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 ${\alpha }^2{E}_{\mathrm{F}}$, 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 ${\alpha }^2{E}_{\mathrm{F}}\left(\frac{m_e}{m_{\mu }}\right)\ln \left(\frac{m_{\mu }}{m_e}\right)$. 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.

• Received 19 July 1983

DOI:https://doi.org/10.1103/PhysRevLett.51.982