Chiral Three-Nucleon Interactions in Light Nuclei, Neutron-$\alpha$ Scattering, and Neutron Matter

We present quantum Monte Carlo calculations of light nuclei, neutron-$\alpha$ scattering, and neutron matter using local two- and three-nucleon ($3N$) interactions derived from chiral effective field theory up to next-to-next-to-leading order (${\mathrm{N}}^2\mathrm{LO}$). The two undetermined $3N$ low-energy couplings are fit to the ${}^4\mathrm{He}$ binding energy and, for the first time, to the spin-orbit splitting in the neutron-$\alpha$ $P$-wave phase shifts. Furthermore, we investigate different choices of local $3N$-operator structures and find that chiral interactions at ${\mathrm{N}}^2\mathrm{LO}$ are able to simultaneously reproduce the properties of $A=3,4,5$ systems and of neutron matter, in contrast to commonly used phenomenological $3N$ interactions.

Figure 1

(a) Couplings $c_E$ vs $c_D$ obtained by fitting the ${}^4\mathrm{He}$ binding energy for different $3N$-operator forms. Triangles are obtained by using $V_{D1}$ and $V_{E\tau }$, while the other symbols are obtained for $V_{D2}$ and three different $V_E$-operator structures. The blue and green lines (lower and upper) correspond to $R_0=1.0\mathrm{fm}$, while the red lines (central) correspond to $R_0=1.2\mathrm{fm}$. The GFMC statistical errors are smaller than the symbols. The stars correspond to the values of $c_D$ and $c_E$ which simultaneously fit the $n\text{−}\alpha$ $P$-wave phase shifts (see Table 1 and the right panel). No fit to both observables can be obtained for the case with $R_0=1.2\mathrm{fm}$ and $V_{D1}$. (b) $P$-wave $n\text{−}\alpha$ elastic scattering phase shifts compared with an $R$-matrix analysis of experimental data. Colors and symbols correspond to the left panel. We also include phase shifts calculated at NLO which clearly indicate the necessity of $3N$ interactions to fit the $P$-wave splitting.

Figure 2

Ground-state energies and point proton radii for $A=3,4$ nuclei calculated at NLO and ${\mathrm{N}}^2\mathrm{LO}$ (with $V_{D2}$ and $V_{E\tau }$) compared with experiment. Blue (red) symbols correspond to $R_0=1.0\mathrm{fm}$ ($R_0=1.2\mathrm{fm}$). The errors are obtained as described in the text and also include the GFMC statistical uncertainties.

Figure 3

The energy per particle in neutron matter as a function of density for the $NN$ and full $3N$ interactions at ${\mathrm{N}}^2\mathrm{LO}$ with $R_0=1.0\mathrm{fm}$. We use $V_{D2}$ and different $3N$ contact structures: The blue band corresponds to $V_{E\tau }$, the red band to $V_{E\mathbb{1}}$, and the green band to $V_{E\mathcal{P}}$. The green band coincides with the $NN+2\pi$-exchange-only result because both $V_D$ and $V_E$ vanish in this case. The bands are calculated as described in the text.