Applications of Nonrelativistic Effective Field Theories to quarkonium systems with a small radius

We are interested here in bound states composed only by heavy quarks (and gluons). In such cases, at least for the lowest states, the characteristic radius r (r being the qq̄ distance) of the system is small, typically too small to probe the confinement effects, which arise at a scale 1/ΛQCD ≫ r. In the following I will call such systems ’Coulombic’ or ‘quasi-Coulombic’. These situations are particularly interesting not only because they have phenomenological relevance, but especially because they allow us to understand much more about QCD. In such cases both the mass scale m and the soft scale 1/r are much bigger than ΛQCD and thus still sit in the perturbative regime. Non-perturbative corrections exist but are not expected to be dominant. As it is apparent from the spectra, heavy quarkonia are non-relativistic systems. Thus, they may be described in first approximation using a Schrödinger equation with a potential interaction. This amounts to saying that the heavy quark bound state is characterized by three energy scales, hierarchically ordered by the quark velocity v ≪ 1: the quark mass m (hard scale), the momentum mv ≃ 1/r (soft scale (S)), and the binding energy mv (ultrasoft scale (US)). In the Coulombic or quasi-Coulombic situation it is v ∼ αs. To address the multiscale dynamics of the heavy quark bound state, the concept of effective field theory turns out to be not only helpful but actually necessary. QCD effective field theories (EFT) with less and less degrees of freedom, can be obtained by systematically integrating out the scales above the energy we aim to describe. This procedure leads ultimately to a field theory derived quantum mechanical description of these systems. The corresponding EFT is called pNRQCD [1]. Here, all the dynamical regimes are organized in a systematic expansion in the ratio of the different scales which is eventually an expansion in v. In practice, by integrating out the hard scale NRQCD is obtained from QCD. After integrating out the soft scale in NRQCD, pNRQCD is obtained. The Lagrangian of pNRQCD is organized in powers of 1/m and r (multipole expansion). The matching is done by comparing appropriate off-shell amplitudes in NRQCD and in pNRQCD, order by order in 1/m, αs and order by order in the multipole expansion. The matching coefficients are non-analytic functions of r and have typically the following structure: V ≃ V(r,p,S1,S2)(A lnmr +B lnμ r + C).