NRQCD, Effective Field Theories and Potential Models

The study of the heavy quark–antiquark system is an old topic (see [1] for earlier references). Here we will concentrate on recent developments based on effective field theories. For large enough masses, these systems can be considered to be non-relativistic (NR) and are then characterized by, at least, three widely separated scales: hard (the mass m, of the heavy quarks), soft (the relative momentum of the heavy-quark–antiquark |p| ∼ mv, v ≪ 1), and ultrasoft (US, the typical kinetic energy E ∼ mv of the heavy quark in the bound state system). In 1986, NRQED [2], an effective field theory for non-relativistic leptons, was presented, providing the first and decisive link in a chain of developments that is still growing. NRQED is obtained from QED by integrating out the hard scale m (see [3] for some applications). It is characterized by an ultraviolet cut-off much smaller than the mass m and much larger than any other scale. NRQCD [4] was born soon afterwards. NRQCD has proved to be extremely successful in studying Q-Q̄ systems near threshold. The Lagrangian of NRQCD can be organized in powers of 1/m, thus making explicit the non-relativistic nature of the physical systems. In order for effective field theories to be useful, a power counting is needed. The first power counting (organized in powers of v and αs) used to assess the relative importance of the NRQCD matrix elements followed from arguments valid in the perturbative regime [5]. In particular, it was assumed that ΛQCD ∼ mv . This implies that dynamics is mainly perturbative, with v ∼ αs, dictated by the solution of the Schrödinger