  Lattice QCD calculations of the hadronic vacuum polarization (HVP) have
reached a precision where the electromagnetic (e.m.) correction can no longer
be neglected. This correction is both computationally challenging and hard to
validate, as it leads to ultraviolet (UV) divergences and to sizeable infrared
(IR) effects associated with the massless photon. While we precisely determine
the UV divergence using the operator-product expansion, we propose to introduce
a separation scale $\Lambda\sim400\;$MeV into the internal photon propagator,
whereby the calculation splits into a short-distance part, regulated in the UV
by the lattice and in the IR by the scale $\Lambda$, and a UV-finite
long-distance part to be treated with coordinate-space methods, thereby
avoiding power-law finite-size effects altogether. In order to predict the
long-distance part, we express the UV-regulated e.m. correction to the HVP via
the forward hadronic light-by-light (HLbL) scattering amplitude and relate the
latter via a dispersive sum rule to $\gamma^*\gamma^*$ fusion cross-sections.
Having tested the relation by reproducing the two-loop QED vacuum polarization
(VP) from the tree-level $\gamma^*\gamma^*\to e^+e^-$ cross-section, we predict
the expected lattice-QCD integrand resulting from the
$\gamma^*\gamma^*\to\pi^0$ process.
