SU(3)-guided realistic nucleon-nucleon interactions for large-scale calculations

We examine nucleon-nucleon realistic interactions, based on their SU(3) decomposition to SU(3)-symmetric components. We find that many of these interaction components are negligible, which, in turn, allows us to identify a subset of physically relevant components that are sufficient to describe the structure of low-lying states in ${}^{12}\mathrm{C}$ and related observables, such as excitation energies, electric quadrupole transitions, and rms radii. We find that paring down the interaction by half or more yields results that practically coincide with the corresponding ab initio calculations with the full interaction. In addition, we show that while various realistic interactions differ in their SU(3) decomposition, their renormalized effective counterparts exhibit a striking similarity and composition that can be linked to dominant nuclear features such as deformation, pairing, clustering, and spin-orbit effect.

Figure 1

Relative strengths $\mathfrak{s}$ (in %) for the $\mathrm{SU}\left(3\right)$-coupled JISP16 (top) and N3LO (bottom) NN interactions and their effective counterparts with $\hslash \mathrm{\Omega }=15$ and 20 MeV, respectively, in the $N_{\max }=6$ model space. The “eff. JISP16” is obtained by the OLS technique for $A=12$, while “eff. N3LO” is by SRG with ${\lambda }_{\mathrm{SRG}}$ = 2.0 ${\mathrm{fm}}^{-1}$. $T$ is the isospin of the two-nucleon system. A set of $\left({\lambda }_0{\mu }_0\right)S_0$ quantum numbers and its conjugate correspond to each of the interaction terms. Only terms with $>1\%$ relative strength for each $T$ are shown; there are more than 120 terms with less than 1% strength for this model space.

Figure 2

Excitation energy of the first $2^{+}$ and $4^{+}$ states in ${}^{12}\mathrm{C}$ from SA-NCSM calculations (connected lines) as a function of the fraction of the terms kept in the interaction, and compared to experiment [35] (labeled as “Expt.”). Results for $N_{\max }=6$, 8, 10, and 12 are shown for various selections of the JISP16 interaction with $\hslash \mathrm{\Omega }=15$ MeV. Specifically, the value 1 on the abscissa indicates that the full interaction (100%) was used, while an abscissa value of 0.4 implies that only the most significant 40% of the tensors were retained, etc.

Figure 3

Same as Fig. 2, but for the rms radius (in fm) of the ${}^{12}\mathrm{C}$ ground state (experimental value from Ref. [42]) and the $B(E2:2_1^{+}\to 0_1^{+})$ value (in $e^2{\mathrm{fm}}^4$) (experimental value from [35]) as a function of the fraction of the terms kept in the interaction. SA-NCSM calculations use various selections for the JISP16 interaction for $\hslash \mathrm{\Omega }=15$ MeV and different $N_{\max }$ model spaces.

Figure 4

Probability amplitudes for the $\left(\lambda \mu \right)S$ configurations that make up the ${}^{12}\mathrm{C}$ ground state ($0_1^{+}$), calculated in $N_{\max }=12$ model space using JISP16 interaction for $\hslash \mathrm{\Omega }=15$ MeV (labeled by “All”) and two selected interactions (labeled by the fraction of the interaction components kept, 46% and 26%). Only states with probability amplitudes $>0.003$ are shown.

Figure 5

${}^{12}\mathrm{C}$ ground state rms radius from SA-NCSM calculations with $N_{\max }=6$ model space vs $\hslash \mathrm{\Omega }$, using the full (“All”) and selected (labeled by the percentage of the tensors kept) JISP16 interaction.

Figure 6

Excitation energies of the first $2^{+}$ and $4^{+}$ states for ${}^{12}\mathrm{C}$ from SA-NCSM calculations with $N_{\max }=6$ and $N_{\max }=8$ model spaces using full JISP16 interaction (“All”) and its selected counterpart (with 37% of the tensors kept), with $\hslash \mathrm{\Omega }=15,20$, and 25 MeV, and compared to experiment.

Figure 7

Energies of the proton-neutron system for the positive-parity lowest-lying states ($<30$ MeV), calculated in the SA-NCSM in $N_{\max }=12$ model space using the JISP16 interaction, with all terms kept (100%) as compared to a selection that keeps only 26% of the terms, for $\hslash \mathrm{\Omega }=15$ MeV.