2-loop short-distance constraints for the $g-2$ HLbL

The recent experimental measurement of the muon $g-2$ at Fermilab National Laboratory, at a $4.2\sigma$ tension with the Standard Model prediction, highlights the need for further improvements on the theoretical uncertainties associated to the hadronic sector. In the framework of the operator product expansion in the presence of a background field, the short-distance behaviour of the hadronic light-by-light contribution was recently studied. The leading term in this expansion is given by the massless quark-loop, which is numerically dominant compared to non-perturbative corrections. Here, we present the perturbative QCD correction to the massless quark-loop and estimate its size numerically. In particular, we find that for scales above 1 GeV it is relatively small, in general roughly $-10\%$ the size of the massless quark-loop. The knowledge of these short-distance constraints will in the future allow to reduce the systematic uncertainties in the Standard Model prediction of the hadronic light-by-light contribution to the $g-2$.