Higher $\eta_c(nS)$ and $\eta_b (nS)$ mesons

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/ψ − η c (1 S ) and ψ (2 S ) − η ′ c (2 S ) splittings and appears to be small, r ss ∼ = 0 . 06 fm. Nevertheless, even with such a small radius an essential suppression of the hyperfine splittings ( ∼ 50%) is observed in bottomonium. For the nS b ¯ b states ( n = 1 , 2 , . . . , 6) we predict the values (in MeV) 28, 12, 10, 6, 6, and 3, respectively. For the 3 S and 4 S charmonium states the splittings 16(2) MeV and 12(4) MeV are obtained.