The Hyperfine Spin Splittings In Heavy Quarkonia

The hyperfine spin splittings in heavy quarkonia are studied using the recently developed renormalization group improved spin-spin potential which is independent of the scale parameter μ. The calculated energy difference between the J/ψ and the ηc fits the experimental data well, while the predicted energy difference ∆Mp between the center of the gravity of 1 P0,1,2 states and the 1P1 state of charmonium has the correct sign but is somewhat larger than the experimental data. This is not surprising since there are several other contributions to ∆Mp, which we discuss, that are of comparable size (∼ 1 MeV) that should be included, before precise agreement with the data can be expected. The mass differences of the ψ − η c, Υ(1S)− ηb, Υ(2S)− η ′ b, and B c −Bc are also predicted. PACS numbers: 12.39.Pn, 12.38.Cy, 12.38.Lg, 12.39.Hg