Microscopic description of translationally invariant core $+N+N$ overlap functions

We derive expressions for $\mathrm{core}+N+N$ overlap integrals starting from microscopic wave functions obtained in the ab initio no-core shell model. These overlap integrals correspond to three-body channel form factors and can be used to investigate the clustering of many-body systems into a core plus two nucleons. We consider the case when the composite system and the core are described in Slater determinant, harmonic oscillator bases, and we show how to remove spurious center-of-mass components exactly in order to derive translationally invariant overlap integrals. We study in particular the Borromean ${}^6$He nucleus using realistic chiral nuclear interactions, and we demonstrate that the observed clusterization in this system is a Pauli focusing effect. The inclusion of three-body forces has a small effect on this structure. In addition, we discuss the issue of absolute normalization for spectroscopic factors, which we show is larger than one. As part of this study we also perform extrapolations of ground-state observables and investigate the dependence of these results on the resolution scale of the interaction.

Figure 1

Extrapolation of ${}^6$He binding energy (a) and point-proton radius (b) as a function of the IR cutoff parameter. Results are obtained with the NCSM using an SRG-evolved chiral two-body interaction (${\Lambda }_{\mathrm{SRG}}=2.0$ fm${}^{-1}$). See text for details on the extrapolation procedure.

Figure 2

Contour plots of the translationally invariant three-body channel form factor $\langle {}^6\text{He}\left(0^{+}\right)|{}^4\text{He}\left(0^{+}\right)+n+n\rangle$ calculated from NCSM wave functions. The two allowed channels for this overlap, $L=S=0$ and $L=S=1$, are shown in panels (a) and (b), respectively.

Figure 3

Model-space dependence of the three-body channel form factor $\langle {}^6\text{He}\left(0^{+}\right)|{}^4\text{He}\left(0^{+}\right)+n+n\rangle$. The main $L=S=0$ channel is shown for a sequence of calculations with increasing $N_{\max }$ performed with ${\Lambda }_{\mathrm{SRG}}=2.0$ fm${}^{-1}$ and $\hslash \Omega =16$ MeV. The lower right panel shows the same contour plot using an SRG-evolved chiral $NN+3N\mathrm{F}$ interaction [43].

Figure 4

Hyperradial functions obtained from NCSM three-body channel form factors calculated with an SRG-evolved chiral two-body interaction (${\Lambda }_{\mathrm{SRG}}=2.0$ fm${}^{-1}$). (a) $N_{\max }$ dependence for a fixed frequency, $\hslash \Omega =20$ MeV. Thick lines correspond to $N_{\max }=14$ results, while dotted ones are $N_{\max }=12$. (b) A fixed model space ($N_{\max }=14$, shaded bands) and a range of HO frequencies, $\hslash \Omega =[16,22]$.