Nucleon-nucleon potentials in phase-space representation

Abstract

A phase-space representation of nuclear interactions, which depends on the distance $\stackrel{P\vec}{r}$ and relative momentum $\stackrel{P\vec}{p}$ of the nucleons, is presented. A method is developed that permits us to extract the interaction $V(\stackrel{P\vec}{r},\stackrel{P\vec}{p})$ from antisymmetrized matrix elements given in a spherical basis with angular momentum quantum numbers, either in momentum- or coordinate-space representation. This representation visualizes in an intuitive way the nonlocal behavior introduced by cutoffs in momentum space or renormalization procedures that are used to adapt the interaction to low-momentum many-body Hilbert spaces, as done in the unitary correlation operator method (UCOM) or with the similarity renormalization group (SRG). It allows us to develop intuition about the various interactions and illustrates how the softened interactions reduce the short-range repulsion in favor of nonlocality or momentum dependence while keeping the scattering phase shifts invariant. It also reveals that these effective interactions can have undesired complicated momentum dependencies at momenta around and above the Fermi momentum. Properties, similarities, and differences of the phase-space representations of the Argonne and the N3LO chiral potential and their UCOM and SRG derivatives are discussed.