We show that some nonrelativistic quantum chromodynamics color-octet matrix elements can be written in terms of (derivatives of) wave functions at the origin and of nonperturbative universal constants once the factorization between the soft and ultrasoft scales is achieved by using an effective field theory where only ultrasoft degrees of freedom are kept as dynamical entities. This allows us to derive a new set of relations between inclusive heavy-quarkonium P-wave decays into light hadrons with different principal quantum numbers and with different heavy flavors. In particular, we can estimate the ratios of the decay widths of bottomonium P-wave states from charmonium data. DOI: 10.1103/PhysRevLett.88.012003 PACS numbers: 12.38. -t, 12.39.Hg, 13.25.Gv Inclusive P-wave decays to light hadrons have proved to be an optimal testing ground of our understanding of heavy quarkonia. The use of nonrelativistic quantum chromodynamics (NRQCD) [1,2] allowed a description of these decays in terms of expectation values of some 4-heavy-quark operators at a quantum-field level in a systematic way. Besides the so-called color-singlet operators, for which their expectation values could be related to wave functions in an intuitive way, there were also color-octet operators. The latter were decisive in solving the infrared sensitivity of earlier calculations [3]. It has been thought so far that these color-octet expectation values could not be related to a Schrödinger-like formulation in any way.We show in this Letter that it is not so. For certain states, the expectation values of color-octet operators can also be written in terms of wave functions and additional bound-state-independent nonperturbative parameters. We will focus on the operators relevant to P-wave decays into light hadrons, but it should become apparent that this is a general feature.The line of developments that has led us to this result is the following. It was pointed out in Ref.[4] that NRQCD still contains dynamical scales, which are not relevant to the kinematical situation of the lower-lying states in heavy quarkonium (energy scales larger than the ultrasoft scale, my 2 , with y being the relative velocity of the heavy quark and m being its mass). Hence, further simplifications occur if we integrate them out. We call potential NRQCD (pNRQCD) the resulting effective field theory [as in [5]; note that in [6], in the situation L QCD ¿ my 2 , the effective field theory (EFT) was called pNRQCD 0 ]. When the typical scale of nonperturbative physics, say, L QCD , is smaller than the soft scale my, and larger than the ultrasoft scale my 2 , the soft scale can be integrated out perturbatively. This leads to an intermediate EFT that also contains, besides the singlet, octet fields and ultrasoft gluons as dynamical degrees of freedom [4,6]. These are eventually integrated out by the (nonperturbative) matching to pNRQCD [6]. When L QCD is of the order of the soft scale, the (nonperturbative) matching to pNRQCD has to be done in one single step. This framework has ...

We derive the imaginary part of the potential nonrelativistic QCD ͑pNRQCD͒ Hamiltonian up to order 1/m 4 , when the typical momentum transfer between the heavy quarks is of the order of ⌳ QCD or greater, and the binding energy E much smaller than ⌳ QCD . We use this result to calculate the inclusive decay widths into light hadrons, photons and lepton pairs, up to O"mv 3 ϫ(⌳ QCD 2 /m 2 ,E/m)… and O(mv 5 ) times a short-distance coefficient, for S-and P-wave heavy quarkonium states, respectively. We achieve a large reduction in the number of unknown nonperturbative parameters and, therefore, we obtain new model-independent QCD predictions. All the NRQCD matrix elements relevant to that order are expressed in terms of the wave functions at the origin and six universal nonperturbative parameters. The wave-function dependence factorizes and drops out in the ratio of hadronic and electromagnetic decay widths. The universal nonperturbative parameters are expressed in terms of gluonic field-strength correlators, which may be fixed by experimental data or, alternatively, by lattice simulations. Our expressions are expected to hold for most of the charmonium and bottomonium states below threshold. The calculations and methodology are explained in detail so that the evaluation of higher order NRQCD matrix elements in this framework should be straightforward. An example is provided.

We present analytic expressions for the vacuum polarization effects due to a light fermion with finite mass in the binding energy and in the wave function at the origin of QED and (weak coupling) QCD non-relativistic bound states. Applications to exotic atoms, Υ(1s) and tt production near threshold are briefly discussed.

The various dynamical scales below the pion mass involved in ϩ Ϫ atoms are sequentially integrated out using nonrelativistic effective field theory techniques. This allows us to systematically organize the corrections to the energy levels and decay width. We present our results in terms of a single unknown constant which may be obtained by matching to the chiral Lagrangian with electromagnetic interactions at two loops.