  Gluon-gluon to photon-photon scattering $gg \to \gamma\gamma$ offers to the
LHC experiments a uniquely powerful probe of dimension-8 operators in the
Standard Model Effective Field Theory (SMEFT) that are quadratic in both the
electromagnetic and gluonic field-strength tensors, such as would appear in the
Born-Infeld extension of the Standard Model (SM). We use 13-TeV ATLAS data on
the production of isolated photon pairs to set lower limits on the scales of
dimension-8 operators $M \gtrsim 1$~TeV, and discuss the prospective
sensitivities of possible future hadron colliders.
