Photoproduction of QED bound states in future electron-ion colliders

In this work we perform an exploratory study of the photoproduction of singlet QED bound states $(l^-l^+)_S$ in electron - ion collisions at the EicC, EIC and LHeC energies. The total cross - sections, event rates per year and rapidity distributions associated with the parapositronium, paramuonium and paratauonium production are estimated. Moreover, we consider the decay of these states in a two - photon system and implement kinematical cuts on the rapidities and energies of the photons in the final state. We demonstrate the paramuonium can, in principle, be observed for the first time in all these colliders and that the EIC is a potential collider to discover the paratauonium state.