Higher ${\eta }_c(nS)$ and ${\eta }_b(nS)$ mesons

Abstract

The hyperfine splittings in heavy quarkonia are studied in a model-independent way using the experimental data on di-electron widths. Relativistic correlations are taken into account together with the smearing of the spin-spin interaction. The radius of smearing is fixed by the known $J/\psi -{\eta }_c(1S)$ and $\psi (2S)-{\eta }_c^{\prime }(2S)$ splittings and appears to be small, $r_{ss}\cong 0.06$ fm. Nevertheless, even with such a small radius an essential suppression of the hyperfine splittings ($\sim 50$ is observed in bottomonium. For the $nSb\bar{b}$ states $(n=1,2,...,6)$ we predict the values (in MeV) 28, 12, 10, 6, 6, and 3, respectively. For the $3S$ and $4S$ charmonium states the splittings 16(2) MeV and 12(4) MeV are obtained.

Comments: 23 pages