Density-dependent nn-potential from subleading chiral three-neutron forces: Long-range terms

The long-range terms of the subleading chiral three-nucleon force [published in Phys.\,Rev.\,C77, 064004 (2008)] are specified to the case of three neutrons. From these $3n$-interactions an effective density-dependent neutron-neutron potential $V_\text{med}$ in pure neutron matter is derived. Following the division of the pertinent 3n-diagrams into two-pion exchange, two-pion-one-pion exchange and ring topology, all self-closings and concatenations of two neutron-lines to an in-medium loop are evaluated. The momentum and $k_n$-dependent potentials associated with the spin-operators $1,\, \vec\sigma_1\!\cdot\!\vec\sigma_2,\, \vec\sigma_1\!\cdot\!\vec q\, \vec\sigma_2\!\cdot\!\vec q,\, i( \vec\sigma_1\!+\!\vec\sigma_2)\!\cdot \! (\vec q\!\times \! \vec p\,),\, (\vec\sigma_1\!\cdot\!\vec p\,\vec\sigma_2\!\cdot\!\vec p+\vec\sigma_1\!\cdot\!\vec p\,'\, \vec\sigma_2\!\cdot\!\vec p\,')$ and $\vec\sigma_1\!\cdot \! (\vec q\!\times \! \vec p\,)\vec\sigma_2\!\cdot \! (\vec q\!\times \! \vec p\,)$ are expressed in terms of functions, which are either given in closed analytical form or require at most one numerical integration. The subsubleading chiral 3N-force is treated in the same way. The obtained results for $V_\text{med}$ are helpful to implement the long-range chiral three-body forces into advanced neutron matter calculations.