Symmetric nuclear matter with chiral three-nucleon forces in the self-consistent Green's functions approach

We present calculations for symmetric nuclear matter using chiral nuclear interactions within the self-consistent Green's functions approach in the ladder approximation. Three-body forces are included via effective one-body and two-body interactions, computed from an uncorrelated average over a third particle. We discuss the effect of the three-body forces on the total energy, computed with an extended Galitskii-Migdal-Koltun sum-rule, as well as on single-particle properties. Saturation properties are substantially improved when three-body forces are included, but there is still some underlying dependence on the similarity renormalization group evolution scale.

Figure 1

The ladder irreducible self-energy can be obtained from the sum of a generalized Hartree-Fock term and a dispersive contribution. Dashed lines represent two-nucleon interactions and double lines with arrows are self-consistently dressed single-particle propagators.

Figure 2

Diagrammatic definition of the one-body (double-dashed line) and two-body (wiggly line) effective interactions. The one-body contribution is the sum of the generalized Hartree-Fock term of the 2NF plus a term containing the two-body propagator, $G_{II}$; the dotted line represents three-nucleon interactions. The $1/4$ factor counts the equivalent lines in the latter diagrams. The two-body effective force is the sum of the 2NF and a 3NF averaged over one particle.

Figure 3

Ladder irreducible self-energy employed in this work, using the effective interactions defined in Fig. 2.

Figure 4

$T$ matrix employed in this work, using the effective interactions defined in Fig. 2.

Figure 5

SCGF (black) and BHF (blue) results for the energy per nucleon of SNM as a function of the density at a temperature of $T=5$ MeV. Diamonds represent calculations including only 2NFs, whereas circles include 2NFs and 3NFs as described in the text. The cross gives the empirical saturation value of nuclear matter.

Figure 6

SCGF results for the energy per nucleon of SNM as a function of the density at a temperature of $T=5$ MeV for different LECs. The different combinations are discussed in the text.

Figure 7

SCGF results for the energy per nucleon of SNM as a function of the density at a temperature of $T=5$ MeV, using N3LO (dotted lines) and the optimized version of N2LO (dashed lines) from [61]. Diamonds represent calculations including only 2NFs, whereas circles include 2NFs and 3NFs as described in the text.

Figure 8

SCGF results for the energy per nucleon of SNM as a function of the density at a temperature of $T=5$ MeV. Different lines represent different choices of cutoffs for the 2NF, $\lambda$, and the 3NF, ${\Lambda }_{3\mathrm{NF}}$.

Figure 9

Comparison of results for the energy per nucleon of SNM obtained with different approaches using the same SRG-evolved 2NF and a 3NF. Circles correspond to extrapolated SCGF results, whereas squares are BHF calculations at $T=0$ MeV. Diamonds correspond to the results of Hebeler et al. [43].

Figure 10

Momentum distribution in SNM using the 2NF only at N3LO (dashed line) and using the density-dependent 2NF (solid line). Calculations are performed at the empirical saturation point (a) and at twice this density (b). The insets focus on the depletion below the Fermi surface.

Figure 11

SP spectral function for SNM using only the 2NF at N3LO (dashed line) and including the density-dependent 2NF (solid line). Calculations are performed at the empirical saturation density (left panels) and at twice this density (right panels). Results are displayed for three characteristic SP momenta.

Figure 12

SP potential for SNM using only the 2NF at N3LO (a) and including the density-dependent 2NF (b). Calculations are performed at the empirical saturation density (solid line) and at twice this density (dotted line).