Radiative-recoil corrections to muonium and positronium hyperfine splitting

We present a detailed description of the calculation of radiative-recoil corrections to the ground-state hyperfine splitting in muonium and positronium to order ${\alpha }^2{E}_F$, the results of which were previously reported. All these corrections involve two-photon exchanges with one-loop radiative corrections to either a photon or a lepton. The QED vacuum polarization corrections are evaluated completely analytically to the order of interest. The hadronic contributions are estimated and found to be very small. As a preliminary to the lepton line calculation, a compact expression is derived for the radiative correction to such lines. This factor is then applied in a number of different contexts: the recalculation of the old nonrecoil result, which is known analytically; the analytic evaluation of terms of order ${\alpha }^2\left(\frac{m_e}{m_{\mu }}\right)\ln \left(\frac{m_{\mu }}{m_e}\right)E_F$, which arise from the electron leg; and the nonlogarithmic terms from both lines, which require numerical calculations. The muonium results are ${\nu }_{{\mu }^{+}{e}^{-}}={\left(\frac{\alpha }{\pi }\right)}^2\left(\frac{m_e}{m_{\mu }}\right)[-2\mathrm{}{\ln }^2(\frac{m_{\mu }}{m_e})+\frac{13}{12}\ln (\frac{m_{\mu }}{m_e})+18.18\mathrm{}\pm 0.58]E_F$ and those for positronium are ${\nu }_{e^{+}{e}^{-}}={\alpha }^2(-1.788\mathrm{}\pm 0.004)E_F$, where $E_F$ for positronium does not include the annihilation contribution.