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We compute the leading QED corrections to the hadronic vacuum polarization(HVP) of the photon, relevant for the determination of leptonic anomalousmagnetic moments, $a_\ell$. We work in the electroquenched approximation anduse dynamical QCD configurations generated by the CLS initiative with twodegenerate flavors of non-perturbatively O($a$)-improved Wilson fermions. Weconsider QED$_{\rm L}$ and QED$_{\rm M}$ to deal with the finite-volume zeromodes. We compare results for the Wilson loops with exact analyticaldeterminations. In addition we make sure that the volumes and photon massesused in QED$_{\rm M}$ are such that the correct dispersion relation isreproduced by the energy levels extracted from the charged pions two-pointfunctions. Finally we compare results for pion masses and the HVP betweenQED$_{\rm L}$ and QED$_{\rm M}$. For the vacuum polarization, corrections withrespect to the pure QCD case, at fixed pion masses, turn out to be at thepercent level.

Electromagnetic corrections to the hadronic vacuum polarization of the photon within QED$_{\rm L}$ and QED$_{\rm M}$