Neutron matter based on consistently evolved chiral three-nucleon interactions

We present the first results for the neutron matter equation of state (EOS) using nucleon-nucleon and three-nucleon chiral effective field theory interactions that are consistently evolved in the framework of the similarity renormalization group (SRG). The dependence of the EOS on the SRG resolution scale is greatly reduced when induced three-nucleon forces (3NF) are included, and the residual variation, which in part is from missing induced four-body interactions, is comparable to estimated many-body perturbation theory truncation errors. The relative growth with decreasing resolution of the 3NF contributions to the energy per neutron is of natural size, but it accelerates at the lowest resolutions where strong renormalization of the long-range 3NF matrix elements is also observed.

Figure 1

Energy per neutron as a function of neutron density for different SRG resolution scales. The results are grouped according to whether no induced 3NF are included (NN-only), the induced 3NF are included but no initial 3NF (3N-induced), or initial and induced 3NF are included (3N-full). The initial interaction is the 500-MeV N${}^3$LO NN potential from Ref. [17] combined with the N${}^2$LO 3NF using the consistent low-energy constants $c_1=-0.81{\mathrm{GeV}}^{-1}$ and $c_3=-3.2{\mathrm{GeV}}^{-1}$ [2].

Figure 2

Partial-wave convergence of the Hartree-Fock 3NF-only energy per neutron as a function of density at two different resolution scales. Panel (a) shows the results before evolution ($\lambda =\infty$) and panel (b) shows them at $\lambda =2.0{\text{fm}}^{-1}$. The Hamiltonian is the same as for the 3N-full results in Fig. 1. For $\lambda =\infty$, only $\mathcal{J}\le 5/2$ give significant contributions so the lines for the higher partial waves are nearly indistinguishable from the exact result.

Figure 3

Energy per neutron as a function of the resolution scale $\lambda$ at nuclear matter saturation density $n_s=0.16{\mathrm{fm}}^{-3}$ for two different interactions. Panel (a) uses the NN and 3N potentials as in Fig. 1 and panel (b) uses the N${}^2$LO potential ($\Lambda /\tilde{\Lambda }=450/500$ MeV) of Ref. [30] plus the consistent 3NF with the low-energy constants $c_1=-0.81{\mathrm{GeV}}^{-1}$ and $c_3=-3.4{\mathrm{GeV}}^{-1}$. The same three calculations (NN-only, 3N-induced, and 3N-full) from Fig. 1 are used here. The solid lines are energies using a resummed particle ladder sum of the NN contributions, with the straight dotted lines marking the energies at $\lambda =3.0{\mathrm{fm}}^{-1}$. The dashed lines are energies including up to second order for the NN contributions. The shaded areas at the right indicate the magnitude of the attractive second-order contribution from unevolved NN and 3N forces.

Figure 4

Scaling of the contributions from NN and 3N forces as a function of $\lambda$. The dashed lines show the ratio of these two contributions based on the 500 MeV N${}^3$LO NN potential [17] and the dashed lines based on the 450/500 MeV N${}^2$LO potential [30] (see also Fig. 3).

Figure 5

Matrix elements of the evolved 3N potential $\langle \xi \alpha =1\left|{\overline{V}}_{123}\right|{\xi }^{\prime }{\alpha }^{\prime }=1\rangle$ (see main text) for $\mathcal{J}=1/2$, $\mathcal{T}=3/2$, and positive total parity ${\pi }_3$ at the hyperangle $\theta =\pi /8$. The interactions are the same as in Fig. 1.