Heavy quarkonium decays and transitions in the language of effective field theories

Heavy quarkonium decays and transitions are discussed in the framework of nonrelativistic effective field theories. Emphasis is put on the matching procedure in the nonperturbative regime. Some exact results valid for the magnetic dipole couplings are discussed. INTRODUCTION In the last years the B factories, CLEO and BES have produced a large amount of new data for heavy quarkonium observables [1]. These data are not only interesting because they may signal new states or new decay or production mechanisms, but also because heavy quarkonium is a system that to a large extent can rigorously be studied in QCD. Therefore, any new understanding of it may potentially provide new insight on the nonperturbative dynamics of QCD. Heavy quarkonium, as a non-relativistic bound state, is characterized by a hierarchy of energy scales: m, mv and mv2, where m is the heavy-quark mass and v ≪ 1 the heavy-quark relative velocity. Whenever a system is described by a hierarchy of scales, observables may be calculated by expanding one scale with respect to the other. An effective field theory (EFT) is a field theory that makes this expansion explicit at the Lagrangian level. To be more precise, let us call H a system described by a fundamental Lagrangian L and suppose it characterized by 2 scales: Λ ≫ λ . The EFT Lagrangian, LEFT , suitable to describe H at scales lower than Λ, is characterized by (1) a cut off Λ ≫ μ ≫ λ ; (2) some degrees of freedom that exist at scales lower than μ . The Lagrangian LEFT is then made of all operators On that may be built from the effective degrees of freedom and are consistent with the symmetries of the original Lagrangian L : LEFT = ∑ n cn(Λ/μ) On(μ,λ ) Λn . (1) The advantage is that, once the scale μ has been run down to λ , the power counting is homogeneous 〈On〉 ∼ λ n, so that the EFT is, indeed, organized as an expansion in λ/Λ. Despite the EFT not being renormalizable in the traditional sense, it is renormalizable order by order in λ/Λ. The matching coefficients cn(Λ/μ) encode the non-analytic behaviour in Λ. They are calculated by imposing that LEFT and L describe the same physics at any finite order in the expansion. The procedure is called matching. Finally, we note that if Λ ≫ ΛQCD, cn(Λ/μ) may be calculated in perturbation theory, if Λ ∼ ΛQCD, the matching must rely on non-perturbative methods. Several effective field theories for heavy quarkonium that take full advantage of the non-relativistic hierarchy of scales have been developed and used over the last decade. For a recent review we refer to [2]. NRQCD is the EFT that exploits the hierarchy Λ = m ≫ λ = mv [3, 4]. Since m ≫ ΛQCD, the matching coefficients of NRQCD may be calculated in perturbation theory. pNRQCD is the EFT that exploits the hierarchy Λ = mv ≫ λ = mv2 [5, 6]. If ΛQCD ∼ mv2, then the matching to pNRQCD may be still done in perturbation theory. We call weak coupling this regime, which may be suited to describe ground-state quarkonium. If ΛQCD ∼ mv, then the matching to pNRQCD is non perturbative. We call strong coupling this regime, which may be suited to describe excited quarkonium states. The fact that EFTs may be built to describe heavy quarkonium in the strong-coupling regime follows from the observation that the non-relativistic hierarchy of scales survives also below ΛQCD [7, 8]. The complication of the strong-coupling regime comes from the non-perturbative matching and from new scales that may arise in loops sensitive to ΛQCD. An example is the three-momentum scale √ mΛQCD discussed in [9]. Nevertheless, many advantages remain in treating even strongly-coupled heavy quarkonium in an EFT framework. In the following we shall sketch a unified framework for the description of inclusive and electromagnetic decays, and radiative transitions in the framework of strongly coupled pNRQCD. For a treatment of inclusive and electromagnetic decay widths in the weak-coupling regime we refer to [2] and references therein. For a treatment of magnetic dipole transitions in the weak-coupling regime we refer to [10]. NRQCD NRQCD is the EFT that follows from QCD when modes of energy or momentum m are integrated out. The structure of the EFT Lagrangian is like Eq. (1) with Λ = m and λ = mv ∼ ΛQCD. The scale mv is sometimes called soft. The degrees of freedom of the EFT Lagrangian are quarks, antiquarks and gluons with energy and momentum lower than m (we neglect light quarks). The NRQCD Lagrangian may be written as LNRQCD = L2− f +L4− f +Llight , (2) where L2− f = ψ† (