Hadronic Light-by-Light Scattering in the Muonium Hyperfine Splitting
  We consider an impact of hadronic light-by-light scattering on the muonium hyperfine structure. A shift of the hyperfine interval $\Delta \nu({\rm Mu}) _{\rm\tiny HLBL}$ is calculated with the light-by-light scattering approximated by exchange of pseudoscalar and pseudovector mesons. Constraints from the operator product expansion in QCD are used to fix parameters of the model similar to the one used earlier for the hadronic light-by-light scattering in calculations of the muon anomalous magnetic moment. The pseudovector exchange is dominant in the resulting shift, $\Delta \nu({\rm Mu})_{\rm\tiny HLBL}= -0.0065(10) {Hz}$. Although the effect is tiny it is useful in understanding the level of hadronic uncertainties. 
S.G. Karshenboim, V.A. Shelyuto, and A.I. Vainshtein