Nuclear-matter equation of state with consistent two- and three-body perturbative chiral interactions

We compute the energy per particle of infinite symmetric nuclear matter from chiral N${}^3\mathrm{LO}$ (next-to-next-to-next-to-leading order) two-body potentials plus N${}^2\mathrm{LO}$ three-body forces. The low-energy constants of the chiral three-nucleon force that cannot be constrained by two-body observables are fitted to reproduce the triton binding energy and the ${}^3\mathrm{H}$-${}^3\mathrm{He}$ Gamow-Teller transition matrix element. In this way, the saturation properties of nuclear matter are reproduced in a parameter-free approach. The equation of state is computed up to third order in many-body perturbation theory, with special emphasis on the role of the third-order particle-hole diagram. The dependence of these results on the cutoff scale and regulator function is studied. We find that the inclusion of three-nucleon forces consistent with the applied two-nucleon interaction leads to a reduced dependence on the choice of the regulator only for lower values of the cutoff.

Figure 1

Neutron-proton phase shifts as predicted by chiral N${}^3\mathrm{LO}$ potentials with different cutoff scale $\Lambda$. Solid (red) curve, $\Lambda =414$ MeV; dashed (blue) curve, $\Lambda =450$ MeV; and dotted (black) curve, $\Lambda =500$ MeV. Partial waves with total angular momentum $J\le 1$ are displayed. The solid dots and open circles are the results from the Nijmegen multienergy $np$ phase shift analysis [30] and the VPI/GWU single-energy $np$ analysis SM99 [31], respectively.

Figure 2

Same as Fig. 1, but for $J=2$ phase shifts and $J\le 2$ mixing parameters.

Figure 3

Same as Fig. 1, but for some representative peripheral partial waves.

Figure 4

The N${}^2\mathrm{LO}$ three-nucleon force contact interactions: $V_{3N}^{1\pi }$ on the left and $V_{3N}^{\mathrm{cont}}$ on the right [see Eqs. (4) and (5), respectively].

Figure 5

$c_D$-$c_E$ trajectories fitted to reproduce the experimental ${}^3\mathrm{H}$ and ${}^3\mathrm{He}$ binding energies. Solid (black) curve for $\Lambda =500$ MeV, dotted-dashed (green) curve for $\Lambda =450$ MeV, and dashed (red) curve for $\Lambda =414$ MeV.

Figure 6

The ratio between the calculated GT value (GT${}^{\mathrm{TH}}$) and the experimental one (GT${}^{\mathrm{EXP}}$) as a function of the LEC $c_D$. Solid (black) curve for $\Lambda =500$ MeV, dotted-dashed (green) curve for $\Lambda =450$ MeV, and dashed (red) curve for $\Lambda =414$ MeV. The shaded stripe represents the experimental uncertainty.

Figure 7

Third-order ring diagram of the Goldstone expansion that we have included in our calculations with $V_{2N}$ and ${\overline{V}}_{3N}$ vertices. Latin-letter subscripts denote particle states; Greek-letter subscripts correspond to hole states.

Figure 8

Nuclear matter energy per particle obtained from the N${}^3\mathrm{LO}$ 2NF with cutoff $\Lambda =500$ MeV. The first, second, and third order in the perturbative expansion and the Padé approximant $\left[2\right|1]$ are shown as a function of the Fermi momentum $k_F$.

Figure 9

Same as in Fig. 8, but including the contribution of the N${}^2\mathrm{LO}$ 3NF.

Figure 10

Same as Fig. 9, but for $\Lambda =450$ MeV.

Figure 11

Results obtained for the g.s.e. per particle of infinite nuclear matter at third-order in perturbation theory for three sets of chiral interactions which differ by the cutoff $\Lambda$.

Figure 12

Second-order Hugenholtz 3p-3h diagram of the Goldstone expansion with two 3NF vertices. Latin-letter subscripts denote particle states; Greek-letter subscripts correspond to hole states.