More nonperturbative corrections to the fine and hyperfine splitting in the heavy quarkonium

Abstract

The leading nonperturbative effects to the fine and hyperfine splitting were calculated some time ago. Recently, they have been used in order to obtain realistic numerical results for the lower levels in bottomonium systems. We point out that a contribution of the same order $O({\Lambda }_{QCD}^4/m^3{\alpha }_s^2)$ has been overlooked. We calculate it in this paper.

Comments: 9 pages, LaTeX, More self-contained and lengthier version without changing physical outputs. To be published in Phys. Rev. D