Benchmark calculations of electromagnetic sum rules with a symmetry-adapted basis and hyperspherical harmonics

We demonstrate the ability to calculate electromagnetic sum rules with the ab initio symmetry-adapted no-core shell model. By implementing the Lanczos algorithm, we compute nonenergy weighted, energy weighted, and inverse energy weighted sum rules for electric monopole, dipole, and quadrupole transitions in ${}^4\mathrm{He}$ using realistic interactions. We benchmark the results with the hyperspherical harmonics method and show agreement within $2\sigma$, where the uncertainties are estimated from the use of the many-body technique. We investigate the dependence of the results on three different interactions, including chiral potentials, and we report on the ${}^4\mathrm{He}$ electric dipole polarizability calculated in the SA-NCSM that reproduces the experimental data and earlier theoretical outcomes. We also detail a novel use of the Lawson procedure to remove the spurious center-of-mass contribution to the sum rules that arises from using laboratory-frame coordinates. We further show that this same technique can be applied in the Lorentz integral transform method, with a view toward studies of electromagnetic reactions for light through medium-mass nuclei.

Figure 1

(a) Ground-state energy and (b) point-proton rms radius for ${}^4\mathrm{He}$ as a function of $K_{\max }$ or $N_{\max }$ with the JISP16 interaction. NCSM and SA-NCSM points are shown for $\hslash \mathrm{\Omega }=25$ MeV, while the extrapolated values are based on a range of $N_{\max }$ and $\hslash \mathrm{\Omega }$ values. Uncertainties of the extrapolated values are smaller than the size of the plot markers.

Figure 2

Nonenergy weighted sum rule as a function of $N_{\max }$ or $K_{\max }$ for ${}^4\mathrm{He}$: (a) total monopole strength ($L=0$) and quadrupole strength ($L=2$), along with (b) dipole strength ($L=1$) and inset showing convergence of three $\hslash \mathrm{\Omega }$ values toward the extrapolated infinite-space value (black solid line). HH, NCSM, and SA-NCSM calculations are performed for the JISP16 interaction; NCSM and SA-NCSM points are shown for $\hslash \mathrm{\Omega }=25$ MeV, while the extrapolated values are based on a range of $N_{\max }$ and $\hslash \mathrm{\Omega }$ values. Uncertainties of the extrapolated values are smaller than the size of the plot markers.

Figure 3

Energy weighted sum rules for (a) monopole and (b) dipole transitions as a function of excitation energy for ${}^4\mathrm{He}$. HH, NCSM, and SA-NCSM calculations are performed for the JISP16 interaction; HH results are shown for $K_{\max }=20$ whereas NCSM and SA-NCSM results are shown for $N_{\max }=16$ and $\langle 6\rangle 16$, respectively, with $\hslash \mathrm{\Omega }=25$ MeV.

Figure 4

(a) Ground-state energy and (b) quadrupole $m_0$ for ${}^4\mathrm{He}$ using the N3LO-EM interaction (NN only) as a function of $K_{\max }$ or $N_{\max }$. NCSM and SA-NCSM results are shown for $\hslash \mathrm{\Omega }=25$ MeV, while the extrapolated values are based on a range of $N_{\max }$ (see Table 1) and a 10% variation in the $\hslash \mathrm{\Omega }$ parameter.

Figure 5

Electric dipole polarizability calculated in the SA-NCSM using the N3LO-EM and ${\mathrm{NNLO}}_{\mathrm{opt}}$ interactions (NN only) for $\langle 7\rangle 17$ model spaces and $\hslash \mathrm{\Omega }=25$ MeV. The values calculated in the HH [45] and the NCSM [59] with N3LO-EM (N3LO $\mathrm{NN}+\mathrm{N}2\mathrm{LO}$ 3N), together with the experimentally deduced value [56, 57], are shown for comparison to the far right of the plot (unrelated to the $E_x$ axis).

Figure 6

Quadrupole $m_0$ sum rule computed in the SA-NCSM with the JISP16 in a $\langle 6\rangle 16$ model space for transitions in ${}^4\mathrm{He}$ as a function of the excitation energy, calculated by using a CM-free pivot, as well as a CM-spurious pivot for different values of the Lawson coefficient $\lambda$. The corresponding HH result is also shown for comparison.

Figure 7

LIT with $\mathrm{\Gamma }=10$ MeV for quadrupole transitions in ${}^4\mathrm{He}$ using the JISP16 potential. Calculations with the SA-NCSM are performed in a $\langle 6\rangle 16$ model space for different values of the Lawson coefficient $\lambda$, and are compared to the HH result.