Quantum Monte Carlo calculations of light nuclei with local chiral two- and three-nucleon interactions

Local chiral effective field theory interactions have recently been developed and used in the context of quantum Monte Carlo few- and many-body methods for nuclear physics. In this work, we go over detailed features of local chiral nucleon-nucleon interactions and examine their effect on properties of the deuteron, paying special attention to the perturbativeness of the expansion. We then turn to three-nucleon interactions, focusing on operator ambiguities and their interplay with regulator effects. We then discuss the nuclear Green's function Monte Carlo method, going over both wave-function correlations and approximations for the two- and three-body propagators. Following this, we present a range of results on light nuclei: Binding energies and distribution functions are contrasted and compared, starting from several different microscopic interactions.

Figure 1

The (normalized) regulator functions for the short-range contact contributions to the local chiral interactions with the typical low (hard, $R_0=1.0$ fm) and high (soft, $R_0=1.2$ fm) coordinate-space cutoffs. In addition, we show the Woods-Saxon core for the central part of the Argonne $v_{18}$ interaction for deuteron pairs. (See text for details.)

Figure 2

The deuteron wave functions with $L=0$ ($S$-wave) and $L=2$ ($D$-wave) at ${\mathrm{N}}^2\mathrm{LO}$ for $R_0=1.0$ fm and $R_0=1.2$ fm. Also shown are the deuteron wave functions for the Argonne $v_{18}$ interaction.

Figure 3

The deuteron momentum distributions at ${\mathrm{N}}^2\mathrm{LO}$ for the two different cutoff scales we use. Also shown is the deuteron momentum distribution for the Argonne $v_{18}$ interaction.

Figure 4

The deuteron tensor polarization at ${\mathrm{N}}^2\mathrm{LO}$ for the two different cutoff scales we use. The bands correspond to an estimate for the uncertainty coming from the truncation of the chiral expansion as described in the text. Also shown is the deuteron tensor polarization for the Argonne $v_{18}$ interaction. The experimental data are from Ref. [46].

Figure 5

The deuteron energy at LO, NLO, and ${\mathrm{N}}^2\mathrm{LO}$ for $R_0=1.0$ fm (1.2 fm) in blue (red). The error bars are the uncertainty estimates coming from the truncation of the chiral expansion as described in the text. Also shown, between the NLO and ${\mathrm{N}}^2\mathrm{LO}$ results, are second- and third-order perturbation theory calculations for the ${\mathrm{N}}^2\mathrm{LO}$ deuteron energies, taking $H_{\text{NLO}}$ as the unperturbed Hamiltonian, and treating $V_{{\text{N}}^2\text{LO}}-V_{\text{NLO}}$ as a perturbation. For the perturbation-theory calculations, we take as the uncertainty the same estimate as for the NLO calculations. The dashed lines serve as guides to the eye. The horizontal dotted line is the experimental binding energy.

Figure 6

The diagrams contributing to $3N$ interactions at ${\mathrm{N}}^2\mathrm{LO}$. Solid lines are nucleons; dashed lines are pions.

Figure 7

The $c_D$-dependent diagram with a fictitious heavy scalar particle $\sigma$ exchanged between two of the nucleons making the participants in the pion exchange explicit. Solid lines are nucleons, the dashed line is a pion, and the dotted line is the fictitious heavy scalar particle.

Figure 8

Correlations of Eqs. (31) and (33) entering the trial wave functions used in the calculations of ${}^4\mathrm{He}$ for the $\text{AV18}+\text{UIX}$ (left panel), ${\mathrm{N}}^2\mathrm{LO} R_0=1.0$ fm (middle panel), and ${\mathrm{N}}^2\mathrm{LO} R_0=1.2$ fm (right panel) interactions.

Figure 9

Energy of ${}^4\mathrm{He}$ as a function of imaginary time in constrained-path GFMC calculations. Past $\tau \sim 0.5{\text{MeV}}^{-1}$ we show the transient estimation. The inset shows the details of the transient estimation and the region used to extract the ground-state energy and uncertainty (light blue band). Note that each point represents an average over a given (varying) imaginary-time interval. The imaginary-time intervals averaged over are shorter at the beginning and end of the propagation in order to show more detail in these intervals.

Figure 10

Energies as calculated using the GFMC method at LO, NLO, and ${\mathrm{N}}^2\mathrm{LO}$ for $A=3,4$ nuclei. The uncertainties include an estimate for the uncertainty coming from the truncation of the chiral expansion. In blue (red) are the energies with the cutoff $R_0=1.0$ fm ($R_0=1.2$ fm). The horizontal lines are the experimental values.

Figure 11

Point-proton radii as calculated using the GFMC method at LO, NLO, and ${\mathrm{N}}^2\mathrm{LO}$ for $A=3,4$ nuclei. The uncertainties include an estimate for the uncertainty coming from the truncation of the chiral expansion. In blue (red) are the energies with the cutoff $R_0=1.0$ fm ($R_0=1.2$ fm). The horizontal bands are the experimental values with uncertainties.

Figure 12

The one-body proton distribution for ${}^4\mathrm{He}$ at ${\mathrm{N}}^2\mathrm{LO}$ with and without $3N$ interactions for the two different cutoffs we consider. The darker (lighter) points include (exclude) $3N$ interactions. The corresponding point-proton radii are shown in a color-coded fashion to the right. The uncertainties quoted for the point-proton radii include only the GFMC statistical uncertainties. See Table 3 for more details on the point-proton radii including uncertainties from the truncation of the chiral expansion.

Figure 13

The one-body proton and neutron distributions for ${}^3\mathrm{He}$ at ${\mathrm{N}}^2\mathrm{LO}$ for the two different cutoffs we consider. The corresponding point-proton radii are shown in a color-coded fashion to the right. The uncertainties quoted for the point-proton radii include only the GFMC statistical uncertainties. See Table 3 for more details on the point-proton radii including uncertainties from the truncation of the chiral expansion.

Figure 14

The ${}^4\mathrm{He}$ longitudinal charge form factor at ${\mathrm{N}}^2\mathrm{LO}$ for both cutoffs and for the $\text{AV}18+\text{UIX}$ interactions. The uncertainty bands include the statistical GFMC uncertainties added in quadrature (for the ${\mathrm{N}}^2\mathrm{LO}$ results) to the uncertainty from the truncation of the chiral expansion as described in the text. The data are from an unpublished compilation by Sick [73] based on Refs. [74, 75, 76, 77, 78].