Continuum and three-nucleon force effects on ${}^9\mathrm{Be}$ energy levels

We extend the recently proposed ab initio no-core shell model with continuum to include three-nucleon (3N) interactions beyond the few-body domain. The extended approach allows for the assessment of effects of continuum degrees of freedom as well as of the 3N force in ab initio calculations of structure and reaction observables of $p$- and lower-$sd$-shell nuclei. As a first application we concentrate on energy levels of the ${}^9\mathrm{Be}$ system for which all excited states lie above the n-${}^8\mathrm{Be}$ threshold. For all energy levels, the inclusion of the continuum significantly improves the agreement with experiment, which was an issue in standard no-core shell model calculations. Furthermore, we find the proper treatment of the continuum indispensable for reliable statements about the quality of the adopted 3N interaction from chiral effective field theory. In particular, we find the ${\frac{1}{2}}^{+}$ resonance energy, which is of astrophysical interest, in good agreement with experiment.

Figure 1

NCSMC n-${}^8\mathrm{Be}$ phase shifts and eigenphase shifts for negative parity at $N_{\text{max}}=8$ computed with NCSM eigenvectors (solid line) compared to the use of IT-NCSM vectors (dashed line). Remaining parameters are $\hslash \Omega =20\text{MeV},\alpha =0.0625{\text{fm}}^4$, and $E_{\text{3max}}=14$. Same colors correspond to identical angular momenta.

Figure 2

$N_{\text{max}}$ dependence of NCSMC n-${}^8\mathrm{Be}$ eigenphase shifts. The left- and right-hand columns show the results for the NN+3N-induced and NN+3N-full Hamiltonian, respectively. Remaining parameters identical to Fig. 1.

Figure 3

SRG flow-parameter dependence of the NCSMC n-${}^8\mathrm{Be}$ eigenphase shifts for the NN+3N-full Hamiltonian at $N_{\text{max}}=10\left(11\right)$ for negative (positive) parity. Remaining parameters identical to Fig. 1.

Figure 4

Negative (a) and positive (b) parity spectrum of ${}^9\mathrm{Be}$ relative to the n-${}^8\mathrm{Be}$ threshold as a function of $N_{\text{max}}$ for the NCSM (left-hand columns) and NCSMC (right-hand columns) compared to experiment [23]. Remaining parameters are $\hslash \Omega =20\text{MeV}$ and $\alpha =0.0625{\text{fm}}^4$. See text for further explanations.

Figure 5

Negative (a) and positive (b) parity spectrum of ${}^9\mathrm{Be}$ relative to the n-${}^8\mathrm{Be}$ threshold at $N_{\text{max}}=12$ and 11, respectively. Shown are NCSM (first two columns) and NCSMC (last two columns) results compared to experiment [23]. First and last columns contain the energies for the NN+3N-induced and the second and fourth column for the NN+3N-full Hamiltonian, respectively. Shaded areas denote the width of the energy levels. Remaining parameters identical to Fig. 4.