Constraints on the hadronic light-by-light tensor in corner kinematics for the muon $g-2$

The dispersive approach to the hadronic light-by-light contribution to the muon $g-2$ involves an integral over three virtual photon momenta appearing in the light-by-light tensor. Building upon previous works, we systematically derive short-distance constraints in the region where two momenta are large compared to the third, the so-called Melnikov-Vainshtein or corner region. We include gluonic corrections for the different scalar functions appearing in the Lorentz decomposition of the underlying tensor, and explicitly check analytic agreement with alternative operator product expansions in overlapping regimes of validity. A very strong pattern of cancellations is observed for the final $g-2$ integrand. The last observation suggests that a very compact expression only containing the axial current form factors can provide a good approximation of the corner region of the hadronic light-by-light tensor.