Computing the UV-finite electromagnetic corrections to the hadronic vacuum polarization in the muon $(g-2)$ from lattice QCD

In order to reach a precision of $0.2\%$ on the hadronic vacuum polarization (HVP) contribution to the anomalous magnetic moment of the muon, $(g-2)_\mu$, such that the full Standard-Model prediction matches in precision the direct experimental measurement, it is crucial to include the leading, O($\alpha$) electromagnetic corrections to HVP. In this work, we determine an important contribution to the latter from a diagram comprised of two two-point quark-loops, connected by the internal photon and gluons. This ultraviolet-finite correction is calculated from lattice QCD using a coordinate-space formalism, where photons are treated in the continuum and infinite volume. Our result amounts to a $(-0.89\pm 0.18)\%$ correction to the leading-order HVP contribution to $(g-2)_\mu$. To overcome the worsening statistical noise at large distances, our analysis is combined with phenomenological models featuring light pseudoscalar mesons with masses below 1 GeV. In particular, our data show a very steep mass dependence as dictated by the charged pion loop. Similarly, the other diagrams appearing in the O$(\alpha)$-corrections to the HVP with the internal photon connecting two separate quark loops are also ultraviolet-finite and can be computed with the same formalism.