Nucleon-nucleon potentials in phase-space representation

TL;DR

This paper introduces a phase-space representation of nucleon-nucleon interactions that visualizes non-local behaviors and helps understand how renormalization techniques modify nuclear potentials for low-momentum many-body calculations.

Contribution

It develops a method to extract and visualize the interaction $V(oldsymbol{r},oldsymbol{p})$ from matrix elements, aiding intuition on the effects of renormalization and softening of nuclear potentials.

Findings

Visualizes non-local effects in nuclear interactions.

Shows how renormalization reduces short-range repulsion.

Reveals complex momentum dependencies at high momenta.

Abstract

A phase-space representation of nuclear interactions, which depends on the distance $\vec{r}$ and relative momentum $\vec{p}$ of the nucleons, is presented. A method is developed that permits to extract the interaction $V(\vec{r},\vec{p})$ 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 non-local 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 or with the similarity renormalization group. It allows to develop intuition about the various interactions and illustrates how the softened interactions reduce the short-range repulsion in favor of non-locality 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.