Halo Nuclei from Ab Initio Nuclear Theory

Abstract

A realistic description of halo nuclei, characterized by low-lying breakup thresholds, requires a proper treatment of continuum effects. We have developed an ab initio approach, the No-Core Shell Model with Continuum (NCSMC), capable of describing both bound and unbound states in light nuclei in a unified way. With chiral two- and three-nucleon interactions as the only input, we can predict the structure and dynamics of halo and other light nuclei and, by comparing to available experimental data, test the quality of chiral nuclear forces. We review NCSMC calculations of weakly bound states and resonances of the exotic halo nuclei ⁶He, ⁸B, ¹¹Be, and ¹⁵C. For the latter, we discuss its production in the capture reaction ¹⁴C(n,$\gamma$)¹⁵C. We highlight the challenges of a description of ⁶He as a Borromean n-n-⁴He system. Finally, we present our calculations of excited states in ¹⁰Be exhibiting a one-neutron halo structure and a large scale No-Core Shell Model investigation of ¹¹Li as a precursor of a full n-n-⁹Li NCSMC study.

1. Introduction

Halo nuclei are exotic weakly bound systems with an extended single-nucleon or two-neutron density well beyond a tightly bound core. They were first discovered in the ¹¹Li nucleus from a series of interaction cross-section measurements four decades ago [1], where the evidence suggested that this unstable nucleus exhibits a two-neutron halo structure around a central ⁹Li core and that the distribution of halo neutrons extends to the size of a nucleus with mass number 200 [2]. Similarly, the ⁶He nucleus is a prominent example of a Borromean quantum halo, i.e., a weakly-bound state of three particles ($\alpha +n+n$) otherwise unbound in pairs, characterized by the large probability of configurations within classically forbidden regions of space [3]. Another prominent example is the ¹¹Be featuring a parity-inverted ground state with an extended neutron-¹⁰Be S-wave halo [4].

There has been a very significant experimental effort, as well as a theoretical effort, to investigate halo nuclei [2,5,6,7,8,9,10,11]. In this paper, we discuss an ab initio, or first-principles, description of halo nuclei. Ab initio methods solve the many-body Schrödinger equation for the system of A nucleons interacting by forces derived within the chiral Effective Field Theory (EFT) formalism [12]. These methods have been applied to halo nuclei in the past, e.g., the No-Core Shell Model (NCSM) [13,14] and the nuclear lattice EFT [15,16]. As halo nuclei are characterized by low-lying breakup thresholds, a proper treatment of continuum effects is essential. Consequently, we focus on applications of the No-Core Shell Model with Continuum (NCSMC) [17,18,19], a method describing bound and unbound states in a unified way.

The paper is organized as follows. In Section 2, we describe the NCSMC method including the underlying NCSM [20] approach. In Section 3, we review NCSMC applications to neutron and proton halo nuclei. In Section 3.1, we discuss the parity-inversion in ¹¹Be, we review the published results for the S-wave halo of its ground state, and we present new results for the P-wave halo of its first excited state. In Section 3.2, we present new results for ¹⁵C, which manifests a neutron S-wave halo ground state, and we discuss its production in the capture reaction ¹⁴C(n,$\gamma$)¹⁵C relevant for astrophysics. The properties of ⁸B with a P-wave proton halo ground state are reviewed in Section 3.3. Our new results for excited neutron halo states in ¹⁰Be are presented in Section 3.4. Applications of the three-body cluster NCSMC to the Borromean two-neutron halo nucleus ⁶He are reviewed in Section 3.5. Large-scale NCSM results for ¹¹Li that also exhibit a Borromean two-neutron halo in its ground state are discussed in Section 3.6. These calculations serve as the prerequisite of a full three-body cluster NCSMC investigation. A concluding discussion is then given in Section 4.

2. Materials and Methods

The starting point of our method is the microscopic Hamiltonian:

$$\hat{H}={\displaystyle \frac{1}{A}}{\displaystyle \sum _{i<j=1}^A}{\displaystyle \frac{{({\hat{\overrightarrow{p}}}_i-{\hat{\overrightarrow{p}}}_j)}^2}{2m}}+{\displaystyle \sum _{i<j=1}^A}{\hat{V}}_{ij}^{NN}+{\displaystyle \sum _{i<j<k=1}^A}{\hat{V}}_{ijk}^{3N},$$

(1)

which describes nuclei as systems of A non-relativistic point-like nucleons interacting through realistic inter-nucleon interactions. The modern theory of nuclear forces is based on the framework of chiral Effective Field Theory (EFT) [12], with the Lagrangian expanded in powers of ${(Q/{\mathsf{\Lambda }}_{\chi })}^n$, where Q is the external momentum and ${\mathsf{\Lambda }}_{\chi }$ represents the hard scale of the theory of the order of 1 GeV. Such an expansion allows a systematic improvement of the interaction and provides a hierarchy of the nucleon–nucleon (NN) and many-nucleon interactions which naturally arise in a consistent scheme [21,22].

As detailed in the next section, in the present work we apply several sets of chiral NN and chiral NN plus three-nucleon (3N) interactions consisting of NN interactions up to the third order (N²LO) [23], fourth order (N³LO) [24] or the fifth order (N⁴LO) [25] in the chiral expansion and a 3N interaction up to the N²LO order, in some cases including a sub-leading contact interaction [26], regulated by local regulators [27], non-local regulators [23] or a combination of both [28]. The interaction parameters, the low-energy constants (LECs), are determined typically in A = 2, 3, 4 nucleon systems, although the properties of medium mass nuclei can also be considered [23].

A faster convergence of our NCSMC calculations is obtained by softening the chiral interaction through the Similarity Renormalization Group (SRG) technique [29,30]. The SRG unitary transformation induces many-body forces that we include up to three-body level. The four- and higher-body induced terms are small at the ${\lambda }_{\mathrm{SRG}}$ ≈ 1.8–2.0 fm⁻¹ range of the resolution scale used in our present calculations.

In the NCSMC [17,18,19], the many-body scattering problem is solved by expanding the wave function on continuous microscopic-cluster states, describing the relative motion between target and projectile nuclei, and discrete square-integrable states, describing the static composite nuclear system. The idea behind this generalized expansion is to augment the microscopic cluster model, which enables the correct treatment of the wave function in the asymptotic region, with short-range many-body correlations that are present at small separations mimicking various deformation effects that might take place during the reaction process. The NCSMC wave function for the A-nucleon system is represented as

$$\begin{array}{cc}\hfill |{\mathsf{\Psi }}_{A,M_T}^{J^{\pi }}\rangle =& \sum _{\lambda }{c}_{\lambda }^{J^{\pi }}|A\lambda J^{\pi }T{M}_T\rangle +\sum _{\nu }\int dr{r}^2{\displaystyle \frac{{\gamma }_{\nu }^{J^{\pi }}\left(r\right)}{r}}{\hat{\mathcal{A}}}_{\nu }|{\mathsf{\Phi }}_{\nu r,M_T}^{J^{\pi }}\rangle .\hfill \end{array}$$

(2)

The first term of Equation (2) consists of an expansion over square-integrable eigenstates of the composite nucleus indexed by $\lambda$. The second term, corresponding to an expansion over the antisymmetrized channel states in the spirit of the resonating group method (RGM) [31,32,33], is given by

$$\begin{array}{cc}\hfill |{\mathsf{\Phi }}_{\nu r,M_T}^{J^{\pi }}\rangle =& {\left[{\left(|A-a{\lambda }_1{J}_1^{{\pi }_1}{T}_1{M}_{T_1}\rangle |a{\lambda }_2{J}_2^{{\pi }_2}{T}_2{M}_{T_2}\rangle \right)}^{\left(s\right)}{Y}_{\ell }\left({\hat{r}}_{A-a,a}\right)\right]}_{M_T}^{\left(J^{\pi }\right)}{\displaystyle \frac{\delta (r-r_{A-a,a})}{r{r}_{A-a,a}}}.\hfill \end{array}$$

(3)

The $\nu$ index represents all the quantum numbers on the right-hand side not appearing on the left-hand side, and the subscript $M_T=M_{T_1}+M_{T_2}$ is the isospin projection, i.e., $(Z-N)/2$. The coordinate ${\overrightarrow{r}}_{A-a,a}$ in Equation (3) is the separation distance between the $(A-a)$-nucleon target and the a-nucleon projectile. It should be noted that the sum in the second term of Equation (2) comprises in general all the mass partitions involved in the formation of the composite system including three- or higher-body clusters (see the discussion below and in Section 3.5). For technical reasons, the NCSMC calculations are typically limited to one or two (e.g., energetically lowest [34] or charge-exchange [35]) mass partitions.

The translationally invariant eigenstates of the aggregate (A), target ($A-a$), and projectile (a) nuclei are all obtained by means of the NCSM [20,36,37] using a basis of many-body harmonic oscillator (HO) wave functions with the same frequency, $\mathsf{\Omega }$, and maximum number of particle excitations $N_{\max }$ from the lowest Pauli-allowed many-body configuration. The case of $a=1$ is trivial; the projectile wave function is simply $|{{\displaystyle \frac{1}{2}}}^{+}{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}\rangle$ ($|{{\displaystyle \frac{1}{2}}}^{+}{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}\rangle$) for proton (neutron).

We note that the approximate isospin quantum number T is included in the NCSM eigenstates in Equations (2) and (3), as they provide useful information. However, in general we do not couple the target and projectile isospins to the total isospin. An exception to this is the ¹¹B calculation discussed in Section 3.1.

The discrete expansion coefficients $c_{\lambda }^{J^{\pi }}$ and the continuous relative-motion amplitudes ${\gamma }_{\nu }^{J^{\pi }}\left(r\right)$ are the solution of the generalized eigenvalue problem derived by representing the Schrödinger equation in the model space of the expansions (2) [19]. The resulting NCSMC equations are solved by the coupled-channel R-matrix method on a Lagrange mesh [38,39].

An intuitive interpretation of the wave functions of halo nuclei is provided by the overlap of the full solution for the eigenstate $|{\mathsf{\Psi }}_{A,M_T}^{J^{\pi }}\rangle$ in Equation (2) with the cluster portion $|{\mathsf{\Phi }}_{\nu r,M_T}^{J^{\pi }}\rangle$ given by

$$r\langle {\mathsf{\Phi }}_{\nu r,M_T}^{J^{\pi }}|{\hat{\mathcal{A}}}_{\nu }|{\mathsf{\Psi }}_{A,M_T}^{J^{\pi }}\rangle$$

(4)

called cluster form factor. Integrating the cluster form factor squared, one obtains the spectroscopic factor. It is straightforward to evaluate cluster form factors within the NCSMC [40] as well as within the NCSM (i.e., with $|{\mathsf{\Psi }}_{A,M_T}^{J^{\pi }}\rangle$ replaced by $|A\lambda J^{\pi }T{M}_T\rangle$ in Equation (4)) [41].

Additional important characteristics of nuclear bound states, and halo states in particular, are Asymptotic Normalization Coefficients (ANCs). In binary cluster scattering, these coefficients parameterize the bound state asymptotics of the nuclear wave function, i.e., the Coulombic tail, and they are accessible quantities in scattering experiments. In a given partial wave channel, these coefficients are defined as

$$\begin{array}{c}\hfill \underset{|\overrightarrow{r}|\to \infty }{lim}{\mathsf{\Psi }}_{lm}\left(\overrightarrow{r}\right)=C_l{\displaystyle \frac{W_{-{\eta }_B,l+1/2}\left(2\kappa \right|\overrightarrow{r}\left|\right)}{|\overrightarrow{r}|}}{Y}_{lm}\left(\hat{r}\right){\eta }_B={\displaystyle \frac{Z_a{Z}_b{e}^2\mu }{{\hslash }^2\kappa }},\end{array}$$

(5)

where $C_l$ is the ANC, W is the Whittaker function, ${\eta }_B$ is the Coulomb–Sommerfeld parameter, $\mu$ is the reduced mass of the two-component system, and $E=-{\hslash }^2{\kappa }^2/2\mu$ is the bound state energy of the system with respect to the breakup threshold.

The NCSMC can also be extended to describe systems dominated by three-body (and, in principle, several-body) breakup channels by coupling discrete NCSM eigenstates with microscopic three-cluster continuum states, enabling a unified treatment of short-range many-body correlations and correct three-body asymptotics. To accomplish this, it is convenient to introduce an appropriate set of Jacobi relative coordinates among the clusters. For a system of three clusters with mass numbers $A-a_{23}$, $a_2$, and $a_3$ (with $a_{23}=a_2+a_3<A$), one possible choice is

$${\overrightarrow{\eta }}_{1,23}=({\eta }_{1,23},{\theta }_{{\eta }_{1,23}},{\varphi }_{{\eta }_{1,23}})=\sqrt{{\displaystyle \frac{a_{23}}{A(A-a_{23})}}}{\displaystyle \sum _{i=1}^{A-a_{23}}}{\overrightarrow{r}}_i-\sqrt{{\displaystyle \frac{A-a_{23}}{A{a}_{23}}}}{\displaystyle \sum _{j=A-a_{23}+1}^A}{\overrightarrow{r}}_j,$$

(6)

which is the relative vector proportional to the displacement between the center of mass of the first cluster and that of the residual two fragments, and

$${\overrightarrow{\eta }}_{23}=({\eta }_{23},{\theta }_{{\eta }_{23}},{\varphi }_{{\eta }_{23}})=\sqrt{{\displaystyle \frac{a_3}{a_{23}{a}_2}}}{\displaystyle \sum _{i=A-a_{23}+1}^{A-a_3}}{\overrightarrow{r}}_i-\sqrt{{\displaystyle \frac{a_2}{a_{23}{a}_3}}}{\displaystyle \sum _{j=A-a_3+1}^A}{\overrightarrow{r}}_j,$$

(7)

which is the relative coordinate proportional to the distance between the centers of mass of clusters 2 and 3. Here, ${\overrightarrow{r}}_i$ denotes the position vector of the ith nucleon.

Within such a three-cluster coordinate system, the ansatz for the many-body wave function of a Borromean halo nucleus such as, e.g., ⁶He can be written analogously to Equation (2) (see Figure 1),

$$\begin{array}{cc}\hfill |{\mathsf{\Psi }}_{A,M_T}^{J^{\pi }}\rangle & =\sum _{\lambda }{c}_{\lambda }^{J^{\pi }}|A\lambda J^{\pi }{M}_T\rangle +\sum _{\nu K}\int d\rho {\rho }^5{\displaystyle \frac{{\gamma }_{K\nu }^{J^{\pi }}\left(\rho \right)}{{\rho }^{5/2}}}{\hat{\mathcal{A}}}_{\nu }|{\mathsf{\Phi }}_{\nu K\rho ,M_T}^{J^{\pi }}\rangle ,\hfill \end{array}$$

(8)

where the RGM channel states for three clusters are given by

$$\begin{array}{cc}\hfill |{\mathsf{\Phi }}_{\nu K\rho ,M_T}^{J^{\pi }}\rangle =& [{\left(|A-a_{23}{\alpha }_1{I}_1^{{\pi }_1}{M}_1^T\rangle {\left(|a_2{\alpha }_2{I}_2^{{\pi }_2}{M}_2^T\rangle |a_3{\alpha }_3{I}_3^{{\pi }_3}{M}_3^T\rangle \right)}_{M_{23}^T}^{(s_{23)}}\right)}_{M_T}^{\left(S\right)}\hfill \\ & \times {\mathcal{Y}}_L^{K{\ell }_x{\ell }_y}\left({\mathsf{\Omega }}_{\eta }\right){]}_{M_T}^{\left(J^{\pi }\right)}{\displaystyle \frac{\delta (\rho -{\rho }_{\eta })}{{\rho }^{5/2}{\rho }_{\eta }^{5/2}}}.\hfill \end{array}$$

(9)

Here, we have introduced the hyperradial and hyperangular coordinates ${\rho }_{\eta }=\sqrt{{\eta }_{23}^2+{\eta }_{1,23}^2}$ and ${\alpha }_{\eta }=arctan\left({\displaystyle \frac{{\eta }_{23}}{{\eta }_{1,23}}}\right)$ and the set of hyperangles ${\mathsf{\Omega }}_{\eta }=({\alpha }_{\eta },{\theta }_{{\eta }_{1,23}},{\varphi }_{{\eta }_{1,23}},{\theta }_{{\eta }_{23}},{\varphi }_{{\eta }_{23}})$. The functions ${\mathcal{Y}}_L^{K{\ell }_x{\ell }_y}\left({\mathsf{\Omega }}_{\eta }\right)$ are hyperspherical harmonics with total orbital angular momentum L and hyperangular momentum K, and ${\gamma }_{K\nu }^{J^{\pi }}\left(\rho \right)$ are unknown hyperradial amplitudes. The remaining notation follows that introduced in Equation (2). For a detailed discussion of the three-cluster RGM formalism, we refer the interested reader to, e.g., Refs. [42,43].

3. Results

3.1. Parity Inversion in ¹¹Be

The theoretical understanding of exotic neutron-rich nuclei constitutes a tremendous challenge. These systems often cannot be explained by mean-field approaches and contradict the regular shell structure. The spectrum of ¹¹Be has some very peculiar features. The $1/2^{+}$ ground state (g.s.) is loosely bound by 502 keV with respect to the n+¹⁰Be threshold and is separated by only 320 keV from its parity-inverted $1/2^{-}$ partner, which would be the expected g.s. in the standard shell-model picture. Both these states exhibit a distinct ¹⁰Be+n halo structure. An accurate description of this complex spectrum is anticipated to be sensitive to the details of the nuclear force, such that a precise knowledge of the NN and also the 3N interaction, desirably obtained from first principles, is crucial. At the same time, an explicit treatment of continuum effects is indispensable.

The first NCSMC investigation of ¹¹Be was reported in Ref. [40] using several sets of chiral NN and 3N interactions. The calculations included the lowest three states of ¹⁰Be ($0_1^{+},2_1^{+},2_2^{+}$) and a range of NCSM eigenstates of ¹¹Be of both parities. A significant sensitivity to the chiral nuclear forces was found. In Figure 2, we compare calculated levels to experimental data, all relative to the n+¹⁰Be($0^{+}$) threshold, for three sets of interactions. First, only the chiral N³LO NN interaction [24] was used with no chiral 3N force (SRG-induced 3N contributions were included in all the calculations). The $1/2^{-}$ ground state was predicted incorrectly, with the $1/2^{+}$ state at the threshold and the $3/2_1^{-}$ and $5/2^{+}$ states inverted as well. Adding the chiral 3N force with a local regulator with a cut-off of 400 MeV [44] improved the spectrum, although the incorrect level orderings remained. On the contrary, the spectrum with the NN + 3N interaction simultaneously fitted to few-nucleon systems and medium mass nuclei, named N²LO_SAT [23], successfully achieving the parity inversions between the $3/2_1^{-}$ and $5/2^{+}$ resonances and, albeit marginally, for the bound states. The low-lying spectrum was significantly improved and agreed well with the experimental data, presumably due to the more accurate description of long-range properties caused by the fit of the interaction to the radii of p-shell nuclei. On the other hand, the strongly overestimated splitting between the $3/2_2^{-}$ and $5/2^{-}$ states hints at the deficiencies of this interaction, which might originate from a too-large splitting of the $p_{1/2}-p_{3/2}$ subshells.

More recently, ¹¹Be has been investigated with the NCSMC in the context of the $\beta$-delayed proton emission [45]. In that work, the chiral N⁴LO NN interaction [25] combined with an N²LO 3N interaction with simultaneous local and non-local regularization was used. Originally introduced in Ref. [46], it is denoted as NN-N⁴LO + 3N_lnl. The SRG evolution was applied with 3N-induced terms included. As shown in the left panel of Figure 3, with this interaction the parity inversion in the ground state of ¹¹Be was comfortably reproduced. Moreover, the isospin analog states in ¹¹B were investigated within the NCSMC, considering the ¹⁰Be+p cluster. As seen in the right panel of Figure 3, the parity inversion was also reproduced in the $T=3/2$ resonances, in agreement with the experimental data.

An insight into the wave functions of the two bound states of ¹¹Be is provided in Figure 4. Cluster form factors calculated according to Equation (4) using the N²LO_SAT interaction are presented for the $1/2^{+}$ ground state (left panel) and the first excited $1/2^{-}$ state (right panel). A clearly extended halo structure beyond 20 fm can be identified for the S-wave and P-wave of the ¹⁰Be$\left(0^{+}\right)$+n relative motion for the $1/2^{+}$ and $1/2^{-}$ states, respectively. The phenomenological energy adjustment to reproduce the ¹⁰Be+n experimental separation energies, the NCSMC-pheno approach [47], only slightly influenced the asymptotic behavior of the S-wave and P-wave, as seen by comparing the solid and dashed black curves, while other partial waves are even indistinguishable on the plot resolution. The corresponding $1/2^{+}$ g.s. spectroscopic factors for the NCSMC-pheno approach, obtained by integrating the squared cluster form factors in the left panel of Figure 4, are $S=0.90$ (S-wave) and $S=0.16$ (D-wave). The S-wave Asymptotic Normalization Coefficient (ANC) is 0.786 fm^−1/2. In

Table 1. ANC² of the S-wave neutron halo $1/2^{+}$ ground state of ¹¹Be and the S- and D-wave spectroscopic factors and ANC of the $1/2^{-}$ halo excited state of ¹¹Be in the ¹⁰Be($0^{+}$)+n P-wave channel with the corresponding spectroscopic factor. Calculations performed with the N²LO_SAT interaction within the NCSMC-pheno are compared to the results from Refs. [48,49].

11Be($1/2^{+}$)	ANC2 $\left[{\mathrm{fm}}^{-1}\right]$	S-wave spectr. factor	D-wave spectr. factor
NCSMC-pheno	0.618	0.90	0.16
Hebborn [48]	$0.62\pm 0.06\pm 0.09$
Yang [49]	$0.616\pm 0.001$
11Be($1/2^{-}$)	ANC $\left[{\mathrm{fm}}^{-1/2}\right]$	10Be($0^{+}$)+n P-wave spectr. factor
NCSMC-pheno	0.129	0.85
Yang [49]	$0.135\pm 0.005$

, we summarize the ANCs and spectroscopic factors of the two bound states and compare them with ANC values from knockout reaction [48] and transfer reaction [49] analyses. The NCSMC cluster form factors can be contrasted with those obtained within the NCSM, computed as discussed below Equation (4), shown by the dotted lines in Figure 4. They drop off to zero at ≈8 fm. Interestingly, the NCSM spectroscopic factors are comparable to the NCSMC ones.

It is known that using the chiral 3N interaction with a non-local regularization improves the description of nuclear radii compared to experiment [50]. Similarly, the NCSMC investigations presented in this subsection suggest the importance of a non-local regularization of the chiral 3N interaction for the reproduction of the parity inversion in the ¹¹Be ground state.

3.2. Halo Ground-State of ¹⁵C

The ¹⁵C is a well-known one-neutron halo nucleus [6,8]. With a rather small one-neutron separation energy of 1.22 MeV, its ground state can be well described as ¹⁴C in its $0^{+}$ ground state and a loosely bound neutron in the $1s_{1/2}$ orbital. It is relevant for nuclear astrophysics. The ¹⁵C synthesis through one-neutron radiative capture, ¹⁴C(n,$\gamma$)¹⁵C, has been suggested to be part of neutron-induced CNO cycles, which take place in the helium-burning zone of asymptotic-giant-branch (AGB) stars [51]. This reaction is also the doorstep to the production of heavy elements in inhomogeneous big-bang nucleosynthesis [52], and it has been shown to be part of possible reaction routes in the nuclear chart during the r process in Type II supernovae [53]. This cross-section is also important as a benchmark both for theories and experiments, as it can be measured directly and used for validation of the Coulomb breakup method for the neutron capture cross-section determination using the ¹⁵C beam [54].

We investigated ¹⁵C within the NCSMC using the NN N³LO+3N_lnl interaction [28] that was SRG-evolved using ${\lambda }_{\mathrm{SRG}}=2$ fm⁻¹ with the 3N-induced terms included. In the basis expansion (2), we considered the ¹⁴C+n cluster including the ¹⁴C $0^{+}$ ground state and the first-excited $2^{+}$ state as well as 7(3) positive(negative)-parity NCSM eigenstates of ¹⁵C. The underlying NCSM calculations were performed up to $N_{\max }=8$ and the NCSMC in $N_{\max }=7$; see Figure 5a, where the energies of low-lying states of both parities are presented. The full basis space was used up to $N_{\max }=5$(6) for ¹⁵C (¹⁴C), while the importance truncated (IT) NCSM [55] was applied for higher $N_{\max }$. One can see the significant increase of binding for positive parity states in the NCSMC compared to the NCSM. The negative-parity states are, on the other hand, basically unchanged when the continuum is taken into account. The $1/2^{+}$ ground state is bound with respect to the ¹⁴C+n threshold in the NCSMC although less than in the experimental data. In contrast, in the NCSM the $5/2^{+}$ state was predicted to be the ground state up to at least $N_{\max }=8$. It gained binding in the NCSMC although it remained unbound in $N_{\max }=7$, contrary to the experimental data.

To gain insight into the structure of the experimentally bound $1/2^{+}$ and $5/2^{+}$ states and to facilitate the calculation of the ¹⁴C(n,$\gamma$)¹⁵C cross-section, we applied the NCSMC-pheno approach [47]. We adjusted the NCSM-calculated excitation energy of the ¹⁴C $2^{+}$ state from the $N_{\max }=6$-calculated 8.73 MeV to the experimental data 7.01 MeV, and we shifted the ¹⁵C NCSM eigenstates to reproduce the experimental energies of the bound states (and thus the experimental threshold) and known low-lying resonances in the NCSMC calculations. As a result, we obtained the $1/2^{+}$ and $5/2^{+}$ states bound as in the experimental data, and we predicted the phase shifts shown in Figure 5b with the broad $3/2^{+}$ resonance and the very narrow $1/2^{-},5/2^{-},3/2^{-}$ resonances matching the experimental centroids. At the same time, we predicted narrow $5/2^{+},3/2^{+},7/2^{+},9/2^{+}$ resonances close to the ¹⁴C($2^{+}$)+n threshold.

In Figure 6, we show the cluster form factors of the two ¹⁵C bound states, the $1/2^{+}$ ground state (a) and the $5/2^{+}$ excited state (b) obtained within the NCSMC-pheno approach. The $1/2^{+}$ state clearly manifests an S-wave neutron halo extending beyond 20 fm. The calculated ANC $C_{1/2^{+}}=1.282$ fm^−1/2 ($C_{1/2^{+}}^2=1.644$ fm⁻¹) is in excellent agreement with that of $C_{1/2^{+}}^2=1.59\pm 0.06$ fm⁻¹ obtained in Ref. [54] using the halo EFT analysis of the ¹⁴C(d,p)¹⁵C transfer reaction. Similarly, it is in agreement with the ANC determination of $C_{1/2^{+}}^2=1.57\pm 0.30\pm 0.18$ fm⁻¹ from ¹⁵C one-neutron knockout reaction data analysis [48]. The corresponding spectroscopic factor obtained by integrating the square of the cluster form factor is 0.96. D-wave contributions from the ¹⁴C($2^{+}$)+n are quite small.

The $5/2^{+}$ state is dominated by ¹⁴C($0^{+}$) and a neutron in a D-wave. Given its weak binding of just 0.48 MeV, the cluster form factor extends beyond 15 fm. The NCSMC-pheno-calculated ANC is $C_{5/2^{+}}=0.048$ fm^−1/2 in good agreement with the the value of $0.0595\left(36\right)$ fm^−1/2 determined from the analysis of the ¹⁴C(d,p)¹⁵C transfer reaction [56]. The corresponding spectroscopic factor is 0.90. There are also small but visible contributions by ¹⁴C($2^{+}$) and a neutron in the S- and D-waves while the G-waves are negligible.

The dotted lines in Figure 6 show the corresponding cluster form factors obtained within the NCSM. While their shapes and the spectroscopic factors are similar to the NCSMC ones, their extent is drastically different as they become negligible beyond $\sim 7$ fm.

Applying the NCSMC-pheno calculations discussed above, we computed the cross-section of the ¹⁴C(n,$\gamma$)¹⁵C radiative capture reaction. The energy-scaled cross-section for energy up to 1 MeV is displayed in Figure 7, showing separately the capture to the ground state and to the $5/2^{+}$ excited state. Overall, the shape and magnitude is in line with recent experimental determinations [57]. Following the pioneering measurement in [58], it is customary to compare theoretical and experimental cross-sections at $E_{\mathrm{c}.\mathrm{m}.}=23.3$ keV. At this energy, we obtained in the present NCSMC-pheno calculations ${\sigma }_{\mathrm{n},\gamma }$($1/2^{+}$) = 4.79 μb, ${\sigma }_{\mathrm{n},\gamma }$($5/2^{+}$) $=0.13$ μb, i.e., the total ${\sigma }_{\mathrm{n},\gamma }=4.92$ μb. This is in good agreement with the value of 4.66(14) μb reported in Ref. [54] (for the capture to the ground state) and the result of 4.75 μb obtained in Ref. [59], while it is slightly higher than the recent measurement by Jiang et al. [60] reporting 3.89(76) μb.

In

Table 2. NCSMC-pheno-calculated ANCs and spectroscopic factors for the $1/2^{+}$ and $5/2^{+}$ bound states of ¹⁵C and the ¹⁴C(n,$\gamma$)¹⁵C cross-section at $E_{\mathrm{c}.\mathrm{m}.}=23.3$ keV for the capture to the $1/2^{+}$ and $5/2^{+}$ final states compared to results obtained by other methods.

15C($1/2^{+}$) S-wave	ANC $\left[{\mathrm{fm}}^{-1/2}\right]$	Spectr. factor
NCSMC-pheno	1.282	0.96
Moschini  [54]	1.26(2)	1
Hebborn [48]	1.25(12)	1
Jiang [60]	1.16(15)	0.68(14)
15C($5/2^{+}$) D-wave	ANC $\left[{\mathrm{fm}}^{-1/2}\right]$	Spectr. factor
NCSMC-pheno	0.048	0.90
Mukhamedzhanov [56]	0.0595(36)	1
${\sigma }_{\mathrm{n},\gamma }$ [μb] at $E_{\mathrm{c}.\mathrm{m}.}=23.3$ keV	$1/2^{+}$	$5/2^{+}$	Total
NCSMC-pheno	4.79	0.13	4.92
Moschini [54]	4.66(14)
Tkachenko [59]			4.75
Jiang [60]	3.89(76)

, we summarize the presently obtained ANCs, spectroscopic factors and cross-sections and compare them to the results in the literature. For a more in-depth review of ¹⁴C(n,$\gamma$)¹⁵C cross-section data, we refer the reader to Ref. [59].

3.3. P-Wave Halo Nucleus ⁸B

⁸B plays an important role in astrophysics, as neutrinos from its $\beta$ decay form the higher-energy part of the solar neutrino flux. It is produced in the solar proton–proton reaction chain through the proton radiative capture on ⁷Be, the ⁷Be(p,$\gamma$)⁸B reaction. The $2^{+}$ ground state of ⁸B is bound by only $\approx 137$ keV with respect to the ⁷Be+p threshold. Consequently, it is anticipated that it manifests a P-wave proton halo. The ⁷Be(p,$\gamma$)⁸B capture reaction and the ⁸B structure have been investigated recently within the NCSMC approach using several sets of chiral NN + 3N interactions. We focus here on the results obtained with the NN-N⁴LO [25] + ${3\mathrm{N}}_{\mathrm{lnl}}^{*}$ [26,61,62] chiral interaction (denoted as NN N⁴LO + 3N_lnlE7 in some of the figures). In this interaction, an additional sub-leading contact term ($E_7$) enhancing the spin-orbit strength [26] has been introduced to the 3N force. It appears to be the best-performing interaction available to us currently. Again, the SRG evolution was applied with 3N-induced terms included.

The positive-parity eigenphase shifts for ⁷Be+p scattering obtained using the NN-N⁴LO + ${3\mathrm{N}}_{\mathrm{lnl}}^{*}$ chiral interaction, presented in Figure 8, show the well-established $1^{+}$ and $3^{+}$ ⁸B resonances as well as several other, yet unobserved, broad resonances. The NCSMC S-wave phase shifts manifest scattering length signs consistent with those determined in recent measurements (negative for ⁵$S_2$, positive for ³$S_1$ [63]). We found that it was difficult to produce a bound ⁸B ground state with this as well as with other chiral interactions [61]. Rather, we obtained a very narrow near-threshold $2^{+}$ resonance that is not visible in the figure.

To investigate the properties of the weakly-bound ⁸B $2^{+}$ ground state, we resorted to the NCSMC-pheno approach [47], i.e., with the ⁷Be(g.s.)+p separation energy adjusted to the experimental value of 137 keV. This was achieved by shifting the NCSM eigenenergies of ⁷Be so that the excitation energies (and therefore the thresholds) matched the experimental ones exactly. Furthermore, the ⁸B NCSM eigenenergies in the $2^{+}$ channel were also modified, bringing the NCSMC states in to the experimentally observed positions.

In Figure 9, we present the cluster form factor for the $2^{+}$ ground state of ⁸B obtained using the NN-N⁴LO+${3\mathrm{N}}_{\mathrm{lnl}}^{*}$ interaction within the NCSMC-pheno approach. The dominant component is clearly the channel-spin $s=2$P-wave of the ⁷Be(g.s.)+p that extends to a distance far beyond the plotted range. The alternative channel spin coupling, $s=1$, P-wave is less pronounced but extends in a similar way. Of a comparable size is the ⁷Be($1/2^{-}$)+p P-wave. Remarkably, we noticed a substantial contribution from the ⁷Be($5/2_2^{-}$)+p P-wave in the channel spin $s=2$. The other possible $s=3$P-wave configuration is negligible. At the same time, the ⁷Be $5/2_2^{-}$ state is dominated by a ⁶Li+p channel-spin $s=3/2$P-wave configuration. Within the NCSM framework relevant to the present calculations, this was shown (for the mirror ⁷Li+n system) in ref. [41]. Therefore, such a large contribution of the $s=2$ ⁷Be($5/2_2^{-}$)+p P-wave to the ⁸B ground state seems to indicate the presence of two anti-parallel protons outside of a ⁶Li core and that their exchanges are important. Clearly, for a realistic description of the ⁸B ground state, this state must be taken into account. Finally, we note that the ⁷Be($7/2^{-}$)+p P-wave component is also substantial. The calculated ANCs, $C_{p1/2}=0.34$ fm^−1/2 and $C_{p3/2}=0.62$ fm^−1/2 are close to the experimental values reported in Ref. [63]; see

Table 3. ANCs, in $\left[{\mathrm{fm}}^{-1/2}\right]$, of the P-wave proton halo $2^{+}$ ground state of ⁸B obtained with the NN-N⁴LO+${3\mathrm{N}}_{\mathrm{lnl}}^{*}$ interaction within the NCSMC-pheno compared to experimental results [63]. Estimates of chiral truncation errors are shown for theoretical values.

8B($2^{+}$)	$C_{p1/2}$	$C_{p3/2}$
NCSMC-pheno	0.34(1)	0.62(2)
Paneru [63]	0.315(9)	0.66(2)

.

3.4. Excited Halo States in ¹⁰Be

The lowest-lying part of the dense spectrum in ¹⁰Be is well understood. What remains however is a clear understanding of the excited $J^{\pi }=1^{-},2^{-}$ states for which there exists inherent nuclear structure interest. These two states have been experimentally measured close to the n + ⁹Be threshold, and both are thus anticipated to have some kind of exotic structure: either strong clustering or S-wave halo formation [64]. The appearance of two exotic, possibly-S-wave halo states so close together is a rather unique feature of the ¹⁰Be spectrum, the classification of which has naturally garnered interest over several decades, partially due to the inconclusive literature on the matter. Recently, there has been novel experimental effort in classifying the nature of these states: an experiment from TRIUMF ISAC-II investigating the angular distributions of both states via the ¹¹$\mathrm{Be}(\mathrm{p},\mathrm{d})$ transfer reaction provided inconclusive information regarding the cluster vs. halo nature of the states [65], while a further proposal to examine the structure of the $2^{-}$ via the one-neutron-transfer reaction ⁹Be(¹¹Be, ¹⁰${\mathrm{Be}}^{*}$[2⁻]) is currently in preparation at ISOLDE [66]. In addition to these exotic bound states, there exists interest in the $J^{\pi }=3^{-}$ resonance state, which sits slightly above the n + ⁹Be threshold. As the isospin mirror of ¹⁰C, characterization of these excited states and resonances can provide insight into isospin symmetry-breaking effects arising from the strong and electromagnetic sectors.

In Figure 10 and Figure 11, we present the NCSMC predictions for the negative- and positive-parity phase shifts, respectively, for the n + ⁹Be scattering process, with the eigenphase shifts shown on the left and the phase shifts shown on the right. These were obtained with the NN-N⁴LO + 3N_lnl chiral interaction and within an $N_{\max }=8,9$ model space depending on the parity. We identified the total spin parity with a given eigenphase shift and the partial wave channel for a given phase shift in the same color as the curves. We used a mass partition, including the ${3/2}^{-}$, ${5/2}^{-}$ and ${1/2}^{-}$ states of ⁹Be and, with this configuration of states, the NCSMC bound four out of the six (both positive- and negative-parity) experimentally observed bound states [67]. We notably missed the excited $J^{\pi }=0^{+}$ state, due to lacking consideration of alpha clustering in the partitions, which has been known to contribute significantly to the structure for some time [68,69].

In Figure 10, dominated by the ³${\mathrm{S}}_1$ partial wave, we find that the $1^{-}$ state is readily bound in the NCSMC calculation, albeit sitting shallow at −0.0295 MeV with respect to the already-quite-shallow experimental observation of −0.8523 MeV. On the other hand, the $2^{-}$ remains unbound in the current model space, but exhibits the expected near-threshold behavior for resonance conversion to a bound state. It is possible that an increase in the model space dimension would drive this state below threshold, but it could be that mixing between additional mass partitions is necessary to describe this state, e.g., ⁶$\mathrm{He}+\alpha$. Moreover, predicated on the inclusion of the full set of low-lying ⁹Be states that could contribute to the formation of these structures, the observed closeness of these states in the spectrum is driven by their sharing of the same underlying ⁹$\mathrm{Be}\left[{3/2}^{-}\right]$ configuration with the neutron spin anti-aligned (³${\mathrm{S}}_1$) or aligned (⁵${\mathrm{S}}_2$) with the nuclear spin—but always with $l=0$ relative orbital momentum. The inclusion of the ${5/2}^{-}$ and ${1/2}^{-}$ proved to be of little consequence to the formation of these states, with stability in the phase shifts as each consecutive state was added. A further improvement to our description of the $1^{-}$ would likely come from inclusion of the ⁶$\mathrm{He}+\alpha$ mass partition, as analysis via microscopic cluster models with a partition of $\alpha +\alpha +n+n$ suggests that a significant contribution to the structure of these states is driven by $\alpha$-clustering effects [70]; notably, the $2^{-}$ is not bound in such an approach, suggesting the reduced effects of $\alpha$-clustering on the structure of said state.

As expected, we further see the appearance of the known $3^{-}$ resonance coming from the ⁵${\mathrm{D}}_2$ partial wave channel, consistent with that which has been similarly seen in calculations of ¹⁰C with the NCSMC [71]. Lastly, a broad $4^{-}$ resonance from the ⁵${\mathrm{D}}_4$ channel appears at relatively low c.m. energies, corresponding to the expected resonance at 2.46 MeV. While the partial waves that dominate the contribution to these states have higher total momentum, it comes primarily from the relative orbital momentum of $l=2$ between the neutron and ⁹Be cluster, i.e., they are still largely built upon the ${3/2}^{-}$ of ⁹Be with neutron spin either anti-aligned or aligned with the nuclear spin.

Before proceeding to the positive parity states, we present the ANCs—see Equation (5) for their definition—for the $1^{-}$ and $2^{-}$ halo states of ¹⁰Be in

Table 4. Asymptotic Normalization Coefficients (ANCs) for the $1^{-}$ halo state of ¹⁰Be, calculated with the same interaction and configuration space as described in Figure 10. The ANCs are presented for a given spin-parity state of ⁹Be, relative orbital momentum of the neutron with respect to the ⁹Be cluster, and total spin of the n + ⁹Be partition. The first column contains the raw ANC from the calculation, whereas the second column contains the ANCs obtained by phenomenological adjustment of the NCSMC eigenenergies to their experimental values.

State of 9Be	l	S	ANC $\left[{\mathrm{fm}}^{-1/2}\right]$	ANC Pheno $\left[{\mathrm{fm}}^{-1/2}\right]$
$3/2^{-}$	0	1	0.363	0.951
$3/2^{-}$	2	1	$0.7\times {10}^{-3}$	$0.392\times {10}^{-1}$
$3/2^{-}$	2	2	$-0.244\times {10}^{-3}$	$-0.137\times {10}^{-1}$
$5/2^{-}$	2	2	$-0.102$	$-0.230$
$5/2^{-}$	2	3	$0.104\times {10}^{-1}$	$0.399\times {10}^{-1}$
$5/2^{-}$	4	3	$-0.603\times {10}^{-4}$	$-3.60\times {10}^{-3}$
$1/2^{-}$	0	1	0.257	0.425
$1/2^{-}$	2	1	$0.184\times {10}^{-1}$	$0.506\times {10}^{-1}$

and

Table 5. Asymptotic Normalization Coefficients (ANCs) for the $2^{-}$ halo state of ¹⁰Be, calculated with the same interaction and configuration space as described in Figure 10. The ANCs are presented for a given spin-parity state of ⁹Be, relative orbital momentum of the neutron with respect to the ⁹Be cluster, and total spin of the n + ⁹Be partition. The only column shown contains the ANCs obtained by phenomenological adjustment of the NCSMC eigenenergies to their experimental values, given that the $2^{-}$ is near-threshold but unbound in the raw calculation.

State of 9Be	l	S	ANC Pheno $\left[{\mathrm{fm}}^{-1/2}\right]$
$3/2^{-}$	2	1	$-0.288\times {10}^{-1}$
$3/2^{-}$	0	2	$-0.756$
$3/2^{-}$	2	2	$-0.103\times {10}^{-1}$
$3/2^{-}$	4	2	$-0.274\times {10}^{-4}$
$5/2^{-}$	0	2	$-0.451$
$5/2^{-}$	2	2	$0.164$
$5/2^{-}$	4	2	$0.849\times {10}^{-4}$
$5/2^{-}$	2	3	$0.126$
$5/2^{-}$	4	3	$-0.128\times {10}^{-3}$
$1/2^{-}$	2	0	$-0.184\times {10}^{-1}$
$1/2^{-}$	2	1	$-0.348\times {10}^{-1}$

, respectively, using the same interaction and configuration space as in determination of the phase shifts in Figure 10. In Table 4, we provide the ANCs for the $1^{-}$ state of ¹⁰Be for each partial wave channel of the n + ⁹Be partition. These are labeled by the spin-parity of the relevant ⁹Be state, the relative orbital momentum of the neutron with respect to the ⁹Be cluster, and the total spin of the coupled n + ⁹Be partition. In the first column, with the ANC results, we show the raw output of the NCSMC from the aforementioned calculations. Due to the dependence of the asymptotic wave function on $\kappa \propto \sqrt{-E}$, the ANCs were incredibly sensitive to the prediction for the energy eigenvalues. Thus, by phenomenologically adjusting the NCSMC energy eigenvalues to match the experimental ones, one can get a more realistic picture of the ANCs. That is what is then shown in the second ANC column, labeled as ANC Pheno. One can obviously see the dramatic difference in values for the predicted ANCs coming from relatively minor shifts in the overall energy eigenvalues, typically of order 1 MeV or so. Identically, in Table 5, we show the ANCs for the $2^{-}$ state of ¹⁰Be. In this case, there were no ANCs for the raw NCSMC calculation, since the state was found to be an unbound, near-threshold resonance, so we show only the ANC Pheno results coming from phenomenological adjustment of the $2^{-}$ state. Based upon the extracted values for the ANCs in both the $1^{-}$ and $2^{-}$ states, we find dominance of the $l=0$ channels, which further supports the interpretation of these states as excited halo candidates, regardless of any additional sub-structure of the ⁹Be partition. While in both states the dominant structure arises from the ⁹Be$[3/2^{-}]\otimes n$ channel, there are still noteworthy contributions from the ⁹Be$[5/2^{-}]\otimes n$ in both states and from ⁹Be$[1/2^{-}]\otimes n$ in the $1^{-}$ state. The resulting picture is therefore not that of a single-channel halo state with no internal clustering, but rather of a multi-channel system whose asymptotics are nevertheless dominated by $\mathrm{S}$-wave neutron motion around a ⁹Be core.

In Figure 11, we find a wealth of positive-parity resonance states compared to the case of negative parity, as is characteristic of our theoretical [72,73] and experimental understanding of the rather dense ¹⁰Be spectrum. Importantly, we find that the three lowest-lying bound states as predicted by the NCSMC—that is, the $0_1^{+}$ sitting at −5.9721 MeV, the $2_1^{+}$ at −2.5104 MeV, and the $2_2^{+}$ at −0.6646 MeV—are notably all bound to the correct experimental values within an MeV or so. For reference, these are, respectively, −6.8122 MeV, −3.444 MeV, and −0.8538 MeV [67]. The corresponding partial wave channels are the ³${\mathrm{P}}_0$, ³${\mathrm{P}}_2$, and ⁵${\mathrm{P}}_2$. As mentioned previously, we did not bind the excited state $0_2^{+}$, due to lacking alpha clustering effects, though we made note of a $0^{+}$ state appearance as a proper resonance of the n + ⁹Be system via the ³${\mathrm{P}}_0$ channel seen on the right-hand side of Figure 11. Beyond inclusion of the additional ⁶$\mathrm{He}+\alpha$ partition, as discussed in Ref. [74], further effects of cluster polarization likely play a significant role in the formation of bound and resonant states in ¹⁰Be.

On to the scattering states: the NCSMC predicted the existence of a near-threshold $1^{+}$ resonance, already seen in earlier NCSM calculations [75], which had yet to be confirmed experimentally. This was consistent with calculations of ¹⁰C and ¹⁰B [71], and it was further anticipated to exist based on the observed isospin analogue states in the spectra of those nuclei. A second narrow and unobserved $1^{+}$ resonance was also predicted about 3 MeV higher in energy. While most of the eigenphase shifts corresponded to a single dominant partial wave channel, the $1_2^{+}$ was an exception and came from stimulation of the same partial wave channel as the $1_1^{+}$ at higher c.m. energy, as can be seen from the behavior of the ⁵${\mathrm{P}}_1$ phase shift. Moving up in spin, we found a $2_3^{+}$ resonance built from the ⁵${\mathrm{P}}_2$ partial wave, which we could not readily match to an experimentally observed resonance state. There existed a resonance with energy 0.7298 MeV, which was presumably related, though our calculation was about 2 MeV above this, and thus we cannot say for certain. We observed two additional $2^{+}$ resonances at higher c.m. energy coming from the ⁷${\mathrm{P}}_2$ and ³${\mathrm{P}}_2$ channels, though they were not readily discerned as any particular state seen in the known experimental spectrum. Lastly, a very narrow $3^{+}$ resonance was observed at quite low c.m. energy in the ⁵${\mathrm{P}}_3$ channel, a place which—when referring to the experimental spectra [67]—was seemingly empty of resonance states.

3.5. Borromean Halo Nucleus ⁶He

A stringent test of any ab initio description of three-cluster dynamics is provided by Borromean halo nuclei, where the bound ground state emerges only through genuine three-body correlations and the wave function exhibits pronounced long-range asymptotics. The ⁶He nucleus is a prototypical example, with a weakly bound ground state and an extended spatial distribution that reflects its dominant $\alpha +n+n$ character. As such, it offered an ideal benchmark to assess whether a unified treatment can simultaneously describe both short-range many-body correlations and the correct three-body continuum behavior.

An initial ab initio description of ⁶He and its $\alpha +n+n$ continuum was obtained in a model space spanned only by microscopic cluster channels, i.e., by omitting discrete NCSM eigenstates of the composite system in the wave function ansatz [42,76]. This three-cluster treatment captured the correct three-body asymptotics and enabled continuum calculations; however, it was clear that additional short-range many-body correlations were missing and that convergence with respect to the model space was comparatively slow. This work was followed by a full NCSMC description, also including square-integrable NCSM eigenstates of the ⁶He system and thus accelerating convergence for bound-state and low-energy continuum observables [43,77]. The three-cluster NCSMC formalism was demonstrated and quantified through a detailed study of the ⁶He ground state and low-lying continuum, using SRG-evolved chiral N³LO NN interactions at two resolution scales, ${\lambda }_{\mathrm{SRG}}=1.5{\mathrm{fm}}^{-1}$ and $2.0{\mathrm{fm}}^{-1}$, while omitting initial and induced 3N forces in those calculations [43,77].

For the ${\lambda }_{\mathrm{SRG}}=2.0{\mathrm{fm}}^{-1}$ interaction, the NCSMC calculation yielded a realistic ⁶He ground-state energy of −29.17 MeV (compared to the experimental value of −29.268 MeV [78]). The corresponding g.s. wave function exhibited the expected dineutron-dominated spatial distribution when analyzed through the three-body probability density constructed from projecting the full NCSMC solution onto the microscopic three-cluster basis (Figure 12). Beyond this qualitative picture, the same continuum-coupled description yielded converged matter ($r_m$) and point-proton radii ($r_{pp}$) in computationally accessible model spaces. More importantly, it enabled simultaneously a description of the small two-neutron separation energy ($S_{2n}=0.94\left(5\right)$ MeV) and the extended spatial size of ⁶He ($r_m=2.46\left(2\right)$ fm, $r_{pp}=1.90\left(2\right)$ fm) broadly consistent with experimental constraints. This is notable, given that in traditional ab initio calculations limited to expansions on square-integrable basis states, including the NCSM, the matter and point-proton radii of ⁶He converge slowly with model-space increase, reflecting the difficulty of representing the long-range halo tail. This difference is exemplified by Figure 13, which compares the hyper-radial components of the $\alpha +n+n$ relative motion in the ⁶He ground state after projection of the full NCSMC wave function and of its NCSM portion onto the orthogonalized microscopic-cluster basis. The comparison makes apparent the deficiency of the square-integrable NCSM component in reproducing the long-range halo tail, and how the inclusion of explicit three-cluster continuum degrees of freedom in the NCSMC restores the correct extended behavior.

As a further illustration of the role of continuum degrees of freedom and many-body correlations, Figure 14 shows the low-lying ⁶He spectrum obtained with the SRG-evolved N³LO NN interaction at $\lambda =1.5$ and $2.0{\mathrm{fm}}^{-1}$. There, we compare the NCSMC results of Ref. [43] with the NCSM spectrum obtained by treating the ⁶He excited states as bound states. Besides the results in the largest accessible HO model space ($N_{max}=12$), for the NCSM we also show the spectrum extrapolated to the infinite-space limit. Because the NCSM is a bound-state technique and does not yield resonance widths, only the excitation energies (with the estimated extrapolation uncertainty) are shown in that case, whereas for the NCSMC the resonances are represented by their centroids (solid lines) and widths (shaded areas). While very narrow resonances such as the first $2^{+}$ can be captured reasonably within the bound-state approximation, the description of broader states generally requires both short-range many-body correlations and explicit coupling to the three-body continuum.

The two SRG resolution scales yielded a qualitatively similar pattern, and their differences provide a rough estimate of the impact of the omitted induced three-nucleon (and higher-body) interactions that are needed to restore the formal unitarity of the SRG transformation. More generally, explicit 3N forces (including the initial chiral 3N interaction) are indispensable for an accurate description of the spectrum as a whole. Indeed, while the SRG-evolved NN interaction at $\lambda =2.0{\mathrm{fm}}^{-1}$ yielded a realistic energy and structure for the ⁶He ground state, neither of the two adopted resolution scales reproduced quantitatively the low-energy excited spectrum reported in Ref. [79].

3.6. Large-Scale NCSM Calculations for ¹¹Li

¹¹Li is the nucleus where a neutron halo was discovered in interaction cross-section measurements more than four decades ago [1]. This drip-line isotope has two weakly bound neutrons in its ground state with separation energy $S_{2n}$ = 369.2(6) keV that have a large spatial extent compared to the core nucleus ⁹Li leading to a ⁹Li + n + n three-body Borromean halo. This exotic structure was postulated to give rise to unconventional excitation modes. It was predicted that a low-energy dipole resonance could arise due to oscillation of the weakly bound halo neutrons against the core [80]. A non-resonant soft electric dipole mode in a Coulomb excitation process was also predicted [5]. There have been several experiments performed aimed at elucidating the nature of its ground state as well as its excitation modes [2,6,81,82,83,84,85,86,87,88,89]. Simultaneously, the properties of ¹¹Li have been subject to numerous theoretical studies [10,11,90,91,92,93,94,95]. Yet, an ab initio description of this complex system is still lacking.

As a step in the direction of remedying this situation, we performed ab initio calculations of the ¹¹Li nuclear structure using the NCSM approach. As input, we employed the chiral EFT NN and 3N interaction NN-N⁴LO + ${3\mathrm{N}}_{\mathrm{lnl}}^{*}$ (denoted as NN N⁴LO + 3N_lnlE7 in the figures). The interaction was softened by the SRG technique with the SRG-induced three-nucleon terms fully included. The evolution parameter ${\lambda }_{\mathrm{SRG}}=1.8$ fm⁻¹ was used primarily, and we checked that the observables were insensitive to the variation of the ${\lambda }_{\mathrm{SRG}}$ parameter between 1.8 and 2.0 fm⁻¹. For earlier NCSM studies reporting some ¹¹Li results obtained using NN interactions only, see Refs. [14,96].

In panel (a) of Figure 15, we present the ¹¹Li $3/2^{-}$ ground-state energy dependence on the NCSM HO frequency in the range of $\hslash \mathsf{\Omega }$ = 14–20 MeV for the basis size up to $N_{\max }=10$. The basis dimension reached 929 million at $N_{\max }=10$. The extrapolated results to $N_{\max }\to \infty$ using the exponential function $E\left(N_{\max }\right)=a+b{e}^{-c{N}_{\max }}$ are shown by the gray band. The uncertainties were obtained by varying the number of extrapolated points, the HO frequencies and the SRG evolution parameter. In addition, we did the same for ⁹Li; see Ref. [97]. The predicted ground-state energy of ¹¹Li was $-43.56\left(35\right)$ MeV, while that of ⁹Li was $-43.73\left(18\right)$ MeV, which can be compared to experimental $-45.709$ MeV and $-45.34$ MeV, respectively. Overall, we found a slight underbinding of ∼1.5–2 MeV and within uncertainties about the same ground-state energy of the two isotopes. We found that the experimentally well-bound ⁹Li ground-state energy converged faster while the very weakly bound ¹¹Li would benefit from an inclusion of three-body cluster components in the trial wave function absent in the present NCSM calculations that might help to bind it with respect to ⁹Li.

In panel (b) of Figure 15, we show the occupations of the major HO shells for the ¹¹Li $3/2^{-}$ ground-state as they evolved with the basis size enlargement. While the proton occupations remain stable, the neutron occupation of the $N=1$ (0p-shell) decreased and the neutron occupation of the higher N shells steadily increased with $N_{\max }$.

The dependence of the lowest calculated excited states on the NCSM basis size is shown in Figure 16 for negative-parity states in panel (a) and for positive-parity states in panel (b). In the latter, the negative-parity $3/2^{-}$ ground state obtained in an $N_{\max }$ basis space is matched with the positive-parity excited states obtained in the $N_{\max }+1$ basis space. For technical reasons, the largest positive-parity space we were able to reach was $N_{\max }=9$ with the dimension of 269 million. While the $1/2_1^{-}$ state excitation energy decreased gradually with $N_{\max }$, the multi-$\hslash \mathsf{\Omega }$ dominated $3/2_2^{-}$ and $1/2_2^{-}$ states manifested a rapid decrease of excitation energies with the basis size correlated with an increase of higher-$\hslash \mathsf{\Omega }$ components in the $3/2_1^{-}$ ground-state wave function, the signature of which is seen in Figure 15b. This trend was also observed in recent large-scale NCSM calculations using NN interactions only [14]. Similarly, the positive-parity state excitation energies decreased steadily with $N_{\max }$. As seen in Figure 16, we found the lowest positive-parity states, $3/2_1^{+}$ and $5/2_1^{+}$, below the lowest negative-parity excited states in the largest spaces we could reach.

It should be noted that in the experimental data, only the ¹¹Li $3/2_1^{-}$ ground state is bound. The excited states calculated within the NCSM could approximate resonances in the continuum. To establish a connection to the experimental observations, we investigated multiple operator transitions from the ground state to excited states, with the details given in Ref. [97].

The present large-scale NCSM calculations are a prerequisite of planned investigation of ¹¹Li within the NCSMC treating this halo nucleus as a three-body cluster system of ⁹Li and two neutrons that will be capable providing a realistic description of the two-neutron halo similarly as accomplished for ⁶He [43]; see Section 3.5.

4. Discussion

In this article, we have provided evidence of the usefulness and power of ab initio nuclear theory for the description and understanding of halo nuclei, exotic weakly bound systems with extended single-nucleon or two-neutron density beyond a tightly bound core. We have reviewed applications of the No-Core Shell Model with Continuum, a method that provides a unified description of bound and unbound nuclear states including precision chiral EFT-based NN + 3N interactions, single-neutron halo nuclei ¹¹Be and ¹⁵C, a single-proton halo nucleus ⁸B, and ⁶He exhibiting two-neutron Borromean halo. Furthermore, we have provided an analysis of excited halo states in ¹⁰Be.

The NCSMC is currently the only method that has successfully reproduced the parity inversion in the ground state of ¹¹Be and, at the same time, obtained wave functions of its $1/2^{+}$ and $1/2^{-}$ bound states demonstrating the neutron+¹⁰Be($0^{+}$) S-wave and P-wave halos, respectively, which extend well beyond 20 fm. Calculated ANCs of the two halo states agree very well with those extracted from the knockout and transfer reaction halo-EFT analyses [48,49].

We have presented new NCSMC results for the single-neutron halo nucleus ¹⁵C. Focusing on the description of its two bound states, the ¹⁴C+n S-wave halo $1/2^{+}$ ground state and the weakly bound $5/2^{+}$ excited state, we have shown good agreement of the calculated ANCs with those obtained by analysis of ¹⁴C(d,p)¹⁵C transfer and knockout reactions [48,54,56]. Also, we have performed calculations of the ¹⁴C(n,$\gamma$)¹⁵C capture reaction cross-section relevant for several astrophysical processes. Our results, obtained for both bound final states, are in line with recent experimental measurements [57] and phenomenological calculations [59], although slightly higher than the most recent experimental determination [60].

We have discussed the structure of the very weakly bound single-proton halo nucleus ⁸B, which plays an important role in astrophysics. We have provided a detailed analysis of its $2^{+}$ halo ground state dominated by the ⁷Be($3/2^{-}$)+proton cluster in the relative P-wave. The ANCs obtained within the NCSMC are in good agreement with the recent experimental determination [63].

While the ¹⁰Be nucleus is well bound, it features two excited states of halo nature just below the ⁹Be+neutron threshold. In new calculations, we have investigated the ¹⁰Be bound and scattering states within the NCSMC, focusing particularly on the structure of the two excited halo states, $1^{-}$ and $2^{-}$, dominated by ⁹Be($3/2^{-}$)+neutron in the S-wave with spins anti-aligned and aligned, respectively. We have provided a detailed analysis of the two states with ANCs for various partial waves. In addition, we have discussed resonances above the ⁹Be+neutron threshold, where we have predicted, e.g., a $1^{+}$ resonance not included in the recent data evaluation [67]. The present calculations will be improved in the future by coupling the ⁶He+$\alpha$ mass partition that lies experimentally just 600 keV above the ⁹Be+neutron threshold, thus impacting the structure of the low-lying positive-parity resonances.

The NCSMC has been extended recently to describe systems dominated by three-body breakup channels. Applying this formalism, we have discussed the structure of the two-neutron Borromean halo nucleus ⁶He described within the NCSMC as an $\alpha$ + n + n system. The weakly bound $0^{+}$ ground states manifested a superposition of the dineutron and cigar configurations as found in earlier cluster model calculations, here obtained microscopically from realistic nucleon–nucleon interactions. The NCSMC calculations also predicted several resonances above the $\alpha$+n+n threshold, starting with a narrow $2^{+}$ state corresponding to the experimentally well-established first exited state of ⁶He. This was then followed by broad $2^{+}$ and $1^{+}$ resonances; the former could be matched to experimentally observed broad $2^{+}$ state [79]. Due to the complexity of the three-body cluster NCSMC calculations, the ⁶He results have so far been obtained using the NN interaction only, with the 3N interaction capability still to be implemented.

The next challenge for the NCSMC and for the ab initio nuclear theory in general is the description of the two-neutron Borromean halo nucleus ¹¹Li, the exotic system discovered four decades ago [1] and later interpreted as the first-ever halo nucleus found [5]: it poses a significant challenge and added complexity compared to the achieved NCSMC ⁶He calculations, due to its heavier mass, non-zero spin, and the need to include excited state(s) of the ⁹Li core. As the required first step, we have presented here large-scale No-Core Shell Model calculations of ¹¹Li reaching basis spaces up to $N_{\max }=10$ that allow extrapolating the total binding energy. We calculated the same for ⁹Li and found that NCSM predicts both nuclei bound by about the same energy close to experimental one. It is reasonable to anticipate that by including the ⁹Li + n + n continuum within the three-body cluster NCSMC, the ¹¹Li would become bound. We also discussed excited states of ¹¹Li predicted by the NCSM. In the largest spaces reached, we found the lowest positive-parity states below the lowest negative-parity excited states; see Ref. [97] for further details. It will be also very important to investigate the excited states within the NCSMC, i.e., including the ⁹Li + n + n continuum.