Applying the density matrix expansion with coordinate-space chiral interactions

We apply the density matrix expansion (DME) at Hartree-Fock level with long-range chiral effective field theory interactions defined in coordinate space up to next-to-next-to-leading order. We consider chiral potentials both with and without explicit $\mathrm{\Delta }$ isobars. The challenging algebra associated with applying the DME to three-nucleon forces is tamed using a new organization scheme, which will also facilitate generalizations. We include local regulators on the interactions to mitigate the effects of singular potentials on the DME couplings and simplify the optimization of generalized Skyrme-like functionals.

Figure 1

Diagrams in the chiral expansion up to $\mathrm{N}{}^2\mathrm{LO}$ for $NN$ and $3N$ forces. Solid lines are nucleons, dashed lines are pions, and double solid lines (red) are $\mathrm{\Delta }$ isobars. Small dots, filled circles, and squares denote vertices of chiral index 0, 1, and 2, respectively [27]. Only one representative example of each type of diagram has been shown. In the $\mathrm{\Delta }$-less theory, only the diagrams in the two corresponding columns contribute to the potential. In the theory with $\mathrm{\Delta }\mathrm{s}$, all the columns contribute.

Figure 2

Diagrams corresponding to the Hartree-Fock topologies absent in our DME couplings with all $3N$ vertices explicitly labeled. Notation is as in Fig. 1. Diagrams without a dashed line have no finite-range radial function.

Figure 3

The $g_t^{\rho \rho }$ couplings from Eq. (70a) are plotted as a function of the isoscalar density ${\rho }_0$ at fixed cutoff $R_{NN}=1.2\text{fm}$ using the regulator in Eq. (40) with $n=6$. The values for the couplings are shown at three different chiral orders up to $\mathrm{N}{}^2\mathrm{LO}$. The isoscalar coupling $g_0^{\rho \rho }$ is shown without (a) and with (c) $\mathrm{\Delta }$ isobars. The isovector coupling $g_1^{\rho \rho }$ is shown without (b) and with (d) $\mathrm{\Delta }$ isobars.

Figure 4

The $g_t^{\rho \rho }$ couplings from Eq. (70a) are plotted as a function of the isoscalar density ${\rho }_0$ at fixed cutoff $R_{NN}=1.0\text{fm}$ using the regulator in Eq. (40) with $n=6$. The values for the couplings are shown at three different chiral orders up to $\mathrm{N}{}^2\mathrm{LO}$. The isoscalar coupling $g_0^{\rho \rho }$ is shown without (a) and with (c) $\mathrm{\Delta }$ isobars. The isovector coupling $g_1^{\rho \rho }$ is shown without (b) and with (d) $\mathrm{\Delta }$ isobars.

Figure 5

The $g^{{\rho }_0^3}$ and $g^{{\rho }_0{\rho }_1^2}$ couplings are plotted as a function of the isoscalar density ${\rho }_0$ at fixed cutoff $R_{3N}=1.2\text{fm}$ using the regulator in Eq. (40) with $n=6$. The values for the couplings are shown at two different chiral orders up to $\mathrm{N}{}^2\mathrm{LO}$. The coupling $g^{{\rho }_0^3}$ is shown without (a) and with (c) $\mathrm{\Delta }$ isobars. The coupling $g^{{\rho }_0{\rho }_1^2}$ is shown without (b) and with (d) $\mathrm{\Delta }$ isobars.

Figure 6

The $g^{{\rho }_0^3}$ and $g^{{\rho }_0{\rho }_1^2}$ couplings are plotted as a function of the isoscalar density ${\rho }_0$ at fixed cutoff $R_{3N}=1.0\text{fm}$ using the regulator in Eq. (40) with $n=6$. The values for the couplings are shown at two different chiral orders up to $\mathrm{N}{}^2\mathrm{LO}$. The coupling $g^{{\rho }_0^3}$ is shown without (a) and with (c) $\mathrm{\Delta }$ isobars. The coupling $g^{{\rho }_0{\rho }_1^2}$ is shown without (b) and with (d) $\mathrm{\Delta }$ isobars.