Hadronic Vacuum Polarization Contribution of Order a Za

In the case of muonic hydrogen rur in (3.30) is the muon-proton reduced mass. 

Due to the analogy between contributions of the diagrams with muon and hadron vacuum polarizations, it is easy to see that insertion of hadron vacuum polarization in one of the exchanged photons in the skeleton diagrams with two-photon exchanges generates a correction of order (x Zotf (see Fig. 7.12). Calculation of this correction is straightforward. One may even take into account the composite nature of the proton and include the proton form factors in photon-proton vertices. Such a calculation was performed in [51, 52] and produced a very small contribution 

3 Contribution of Order cx Zcx) m Induced by Insertion of the Hadron Polarization in the Radiative Photon 

The muon mass is only slightly lower than the pion mass, and we should expect that insertion of hadronic vacuum polarization in the radiative photon in Fig. 7.13 will give a contribution to the anomalous magnetic moment comparable with the contribution induced by insertion of the muon vacuum polarization. 

Respective corrections are written via the slope of the Dirac form factor and the anomalous magnetic moment exactly as in Subsect. 7.3.4. The only difference is that the contributions to the form factors are produced by the hadronic vacuum polarization. 

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