Hyperfine Splitting in Non-Relativistic Bound States

Abstract

We study the mass difference between the spin singlet and spin triplet states of positronium and heavy quarkonium, an effect which is referred to as the hyperfine splitting. For positronium, a bound state of an electron and a positron, we analyze the one-loop nonrelativistic effective Hamiltonian in d-dimensions. Our result constitutes an important part of the analysis in [1], which studies positronium’s hyperfine splitting to order alpha to the seventh, and substantially reduces the overall theoretical uncertainty. This is crucial for comparing high precision predictions of quantum electrodynamics with the results of modern experiments. For quarkonium, a non-relativistic flavourless quark-antiquark bound-state, we set up a matching procedure between the perturbative analysis of the short-distance interactions and the nonperturbative lattice analysis of the long-distance effects. In particular, our result is used in [2], and it corrects an error in the previous matching calculation of Ref. [3], which was subsequently used in the analyses [4] and [5]. Combined with the one-loop perturbative lattice calculation, our result brings theory and experiment into agreement and effectively solves the eta-b mass puzzle.