  The $\gamma^\star \gamma^\star \to \pi \pi$ scattering amplitude plays a key
role in a wide range of phenomena, including understanding the inner structure
of scalar resonances as well as constraining the hadronic contributions to the
anomalous magnetic moment of the muon. In this work, we explain how the
infinite-volume Minkowski amplitude can be constrained from finite-volume
Euclidean correlation functions. The relationship between the finite-volume
Euclidean correlation functions and the desired amplitude holds up to energies
where $3\pi$ states can go on shell, and is exact up to exponentially small
corrections that scale like $\mathcal{O}(e^{-m_\pi L})$, where $L$ is the
spatial extent of the cubic volume and $m_\pi$ is the pion mass. In order to
implement this formalism and remove all power-law finite volume errors, it is
necessary to first obtain $\pi \pi \to \pi\pi$, $\pi \gamma^\star \to \pi$,
$\gamma^\star \to\pi\pi$, and $\pi\pi\gamma^\star \to\pi\pi$ amplitudes; all of
which can be determined via lattice quantum chromodynamic calculations.
