No-Core Shell Model with Continuum (NCSM/HORSE)

• No-Core Shell Model with Continuum is an ab initio framework that unifies harmonic oscillator calculations with continuum scattering techniques to study resonances.
• It employs the J-matrix/HORSE formalism to analytically map eigenvalues to phase shifts, enabling precise determination of resonance energies and widths.
• Applications include investigations of light nuclei, trineutron resonances, and n-α systems, offering critical insights into nuclear structure and decay.

The No-Core Shell Model with Continuum (NCSM/HORSE/J-Matrix) refers to a set of ab initio computational frameworks developed to unify discrete shell-model calculations in harmonic oscillator (HO) bases with the treatment of nuclear continuum dynamics, specifically resonances and scattering. This approach allows for the extraction of physically meaningful continuum observables—such as phase shifts, resonance energies, and widths—directly from variational diagonalizations in a truncated oscillator basis, leveraging the J-matrix/HORSE (Harmonic Oscillator Representation of Scattering Equations) formalism. The framework is particularly valuable for investigating light nuclei, unbound states, and cluster-decay phenomena, where coupling between bound and continuum degrees of freedom is essential.

1. Foundations and Theoretical Framework

The core of the NCSM/HORSE/J-Matrix approach begins with the ab initio No-Core Shell Model (NCSM), which expands the $A$-body nuclear wave function in a complete many-body HO basis, truncated at a maximum excitation quantum number $N_{\max}$. The inclusion of continuum effects hinges on the J-matrix or HORSE approach, which partitions the Hilbert space into a finite “P” space (fully interacting states up to $N_{\max}$) and an infinite “Q” space (states with $N > N_{\max}$ where only the kinetic energy is retained and the interaction is neglected) (Shirokov et al., 2016, Mazur et al., 2024).

The matching condition at the $P$/$Q$ boundary maps discrete NCSM eigenenergies $\{E_d\}$ (as functions of HO parameter $\hbar\omega$) onto scattering phase shifts via analytic formulas derived in the J-matrix formalism, enabling access to continuum behavior without explicit continuum basis construction.

2. HORSE Formalism and Key Equations

The HORSE/J-matrix formalism expands single-particle radial wavefunctions as

$$u_\ell(k,r) = \sum_{N=N_0,N_0+2,\ldots} a_{N\ell}(k)\, R_{N\ell}(r)$$

where $R_{N\ell}(r)$ are HO radial functions with $N_0=\ell$ (Shirokov et al., 2016). For $N\gg N_{\max}$, the potential vanishes and the basis states are free. The matching between model space and continuum is exact in this representation for both two- and many-body systems.

At a model-space truncation $N_{\max}$, the phase shift for eigenenergies $E_d$ is given by

$$\tan\delta(E_d) = -\frac{S_{N_{\max}+N_{\min}+2,\,\mathcal{L}}(E_d;\hbar\omega_d)}{C_{N_{\max}+N_{\min}+2,\,\mathcal{L}}(E_d;\hbar\omega_d)}$$

where $S$ and $C$ are regular and irregular HO J-matrix solutions, $N_{\min}$ is the minimum quanta allowed by antisymmetry, and $\mathcal{L}$ encodes the relevant hyperangular momentum (Mazur et al., 2024).

For odd $A$, half-integer $\mathcal{L}$ values arise, requiring a specialized analytic continuation of the S-matrix:

$S(k) = \exp\left[2i\delta(E)\right]$

or, for explicit branch cut structure,

$$S(k) = \frac{X(k) + 2k^{2\mathcal{L}+1}\ln(k/q_0) + i\pi k^{2\mathcal{L}+1}}{X(k) + 2k^{2\mathcal{L}+1}\ln(k/q_0) - i\pi k^{2\mathcal{L}+1}}$$

with $X(k)$ a real, even polynomial and $q_0$ a scale parameter (Mazur et al., 2024).

Extraction of resonance parameters proceeds by parametrizing $X(k)$ and finding S-matrix poles in the complex-$k$ plane, yielding resonance energy $E_r$ and width $\Gamma$:

$$X(k) + 2k^{2\mathcal{L}+1}\ln(k/q_0) - i\pi k^{2\mathcal{L}+1} = 0$$

with $E_r = \hbar^2 k_r^2/2M$, $\Gamma = \hbar^2 k_r\gamma/M$.

3. Practical Implementation in Ab Initio Calculations

Numerical implementation in the SS-HORSE–NCSM (Single-State HORSE extension of NCSM) proceeds as follows (Mazur et al., 2024, Shirokov et al., 2016):

• Diagonalize the $A$-body Hamiltonian with realistic NN and, optionally, 3N interactions in a HO basis for various $N_{\max}$ and $\hbar\omega$.
• Obtain a set of eigenenergies $\{E_d\}$ as functions of $\hbar\omega$ and $N_{\max}$, with typical ranges $N_{\max}\leq20$ and $\hbar\omega=2$–$50$ MeV.
• Remove center-of-mass motion from the spectrum.
• Apply the HORSE matching formula to map each $E_d$ to a phase shift point $(E_d,\delta(E_d))$ in the continuum.
• Fit the resulting discrete phase-shift points with an analytic S-matrix parametrization incorporating branch cuts and resonance poles.
• For each model space, resonance pole extraction and model convergence are monitored (stable pole locations are observed for $N_{\max}=16,18,20$ and polynomial orders $W=4,5$, with residual r.m.s.\ errors at $\sim$keV).

Soft NN potentials (e.g., Daejeon16, JISP16, SRG-evolved chiral) and bare chiral EFT potentials (Idaho N$^3$LO, LENPIC N$^4$LO) are both supported, revealing sensitivity to three-body/induced corrections as discussed below.

4. Applications: Case Studies and Results

Three-Neutron (Trineutron) System

SS-HORSE–NCSM applied to the trineutron system using soft NN potentials predicts two nearly degenerate, overlapping resonances in the $3/2^-$ and $1/2^-$ channels (Mazur et al., 2024):

Potential	$J^\pi$	$E_r$ (MeV)	$\Gamma$ (MeV)
Daejeon16	$3/2^-$	$0.48\pm0.06$	$0.96\pm0.21$
Daejeon16	$1/2^-$	$0.48\pm0.08$	$0.96\pm0.17$
JISP16	$3/2^-$	$0.35\pm0.08$	$0.70\pm0.09$
JISP16	$1/2^-$	$0.35\pm0.11$	$0.67\pm0.22$
SRG\,(2 fm$^{-1}$) N$^3$LO	$3/2^-$	$0.34\pm0.08$	$0.70\pm0.19$
SRG\,(2 fm$^{-1}$) N$^3$LO	$1/2^-$	$0.35\pm0.09$	$0.68\pm0.16$

Bare chiral EFT potentials (Idaho N$^3$LO, LENPIC N$^4$LO) yield no resonance-type phase shift rise, showing no S-matrix poles in the physical sheet for these systems. No virtual states appear for odd $A$ due to the topological structure of the S-matrix in these cases (Mazur et al., 2024).

Two-Body and $n\alpha$ Systems

The HORSE/SS-HORSE approach is validated for two-body and cluster systems such as neutron–alpha ($n\alpha$) resonant and non-resonant scattering. For $^5$He, JISP16-based SS-HORSE calculations produce resonance parameters in reasonable agreement with empirical R-matrix and direct calculations, demonstrating the method’s reliability for extracting continuum observables from variational calculations (Shirokov et al., 2016).

Three-Cluster NCSMC and $^6$He

NCSMC extended to three-cluster systems (Romero-Redondo et al., 2015) achieves rapid and accurate convergence to both many-body bound-state energies and halo structures, as illustrated in $^6$He. Full antisymmetrization between all clusters and coupling to discrete (NCSM) and continuum (RGM-type) sectors allow accurate reproduction of asymptotics, binding, and clustering, outperforming standard NCSM and NCSM/RGM.

5. Methodological Comparison and Theoretical Context

SS-HORSE–NCSM offers several operational advantages over other continuum methods:

• No explicit continuum basis required: Continuum observables are obtained by analytic continuation and mapping from discrete eigenvalues, eliminating the need for large-scale coupled-channel or complex-scaling solutions as in NCSM-with-continuum (NCSMC) and no-core Gamow Shell Model (GSM) (Mazur et al., 2024, Shirokov et al., 2016).
• Transparent extraction of resonances: Resonance locations and widths are calculable directly from S-matrix pole analysis without auxiliary boundary-condition matching.
• Rapid convergence: Stable resonance parameters are obtained from modest-sized model spaces using soft NN forces, allowing for systematic uncertainty assessment via $N_{\max}$ and parametrization order.

NCSMC remains the method of choice for problems requiring explicit coupling to multiple continuum channels and extended spatial asymptotics (e.g., two-neutron halos [$^6$He]), incorporating both shell-model and antisymmetrized microscopic-cluster channels (Romero-Redondo et al., 2015).

6. Physical Insights and Sensitivity to Nuclear Forces

The emergent picture from SS-HORSE–NCSM and comparable ab initio continuum approaches is that the appearance and structure of few-neutron resonances are highly sensitive to the choice of nuclear interaction:

• Soft, off-shell-tuned NN interactions (Daejeon16, JISP16, SRG-evolved chiral) favor the formation of three-neutron resonances just above threshold, with energies $E_r\approx 0.3$–$0.5$ MeV and widths $\Gamma\approx 0.7$–$1.0$ MeV.
• Bare chiral EFT two-body forces do not produce such resonances, suggesting that effective inclusion of three-body physics (or off-shell effects) is crucial for resonance formation in these systems (Mazur et al., 2024).
• Resonance energies for the trineutron from SS-HORSE–NCSM systematically lie below quantum Monte Carlo trap-extrapolated and GSM predictions, though resonance widths are similar (e.g., GSM $E_r=1.29$ MeV, $\Gamma=0.91$ MeV) (Mazur et al., 2024).

A plausible implication is that three-body forces or off-shell effects, implicitly or explicitly included in soft interactions, play a pivotal role in stabilizing neutron-rich few-body resonant states.

7. Limitations and Outlook

There are intrinsic and methodological constraints:

• Physical sheet virtual states: The democratic-decay S-matrix formulation for odd-$A$ fragment systems forbids virtual states as poles on the negative imaginary $k$ axis when effective angular momentum is half-integer, constraining possible interpretations for peak-like enhancements (Mazur et al., 2024).
• Overlap of spin–parity channels: The prediction of strongly overlapping $3/2^-$ and $1/2^-$ resonances suggests that experimental isolation of individual channels in trineutron decay is intrinsically challenging, requiring reaction mechanisms that populate all neutrons simultaneously.

Future theoretical directions include systematic exploration of genuine three-body forces, further validation against direct-continuum approaches (e.g., GSM, NCSMC), and the development of improved S-matrix parametrizations for complex clustering and decay scenarios.

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References:

• (Mazur et al., 2024): "Trineutron resonances in the SS-HORSE-NCSM approach"
• (Shirokov et al., 2016): "Shell Model States in the Continuum"
• (Romero-Redondo et al., 2015): "Advances in the ab initio description of nuclear three-cluster systems"