Hyperfine splittings in heavy-quark systems

The hyperfine splittings in heavy-quark systems are calculated to fourth order in the strong coupling constant. The $\psi -{\eta }_c$ mass difference gives ${\Lambda }_{\overline{\mathrm{MS}}}=0.16\mathrm{}\pm 0.02\pm 0.07$ GeV ($\overline{\mathrm{MS}}$ refers to the modified minimal-subtraction scheme) where the first uncertainty is experimental and the second is a crude estimate of the uncertainty arising from scheme dependence. From the $\Upsilon -{\eta }_b$ mass difference we expect a more reliable and accurate determination of $\Lambda$.