The ratio of reduced cross-sections in eA processes at Electron-Ion Colliders at x_(min)=Q²/s

We study the predictions of saturation effects in electron-ion colliders at high inelasticity, using a generalization for nuclear targets of the ASW and GBW models where the saturation scale, $Q_{\mathrm{sat}}$, drives the energy dependence and the corresponding nuclear effects. We expect to observe an enhancement of the ratio of nuclear reduced cross sections in the saturation region in future electron-ion colliders. The ratio $R^{A}_{\sigma_{r}}$ is discussed in the kinematic range of the electron-Ion collider with center-of-mass energy $\sqrt{s}=89~\mathrm{GeV}$ at high inelasticity $y{=}1$. The importance of the nuclear longitudinal structure function $F^{A}_{L}$ in the ratio $R^{A}_{\sigma_{r}}$ for the heavy nucleus lead and light nucleus deuteron at $x_{\mathrm{min}}=Q^2/s$ is discussed. This enhancement in the range $Q^2\sim(1-4~\mathrm{GeV}^2)$ in the ratio $R^{A}_{\sigma_{r}}$ is not observed in the ratio $R^{A}_{F_{2}}$ which is comparable with the nuclear ratio of the nuclear parton distribution functions. We demonstrate that the study of nuclear charm structure function allows us to estimate the magnitude of shadowing effects in high inelasticity in the nuclear gluon distribution.