Field-theoretic versus data-driven evaluations of electromagnetic corrections to hadronic vacuum polarization in $(g-2)_\mu$

The Standard Model prediction of the muon $g-2$ increasingly depends on lattice QCD computations of the hadronic vacuum polarization (HVP), where the isospin-breaking (IB) effects remain a significant source of uncertainty. To complement the lattice QCD evaluations, the data-driven approach to HVP has been used to assess some of the electromagnetic IB effects, in particular from the channels with a photon in the final state, e.g., $e^+e^-\to\pi^0 \gamma$. Here we argue that such contributions are largely canceled by virtual electromagnetic corrections to the purely hadronic channels: $\pi^+ \pi^-$, $\pi^+ \pi^- \pi^0$, etc. We identify these leading corrections by performing a field-theoretical calculation in a vector-meson dominance model, thereby reconciling the timelike and spacelike approaches to electromagnetic effects. Although these virtual corrections are more difficult to extract in a systematic manner, addressing them is essential for the data-driven method to consistently complement the lattice QCD program.