On the two-photon contributions to $e^+e^-\to\eta\gamma$ and $e^+e^-\to\eta^\prime\gamma$

Motivated by recent BABAR measurements of the $\gamma^*\to\gamma \eta$ and $\gamma^*\to\gamma \eta^\prime$ transition form factors, we estimate two-photon exchange contributions to the corresponding cross sections. By using a phenomenological model, based on the vector meson dominance, it is argued that the expected contributions are small enough not to effect the BABAR results. As a by product we predict $Br(\eta^\prime\to\mu^+\mu^-)= (1.4\pm 0.2)\times 10^{-7}$. Our results might be useful also in high precision calculations of radiative corrections to the Dalitz decays of pseudoscalar mesons.