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Following the publication of the new measurement of the anomalous magneticmoment of the muon, the discrepancy between experiment and the theoryprediction from the $g-2$ theory initiative has increased to $4.2\,\sigma$.Recent lattice QCD calculations predict values for the hadronic vacuumpolarization contribution that are larger than the data-driven estimates,bringing the Standard Model prediction closer to the experimental measurement.Euclidean time windows in the time-momentum representation of the hadronicvacuum polarization contribution to the muon $g-2$ can help clarify thediscrepancy between the phenomenological and lattice predictions. We presentour calculation of the intermediate distance window contribution using$N_\mathrm{f}=2+1$ flavors of O$(a)$ improved Wilson quarks. We employensembles at six lattice spacings below $0.1\,$fm and pion masses down to thephysical value. We present a detailed study of the continuum limit, using twodiscretizations of the vector current and two independent sets of improvementcoefficients. Our result at the physical point displays a tension of$3.9\,\sigma$ with a recent evaluation of the intermediate window based on thedata-driven method.

Intermediate window observable for the hadronic vacuum polarization contribution to the muon $g-2$ from O$(a)$ improved Wilson quarks