Light-by-light scattering at future $$e^+e^-$$ e + e - colliders

We study the sensitivity of possible CLIC and FCC-ee measurements of light-by-light scattering to old and new physics, including the Heisenberg–Euler Lagrangian in the Standard Model with possible contributions from loops of additional charged particles or magnetic monopoles, the Born–Infeld extension of QED, and effective dimension-8 operators involving four electromagnetic field strengths as could appear in the Standard Model Effective Field Theory. We find that FCC-ee measurements at 365 GeV and CLIC measurements at 350 GeV would be sensitive to new physics scales of half a TeV in the dimension-8 operator coefficients, and that CLIC measurements at 1.4 TeV or 3 TeV would be sensitive to new physics scales $$\sim 2$$ TeV or 5 TeV at 95% CL, corresponding to probing loops of new particles with masses up to $$\sim 3.7$$ TeV for large charges and/or multiple species. Within Born–Infeld theory, the $$95\%$$ CL sensitivities would range from $$\sim 300$$  GeV to 1.3 or 2.8 TeV for the high-energy CLIC options. Measurements of light-by-light scattering would not exclude monopole production at FCC-hh, except in the context of Born–Infeld theory.