The Euclidean Adler Function and its Interplay with $\Delta\alpha^{\mathrm{had}}_{\mathrm{QED}}$ and $\alpha_s$

TL;DR

This paper compares three methods—dispersion relations, lattice simulations, and perturbative QCD—for accurately describing the Euclidean Adler function around 2 GeV, assessing uncertainties and the impact of the strong coupling constant.

Contribution

It provides a comprehensive analysis of the perturbative QCD approach with power corrections, comparing it with lattice and dispersive data, and evaluates the sensitivity to the strong coupling constant.

Findings

pQCD predictions agree well with lattice data when using FLAG's alpha_s

Dispersive results are systematically lower than lattice and pQCD

The study assesses the precision of testing the renormalisation group equation

Abstract

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 $\Delta\alpha^{\mathrm{had}}_{\mathrm{QED}}(Q^2)$, using both the DHMZ compilation of $e^+e^-$ data and published lattice results. Taking as input the FLAG value of $\alpha_s$, 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.