The energy dependency (running) of the strength of electromagnetic interactions and of the mixing with weak interactions plays an important role in precision tests of the SM. The running of the former to the Z pole is an input quantity for global electroweak fits, while the running of the mixing angle is susceptible to the effects of BSM physics, particularly at low energies.
I will discuss a recent computation of the hadronic vacuum polarization (HVP) contribution to the running of these electroweak couplings at the non-perturbative level in lattice QCD, in the space-like regime up to momentum transfers of 7 GeV^2. The result of the lattice computation shows a tension of up to 3.5 standard deviations with estimates based on the R-ratio in the 3 to 7 GeV^2range. The tension is, however, strongly diminished when translating our result to the Z pole, by employing the Euclidean split technique and perturbative QCD.
This quantity is closely related with the HVP contribution to the g-2 of the muon, and the observed tension is consistent with the one recently obtained by multiple lattice collaborations in the so-called intermediate window of the muon g-2 HVP. This hints to a scenario in which the hadronic cross-section has to be larger mainly at low energies to reproduce the lattice results.