Towards an improved understanding of $\eta \to \gamma^* \gamma^*$

We argue that high-quality data on the reaction $e^+e^-\to \pi^+\pi^-\eta$ will allow one to determine the doubly-virtual form factor $\eta\to \gamma^*\gamma^*$ in a model-independent way with controlled accuracy. This is an important step towards a reliable evaluation of the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon. When analyzing the existing data for $e^+e^-\to\pi^+\pi^-\eta$ for total energies squared $k^2>1\text{GeV}^2$, we demonstrate that the effect of the $a_2$ meson provides a natural breaking mechanism for the commonly employed factorization ansatz in the doubly-virtual form factor $F_{\eta\gamma^*\gamma^*}(q^2,k^2)$. However, better data are needed to draw firm conclusions.