Nuclear structure with accurate chiral perturbation theory nucleon-nucleon potential: Application to ${}^6\mathrm{Li}$ and ${}^{10}\mathrm{B}$

We calculate properties of the $A=6$ system using the accurate charge-dependent nucleon-nucleon (NN) potential at fourth order of chiral perturbation theory. By application of the ab initio no-core shell model and a variational calculation in the harmonic-oscillator basis with basis size up to $16\hslash \Omega$ we obtain the ${}^6\mathrm{Li}$ binding energy of $28.5\left(5\right)\text{MeV}$ and a converged excitation spectrum. Also, we calculate properties of ${}^{10}\mathrm{B}$ using the same NN potential in a basis space of up to $8\hslash \Omega$. Our results are consistent with results obtained by standard accurate NN potentials and demonstrate a deficiency of Hamiltonians consisting of only two-body terms. At this order of chiral perturbation theory three-body terms appear. It is expected that inclusion of such terms in the Hamiltonian will improve agreement with experiment.

Figure 1

(Color online) Harmonic-oscillator frequency dependence of the ${}^4\mathrm{He}$ ground-state energy obtained in $12\hslash \Omega –18\hslash \Omega$ $\left(N_{\max }=12–18\right)$ basis spaces using the bare ${\mathrm{N}}^3\text{LO}$ NN potential (dotted lines) and two-body effective interactions derived from the ${\mathrm{N}}^3\text{LO}$ NN potential (full lines). The Coulomb potential is included.

Figure 2

(Color online) The same as Fig. 1, but for the point-nucleon radius.

Figure 3

(Color online) Basis size dependence of the ${}^4\mathrm{He}$ point-nucleon radius obtained using the two-body effective interactions derived from the ${\mathrm{N}}^3\text{LO}$ NN potential. Results for the harmonic-oscillator frequencies $\hslash \Omega =24,28,32,36,40\text{MeV}$ are presented.

Figure 4

(Color online) Harmonic-oscillator frequency dependence of the ${}^6\mathrm{Li}$ ground-state energy obtained in $0\hslash \Omega –14\hslash \Omega$ basis spaces using two-body effective interactions derived from the Minnesota NN potential. Coulomb potential is not included. The dotted line corresponds to the stochastic variational method result [29].

Figure 5

(Color online) Harmonic-oscillator frequency dependence of the ${}^6\mathrm{Li}$ ground-state energy obtained in $2\hslash \Omega –14\hslash \Omega$ and $8\hslash \Omega –16\hslash \Omega$ basis spaces using two-body effective interactions derived from the ${\mathrm{N}}^3\text{LO}$ NN potential (full lines) and the bare ${\mathrm{N}}^3\text{LO}$ NN potential (dotted lines), respectively. The Coulomb potential is included.

Figure 6

(Color online) Basis size dependence of the ${}^6\mathrm{Li}$ ground-state energy obtained in $0\hslash \Omega –14\hslash \Omega$ and $0\hslash \Omega –16\hslash \Omega$ basis spaces using two-body effective interactions derived from the ${\mathrm{N}}^3\text{LO}$ NN potential and the bare ${\mathrm{N}}^3\text{LO}$ NN potential, respectively. The Coulomb potential is included. The two-body effective interaction results correspond to the ground-state energy minimum for each model space. For details on the extrapolation see the text.

Figure 7

(Color online) Basis size dependence of the ${}^6\mathrm{Li}$ ground-state energy obtained in $0\hslash \Omega –14\hslash \Omega$ ($16\hslash \Omega$ for MN) basis spaces using two-body effective interactions derived from the $\text{AV}{8}^{\prime }$, ${\mathrm{N}}^3\text{LO}$, CD-Bonn 2000, and the Minnesota NN potentials. The $\text{AV}{8}^{\prime }$ result obtained by the GFMC method [10] and the Minnesota results obtained by the SVM [29] and the EIHH methods [31] are shown for a comparison.

Figure 8

(Color online) Calculated positive-parity excitation spectra of ${}^6\mathrm{Li}$ obtained in $0\hslash \Omega –14\hslash \Omega$ ($12\hslash \Omega$ for the $2^{+}1$ and $1_2^{+}0$ states) basis spaces using two-body effective interactions derived from the ${\mathrm{N}}^3\text{LO}$ NN potential are compared to experiment. The HO frequency of $\hslash \Omega =8\text{MeV}$ was used. The experimental values are from Ref. 39.

Figure 9

(Color online) The same as Fig. 8, but for the HO frequency of $\hslash \Omega =10\text{MeV}$.

Figure 10

(Color online) The same as Fig. 8, but for the HO frequency of $\hslash \Omega =13\text{MeV}$.

Figure 11

(Color online) Calculated positive-parity excitation spectra of ${}^6\mathrm{Li}$ obtained in the $14\hslash \Omega$ ($12\hslash \Omega$ for the $2^{+}1$ and $1_2^{+}0$ states) basis space using two-body effective interactions derived from the ${\mathrm{N}}^3\text{LO}$ NN potential are compared to experiment. The HO frequency dependence in the range of $\hslash \Omega =8–13\text{MeV}$ is presented. The experimental values are from Ref. 39.

Figure 12

(Color online) Harmonic-oscillator frequency dependence of the ${}^6\mathrm{Li}$ quadrupole moment obtained in $0\hslash \Omega –14\hslash \Omega$ $\left(N_{\max }=0–14\right)$ basis spaces using two-body effective interactions derived from the ${\mathrm{N}}^3\text{LO}$ NN potential.

Figure 13

(Color online) The dependence of the ${}^{10}\mathrm{B}$ $1^{+}0$ and $3^{+}0$ state energy on the HO frequency for the $4\hslash \Omega$, $6\hslash \Omega$, and $8\hslash \Omega$ model spaces calculated using the ${\mathrm{N}}^3\text{LO}$ NN potential. In the inset, the $1^{+}0$ and $3^{+}0$ state energies at their respective HO frequency minima are plotted as a function of $N_{\max }$.

Figure 14

(Color online) Calculated positive-parity excitation spectra of ${}^{10}\mathrm{B}$ obtained in $0\hslash \Omega –8\hslash \Omega$ basis spaces using two-body effective interactions derived from the ${\mathrm{N}}^3\text{LO}$ NN potential are compared to experiment. The HO frequency of $\hslash \Omega =12\text{MeV}$ where the $8\hslash \Omega$ ground state is at the minimum was used. The experimental values are from Ref. 39.