  The hadronic light-by-light contribution to the muon anomalous magnetic
moment depends on an integration over three off-shell momenta squared ($Q_i^2$)
of the correlator of four electromagnetic currents and the fourth leg at zero
momentum. We derive the short-distance expansion of this correlator in the
limit where all three $Q_i^2$ are large and in the Euclidean domain in QCD.
This is done via a systematic operator product expansion (OPE) in a background
field which we construct. The leading order term in the expansion is the
massless quark loop. We also compute the non-perturbative part of the
next-to-leading contribution, which is suppressed by quark masses, and the
chiral limit part of the next-to-next-to leading contributions to the OPE. We
build a renormalisation program for the OPE. The numerical role of the
higher-order contributions is estimated and found to be small.
