Density-dependent NN interaction from subleading chiral three-nucleon forces: Long-range terms

Abstract

We derive from the subleading contributions to the chiral three-nucleon (3N) force [long-range terms; published by V. Bernard et al., 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 ${\stackrel{P\vec}{\ensuremath{\tau}}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{P\vec}{\ensuremath{\tau}}}_{2}$) and five independent spin structures are expressed in terms of functions which either are given in closed analytical form or require at most one numerical integration. In the same way we treat the $2\ensuremath{\pi}$-exchange 3N force at sub-subleading order. Our results for ${V}_{\text{med}}$ are most helpful to implement long-range subleading chiral 3N forces into nuclear many-body calculations.