The QCD potential at O(1/m^2): Complete spin-dependent and spin-independent result

TL;DR

The paper addresses deriving a complete non-perturbative QCD potential for heavy quarkonia at $O(1/m^2)$ using NRQCD and pNRQCD, focusing on pure gluodynamics. It provides a Hamiltonian formulation of gluonic excitations, and expresses the heavy-quarkonium potential in terms of Wilson loops with field-strength insertions, including detailed spin-dependent and spin-independent pieces. The work corrects earlier results (notably the Eichten–Feinberg expressions) and clarifies the ultraviolet behavior and power counting in the non-perturbative regime, offering formulas that can be evaluated on the lattice or via QCD vacuum models. These results deepen the connection between QCD dynamics and the potential picture, enabling more precise studies of quarkonium spectra and vacuum structure in the non-perturbative domain.

Abstract

Within an effective field theory framework, we obtain an expression, with O(1/m^2) accuracy, for the energies of the gluonic excitations between heavy quarks, which holds beyond perturbation theory. For the singlet heavy quark--antiquark energy, in particular, we also obtain an expression in terms of Wilson loops. This provides, twenty years after the seminal work of Eichten and Feinberg, the first complete expression for the heavy quarkonium potential up to O(1/m^2) for pure gluodynamics. Several errors present in the previous literature (also in the work of Eichten and Feinberg) have been corrected. We also briefly discuss the power counting of NRQCD in the non-perturbative regime.