Study of the Couplings of QED and QCD from the Adler Function

The contribution from hadronic vacuum polarisation effects is responsible for a large fraction of the theoretical uncertainty in the running of the QED coupling. The current level of uncertainty has become a limitation for electroweak precision tests. We use lattice QCD simulations with two flavours of O$(a)$ improved Wilson fermions to determine the Adler function in a broad range of the momentum transfer $Q^2$. The running of the QED coupling, including valence contributions from $u$, $d$, $s$ and $c$ quarks, is compared to phenomenological results at intermediate $Q^2$ values. In the large $Q^2$ regime, the lattice determination of the Adler function is fitted to perturbation theory in order to examine the feasibility of a determination of the strong coupling constant.