Three-cluster dynamics within the ab initio no-core shell model with continuum: How many-body correlations and $\alpha$ clustering shape ${}^6\mathrm{He}$

We realize the treatment of bound and continuum nuclear systems in the proximity of a three-body breakup threshold within the ab initio framework of the no-core shell model with continuum. Many-body eigenstates obtained from the diagonalization of the Hamiltonian within the harmonic-oscillator expansion of the no-core shell model are coupled with continuous microscopic three-cluster states to correctly describe the nuclear wave function both in the interior and asymptotic regions. We discuss the formalism in detail and give algebraic expressions for the case of core + $n+n$ systems. Using similarity-renormalization-group evolved nucleon-nucleon interactions, we analyze the role of ${}^4\mathrm{He}+n+n$ clustering and many-body correlations in the ground and low-lying continuum states of the Borromean ${}^6\mathrm{He}$ nucleus, and study the dependence of the energy spectrum on the resolution scale of the interaction. We show that ${}^6\mathrm{He}$ small binding energy and extended radii compatible with experiment can be obtained simultaneously, without resorting to extrapolations. We also find that a significant portion of the ground-state energy and the narrow width of the first $2^{+}$ resonance stem from many-body correlations that can be interpreted as core-excitation effects.

Figure 1

We show the Jacobi coordinates ${\vec{\eta }}_{1,23}$ (proportional to the vector between the c.m. of the first cluster and that of the residual two fragments) and ${\vec{\eta }}_{23}$ (proportional to the vector between the c.m. of clusters 2 and 3). In the figure, a case with three clusters of four, two, and one nucleons are shown; however, the formalism is completely general and can be used to describe any three cluster configuration.

Figure 2

Dependence of the NCSM ${}^6\mathrm{He}$ and ${}^4\mathrm{He}{J}^{\pi }=0^{+}$ ground state energies $E$(g.s.) on the HO model space size $N_{\max }$ for the SRG-$\mathrm{N}{}^3\mathrm{LO}NN$ potential with (a) ${\lambda }_{\mathrm{SRG}}=1.5{\mathrm{fm}}^{-1}$ and $\hslash \mathrm{\Omega }=14$ MeV and (b) ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$ and $\hslash \mathrm{\Omega }=20$ MeV.

Figure 3

Dependence of the NCSM ${}^6\mathrm{He}{J}^{\pi }=0^{+}$ excitation energies ($E_{\mathrm{x}}$) on the HO model space size $N_{\max }$ for the SRG-$\mathrm{N}{}^3\mathrm{LO}NN$ potential with (a) ${\lambda }_{\mathrm{SRG}}=1.5{\mathrm{fm}}^{-1}$ and $\hslash \mathrm{\Omega }=14$ MeV and (b) ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$ and $\hslash \mathrm{\Omega }=20$ MeV.

Figure 4

Same as Fig. 2 including also the dependence of the NCSMC ${}^6\mathrm{He}{J}^{\pi }=0^{+}$ ground-state energy ($E$) on the HO model space size $N_{\max }$ for the SRG-evolved $\mathrm{N}{}^3\mathrm{LO}NN$ potential with (a) ${\lambda }_{\mathrm{SRG}}=1.5{\mathrm{fm}}^{-1}$ and $\hslash \mathrm{\Omega }=14$ MeV, and (b) ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$ and $\hslash \mathrm{\Omega }=20$ MeV. The extrapolated $N_{\max }\to \infty$ NCSM ${}^6\mathrm{He}$ is shown as a band of which the width represents the extrapolation uncertainty.

Figure 5

Probability distribution the $J^{\pi }=0^{+}$ ground state of the ${}^6\mathrm{He}$. Here $r_{nn}=\sqrt{2}{\eta }_{nn}$ and $r_{\alpha ,nn}=\sqrt{3/4}{\eta }_{\alpha ,nn}$ are, respectively, the distance between the two neutrons and the distance between the c.m. of ${}^4\mathrm{He}$ and that of the two neutrons.

Figure 6

(a) Most relevant hyper-radial components ${\tilde{u}}_{\nu K}\left(\rho \right)$ [see text] of the $\alpha +n+n$ relative motion within the ${}^6\mathrm{He}$ g.s. after projection of the ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$ full NCSMC wave function in the largest model space (blue solid lines) as well as of its NCSM portion (red dashed lines) into the orthogonalized microscopic-cluster basis. [(b), (c)] Contour plots of the probability distribution obtained from the projection of the full NCSMC wave function of panel (a) and its NCSM component, respectively, as a function of the relative coordinates $r_{nn}=\sqrt{2}{\eta }_{nn}$ and $r_{\alpha ,nn}=\sqrt{3/4}{\eta }_{\alpha ,nn}$.

Figure 7

NCSM (green symbols) and NCSMC (blue symbols) rms matter (triangles) and point-proton (squares) radii, and two-neutron separation energy (circles), obtained using the SRG-$\mathrm{N}{}^3\mathrm{LO}NN$ interaction with ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$ in the largest HO model space ($N_{\max }=12$). Also shown are the infinite-basis ($\infty$) extrapolations from Sääf et al. [39] (red symbols) and the EIHH results from Ref. [38] (indigo symbols) based on the $V_{\mathrm{low}k}$($\mathrm{N}{}^3\mathrm{LO}$) $NN$ interaction at the resolution scales ${\mathrm{\Lambda }}_{\mathrm{low}k}=1.8$ and 2.0 ${\mathrm{fm}}^{-1}$. The range of experimental values are represented by horizontal bands (see text for more details).

Figure 8

Most relevant attractive eigenphase shifts below 6 MeV above the two-neutron emission threshold $[E_{\mathrm{th}}$($\alpha +n+n$)] computed within the NCSMC [panels (a) and (b)] and within the more limited model space spanned by the ${}^4\mathrm{He}$ (g.s.) + $n+n$ cluster basis alone of Ref. [32] [panels (c) and (d)], using the SRG-evolved $\mathrm{N}{}^3\mathrm{LO}NN$ potential with ${\lambda }_{\mathrm{SRG}}=1.5{\mathrm{fm}}^{-1}$. We show positive-parity states in panels (a) and (c) and negative-parity states in panels (b) and (d).

Figure 9

Lowest-lying attractive eigenphase shifts for the $J^{\pi }=1^{\pm }$ and $2^{+}$ channels computed within the NCSMC (blue solid line) and within the more limited model space spanned by the ${}^4\mathrm{He}$ (g.s.) + $n+n$ cluster basis alone of Ref. [32] (red dot-dashed line) using the SRG-evolved $\mathrm{N}{}^3\mathrm{LO}NN$ potential with ${\lambda }_{\mathrm{SRG}}=1.5{\mathrm{fm}}^{-1}$.

Figure 10

Most relevant eigenphase shifts for the $J^{\pi }=0^{+}$ channel below 6 MeV above the two-neutron emission threshold $[E_{\mathrm{th}}$($\alpha +n+n$)] computed within the NCSMC (blue solid line) and within the more limited model space spanned by the ${}^4\mathrm{He}$ (g.s.) + $n+n$ cluster basis alone of Ref. [32] (red dot-dashed line) using the SRG-evolved $\mathrm{N}{}^3\mathrm{LO}NN$ potential with ${\lambda }_{\mathrm{SRG}}=1.5{\mathrm{fm}}^{-1}$.

Figure 11

Convergence with respect to the number ($N_{\lambda }$) of square-integrable eigenstates of the composite system included in the NCSMC calculation of the most relevant eigenphase shifts for the $0^{+}$ channel below 8 MeV above the two-neutron emission threshold $[E_{\mathrm{th}}$($\alpha +n+n$)], using the SRG-evolved $\mathrm{N}{}^3\mathrm{LO}NN$ potential with ${\lambda }_{\mathrm{SRG}}=1.5{\mathrm{fm}}^{-1}$. Also shown are the results from Ref. [32], corresponding to the inclusion of zero composite states (cluster basis).

Figure 12

Spectrum of low-lying energy levels of the ${}^6\mathrm{He}$ nucleus for ${\lambda }_{\mathrm{SRG}}=1.5$ as obtained within the ${}^4\mathrm{He}$ (g.s.) + $n+n$ cluster basis of Ref. [32], the NCSMC, and within the NCSM by treating the ${}^6\mathrm{He}$ excited states as bound states at $N_{\max }=12$. For the NCSM, we show both the energy levels at $N_{\max }=12$ and the results of an extrapolation to the infinite model space ($\infty$), performed through the exponential fit function from Eq. (53).

Figure 13

Lowest-lying attractive eigenphase shifts for the $J^{\pi }=1^{\pm }$ and $2^{+}$ channels computed within the NCSMC using the SRG-evolved $\mathrm{N}{}^3\mathrm{LO}NN$ potential with ${\lambda }_{\mathrm{SRG}}=1.5{\mathrm{fm}}^{-1}$ (solid lines) and ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$ (dashed lines).

Figure 14

Spectra of low-lying energy levels of the ${}^6\mathrm{He}$ nucleus computed within the NCSMC using the SRG-evolved $\mathrm{N}{}^3\mathrm{LO}NN$ potential with ${\lambda }_{\mathrm{SRG}}=1.5$ and ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$ compared to the most recent experimental spectrum of Ref. [27].