Neutron matter from chiral effective field theory interactions

The neutron-matter equation of state constrains the properties of many physical systems over a wide density range and can be studied systematically using chiral effective field theory (EFT). In chiral EFT, all many-body forces among neutrons are predicted to next-to-next-to-next-to-leading order (N${}^3$LO). We present details and additional results of the first complete N${}^3$LO calculation of the neutron-matter energy, which includes the subleading three-nucleon as well as the leading four-nucleon forces, and provides theoretical uncertainties. In addition, we discuss the impact of our results for astrophysics: for the supernova equation of state, the symmetry energy and its density derivative, and for the structure of neutron stars. Finally, we give a first estimate for the size of the N${}^3$LO many-body contributions to the energy of symmetric nuclear matter, which shows that their inclusion will be important in nuclear structure calculations.

Figure 1

Diagrams contributing to the Hartree-Fock energy. These include the kinetic energy $E_{\mathrm{kin}}$ and the first-order $NN$, $3N$, and $4N$ interaction energies $E_{NN}^{\left(1\right)}$, $E_{3N}^{\left(1\right)}$, and $E_{4N}^{\left(1\right)}$.

Figure 2

Second-order contributions to the energy due to $NN$-$NN$ correlations, $E_1^{\left(2\right)}$, and $NN$-$3N$ and $3N$-$3N$ correlations, $E_{2/3}^{\left(2\right)}$ and $E_4^{\left(2\right)}$, where the $3N$ forces enter as density-dependent two-body interactions, as well as the residual $3N$-$3N$ contribution $E_5^{\left(2\right)}$ not considered here.

Figure 3

Energy per particle as a function of density for the different N${}^3$LO $NN$ potentials of Refs. [35, 36, 37, 38]. The dashed lines are Hartree-Fock results only. The filled and shaded bands are second- and third-order energies, respectively, where at each order the band ranges from using a free to a Hartree-Fock spectrum.

Figure 4

Energy per particle as a function of density for the different N${}^3$LO $NN$ potentials of Refs. [35, 36, 37, 38] and including the leading N${}^2$LO $3N$ forces. The dashed lines are Hartree-Fock results only. The filled and shaded bands are second- and third-order energies, respectively, where at each order the band ranges from using a free to a Hartree-Fock spectrum. All calculations are performed for a $3N$ cutoff $\Lambda =2.0{\mathrm{fm}}^{-1}$ and low-energy couplings $c_1=0.75{\mathrm{GeV}}^{-1}$ and $c_3=4.77{\mathrm{GeV}}^{-1}$.

Figure 5

Energy per particle as a function of density for all individual N${}^3$LO $3N$- and $4N$-force contributions to neutron matter at the Hartree-Fock level. All bands are obtained by varying the $3N$/$4N$ cutoffs $\Lambda =2$–2.5 ${\mathrm{fm}}^{-1}$. For the two-pion-exchange–contact and the relativistic-corrections $3N$ forces, the different bands correspond to the different $NN$ contacts, $C_T$ and $C_S$, determined consistently for the N${}^3$LO EM/EGM potentials. The inset diagram illustrates the $3N/4N$ force topology of the particular contributions.

Figure 6

Contributions from $3N$ forces in the Hartree-Fock approximation at N${}^2$LO plus N${}^3$LO (first three panels) in comparison with the $3N$ contribution in a N${}^2$LO calculation (fourth panel). The first panel shows the N${}^2$LO $3N$ contribution in the N${}^3$LO calculation, using the N${}^3$LO values of the $c_i$ couplings, and the second panel gives the N${}^3$LO $3N$ contribution. The third panel shows the total $3N$ contribution at N${}^3$LO (the sum of the first two panels). This is compared in the fourth panel to the $3N$ contribution at N${}^2$LO, using the $c_i$ values recommended for an N${}^2$LO calculation (see Table ). For the EGM potentials the total $3N$ contribution at N${}^3$LO differs by less than 1 MeV compared to the N${}^2$LO results. However, for the EM potential, the result changes by almost 3 MeV. All bands include the $c_i$ range from Table and the $3N$ cutoff variation.

Figure 7

Neutron-matter energy per particle as a function of density including $NN$, $3N$, and $4N$ forces to N${}^3$LO. The three overlapping bands are labeled by the different $NN$ potentials and include uncertainty estimates due to the many-body calculation, the low-energy $c_i$ constants, and by varying the $3N/4N$ cutoffs (see text for details). For comparison, we show the results for the RG-evolved $NN$ EM 500 MeV potential including only N${}^2$LO $3N$ forces from Ref. [4].

Figure 8

Neutron-matter energy per particle as a function of density at N${}^2$LO (upper blue band that extends to the dashed line) and N${}^3$LO (lower red band). The bands are based on the EGM $NN$ potentials and include uncertainty estimates as in Fig. 7.

Figure 9

Comparison of the neutron-matter energy at N${}^3$LO of Fig. 7 (red band) with equations of state for core-collapse supernova simulations provided by Lattimer-Swesty (LS [55] with different incompressibilities, 180, 220, and $375\mathrm{MeV}$), G. Shen (FSU2.1, NL3 [61]), Hempel (TM1, SFHo, SFHx [62]), and Typel (DD2 [56]).

Figure 10

Constraints on the mass-radius diagram of neutron stars based on our neutron-matter results at N${}^3$LO following Ref. [52] for the extension to neutron-star matter and to high densities (red band), in comparison to the constraints from calculations based on RG-evolved $NN$ interactions (thick dashed blue lines) [52]. We also show the mass-radius relations obtained from the equations of state for core-collapse supernova simulations shown in Fig. 9 [55, 57, 58, 59, 60, 64, 65]. The legend for the thin lines is as described in the caption to Fig. 9.

Figure 11

Energy per particle versus density for all individual N${}^3$LO $3N$- and $4N$-force contributions to symmetric nuclear matter at the Hartree-Fock level. All bands are obtained by varying the $3N/4N$ cutoff $\Lambda =2$–2.5 ${\mathrm{fm}}^{-1}$. For the two-pion-exchange–contact, the relativistic-corrections $3N$ forces, and the short-range $4N$ forces, the different bands correspond to the different $NN$ contacts, $C_T$ and $C_S$, determined consistently for the N${}^3$LO EM/EGM potentials. The inset diagram illustrates the $3N/4N$ force topology of the particular contribution.