The Euclidean Adler Function and its Interplay with $Δα^{\mathrm{had}}_{\mathrm{QED}}$ and $α_s$
Three different approaches to precisely describe the Adler function in the Euclidean regime at around $2\, \mathrm{GeVs}$ 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 $Δα^{\mathrm{had}}_{\mathrm{QED}}(Q^2)$, 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.