Radiative-recoil corrections to muonium and positronium hyperfine splitting

Abstract

We present a detailed description of the calculation of radiative-recoil corrections to the ground-state hyperfine splitting in muonium and positronium to order ${\ensuremath{\alpha}}^{2}{E}_{F}$, 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 ${\ensuremath{\alpha}}^{2}(\frac{{m}_{e}}{{m}_{\ensuremath{\mu}}})\mathrm{ln}(\frac{{m}_{\ensuremath{\mu}}}{{m}_{e}}){E}_{F}$, which arise from the electron leg; and the nonlogarithmic terms from both lines, which require numerical calculations. The muonium results are ${\ensuremath{\nu}}_{{\ensuremath{\mu}}^{+}{e}^{\ensuremath{-}}}={(\frac{\ensuremath{\alpha}}{\ensuremath{\pi}})}^{2}(\frac{{m}_{e}}{{m}_{\ensuremath{\mu}}})[\ensuremath{-}2{\mathrm{ln}}^{2}(\frac{{m}_{\ensuremath{\mu}}}{{m}_{e}})+\frac{13}{12}\mathrm{ln}(\frac{{m}_{\ensuremath{\mu}}}{{m}_{e}})+18.18\ifmmode\pm\else\textpm\fi{}0.58]{E}_{F}$ and those for positronium are ${\ensuremath{\nu}}_{{e}^{+}{e}^{\ensuremath{-}}}={\ensuremath{\alpha}}^{2}(\ensuremath{-}1.788\ifmmode\pm\else\textpm\fi{}0.004){E}_{F}$, where ${E}_{F}$ for positronium does not include the annihilation contribution.