Density-dependent $NN$ interaction from subleading chiral $3N$ forces: Short-range terms and relativistic corrections

We derive from the subleading contributions to the chiral three-nucleon force [short-range terms and relativistic corrections, published in Phys. Rev. C 84, 054001 (2011)] a density-dependent two-nucleon interaction $V_{\text{med}}$ in isospin-symmetric nuclear matter. The momentum and $k_f$-dependent potentials associated with the isospin operators (1 and ${\vec{\tau }}_1·{\vec{\tau }}_2$) and five independent spin structures are expressed in terms of loop functions, which are either given in closed analytical form or require at most one numerical integration. Our results for $V_{\text{med}}$ are most helpful to implement subleading chiral $3N$ forces into nuclear many-body calculations.

Figure 1

Generic form of the $3N$ interaction. The dashed line symbolizes pion exchange and the wiggly line represents a short-range interaction.

Figure 2

Self-closings of a nucleon line generating the contribution $V_{\text{med}}^{\left(0\right)}$ to the in-medium $NN$ interaction. Mirror graphs $N_1\leftrightarrow N_2$ have to be supplemented.

Figure 3

Vertex corrections by the short-range interaction generating the contribution $V_{\text{med}}^{\left(1\right)}$. Mirror graphs $N_1\leftrightarrow N_2$ have to be supplemented.

Figure 4

Vertex corrections by pion-exchange generating the contribution $V_{\text{med}}^{\left(2\right)}$. Mirror graphs $N_1\leftrightarrow N_2$ have to be supplemented.

Figure 5

Double exchanges generating the contribution $V_{\text{med}}^{\left(3\right)}$ to the in-medium $NN$ interaction. Mirror graphs $N_1\leftrightarrow N_2$ have to be supplemented.