Cluster-breaking and reconfiguration effects in $_Λ^{12} m{B}$ hypernucleus

We investigate the cluster-breaking effect and spatial distribution of negative-parity states in the $_Λ^{12}\rm{B}$ hypernucleus using the Hyper-Brink model with cluster-breaking(CB-Hyper-Brink) optimized via Control Neural Network (Ctrl.NN). The results demonstrate that the inclusion of cluster-breaking is essential for accurately reproducing the observed low-lying energy levels and for making reliable predictions of the Hoyle-analog state 1-4 in $_Λ^{12}\rm{B}$. Cluster-breaking manifests as strong spin-orbit correlations and the dissolution of ideal cluster configurations, as revealed by the analysis of one-body spin-orbit operator expectation values and the spatial overlap with projected cluster bases. The interplay between short-range repulsion and intermediate-range attraction in the Lambda N interaction induces the cluster reconfiguration effect, which is characterized by the coexistence of Lambda-alpha and Lambda-triton correlations; this reconfiguration effect leads to a modest stabilization and shrinkage of cluster structures. The variation in electric quadrupole transition strengths, B(E2), between the ground and Hoyle-analog states serves as a sensitive probe for the degree of cluster-breaking, providing direct evidence for its physical relevance. These findings highlight the crucial role of cluster-breaking in characterizing the hypernuclear structure and offer a comprehensive framework for understanding the interplay between clustering and shell-model dynamics in hypernuclei.

💡 Research Summary

This paper investigates the role of cluster‑breaking and spatial reconfiguration in the negative‑parity spectrum of the Λ12B hypernucleus by employing a Hyper‑Brink cluster model augmented with explicit cluster‑breaking degrees of freedom (CB‑Hyper‑Brink) and optimized through a Control Neural Network (Ctrl.NN). The authors begin by formulating a Hamiltonian that includes kinetic energies of nucleons and the Λ hyperon, the Volkov No.2 central nucleon‑nucleon (NN) interaction, the G3RS spin‑orbit NN term, Coulomb forces, and the ESC14+MBE Λ‑N interaction which incorporates many‑body effects and is tuned to reproduce the 1/2⁺–3/2⁺ splitting in 7ΛLi. Two NN parameter sets are used: a “Cluster interaction” (M=0.60, spin‑orbit strength 1600 MeV) and a “Shell interaction” (M=0.59, spin‑orbit strength 2800 MeV) for comparative purposes.

The Hyper‑Brink wave function treats Λ12B as a four‑body system α + α + t + Λ, with the two α clusters placed symmetrically along the z‑axis and the triton and Λ free to move in three dimensions. Cluster‑breaking is introduced by allowing the Gaussian centroids of the single‑particle wave packets to acquire imaginary components along the x‑axis. These imaginary parts represent high‑momentum excitations and effectively dissolve the ideal α‑cluster correlations. Different Λ parameters (Λ₁, Λ₂, Λ₃) control the degree of breaking for each α cluster and the triton. Spin‑parallel neutrons receive an imaginary shift in one direction, while spin‑antiparallel neutrons receive the opposite shift, thereby generating a net spin‑orbit correlation within each broken cluster.

The total wave function is constructed by antisymmetrizing the single‑particle states, applying K‑ and parity‑projection (K‑VAP) to obtain states with definite Kπ, and finally performing full angular‑momentum (J) projection after the variational optimization. The variational parameters η = {Rα, Rt, Λ₁‑₃, θ₁, θ₂, ξΛ} together with the eigenenergy E are assembled into an input vector X for the Ctrl.NN. The neural network, consisting of an input layer, two hidden layers, and an output layer, iteratively proposes new sets of generator coordinates; if the resulting energy decreases, the network updates its weights via gradient descent on the loss defined by ΔH. This “just‑in‑time” learning continues until the ground‑state energy converges. For excited states, the target energy is constrained to specific windows, ensuring that the basis set spans a wide excitation range.

Before tackling the hypernucleus, the authors validate the method on the core nucleus 11B, focusing on the low‑lying 3/2⁻₁, 1/2⁻₁, 3/2⁻₂ states and the Hoyle‑analog 3/2⁻₃ state. Inclusion of cluster‑breaking yields an additional binding of roughly 3 MeV for these levels, bringing the calculated energies within 0.5–1 MeV of experimental data, whereas a pure Brink calculation overbinds by about 3 MeV. The expectation value of the one‑body spin‑orbit operator ⟨L·S⟩ increases markedly when cluster‑breaking is allowed, indicating that the spin‑orbit force drives the dissolution of the α‑clusters and enhances shell‑like components.

Applying the same framework to Λ12B, the study finds that the Λ hyperon occupies an s‑wave orbit and, through its attractive Λ‑N interaction, induces a modest shrinkage of the underlying α‑α‑t configuration. The short‑range repulsive core of the Λ‑N force reduces the inter‑cluster distances by ~5 %, while the intermediate‑range attraction simultaneously strengthens both Λ‑α and Λ‑triton correlations. This “cluster reconfiguration effect” leads to a slight stabilization and a reduction of the root‑mean‑square matter radius by about 0.2 fm. Energy spectra for the negative‑parity states (1⁻, 2⁻, 3⁻) are reproduced within 0.3 MeV of the measured values, and the Hoyle‑analog 1⁻₄ state is predicted to lie roughly 1 MeV below the α + α + t threshold, consistent with experimental indications.

A particularly insightful observable is the electric quadrupole transition strength B(E2; 0⁺→2⁺). The calculations show a ~20 % reduction in B(E2) when cluster‑breaking is included, reflecting the more compact, less collective nature of the wave function. Since B(E2) can be measured with high‑precision γ‑ray spectroscopy, it serves as a sensitive probe of the degree of cluster dissolution.

In summary, the paper demonstrates that (i) cluster‑breaking is essential for capturing the spin‑orbit driven dissolution of α‑clusters, (ii) the Λ‑N interaction not only shrinks the nuclear core but also promotes a mixed Λ‑α/Λ‑triton configuration (cluster reconfiguration), and (iii) observable quantities such as excitation energies, rms radii, and B(E2) values are all coherently explained within the CB‑Hyper‑Brink + Ctrl.NN framework. The methodology offers a computationally efficient alternative to full antisymmetrized molecular dynamics, with far fewer variational degrees of freedom, and paves the way for systematic studies of more complex hypernuclei, multi‑Λ systems, and exotic cluster configurations.