New Results on Inclusive Quarkonium Decays

Considered as bound states, heavy quarkonia (e.g. Υ, χb, ψ, χc, ...) have two main characteristics: 1) a mass scale, which is in the perturbative regime: mb ≃ 5 GeV, mc ≃ 1.5 GeV; 2) level splittings of a relative size that is typical of non-relativistic systems. Being non-relativistic bound states, heavy quarkonia are characterized by at least three hierarchically ordered energy scales: the hard scale m, the momentum scale mv and the bound-state energy scale mv2, where v ≪ 1 is the heavy-quark velocity inside the bound state. 1 The different scales entering the quarkonium dynamics may be systematically integrated out, leading from QCD to simpler but equivalent effective field theories (EFTs). NRQCD is the EFT obtained by integrating out the hard scale m. 1,2 This m being larger than the scale of nonperturbative physics, ΛQCD, the matching to NRQCD can be done order by order in αs. Hence, the NRQCD Lagrangian can be written as a sum of terms like fnO (dn) n /md−4, ordered in powers of αs and v. More specifically, the Wilson coefficients fn are series in αs(m) and encode the ultraviolet physics that has been integrated out from QCD. The operators O (dn) n of dimension dn describe the low-energy dynamics and are counted in powers of v. Heavy quarkonium decays are processes that take place at an energy-transfer scale of the order of the heavy-quark mass. Their signature is encoded in the imaginary part of the Wilson coefficients of the 4-fermion operators (O (dn) n = ψKnχχ K ′ nψ) in the NRQCD Lagrangian. The NRQCD factorization formula for quarkonium inclusive decay widths into light hadrons (LH) reads 2