Nuclear properties with semilocal momentum-space regularized chiral interactions beyond ${\mathrm{N}}^2\mathrm{LO}$

We present a comprehensive investigation of few-nucleon systems as well as light and medium-mass nuclei up to $A=48$ using the current Low Energy Nuclear Physics International Collaboration two-nucleon interactions in combination with the third-order (${\mathrm{N}}^2\mathrm{LO}$) three-nucleon forces. To address the systematic overbinding of nuclei starting from $A\approx 10$ found in our earlier study utilizing the ${\mathrm{N}}^2\mathrm{LO}$ two- and three-nucleon forces, we take into account higher-order corrections to the two-nucleon potentials up through fifth order in chiral effective field theory. The resulting Hamiltonian can be completely determined using the $A=3$ binding energies and selected nucleon-deuteron cross sections as input. It is then shown to predict other nucleon-deuteron scattering observables and spectra of light $p$-shell nuclei, for which a detailed correlated truncation error analysis is performed, in agreement with experimental data. Moreover, the predicted ground state energies of nuclei in the oxygen isotopic chain from ${}^{14}\mathrm{O}$ to ${}^{26}\mathrm{O}$ as well as ${}^{40}\mathrm{Ca}$ and ${}^{48}\mathrm{Ca}$ show a remarkably good agreement with experimental values, given that the Hamiltonian is fixed completely from the $A\le 3$ data, once the fourth-order (${\mathrm{N}}^3\mathrm{LO}$) corrections to the two-nucleon interactions are taken into account. On the other hand, the charge radii are found to be underpredicted by $\approx 10\%$ for the oxygen isotopes and by almost $20\%$ for ${}^{40}\mathrm{Ca}$ and ${}^{48}\mathrm{Ca}$.

Figure 1

Predictions for the neutron-deuteron total cross section at 70 MeV (left panel) and 135 MeV (right panel) based on the semilocal momentum-space regularized chiral interactions at different orders (shown by solid symbols with error bars). Three-nucleon force is included at ${\mathrm{N}}^2\mathrm{LO}$ only. Error bars show the EFT truncation uncertainty calculated using the Bayesian model ${\overline{C}}_{0.5-10}^{650}$ from Ref. [26] [68% degree of belief (DoB) intervals]. For the incomplete calculations at ${\mathrm{N}}^3\mathrm{LO}$ and ${\mathrm{N}}^4\mathrm{LO}$, the quoted errors correspond to the ${\mathrm{N}}^2\mathrm{LO}$ truncation uncertainties. Gray open symbols without error bars show the results based on the two-nucleon forces only. Horizontal bands are experimental data from Ref. [30].

Figure 2

The center-of-mass differential cross section $\frac{d\sigma }{d\mathrm{\Omega }}$, the deuteron vector analyzing power $A_Y\left(\mathrm{d}\right)$ and the the tensor analyzing power $A_{XX}-A_{YY}$ for the neutron-deuteron elastic scattering at incoming neutron laboratory energy $E=200$ MeV. In the top panels the dashed blue (red) curve represents predictions based on the two-nucleon ${\mathrm{N}}^2\mathrm{LO}$ (${\mathrm{N}}^4{\mathrm{LO}}^{+}$) forces. The solid blue curve represents complete results at ${\mathrm{N}}^2\mathrm{LO}$ and the solid red curve stands for predictions of ${\mathrm{N}}^4{\mathrm{LO}}^{+}$ NN interaction supplemented by ${\mathrm{N}}^2\mathrm{LO}$ 3NF. In all cases, the cutoff $\mathrm{\Lambda }=450$ MeV is used. In the bottom panels, the light (dark) green band shows the size of the truncation error at 95% (68%) DoB. The grey band shows a spread of the ${\mathrm{N}}^4{\mathrm{LO}}^{+}$ NN $+ {\mathrm{N}}^2\mathrm{LO}$ 3N force based predictions due to the value of $\mathrm{\Lambda }$ regulator, in the range of $\mathrm{\Lambda }\in [400,550]$ MeV. The red curve is the same as in the upper panels. In (e) and (f), the dashed dark-green curve shows the borders of the dark green band. Data in (a) and (d) are from [34]: black circles for $E=181$ MeV and orange triangles for $E=216.5$ MeV. Data in (b), (c), (e), and (f) are from [35].

Figure 3

The differential cross section and the nucleon analyzing power $A_Y$(N) at incoming neutron laboratory energy $E=135$ MeV. The directions of momenta of outgoing neutrons are in (a) and (d) ${\theta }_1={20}^{\circ },{\theta }_2={16}^{\circ },$ and ${\varphi }_{12}={180}^{\circ }$; in (b) and (e) ${\theta }_1={28}^{\circ },{\theta }_2={28}^{\circ },$ and ${\varphi }_{12}={180}^{\circ }$; and in (c) and (f) ${\theta }_1={20}^{\circ },{\theta }_2={16}^{\circ },$ and ${\varphi }_{12}={20}^{\circ }$. In the top panels the dashed blue (red) curve represents predictions based on the two-nucleon ${\mathrm{N}}^2\mathrm{LO}$ (${\mathrm{N}}^4{\mathrm{LO}}^{+}$) forces. The solid blue curve represents complete results at ${\mathrm{N}}^2\mathrm{LO}$ and the solid red curve stands for predictions of ${\mathrm{N}}^4{\mathrm{LO}}^{+}$ NN interaction supplemented by ${\mathrm{N}}^2\mathrm{LO}$ 3NF. In all cases, the cutoff $\mathrm{\Lambda }=450$ MeV is used. In the bottom panels, the light (dark) green band shows size of truncation error at 95% (68%) DoB at $\mathrm{\Lambda }=450$ MeV. The grey band shows a spread of the ${\mathrm{N}}^4{\mathrm{LO}}^{+}$ NN $+ {\mathrm{N}}^2\mathrm{LO}$ 3NF based predictions due to the variation of $\mathrm{\Lambda }$ in the range $\mathrm{\Lambda }\in [400,550]$ MeV. The red curve is the same as in the upper panels. Data are from Ref. [37].

Figure 4

The differential cross section and the deuteron analyzing power $A_{XX}$ at incoming neutron laboratory energy $E=200$ MeV. The directions of momenta of outgoing neutrons are in (a) and (d) ${\theta }_1={100}^{\circ },{\theta }_2={25}^{\circ },$ and ${\varphi }_{12}={180}^{\circ }$; in (b) and (e) ${\theta }_1={15}^{\circ },{\theta }_2={10}^{\circ },$ and ${\varphi }_{12}=0^{\circ }$; and in (c) and (f) ${\theta }_1={50}^{\circ },{\theta }_2={50}^{\circ },$ and ${\varphi }_{12}={160}^{\circ }$. The curves and bands are the same as in Fig. 4.

Figure 5

Expansion coefficients defined as in Eq. (2) for the individual of the light nuclei listed with $\alpha =0.08{\mathrm{fm}}^4$ and $\mathrm{\Lambda }=450$ MeV. These are extracted with a fixed value of $Q\approx 0.31$ and $X_{\mathrm{ref}}$ taken from experiment [51, 52, 53] (or the ${\mathrm{N}}^2\mathrm{LO}$ result for the $0^{+}$ in ${}^8\mathrm{Li}$).

Figure 6

Posteriors for the dimensionless expansion parameter $Q$ learned from the order-by-order coefficients for the ground and excited-state energies of the nuclei in Tables 3 and 4. These coefficients encode the convergence pattern of the chiral expansion for these nuclei. The posteriors were extracted separately for nuclei with $A<8$ and $A\ge 8$ because of the differing degree of correlation between the coefficients in these two groups (see Fig. 5). The top panel is for $\mathrm{\Lambda }=450\text{MeV}$ and the bottom panel for $\mathrm{\Lambda }=500\text{MeV}$, both using $\alpha =0.08{\text{fm}}^4$.

Figure 7

Ground-state energies for ${}^4\mathrm{He}, {}^6\mathrm{He}, {}^8\mathrm{He}$, and ${}^6\mathrm{Li}$ with SMS interactions from NLO to ${\mathrm{N}}^4{\mathrm{LO}}^{+}$ with $\mathrm{\Lambda }=450\text{MeV}$ (left-hand panel) and $\mathrm{\Lambda }=500\text{MeV}$ (right-hand panel), both using $\alpha =0.08{\text{fm}}^4$. Error bars indicate the NCSM model-space uncertainties and shaded bands indicate the chiral truncation uncertainties at the 95% confidence level for the ${\mathrm{N}}^2\mathrm{LO}$ (blue) and ${\mathrm{N}}^4{\mathrm{LO}}^{+}$ (purple) NN potentials. Horizontal bars show the experimental ground-state energies.

Figure 8

Ground-state energies for ${}^{10}\mathrm{Be}, {}^{10}\mathrm{B}, {}^{12}\mathrm{B}$, and ${}^{12}\mathrm{C}$ with SMS interactions from NLO to ${\mathrm{N}}^4{\mathrm{LO}}^{+}$ with $\mathrm{\Lambda }=450\text{MeV}$ (left-hand panel) and $\mathrm{\Lambda }=500\text{MeV}$ (right-hand panel), both using $\alpha =0.08{\text{fm}}^4$. Error bars and bands are the same as in Fig. 7. Horizontal bars show the experimental ground-state energies.

Figure 9

Excitation energies of low-lying states in ${}^{10}\mathrm{B}$ with SMS interactions from NLO (gray) to ${\mathrm{N}}^4{\mathrm{LO}}^{+}$ (purple) with $\mathrm{\Lambda }=450\text{MeV}$ (left-hand panel) and $\mathrm{\Lambda }=500\text{MeV}$ (right-hand panel), both using $\alpha =0.08{\text{fm}}^4$. Error bars indicate the NCSM model-space uncertainties and shaded bands indicate the chiral truncation uncertainties at the 95% confidence level. Horizontal lines show the experimental excitation energies.

Figure 10

Excitation energies of low-lying states in ${}^{12}\mathrm{B}$ with SMS interactions from NLO (gray) to ${\mathrm{N}}^4{\mathrm{LO}}^{+}$ (purple) with $\mathrm{\Lambda }=450\text{MeV}$ (left-hand panel) and $\mathrm{\Lambda }=500\text{MeV}$ (right-hand panel), both using $\alpha =0.08{\text{fm}}^4$. Error bars and bands are the same as in Fig. 9. Horizontal lines show the experimental excitation energies.

Figure 11

Excitation energies of low-lying states in ${}^{12}\mathrm{C}$ with SMS interactions from NLO (gray) to ${\mathrm{N}}^4{\mathrm{LO}}^{+}$ (purple) with $\mathrm{\Lambda }=450\text{MeV}$ (left-hand panel) and $\mathrm{\Lambda }=500\text{MeV}$ (right-hand panel), both using $\alpha =0.08{\text{fm}}^4$. Error bars and bands are the same as in Fig. 9. Horizontal lines show the experimental excitation energies.

Figure 12

Ground-state energies and point-proton radii for even oxygen isotopes obtained in the IM-NCSM with SMS interactions from ${\mathrm{N}}^2\mathrm{LO}$ to ${\mathrm{N}}^4{\mathrm{LO}}^{+}$ with $\mathrm{\Lambda }=450\text{MeV}$ (left-hand panels) and $\mathrm{\Lambda }=500\text{MeV}$ (right-hand panels) for flow-parameter $\alpha =0.08{\text{fm}}^4$. The error bands show the chiral truncation uncertainties at the 95% confidence level obtained with the Bayesian model for ${\mathrm{N}}^2\mathrm{LO}$ and ${\mathrm{N}}^4{\mathrm{LO}}^{+}$.

Figure 13

Ground-state energies and point-proton radii for doubly magic oxygen and calcium isotopes obtained in the IM-SRG with SMS interactions from NLO to ${\mathrm{N}}^4{\mathrm{LO}}^{+}$ for $\mathrm{\Lambda }=450\text{MeV}$ (left-hand panels) and $\mathrm{\Lambda }=500\text{MeV}$ (right-hand panels) with SRG flow parameter $\alpha =0.08{\text{fm}}^4$. The error bands show the chiral truncation uncertainties at the 95% confidence level obtained with the pointwise Bayesian model for ${\mathrm{N}}^2\mathrm{LO}$ and ${\mathrm{N}}^4{\mathrm{LO}}^{+}$.

Figure 14

Ground-state energies and point-proton radii for even oxygen isotopes obtained in the IM-NCSM with the SMS interaction at ${\mathrm{N}}^4{\mathrm{LO}}^{+}$ for $\mathrm{\Lambda }=450\text{MeV}$, supplemented by the $E1$ three-nucleon contact term at ${\mathrm{N}}^4\mathrm{LO}$ with LEC values $c_{E1}=0,\pm 1$.