  The recent experimental observation of Light-by-Light (LbL) scattering at the
Large Hadron Collider has revived interest in this fundamental process, and
especially of the accurate prediction of its cross-section, which we present
here for the first time at Next-to-Leading Order (NLO) in both QCD and QED. We
compare two radically different computational approaches, both exact in the
fermion mass dependence, thus offering a strong cross-check of our results. The
first approach is a fully analytic method to calculate compact and
well-organized two-loop helicity amplitudes. The second one is entirely
numerical and leverages the Local Unitarity construction. Our two calculations
agree with each other and conclude that including the exact fermion mass
contribution typically increases the size of the NLO corrections. Moreover, we
find that the exact result converges slowly to the massless limit of the
high-energy regime, thus emphasizing the importance of including the full mass
dependence at NLO. We also compare our results with the ATLAS measurement of
LbL in ultra-peripheral lead-lead collisions, and find that the inclusion of
exact NLO corrections reduces, but does not eliminate, the existing tension
with theoretical predictions.
