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

Abstract

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 $1,{\vec{\sigma }}_1\cdot {\vec{\sigma }}_2,{\vec{\sigma }}_1\cdot \vec{q}{\vec{\sigma }}_2\cdot \vec{q},i({\vec{\sigma }}_1+{\vec{\sigma }}_2)\cdot (\vec{q}\times \vec{p}),({\vec{\sigma }}_1\cdot \vec{p}{\vec{\sigma }}_2\cdot \vec{p}+{\vec{\sigma }}_1\cdot \vec{p}{}^{\prime }{\vec{\sigma }}_2\cdot \vec{p}{}^{\prime })$ 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.

Comments: 14 pages, 1 figure, section 7 updated. arXiv admin note: text overlap with arXiv:1903.03183