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

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