The LHeC and the FCC-he will measure DIS cross sections and the partonic structure of protons and nuclei in an unprecedented range of small $𝑥$. In this kinematic region the non-linear dynamics expected in the high energy regime of QCD should be relevant in a region of small coupling. In this talk we will demonstrate the unique capability of these high-energy colliders for unravelling dynamics beyond fixed-order perturbation theory, proving the non-linear regime of QCD, saturation, to exist (or to disprove). This is enabled through the simultaneous measurements, of similar high precision and range, of $𝑒𝑝$ and $e$A collisions which will eventually disentangle nonlinear parton-parton interactions from nuclear environment effects.
Reference: P. Agostini et al. (LHeC Study Group), The Large Hadron-Electron Collider at the HL-LHC, J. Phys. G 48 (2021) 11, 110501, e-Print: 2007.14491 [hep-ex].