Chiral three-nucleon forces and neutron matter

We calculate the properties of neutron matter and highlight the physics of chiral three-nucleon forces. For neutrons, only the long-range $2\pi$-exchange interactions of the leading chiral three-nucleon forces contribute, and we derive density-dependent two-body interactions by summing the third particle over occupied states in the Fermi sea. Our results for the energy suggest that neutron matter is perturbative at nuclear densities. We study in detail the theoretical uncertainties of the neutron matter energy, provide constraints for the symmetry energy and its density dependence, and explore the impact of chiral three-nucleon forces on the $S$-wave superfluid pairing gap.

Figure 1

Diagrammatic representation of the antisymmetrized ${\mathrm{N}}^2\mathrm{LO}$ 3N force in neutron matter. The $c_4$ term in $V_c$ and the shorter-range parts $V_D$ and $V_E$ vanish in neutron matter.

Figure 2

Diagonal momentum-space matrix elements of the density-dependent two-body interaction ${\overline{V}}_{3\mathrm{N}}$ for $P=0$ in the ${}^1{\mathrm{S}}_0$ channel. Results with ${\Lambda }_{3\mathrm{NF}}=2.0\hspace{0.5em}{\mathrm{fm}}^{-1}$ are shown versus relative momentum $k$ for different Fermi momenta $k_{\mathrm{F}}=1.0,1.2,1.4$, and $1.6\hspace{0.5em}{\mathrm{fm}}^{-1}$ (which increase in strength) in neutron matter.

Figure 3

Diagonal momentum-space matrix elements of the density-dependent two-body interaction ${\overline{V}}_{3\mathrm{N}}$ for $P=0$ in the spin-triplet $P$-wave channels. Results with ${\Lambda }_{3\mathrm{NF}}=2.0\hspace{0.5em}{\mathrm{fm}}^{-1}$ are shown versus relative momentum $k$ for different Fermi momenta $k_{\mathrm{F}}=1.0,1.2,1.4$, and $1.6\hspace{0.5em}{\mathrm{fm}}^{-1}$ (which increase in strength). For $k_{\mathrm{F}}=1.6\hspace{0.5em}{\mathrm{fm}}^{-1}$, the dotted lines represent the central parts (degenerate in $J$) of ${\overline{V}}_{3\mathrm{N}}$, whereas the dashed lines include the central plus tensor interactions [without the $c_a^n$ terms in Eqs. (16) and (17)].

Figure 4

Top row: Diagrams contributing to the Hartree-Fock energy. These include the kinetic energy $E_{\mathrm{kin}}$ and the first-order NN and 3N interaction energies $E_{\mathrm{NN}}^{(1)}$ and $E_{3\mathrm{N}}^{(1)}$. Middle and bottom rows: Second-order contributions to the energy due to NN-NN interactions $E_1^{(2)}$, NN-3N and 3N-3N interactions, where 3N forces enter as density-dependent two-body interactions $E_{2,3}^{(2)}$ and $E_4^{(2)}$, respectively, and the remaining 3N-3N diagram $E_5^{(2)}$.

Figure 5

Energy per particle $E/N$ of neutron matter as a function of density $\rho$ at the Hartree-Fock level, based on low-momentum chiral NN and 3N interactions with $\Lambda /{\Lambda }_{3\mathrm{NF}}=2.0\hspace{0.5em}{\mathrm{fm}}^{-1}$. The first-order energy $E_{3\mathrm{N},\mathrm{eff}}^{(1)}$, which uses $V_{3\mathrm{N}}$ with $P=0$, is compared to the exact Hartree-Fock energy $E_{3\mathrm{N},\mathrm{full}}^{(1)}$. To show the impact of chiral 3N forces, we also give the first-order energy based only on NN interactions.

Figure 6

Effective mass $m^{*}(k_{\mathrm{F}})/m$ at the Fermi surface as a function of Fermi momentum $k_{\mathrm{F}}$ in neutron matter. Results for $\Lambda /{\Lambda }_{3\mathrm{NF}}=2.0\hspace{0.5em}{\mathrm{fm}}^{-1}$ are shown at the Hartree-Fock level, plus second-order contributions, and based only on NN interactions for comparison. At second order, the effective mass includes $k$-mass and $e$-mass effects.

Figure 7

Energy per particle $E/N$ of neutron matter as a function of density $\rho$ at the Hartree-Fock level (left) and including second-order contributions (right). The results are based on evolved ${\mathrm{N}}^3\mathrm{LO}$ NN potentials and ${\mathrm{N}}^2\mathrm{LO}$ 3N forces. Theoretical uncertainties are estimated by varying the NN cutoff (lines) and the 3N cutoff (band for fixed $\Lambda =2.0\hspace{0.5em}{\mathrm{fm}}^{-1}$).

Figure 8

Theoretical uncertainties of the second-order energy with $\Lambda /{\Lambda }_{3\mathrm{NF}}=2.0\hspace{0.5em}{\mathrm{fm}}^{-1}$ as a function of density due to the uncertainties in the $c_1$ and $c_3$ coefficients of 3N forces.

Figure 9

Comparison of the second-order energy with the $c_i$ uncertainty band of Fig. 8 to other neutron matter results (see text for details).

Figure 10

The neutron-neutron ${}^1{\mathrm{S}}_0$ superfluid pairing gap on the Fermi surface $\Delta (k_{\mathrm{F}})$ as a function of Fermi momentum $k_{\mathrm{F}}$. The results are based on evolved ${\mathrm{N}}^3\mathrm{LO}$ NN potentials and ${\mathrm{N}}^2\mathrm{LO}$ 3N forces with $\Lambda /{\Lambda }_{3\mathrm{NF}}=2.0\hspace{0.5em}{\mathrm{fm}}^{-1}$, using free and Hartree-Fock single-particle energies.