavy-quark–antiquark system. However, in dependence of the specific system, the scale of non-perturbative physics, ΛQCD, may turn out to be close to some of its dynamical scales. The physical picture, which then arises, may be quite different from the perturbative situation. What remains guaranteed, also for heavy quarkonia, is that m≫ ΛQCD and that at least the mass scale can be treated perturbatively, i.e. integrated out from QCD order by order in the coupling constant. The resulting EFT is called NRQCD . A lot of effort has been put over the last two decades in order to find the relevant operators, which parameterize the non-perturbative heavy-quark–