Density-dependent NN interaction from subsubleading chiral 3N forces: Intermediate-range contributions

Abstract

From the subsubleading chiral three-nucleon force [intermediate-range contributions, published in Phys. Rev. C\,87, 054007 (2013)] a density-dependent NN-interaction $V_\text{med}$ is derived in isospin-symmetric nuclear matter. Following the division of the pertinent 3N-diagrams into two-pion-one-pion exchange topology and ring topology, one evaluates for these all selfclosings and concatenations of nucleon-lines to an in-medium loop. In the case of the $2\pi 1\pi$-exchange topology, 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 require at most one numerical integration. For the more challenging (concatenations of the) ring diagrams proportional to $c_{1,2,3,4}$, one ends up with regularized double-integrals $\int_0^\lambda dr\,r \int_0^{\pi/2} d\psi$ from which the $\lambda^2$-divergence has been subtracted and the logarithmic piece $\sim \ln (m_\pi/\lambda)$ is isolated. The derived semi-analytical results are most helpful to implement the subsubleading chiral 3N-forces into nuclear many-body calculations.