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 $\Delta \nu (\mathrm{Mu}{)}_{\mathrm{HLBL}}$ 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, $\Delta \nu (\mathrm{Mu}{)}_{\mathrm{HLBL}}=-0.0065(10)\mathrm{Hz}$. Although the effect is tiny it is useful in understanding the level of hadronic uncertainties.

Figure 1

The hadronic vacuum polarization contribution to $a_{\mu }$.

Figure 2

The light-by-light scattering contribution to $a_{\mu }$.

Figure 3

Meson exchanges in the light-by-light scattering. Summation goes over interchanges of photons and over $C$-even neutral mesons.

Figure 4

The hadronic light-by-light scattering in the electron-muon interaction.

Figure 5

The characteristic diagram for the hadronic vacuum polarization contribution to the muonium HFS interval.

Figure 6

The vertical exchange of the pseudovector meson $a$ in $e\mu$ scattering.

Figure 7

The horizontal exchange of the pseudoscalar $\pi$ and pseudovector mesons $a$ in the $e\mu$ scattering.