  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 $g-2$.
