Neutron matter from chiral two- and three-nucleon calculations up to ${\mathrm{N}}^3\mathrm{LO}$

Neutron matter is an ideal laboratory for nuclear interactions derived from chiral effective field theory since all contributions are predicted up to next-to-next-to-next-to-leading order (${\mathrm{N}}^3\mathrm{LO}$) in the chiral expansion. By making use of recent advances in the partial-wave decomposition of three-nucleon ($3N$) forces, we include for the first time ${\mathrm{N}}^3\mathrm{LO} 3N$ interactions in many-body perturbation theory (MBPT) up to third order and in self-consistent Green's function theory (SCGF). Using these two complementary many-body frameworks we provide improved predictions for the equation of state of neutron matter at zero temperature and also analyze systematically the many-body convergence for different chiral EFT interactions. Furthermore, we present an extension of the normal-ordering framework to finite temperatures. These developments open the way to improved calculations of neutron-rich matter including estimates of theoretical uncertainties for astrophysical applications.

Figure 1

The energy per particle in neutron matter for three different ${\mathrm{N}}^3\mathrm{LO}NN$ potentials with ${\mathrm{N}}^2\mathrm{LO}$ (top) and ${\mathrm{N}}^3\mathrm{LO}$ (bottom) $3N$ forces, respectively. The uncertainty bands are due to the given $c_i$ and $3N$ cutoff variation. For the MBPT results, we consider in addition the maximum range of third-order calculations with a free and a Hartree-Fock spectrum (dark-blue band) plus the change from a second-order calculation with a Hartree-Fock spectrum, which is indicated by the light-blue extension of the pure third-order uncertainty band. The two bands together define the total uncertainty estimate of MBPT. The region between the two red-dashed lines denotes the uncertainty band of the SCGF method, which we do not fill for a better view. In each panel the energy range at saturation density obtained in MBPT is given.

Figure 2

The energy per particle at different orders of MBPT is shown, up to Hartree-Fock ($E_{\text{tot}}^{\text{HF}}/N$), second order ($E_{\text{tot}}^{\left(2\right)}/N$), and third order ($E_{\text{tot}}^{\left(3\right)}/N$), respectively, in comparison to the energies obtained from the SCGF method ($E_{\mathrm{SCGF}}/N$) at $n_0$ (first row) and $n_0/2$ (second row), respectively. The ${\mathrm{N}}^3\mathrm{LO} NN$ potentials are given in each panel. Three-body effects are included at ${\mathrm{N}}^2\mathrm{LO}$ (blue) and at ${\mathrm{N}}^3\mathrm{LO}$ (red), respectively. The dashed lines connecting the data points are in order to guide the eyes. The error bars are due to the $c_i$ and ${\mathrm{\Lambda }}_{3N}$ variations. In this plot, the third-order calculation does not include the additional many-body uncertainty (the light-blue band in Fig. 1).

Figure 3

The energy per particle of neutron matter at ${\mathrm{N}}^3\mathrm{LO}$ for the three different $NN$ potentials (this work: blue bands) in comparison to Ref. [29] [Krüger et al. (2013): black lines]. The second row combines the results of the first row. In each panel, we give the energy range at saturation density obtained within the improved calculations presented in this work. See text for details.

Figure 4

Comparison of the leading $3N$ Hartree-Fock energies at saturation density for several temperatures obtained using the effective $NN$ potential in terms of $3N$ operators (blue) and the partial-wave approach (red). We include $3N$ matrix elements up to $\mathcal{J}\le 9/2$ and $J\le 6$. For the uncertainty estimate we use the same parameter variation in the $3N$ forces as in Fig. 1.

Figure 5

Momentum-space diagonal matrix elements of the density-dependent effective $NN$ potentials at ${\mathrm{N}}^2\mathrm{LO}$ for a selection of four partial-wave channels and two temperatures.