Long-range spin-orbit potential, Gromes's consistency condition, Richardson's potential, and properties of heavy quarkonium systems

Abstract

The energies, fine-structure splittings, leptonic widths, and dipole electromagnetic transition rates are calculated for the $\ensuremath{\Upsilon}$ and charmonium systems using a potential model based on Richardson's interpolating form for the running coupling constant. Using the definitions of the spin-dependent potentials of Eichten and Feinberg and arguments based on lattice pauge calculations, we have determined the long-range spin-orbit potential from Gromes's consistency condition. It is also shown that the sign difference between the long-range spin-orbit potential and the perturbative spin-orbit potential is important in explaining the measured values of the fine-structure ratio in both the $\ensuremath{\Upsilon}$ and charmonium $P$ states. The question of whether the parameter $\ensuremath{\Lambda}$ of Richardson's potential is supposed to play the role of the universal QCD scale is addressed. It is argued that the agreement of the $\ensuremath{\Upsilon}$ and charmonium leptonic widths with experiment is convincing support for Richardson's form for the running coupling constant since these numbers reflect directly the values of the various $S$ state wave functions at the origin. Certain problems with the magnitudes of the hyperfine splittings and dipole transition rates of charmonium are discussed.