Hadronic light-by-light scattering in muonium hyperfine splitting

We consider an impact of hadronic light-by-light scattering on the muonium hyperfine structure. A shift of the hyperfine interval Δν(Mu)HLBL is calculated with the light-by-light scattering approximated by the 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, Δν(Mu)HLBL=-0.0065(10)Hz. Although the effect is tiny it is useful in understanding the level of hadronic uncertainties.