Precision studies of quantum electrodynamics at future $e^+e^-$ colliders

We classify the possible deviations from the Standard Model in the QED-dominated $\mathrm{e^+e^-}\rightarrow\gamma\gamma$ process under the assumption of a preserved $SU(2)_L \times U(1)_Y$ symmetry. We find that the only deviations really observable in practice correspond to a correction of the differential cross section by a factor $(1 + \frac{c_8~s^2}{8\pi\alpha\Lambda^4}~\sin^2\theta)$, where $\Lambda$ is the scale of new physics, $\theta$ is the polar angle of any of the final state photons and $c_8$ is a constant of order 1. We also provide sensitivity estimates for QED deviations at future $\mathrm{e^+e^-}$ facilities. An $\mathrm{e^+e^-}$ collider operating at $\sqrt{s}=3$ TeV could provide sensitivity to $\Lambda$ scales as large as 15 TeV, provided that acceptances and efficiencies are controlled at the per mille level. Finally, we also discuss the possibility of a measurement of the luminosity at the FCC-ee with $\lesssim 10^{-4}$ precision, using analyses of the $\mathrm{e^+e^-}\rightarrow\gamma\gamma$ process at $\sqrt{s}\approx m_\mathrm{Z}$ energies.