Parallel scattering, saturation, and generalized Abramovskii-Gribov-Kancheli (AGK) theorem in the EPOS4 framework, with applications for heavy-ion collisions at $\sqrt{s_{NN}}$ of 5.02 TeV and 200 GeV

Ultrarelativistic heavy-ion collisions will first realize many nucleon-nucleon scatterings, happening instantaneously and therefore necessarily in parallel, due to the short collision time. An appropriate quantum mechanical tool to treat that problem is S-matrix theory, and it has been known for a long time how to derive a simple geometric probabilistic picture, still widely used, and here the Abramovskii-Gribov-Kancheli (AGK) theorem plays a crucial role. All this is done in a scenario where energy conservation is not taken care of, but this is needed, in particular for Monte Carlo simulations. When introducing energy-momentum sharing properly, the AGK theorem does not apply anymore, nor do simple geometric concepts such as binary scaling. I will discuss this (very serious) problem, and how it can be solved, in the EPOS4 framework. When connecting the multiple Pomeron approach (for parallel scatterings) and perturbative QCD, one is actually forced to implement in a very particular way saturation scales, in order to get an approach free of contradictions. One recovers a generalized AGK theorem (gAGK), valid at large $p_{t}$ (larger than the relevant saturation scales). I discuss how gAGK is related to factorization (in proton-proton scatterings) and binary scaling (in heavy-ion collisions). I will show some applications, using this new approach as an initial condition for hydrodynamical evolutions, for heavy-ion collisions at $\sqrt{s_{NN}}$ of 5.02 TeV and 200 GeV, to get some idea about the energy dependence.