Hyperfine splitting in positronium and muonium

The authors calculate corrections to the ground state hyperfine structure of relative order α2ln α-1 in positronium and relative order (msub(e)/msub(μ))α2 ln α-1 in muonium due to the exchange of virtual photons. Their results are in agreement with those of Lepage. Contributions arising from the Coulomb potential and from the exchange of one transverse photon along with any number of ladder Coulomb photons are discussed in detail. In treating the single transverse photon-multiple Coulomb photon exchanges, the contributions involving different numbers of Coulomb photons are summed and resulting expression is reexpanded in terms of a quantity that is inherently smaller than the Coulomb potential in the non-relativistic region. This procedure enables the authors to take into account from the beginning important cancellations that occur between the various terms in an expansion in powers of the Coulomb potential. The techniques developed here may be useful in calculating higher order corrections. (Auth.)