  We compute a theoretically driven prediction for the hadronic contribution to
the electromagnetic running coupling at the $Z$ scale using lattice QCD and
state-of-the-art perturbative QCD. We
obtain$$\Delta\alpha^{(5)}(M^2_Z)=\left[279.5\pm0.9\pm0.59\right]\times10^{-4}\quad\quad\,\,\,\,\,\,(\mathrm{Mainz
\,\,\,Collaboration})$$$$\Delta\alpha^{(5)}(M^2_Z)=\left[278.42\pm0.22\pm0.59\right]\times10^{-4}\,\,\,\,\,\,\,\,\quad(\mathrm{
BMW \,\,\,Collaboration}),$$
  where the first error is the quoted lattice uncertainty. The second is due to
perturbative QCD, and is dominated by the parametric uncertainty on
$\hat{\alpha}_s$, which is based on a rather conservative error. Using instead
the PDG average, we find a total error on $\Delta\alpha^{(5)}(M^2_Z)$ of
$0.4\times10^{-4}$. Furthermore, with a particular emphasis on the charm quark
contributions, we also update $\Delta\alpha^{(5)}(M^2_Z)$ when low-energy
cross-section data is used as an input, obtaining $\Delta\alpha^{(5)}(M^2_Z) =
\left[276.29 \pm 0.38 \pm 0.62\right] \times 10^{-4}$. The difference between
lattice QCD and cross-section-driven results reflects the known tension between
both methods in the computation of the anomalous magnetic moment of the muon.
Our results are expressed in a way that will allow straightforward
modifications and an easy implementation in electroweak global fits.

High Energy Physics - Phenomenology;High Energy Physics - Lattice