Study of the hadronic contributions to the running of the QED coupling and the weak mixing angle

The electromagnetic coupling receives significant contributions to its running from non-perturbative QCD effects. We present an update of a lattice QCD study of the Adler function and of its application to the determination of the running of the QED coupling. We perform a high-statistics computation with two flavours of O$(a)$ improved Wilson fermions in a large range of momentum transfer $Q^2$. The running of the electromagnetic coupling, including contributions from $u$, $d$, $s$ and $c$ valence quarks, is compared to phenomenological determinations at intermediate $Q^2$ values. An extension of this study to the determination of the hadronic contributions to the running of the weak mixing angle is also described.