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 $η_c$ fits the experimental data well, while the predicted energy difference $ΔM_p$ between the center of the gravity of $1^3P_{0,1,2}$ states and the $1^1P_1$ 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 $ΔM_p$, which we discuss, that are of comparable size ($\sim 1$ MeV) that should be included, before precise agreement with the data can be expected. The mass differences of the $ψ'-η_c'$, $Υ(1S)-η_b$, $Υ(2S)-η_b'$, and $B_c^*-B_c$ are also predicted.

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