  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.

High Energy Physics - Phenomenology;High Energy Physics - Experiment