The Euclidean Adler function and its interplay with Δα_QED^had and α s

Three different approaches to precisely describe the Adler function in the Euclidean regime at around 2 GeVs are available: dispersion relations based on the hadronic production data in e + e − annihilation, lattice simulations and perturbative QCD (pQCD). We make a comprehensive study of the perturbative approach, supplemented with the leading power corrections in the operator product expansion. All known contributions are included, with a careful assessment of uncertainties. The pQCD predictions are compared with the Adler functions extracted from Δα_QED^had ( Q 2 ), using both the DHMZ compilation of e + e − data and published lattice results. Taking as input the FLAG value of α s , the pQCD Adler function turns out to be in good agreement with the lattice data, while the dispersive results lie systematically below them. Finally, we explore the sensitivity to α s of the direct comparison between the data-driven, lattice and QCD Euclidean Adler functions. The precision with which the renormalisation group equation can be tested is also evaluated.