Leading isospin breaking effects in the HVP contribution to a${}_{\mu }$ and to the running of α

The anomalous magnetic moment of the muon aμ and the running of the electromagnetic coupling α play a fundamental role in beyond Standard Model (SM) physics searches. Non-perturbative hadronic contributions to both quantities, which are related to the hadronic vacuum polarization (HVP) function consisting of two electromagnetic currents, are a main source of uncertainty in the SM prediction. We compute the HVP function in lattice QCD+QED applying the time-momentum representation method. We expand the relevant correlation functions around the isosymmetric limit. In particular, we focus on leading isospin breaking effects taking quark-connected contributions into account, which we evaluate on isosymmetric N${}_f$=2+1 QCD gauge ensembles generated by the CLS initiative with non-perturbatively O(a)-improved Wilson fermions.