Correlated density-dependent chiral forces for infinite-matter calculations within the Green's function approach

The properties of symmetric nuclear and pure neutron matter are investigated within an extended self-consistent Green's function method that includes the effects of three-body forces. We use the ladder approximation for the study of infinite nuclear matter and incorporate the three-body interaction by means of a density-dependent two-body force. This force is obtained via a correlated average over the third particle, with an in-medium propagator consistent with the many-body calculation we perform. We analyze different prescriptions in the construction of the average and conclude that correlations provide small modifications at the level of the density-dependent force. Microscopic as well as bulk properties are studied, focusing on the changes introduced by the density-dependent two-body force. The total energy of the system is obtained by means of a modified Galitskii-Migdal-Koltun sum rule. Our results validate previously used uncorrelated averages and extend the availability of chirally motivated forces to a larger density regime.

Figure 1

Diagrammatic representation of the effective 1B and 2B interactions given, respectively, in Eqs. (2) and (3). The 1B effective operator, (a), is given by the sum of the 2B interaction (dashed line) contracted with a dressed SP propagator, $G$ (double line with arrow), and the 3B interaction (long-dashed line) contracted with the Hartree-Fock approximation of a dressed 2B propagator $G^{\mathrm{II}}$. The correct symmetry factor of 1/2 in the last term is also shown explicitly. The effective 2B operator, (b), is given by the sum of the original 2B interaction (dashed line) and the 3B interaction (long-dashed line) contracted with a dressed SP propagator, $G$.

Figure 2

Diagrammatic representation of the irreducible self-energy ${\Sigma }^{☆}$. The first term is energy independent and it is computed from the effective 1B interaction of Fig. 1. The second term is a dynamical contribution, consisting of excited configurations generated through the 2B effective interactions of Fig. 1.

Figure 3

Diagrammatic representation of the $T$-matrix interacting vertex function, constructed with the 2B effective interaction of Fig. 1.

Figure 4

3NF terms appearing at N2LO in the chiral expansion. Diagram (a) corresponds to the TPE term given in Eq. (7). Diagram (b) is the OPE term of Eq. (8). Diagram (c) yields the contact term of Eq. (9). Dashed lines define pions. Small dots, big dots, and squares define the nature of the vertices [5].

Figure 5

Six density-dependent contributions arising from contractions of the three 3NF terms appearing at N2LO in the chiral expansion. Dashed lines define pions; double arrowed lines correspond to a dressed single-particle propagator. Diagrams (a), (b), and (c) arise form contraction of the long-range 3NF TPE term given in Eq. (7) and correspond, respectively, to the formal expressions given in the appendix in Eqs. (A2), (A5), and (A11). Diagrams (d) and (e) are obtained from averaging the medium-range 3NF OPE term, Eq. (8), and correspond, respectively, to Eq. (A17) and Eq. (A18). Diagram (f) is the result of contracting the contact 3NF contribution given in Eq. (9) and is defined in Eq. (A19). Small and big dots and squares define the nature of the vertices in the chiral expansion.

Figure 6

$S,D$, and $S-D$ mixing [top panels: (a), (b), (c) and (d)] and $P$ [bottom panels: (e), (f), (g), and (h)] partial-wave matrix elements of the two-body effective interaction. Black solid lines depict the bare 2B N3LO potential. Red dashed, green dot-dashed, and orange double-dot-dashed curves include the 2B density-dependent force obtained, respectively, in the correlated and free versions with full regulator and in the correlated version with the external regulator (see explanation in the text). Density-dependent terms are obtained at the empirical saturation density of symmetric nuclear matter, ${\rho }_0=0.16$ ${\mathrm{fm}}^{-3}$, and for a temperature of $T=5$ MeV.

Figure 7

Momentum distribution in SNM using 2NF at N3LO (black solid lines) and including the density-dependent 2NF via different averaging procedures. Red dashed, green dot-dashed, and orange double-dot-dashed curves correspond to results with 2B density-dependent forces obtained, respectively, with a correlated and uncorrelated distribution with full regulator and with the correlated version with the external regulator. Calculations are performed at $\rho =0.08,0.16$, and $0.32$ ${\mathrm{fm}}^{-3}$ in panels going from left to right.

Figure 8

SNM energy per nucleon as a function of density at $T=0$ MeV. The black solid line shows the 2B N3LO calculation. Red dashed, green dot-dashed, and orange double-dot-dashed curves correspond to results with 2B density-dependent forces obtained, respectively, with a correlated and uncorrelated distribution with full regulator and with the correlated version with the external regulator. The orange cross represents the empirical saturation point.

Figure 9

PNM energy per nucleon as a function of density at $T=0$ MeV. The black solid line shows the 2B N3LO calculation. Red dashed, green dot-dashed, and orange double-dot-dashed curves correspond to results with 2B density-dependent forces obtained, respectively, with a correlated and uncorrelated distribution with full regulator and with the correlated version with the external regulator.

Figure 10

PNM energy per nucleon as a function of density at $T=0$ MeV including errors from LECs. Red dashed lines show the calculation for the 2B N3LO plus the density-dependent 2NF obtained with a correlated momentum distribution and a full regulator. The blue solid band corresponds to the calculation where a SRG evolution is applied on the 2B N3LO potential down to ${\Lambda }_{\mathrm{SRG}}=2.0$ ${\mathrm{fm}}^{-1}$. The hatched band shows results obtained by Drischler et al. [21, 62] in a perturbative calculation up to second order with the same SRG-evolved input 2B potential and a density-dependent 2NF obtained from 3NFs at N2LO. The bands reflect the uncertainty on the underlying LECs, as explained in the text.

Figure 11

PNM energy per nucleon as a function of density at $T=0$ MeV for different chiral interactions. The black solid line and the green dot-dashed lines show results with 2NF only, with either N3LO or N2LOopt potentials. The red dashed line corresponds to calculation performed with 2B N3LO plus the N2LO density-dependent force obtained in the correlated version with a full regulator. The blue double-dot-dashed line is obtained from a 2B N2LOopt potential and a density-dependent 2B force with the corresponding $c_1,c_3$ of N2LOopt [47]. The arrows connect the curves obtained with the N2LO (green dot-dashed) and with the N3LO (black solid) 2NFs with the respective curves including 3NFs. The symbols are data from the pp-hh coupled-cluster calculations of Baardsen et al. [35] and of Hagen et al. [36].