Perturbative contributions to $$ \Delta {\alpha}^{(5)}\left({M}_Z^2\right) $$

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$$ {\displaystyle \begin{array}{cc}\Delta {\alpha}^{(5)}\left({M}_Z^2\right)=\left[279.5\pm 0.9\pm 0.59\right]\times {10}^{-4}& \left(\textrm{Mainz}\ \textrm{Collaboration}\right)\\ {}\Delta {\alpha}^{(5)}\left({M}_Z^2\right)=\left[278.42\pm 0.22\pm 0.59\right]\times {10}^{-4}& \left(\textrm{BMW}\ \textrm{Collaboration}\right),\end{array}} $$ Δ α 5 M Z 2 = 279.5 ± 0.9 ± 0.59 × 10 − 4 Mainz Collaboration Δ α 5 M Z 2 = 278.42 ± 0.22 ± 0.59 × 10 − 4 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 $$ {\hat{\alpha}}_s $$ α ̂ s , which is based on a rather conservative error. Using instead the PDG average, we find a total error on $$ \Delta {\alpha}^{(5)}\left({M}_Z^2\right) $$ Δ α 5 M Z 2 of 0.4 × 10−4. Furthermore, with a particular emphasis on the charm quark contributions, we also update $$ \Delta {\alpha}^{(5)}\left({M}_Z^2\right) $$ Δ α 5 M Z 2 when low-energy cross-section data is used as an input, obtaining $$ \Delta {\alpha}^{(5)}\left({M}_Z^2\right) $$ Δ α 5 M Z 2 = [276.29 ± 0.38 ± 0.62] × 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.