A Theoretical Review of Heavy Quarkonium Inclusive Decays

Heavy quarkonia (charmonium, bottomonium, ...) provide an ideal set of observables to probe properties of low-energy QCD in a controlled way. The reason is the following. Heavy quarkonia are non-relativistic bound states and, therefore, characterized by a set of energy scales hierarchically ordered: m, mv, mv, ... where m is the heavy-quark mass and v ≪ 1 the relative heavy-quark velocity. For heavy quarkonia,m is much larger than the scale of non-perturbative physics, ΛQCD, and, therefore, degrees of freedom associated with that scale can be treated perturbatively and calculations done order by order in αs. The non-relativistic hierarchy of scales also survives below ΛQCD. Therefore, for any heavy quarkonium state the low-energy dynamics is organized in matrix elements ordered in powers of v (and, in general, ΛQCD/m). To any given order in αs and v, only a finite number of Feynman diagrams and matrix elements respectively have to be calculated. The way to implement rigorously these expansions in QCD is provided by the non-relativistic effective field theories (EFTs) of QCD. The first has been NonRelativistic QCD, NRQCD1,2. It is obtained from QCD by integrating out degrees of freedom of energy m. NRQCD still contains the lower energy scales as dynamical degrees of freedom. In the last few years, the problem of integrating out the remaining dynamical scales has been addressed by several groups and has now reached a solid level of understanding (lists of references may be found in3). The ultimate EFT obtained by subsequent matchings from QCD, where only the lightest degrees of freedom of energy mv are left dynamical, is potential NRQCD, pNRQCD4,5. This EFT is close to a quantum-mechanical description of the bound system and,