# Density-dependent NN-interaction from subleading chiral 3N-forces: Long-range terms

## TL;DR

This paper derives a density-dependent two-nucleon interaction from subleading chiral three-nucleon forces, providing analytical and numerical tools for incorporating long-range 3N forces into nuclear many-body calculations.

## Contribution

It presents a detailed derivation of the in-medium two-nucleon potential from subleading chiral 3N forces, including analytical expressions and numerical methods.

## Key findings

- Derived explicit density-dependent potentials for various isospin and spin structures.
- Expressed potentials in analytical form or with minimal numerical integration.
- Facilitates implementation of long-range chiral 3N forces in nuclear many-body models.

## Abstract

We derive from the subleading contributions to the chiral three-nucleon force (long-range terms, published in Phys.\,Rev.\,C\,77, 064004 (2008)) a density-dependent two-nucleon interaction $V_\text{med}$ in isospin-symmetric, spin-saturated nuclear matter. Following the division of the pertinent 3N-diagrams into two-pion exchange topology, two-pion-one-pion exchange topology and ring topology, we evaluate for these all self-closings and concatenations of nucleon-lines to an in-medium loop. The momentum and $k_f$-dependent potentials associated with the isospin operators ($1$ and $\vec\tau_1\!\cdot\!\vec\tau_2$) and five independent spin-structures are expressed in terms of functions, which are either given in closed analytical form or require at most one numerical integration. In the same way we treat the $2\pi$-exchange 3N-force up to fourth order. Our results for $V_\text{med}$ are most helpful to implement the long-range subleading chiral 3N-forces into nuclear many-body calculations.