Perturbative contributions to Δα5MZ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Delta {\\alpha}^{(5)}\\left({M}_Z^2\\right) $$\\end{document}

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 obtainΔα5MZ2=279.5±0.9±0.59×10−4Mainz CollaborationΔα5MZ2=278.42±0.22±0.59×10−4BMWCollaboration,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\displaystyle \\begin{array}{cc}\\Delta {\\alpha}^{(5)}\\left({M}_Z^2\\right)=\\left[279.5\\pm 0.9\\pm 0.59\\right]\ imes {10}^{-4}& \\left(\ extrm{Mainz}\\ \ extrm{Collaboration}\\right)\\\\ {}\\Delta {\\alpha}^{(5)}\\left({M}_Z^2\\right)=\\left[278.42\\pm 0.22\\pm 0.59\\right]\ imes {10}^{-4}& \\left(\ extrm{BMW}\\ \ extrm{Collaboration}\\right),\\end{array}} $$\\end{document}where the first error is the quoted lattice uncertainty. The second is due to perturbative QCD, and is dominated by the parametric uncertainty on α̂s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\hat{\\alpha}}_s $$\\end{document}, which is based on a rather conservative error. Using instead the PDG average, we find a total error on Δα5MZ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Delta {\\alpha}^{(5)}\\left({M}_Z^2\\right) $$\\end{document} of 0.4 × 10−4. Furthermore, with a particular emphasis on the charm quark contributions, we also update Δα5MZ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Delta {\\alpha}^{(5)}\\left({M}_Z^2\\right) $$\\end{document} when low-energy cross-section data is used as an input, obtaining Δα5MZ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Delta {\\alpha}^{(5)}\\left({M}_Z^2\\right) $$\\end{document} = [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.