Towards an improved understanding of $$\varvec{\eta \rightarrow \gamma ^*\gamma ^*}$$ η → γ ∗ γ ∗

We argue that high-quality data on the reaction $$e^+e^-\rightarrow \pi ^+\pi ^-\eta $$ will allow one to determine the doubly-virtual form factor $$\eta \rightarrow \gamma ^*\gamma ^*$$ 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 $$e^+e^-\rightarrow \pi ^+\pi ^-\eta $$ 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.