Decay of three-body resonances in a discrete basis

We present a theoretical framework for calculating the asymptotic properties and decay dynamics of three-body resonances described in a discrete basis. The method involves solving an inhomogeneous Schrödinger equation to determine the non-normalizable resonant state by identifying a normalizable source state, which captures the short-range internal structure. The long-range behavior is then calculated using the free three-body propagator, providing accurate asymptotic coefficients necessary for describing decay correlations. We apply this formalism to the two-neutron decay of the 0$^{+}$ ground-state and the 2$^{+}$ excited-state resonances of $^{16}\text{Be}$ ($^{14}\text{Be}+n+n$), working within the hyperspherical expansion method with an analytical transformed harmonic oscillator basis. Our results show that the decay is strongly dominated by the lowest hypermomentum components at large separations, reflecting effective three-body barrier penetration dynamics that shape the final state. The calculated relative-energy distributions exhibit clear neutron-neutron correlations for both states, arising from mixing between different asymptotic channels, and are consistent with a direct two-neutron emission mechanism, in agreement with recent experimental observations. This work provides a reliable tool for linking the internal structure of three-body resonances to their decay properties.

💡 Research Summary

In this paper the authors develop a comprehensive theoretical framework for describing the asymptotic properties and decay dynamics of three‑body resonances when the system is represented in a discrete basis. The central idea is to treat the resonant state as a solution of an inhomogeneous Schrödinger equation, (H − E_r) |ψ_r⟩ = λ |ϕ_r⟩, where |ψ_r⟩ is the non‑normalizable resonant wave function, E_r = ε_r − iΓ/2 is the complex resonance energy, λ is a strength factor that encodes the details of the reaction (projectile, target, total energy, etc.), and |ϕ_r⟩ is a normalizable “source state” that contains the short‑range structure of the resonance. By projecting the full Hilbert space onto a finite set of square‑integrable basis functions {|b⟩} the authors introduce projection operators P (short‑range subspace) and Q = 1 − P (long‑range subspace). Using the Feshbach formalism they construct an effective Hamiltonian H_eff acting only within the P‑space, leading to the compact relation P|ψ_r⟩ = λ (H_eff − E_r)⁻¹ |ϕ_r⟩. The long‑range part of the wave function is then obtained analytically by applying the free three‑body propagator in hyperspherical coordinates; the asymptotic behavior is expressed through Bessel functions J_{K+2}(kρ) multiplied by coefficients A_K that depend on the overlap of the source state with the eigenstates of H_eff. These coefficients encode the contribution of each hypermomentum K (the hyperspherical analogue of angular momentum) to the decay.

For the practical implementation the authors adopt the hyperspherical expansion method combined with an analytical transformed harmonic oscillator (THO) basis. The THO transformation stretches the radial coordinate so that the discrete pseudostates generated from a finite basis accurately represent the continuum, while preserving the correct asymptotic fall‑off. The three‑body Hamiltonian is written as H = H_0 + V, where H_0 contains the kinetic energy (and possibly a Coulomb term) and V is a short‑range interaction restricted to the P‑space. The source state |ϕ_r⟩ is constructed to maximize the modified source term λ|ϕ_r⟩, which effectively defines a resonance operator whose eigenvectors serve as optimal source states.

The formalism is applied to the unbound nucleus ¹⁶Be, modeled as ¹⁴Be + n + n. Both the 0⁺ ground‑state resonance and the first 2⁺ excited resonance are investigated. Using realistic core‑neutron potentials and a neutron‑neutron interaction reproducing the low‑energy nn scattering length, the authors compute the complex resonance energies, obtaining widths and centroids that agree with recent experimental measurements. By evaluating the asymptotic coefficients A_K they find that at large hyperradius the lowest hypermomentum component (K = 0) dominates the decay amplitude for both resonances. This dominance reflects an effective three‑body barrier penetration that preferentially selects the lowest‑K channel, a feature that is robust against variations of the short‑range interaction.

The calculated relative‑energy distributions of the two emitted neutrons display a pronounced enhancement at low nn relative energy, indicative of strong nn correlations often described as a “dineutron” configuration. The authors demonstrate that this correlation arises from the mixing of several asymptotic channels (different K values) and is a direct consequence of the structure encoded in the source state. The resulting energy and angular correlations match the patterns observed in recent invariant‑mass spectroscopy experiments, supporting a direct two‑neutron emission mechanism rather than a sequential decay through an intermediate ¹⁵Be state.

Finally, the paper discusses the role of the strength factor λ, emphasizing that while λ influences the absolute magnitude of the production cross section, the shape of the observable correlations is governed solely by the asymptotic coefficients and therefore is largely independent of the specific reaction conditions. The authors conclude that their discrete‑basis, source‑state approach provides a reliable bridge between the internal structure of three‑body resonances and the measurable decay observables, and they suggest that the method can be extended to other exotic decay modes such as two‑proton radioactivity or three‑α clustering phenomena.