Light-by-Light Scattering Single-Logarithmic Corrections to Hyperfine   Splitting in Muonium
  We consider three-loop radiative-recoil corrections to hyperfine splitting in muonium generated by the gauge invariant set of diagrams with virtual light-by-light scattering block. These corrections are enhanced by the large logarithms of the electron-muon mass ratio. We present the results of an analytic calculation of the single-logarithmic radiative-recoil corrections of order $\alpha^2(Z\alpha)(m/M)E_F$ to hyperfine splitting in muonium generated by these diagrams. 
Michael I. Eides and Valery A. Shelyuto
hep-ph physics.atom-ph