The Euclidean Adler Function and its Interplay with $\mathrm{\Delta }{\alpha }_{\mathrm{Q}\mathrm{E}\mathrm{D}}^{\mathrm{h}\mathrm{a}\mathrm{d}}$ and ${\alpha }_s$

Three different approaches to precisely describe the Adler function in the Euclidean regime at around $2\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{s}$ 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 $\mathrm{\Delta }{\alpha }_{\mathrm{Q}\mathrm{E}\mathrm{D}}^{\mathrm{h}\mathrm{a}\mathrm{d}}(Q^2)$, using both the DHMZ compilation of $e^{+}{e}^{-}$ data and published lattice results. Taking as input the FLAG value of ${\alpha }_s$, 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 ${\alpha }_s$ 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.