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 α²EF, 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 α²(mₑm_μ)ln(m_μmₑ)EF, which arise from the electron leg; and the nonlogarithmic terms from both lines, which require numerical calculations. The muonium results are ${{ν}}_{{{μ}}⁺{e}^{{-}}}={({{α}}{{π}})}²({{m}ₑ}{{m}_{{μ}}})[{-}2{ln}²({{m}_{{μ}}}{{m}ₑ})+13/12ln({{m}_{{μ}}}{{m}ₑ})+18.18±{}0.58]{E}F$ and those for positronium are ${{ν}}_{{e}⁺{e}^{{-}}}={{α}}²({-}1.788±{}0.004){E}F$, where ${E}F$ for positronium does not include the annihilation contribution.