Quantum correlation dynamics and in-medium 3$\leftrightarrow$3 collisions of fermions

In this study we aim for quantifying the role of in-medium 3$\leftrightarrow$3 collisions for systems of fermions which initially are out-off equilibrium. The formulation of the 3-body dynamics is based on the equations of motion method for identical fermions -- also denoted as quantum correlation dynamics -- and presented in detail. The on-shell 2-body collision integral is briefly reviewed and the on-shell 3-body collision integral is derived on the basis of the same two-body interaction in leading order. The resulting equations obey particle number as well as energy-momentum conservation. For a quantification of the relative impact of 3-body interactions we employ a model study for a homogeneous system in space in a finite box with periodic boundary conditions. We address spin-isospin symmetric nuclear matter systems with momentum distributions that are given by shifted Fermi spheres (without overlap) as encountered in the initial phase of nucleus-nucleus collisions after contact. The results for the relaxation times -- employing an effective 2-body interaction -- are compared to Boltzmann-Uehling-Uhlenbeck (BUU) transport calculations in the continuum limit for the same bombarding energies and are found to agree on the level of a few percent. We find that the additional 3-body interactions reduce the relaxation times up to a factor of 3 at 130 A$\cdot$MeV. Furthermore, it is shown in BUU transport calculations that an enhanced stopping by 3$\leftrightarrow$3 collisions shows up in the angular distribution of energetic nucleons ($>$ 60 MeV) e.g. in central $^{97}_{45}Rh$ collisions at 40 A$\cdot$MeV that lead to the formation of a compound nucleus. The angular distribution of the energetic nucleons changes from a slightly forward peaked angular distribution to a slightly sidewards peaked angular distribution which might be controlled experimentally.