$\mathrm{\Delta }$ isobars and nuclear saturation

We construct a nuclear interaction in chiral effective field theory with explicit inclusion of the $\mathrm{\Delta }$-isobar $\mathrm{\Delta }\left(1232\right)$ degree of freedom at all orders up to next-to-next-to-leading order (NNLO). We use pion-nucleon ($\pi N$) low-energy constants (LECs) from a Roy-Steiner analysis of $\pi N$ scattering data, optimize the LECs in the contact potentials up to NNLO to reproduce low-energy nucleon-nucleon scattering phase shifts, and constrain the three-nucleon interaction at NNLO to reproduce the binding energy and point-proton radius of ${}^4\mathrm{He}$. For heavier nuclei we use the coupled-cluster method to compute binding energies, radii, and neutron skins. We find that radii and binding energies are much improved for interactions with explicit inclusion of $\mathrm{\Delta }\left(1232\right)$, while $\mathrm{\Delta }$-less interactions produce nuclei that are not bound with respect to breakup into $\alpha$ particles. The saturation of nuclear matter is significantly improved, and its symmetry energy is consistent with empirical estimates.

Figure 1

Schematic figure of relevant diagrams that enter in $\mathrm{\Delta }$-full $\chi \mathrm{EFT}$ at leading order (LO), next-to-leading order (NLO), and next-to-next-to-leading order (NNLO). The leading $\pi N$ and $\pi N\mathrm{\Delta }$ axial couplings are denoted by $g_A$ and $h_A$, respectively. Note that there are no $\mathrm{\Delta }$ contributions at LO. At NLO, the $\mathit{NN}$ contact interactions also remain unchanged. However, the leading-order $NNN$ interaction, i.e., the well-known Fujita-Miyazawa term [35], appears at this order; see also Ref. [36]. The $\mathrm{\Delta }$ contributions to the $NNN$ interaction at NNLO vanish because of the Pauli principle or are suppressed and demoted to a higher chiral order. In addition to the subleading $\pi N$ LECs $c_1,c_2,c_3,c_4$, the $NNN$ diagrams contain two additional LECs: $c_D$ and $c_E$.

Figure 2

Neutron-proton scattering phase shifts for the contact partial waves using the $\mathrm{\Delta }$-full and $\mathrm{\Delta }$-less $\chi \mathrm{EFT}$ potentials with nonlocal cutoff $\mathrm{\Lambda }=450$ MeV. All phases are compared to the results from the Granada phase shift analysis [43]. The bands correspond to the order-by-order EFT truncation error in the $\mathrm{\Delta }$-full approach, as described in the main text.

Figure 3

Neutron-proton scattering phase shifts for selected peripheral partial waves using the $\mathrm{\Delta }$-full and $\mathrm{\Delta }$-less $\chi \mathrm{EFT}$ potentials with nonlocal cutoff $\mathrm{\Lambda }=450$ MeV. All phases are compared to the results from the Granada phase shift analysis [43]. The bands correspond to the order-by-order EFT truncation error in the $\mathrm{\Delta }$-full approach, as described in the main text.

Figure 4

Proton-proton scattering phase shifts for the contact and selected peripheral partial waves using the $\mathrm{\Delta }$-full and $\mathrm{\Delta }$-less $\chi \mathrm{EFT}$ potentials with nonlocal cutoff $\mathrm{\Lambda }=450$ MeV. All phases are compared to the results from the Granada phase shift analysis [43]. The bands correspond to the order-by-order EFT truncation error in the $\mathrm{\Delta }$-full approach, as described in the main text.

Figure 5

Ground-state energy (negative of binding energy) per nucleon and charge radii for selected nuclei computed with coupled cluster theory and the $\mathrm{\Delta }$-full potential ($\mathrm{\Lambda }=450$ MeV). For each nucleus, from left to right as follows: LO (red triangle), NLO (green square), and NNLO (blue circle). The black bars are data. Vertical bars estimate uncertainties from the order-by-order EFT truncation errors $\sigma$(LO), $\sigma$(NLO), and $\sigma$(NNLO). At NLO and NNLO we estimate a conservative 95% confidence interval, i.e., $1.96\times \sigma$. See the text for details.

Figure 6

Elastic charge form factor of ${}^{48}\mathrm{Ca}$ from ${\mathrm{NNLO}}_{\mathrm{sat}}$ (gray dash-dotted), $\mathrm{\Delta }\mathrm{NLO}$ (green dashed), and $\mathrm{\Delta }\mathrm{NNLO}$ (blue solid line) compared to experimental data (black).

Figure 7

Energy per nucleon (in MeV) of symmetric nuclear matter (upper row) and pure neutron matter (lower row) up to third order in $\chi \mathrm{EFT}$ without (left column) and with (right column) explicit inclusion of the $\mathrm{\Delta }$ isobar in $\chi \mathrm{EFT}$. All interaction employ a momentum cutoff $\mathrm{\Lambda }=450$ MeV. Shaded areas indicate the estimated EFT-truncation errors, and the (square) diamond marks the saturation point in symmetric nuclear matter for ($\mathrm{\Delta }\mathrm{NLO}$) $\mathrm{\Delta }\mathrm{NNLO}$. The black rectangle indicates the region $E/A=-16\pm 0.5$ MeV and $\rho =0.16\pm 0.01{\mathrm{fm}}^{-3}$.

Figure 8

The symmetry energy as a function of the density for $\mathrm{\Delta }\mathrm{NLO}$ (green), and $\mathrm{\Delta }\mathrm{NNLO}$ (blue) with $\mathrm{\Delta }$-full interaction at cutoff of 450 MeV, with uncertainties shown as shaded areas.