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 α²EF, 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 α²EF(mₑm_μ)ln(m_μmₑ). Nonlogarithmic radiative-recoil corrections obtained by numerical evaluation of this expression yield new contributions of 2.64 ±{} 0.07 kHz for muonium and -11.12 ±{} 0.02 MHz positronium.
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Sapirstein et al. (1983) studied this question.
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