Convergence of the hole-line expansion with modern nucleon-nucleon potentials

We calculate the three-hole-line contributions to the binding energy of symmetric nuclear matter in the Brueckner-Bethe-Goldstone expansion using various modern nucleon-nucleon potentials of high precision. The relation with the correlation parameter $\kappa =\rho V_{\mathrm{core}}$ is examined. In all cases the three-hole-line contributions turn out to be sufficiently small, but no satisfactory saturation is obtained. This means that three-nucleon forces are essential for all considered potentials.

Figure 1

Different Goldstone diagrams contributing to the binding energy of nuclear matter. Diagrams (a) and (b) correspond to the BHF calculation. The sum of the other diagrams (c)–(f) gives the three-hole-line contribution. The expansion of the Bethe-Fadeev integral equation is sketched in (g). For more details, please see the text.

Figure 2

Saturation curves of symmetric nuclear matter for different $NN$ potentials in the 2HL (left panel) and 2HL + 3HL (right panel) approximations with continuous (bold curves) or gap-choice (thin curves) s.p. potentials. The markers indicate the empirical saturation point.

Figure 3

(a) Continuous-choice 3HL contributions $E_3$ to the binding energy of SNM. (b) Ratio $E_3/\left|E_2\right|$ of 3HL-to-2HL contributions to the binding energy. (c) Ratio $E_{\mathrm{Bubble}}/\left|E_2\right|$. (d) Correlation parameter $\kappa$ [7] of SNM as a function of density for different potentials.

Figure 4

Contributions of individual 3HL diagrams to the binding energy per nucleon, see Fig. 1: bubble [dashed lines, Fig. 1 + Fig. 1], ring [dashed-dotted lines, Fig. 1], higher-order [dotted lines, Fig. 1], and total $E_3$ (solid lines) contributions with the continuous (bold curves) and gap (thin curves) choices for different potentials. Note the different energy scales.