Quantitative evaluation of first-order retardation corrections to the quarkonium spectrum

We evaluate numerically first-order retardation corrections for some charmonium and bottomonium masses under the usual assumption of a Bethe-Salpeter purely scalar confinement kernel. The result depends strictly on the use of an additional effective potential to express the corrections (rather than to resort to Kato perturbation theory) and on an appropriate regularization prescription. The kernel has been chosen in order to reproduce in the instantaneous approximation a semirelativistic potential suggested by the Wilson loop method. The calculations are performed for two sets of parameters determined by fits in potential theory. The corrections turn out to be typically of the order of a few hundred MeV and depend on an additional scale parameter introduced in the regularization. A conjecture existing in the literature on the origin of the constant term in the potential is also discussed.