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One of the main goals of future electron-ion colliders is to improve ourunderstanding of the structure of hadrons. In this letter, we study theexclusive $\eta_c$ production by $\gamma^{*} \gamma$ interactions in $eA$collisions and demonstrate that future experimental analysis of this processcan be used to improve the description of the $\eta_c$ transition form factor.The rapidity, transverse momentum and photon virtuality distributions areestimated considering the energy and target configurations expected to bepresent at the EIC, EicC and LHeC and assuming different predictions for thelight-front wave function of the $\eta_c$ meson. Our results indicate that theelectron-ion colliders can be considered an alternative to providingsupplementary data to those obtained in $e^- e^+$ colliders.

Exclusive $\eta_c$ production by $\gamma^{*} \gamma$ interactions in electron-ion collisions