Density-dependent effective nucleon-nucleon interaction from chiral three-nucleon forces

We derive density-dependent corrections to the in-medium nucleon-nucleon interaction from the leading-order chiral three-nucleon force. To this order there are six distinct one-loop diagrams contributing to the in-medium nucleon-nucleon scattering $T$ matrix. Analytic expressions are presented for each of these in both isospin-symmetric nuclear matter and nuclear matter with a small isospin asymmetry. The results are combined with the low-momentum nucleon-nucleon potential $V_{\mathrm{low}\hspace{0.5em}k}$ to obtain an effective density-dependent interaction suitable for nuclear structure calculations. The in-medium interaction is decomposed into partial waves up to orbital angular momentum $L=2$. Our results should be particularly useful in calculations where an exact treatment of the chiral three-nucleon force would otherwise be computationally prohibitive.

Figure 1

Leading-order contributions to the chiral three-nucleon force: (a) the long-range $2\pi$-exchange force $V_{3N}^{(2\pi )}$, (b) the medium-range $1\pi$-exchange force $V_{3N}^{(1\pi )}$, and (c) the short-range contact interaction $V_{3N}^{(\mathrm{ct})}$.

Figure 2

In-medium $\mathit{NN}$ interaction generated by the two-pion exchange component ($~c_{1,3,4}$) of the chiral three-nucleon interaction. The short double-line symbolizes the filled Fermi sea of nucleons, that is, the medium insertion $-2\pi \delta (k_0)\theta (k_f-|\mathrm{k⃗}|)$ in the in-medium nucleon propagator. Reflected diagrams of (2) and (3) are not shown.

Figure 3

In-medium $\mathit{NN}$ interaction generated by the one-pion exchange [diagrams (4) and (5); $~c_D$] and the short-range component [diagram (6); $~c_E$] of the chiral three-nucleon interaction. Reflected diagrams of (4) and (5) are not shown, and diagram (6) represents all possible ways to contract a nucleon line.

Figure 4

Modifications to the ${}^1{S}_0$, ${}^3{S}_1$, ${}^3{D}_1$, and ${}^3{S}_1$–${}^3{D}_1$ partial-wave amplitudes of $V_{\mathrm{low}\hspace{0.5em}k}$ (solid curve) due to the first three density-dependent contributions $V_{\mathit{NN}}^{\mathrm{med};1,2,3}$ at nuclear matter saturation density ${\rho }_0=0.16$ fm${}^{-3}$.

Figure 5

Modifications to the ${}^1{S}_0$, ${}^3{S}_1$, ${}^3{D}_1$, and ${}^3{S}_1$–${}^3{D}_1$ partial-wave amplitudes of $V_{\mathrm{low}\hspace{0.5em}k}$ (solid curve) due to the three density-dependent contributions $V_{\mathit{NN}}^{\mathrm{med};4,5,6}$ at nuclear matter saturation density ${\rho }_0=0.16$ fm${}^{-3}$.

Figure 6

Dependence of the low-momentum in-medium $\mathit{NN}$ interaction on the nuclear density $\rho$. Shown are the momentum space matrix elements in the ${}^1{S}_0$, ${}^3{S}_1$, and ${}^3{D}_1$ partial waves and the ${}^3{S}_1$–${}^3{D}_1$ mixing matrix element.

Figure 7

Dependence of the low-momentum in-medium nucleon-nucleon interaction on the isospin asymmetry ${\delta }_{\mathit{np}}=({\rho }_n-{\rho }_p)/\rho$ at ${\rho }_0=0.16$ fm${}^{-3}$. Shown are the momentum space matrix elements in the ${}^1{S}_0$, ${}^3{S}_1$, and ${}^3{D}_1$ partial waves and the ${}^3{S}_1$–${}^3{D}_1$ mixing matrix element.

Figure 8

Modifications to the ${}^1{P}_1$, ${}^3{P}_0$, ${}^3{P}_1$, and ${}^3{P}_2$ partial-wave amplitudes of $V_{\mathrm{low}\hspace{0.5em}k}$(solid line) due to the first three density-dependent contributions $V_{\mathit{NN}}^{\mathrm{med};1,2,3}$ at a nuclear density $\rho ={\rho }_0$.

Figure 9

Modifications to the ${}^1{P}_1$, ${}^3{P}_0$, ${}^3{P}_1$, and ${}^3{P}_2$ partial-wave amplitudes of $V_{\mathrm{low}\hspace{0.5em}k}$ (solid line) due to $V_{\mathit{NN}}^{\mathrm{med};4,5}$ at nuclear density $\rho ={\rho }_0$.

Figure 10

Dependence of the low-momentum in-medium nucleon-nucleon interaction on the nuclear density $\rho$. Shown are the momentum space matrix elements in the ${}^1{P}_1$, ${}^3{P}_0$, ${}^3{P}_1$, and ${}^3{P}_2$ partial waves.

Figure 11

Dependence of the low-momentum in-medium nucleon-nucleon interaction on the isospin asymmetry ${\delta }_{\mathit{np}}=({\rho }_n-{\rho }_p)/\rho$ at ${\rho }_0=0.16$ fm${}^{-3}$. Shown are the momentum space matrix elements in the ${}^1{P}_1$, ${}^3{P}_0$, ${}^3{P}_1$, and ${}^3{P}_2$ partial waves.

Figure 12

Modifications to the ${}^1{D}_2$, ${}^3{D}_2$, and ${}^3{D}_3$ partial-wave amplitudes and the ${}^3{D}_3$–${}^3{G}_3$ mixing matrix element of $V_{\mathrm{low}\hspace{0.5em}k}$ (solid line) due to the first three density-dependent contributions $V_{\mathit{NN}}^{\mathrm{med};1,2,3}$ at saturation density ${\rho }_0$.

Figure 13

Modifications to the ${}^1{D}_2$, ${}^3{D}_2$, and ${}^3{D}_3$ partial-wave amplitudes and the ${}^3{D}_3$–${}^3{G}_3$ mixing matrix element of $V_{\mathrm{low}\hspace{0.5em}k}$ (solid line) due to $V_{\mathit{NN}}^{\mathrm{med};4,5}$ at saturation density ${\rho }_0$.

Figure 14

Dependence of the low-momentum in-medium nucleon-nucleon interaction on the nuclear density $\rho$. Shown are the momentum space matrix elements in the ${}^1{D}_2$, ${}^3{D}_2$, and ${}^3{D}_3$ partial waves as well as the ${}^3{D}_3$–${}^3{G}_3$ mixing matrix element.

Figure 15

Dependence of the low-momentum in-medium $\mathit{NN}$ interaction on the isospin asymmetry ${\delta }_{\mathit{np}}=({\rho }_n-{\rho }_p)/\rho$ at ${\rho }_0=016$ fm${}^{-3}$. Shown are the momentum space matrix elements in the ${}^1{D}_2$, ${}^3{D}_2$, and ${}^3{D}_3$ channels and the ${}^3{D}_3$–${}^3{G}_3$ mixing matrix element.

Figure 16

Partial-wave matrix elements of $V_{\mathit{NN}}^{\mathrm{med}}$ in the allowed channels with $L⩽2$ for two interacting neutrons in a medium of pure neutron matter with density ${\rho }_n\simeq 0.14$ fm${}^{-3}$. For these plots $c_1=-0.81$ GeV${}^{-1}$ and $c_3=-3.2$ GeV${}^{-1}$.