Impact of correlations between $a_\mu$ and $\alpha_{QED}$ on the EW fit

We study the potential impact on the electroweak (EW) fits due to the tensions between the current determinations of the hadronic vacuum polarisation (HVP) contributions to the anomalous magnetic moment of the muon ($a_\mu$), based on either phenomenological dispersion integrals using measured hadronic spectra or on Lattice QCD calculations. The impact of the current tension between the experimental measurement of $a_\mu$ and the total theoretical prediction based on the phenomenological calculations of the HVP are also studied. The correlations between the uncertainties of the theoretical predictions of $a_\mu$ and of the running of $\alpha_{QED}$ are taken into account in the studies. We conclude that the impact on the EW fit can be large in improbable scenarios involving global shifts of the full HVP contribution, while it is much smaller if the shift is restricted to a lower mass range and/or if the shift in $\alpha_{QED}$ is obtained from that in $a_\mu$ through appropriate use of the correlations. Indeed, the latter scenarios only imply at most a 2.6/16 increase in the $\chi^2/n.d.f.$ of the EW fits and relatively small changes for the resulting fit parameter values.