Perturbative contributions to $\Delta {\alpha }^{(5)}(M_Z^2)$

Abstract

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$$\Delta {\alpha }^{(5)}(M_Z^2)=\left[279.5\pm 0.9\pm 0.59\right]\times 10^{-4}(\mathrm{Mainz}\mathrm{Collaboration})$$$$\Delta {\alpha }^{(5)}(M_Z^2)=\left[278.42\pm 0.22\pm 0.59\right]\times 10^{-4}(\mathrm{BMW}\mathrm{Collaboration}),$$ 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 $\Delta {\alpha }^{(5)}(M_Z^2)$ of $0.4\times 10^{-4}$. Furthermore, with a particular emphasis on the charm quark contributions, we also update $\Delta {\alpha }^{(5)}(M_Z^2)$ when low-energy cross-section data is used as an input, obtaining $\Delta {\alpha }^{(5)}(M_Z^2)=\left[276.29\pm 0.38\pm 0.62\right]\times 10^{-4}$. 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.