Abstract
A compact expression for one-loop radiative corrections to lepton lines in positronium and muonium hyperfine splitting is presented. It is valid for hyperfine-splitting contributions of order ${\ensuremath{\alpha}}^{2}{E}_{\mathrm{F}}$, including the effects of recoil to all orders. For muonium, known nonrecoil parts of this contribution are easily evaluated analytically, as is a previously calculated radiative-recoil contribution of order ${\ensuremath{\alpha}}^{2}{E}_{\mathrm{F}}(\frac{{m}_{e}}{{m}_{\ensuremath{\mu}}})\mathrm{ln}(\frac{{m}_{\ensuremath{\mu}}}{{m}_{e}})$. Nonlogarithmic radiative-recoil corrections obtained by numerical evaluation of this expression yield new contributions of 2.64 \ifmmode\pm\else\textpm\fi{} 0.07 kHz for muonium and -11.12 \ifmmode\pm\else\textpm\fi{} 0.02 MHz positronium.
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