We compute the leading $QED$ corrections to the hadronic vacuum polarization (HVP) of the photon, relevant for the determination of leptonic anomalous magnetic moments. We work in the electroquenched approximation and use dynamical $QCD$ configurations generated by the CLS initiative with two degenerate flavors of non-perturbatively O($a$) improved Wilson fermions. We consider $QED_{\rm L}$ and $QED_{\rm M}$ to deal with the finite-volume zero modes. We compare results for the Wilson loops with exact analytical determinations. In addition we make sure that the volumes and photon masses used in $QED_{\rm M}$ are such that the correct dispersion relation is reproduced by the energy levels extracted from the charged pions two-point functions. Finally we compare results for pion masses and the HVP between $QED_{\rm L}$ and $QED_{\rm M}$. For the vacuum polarization, corrections with respect to the pure $QCD$ case, at fixed pion masses, turn out to be at the percent level.

Hadron Structure