Weakly bound nuclear states — halo nuclei and drip-line systems with keV-scale binding — sit so close to reaction thresholds that their coupling to the continuum of scattering states cannot be neglected. Nuclear lattice effective field theory (NLEFT), which places nucleons on a finite periodic spatial lattice with chiral effective-field-theory interactions, currently has no explicit, practicable treatment of this continuum coupling. Wu and Meißner (2020) took a first step with a schematic model: a heavy core A, nucleons N, and a very weakly bound AN state B, studied in a finite-volume momentum-lattice Hamiltonian with B held fixed as a stable dimer. The open problem is to carry through the refinements the authors stated as necessary for their model: treat the full three-body ANN system in the finite-volume lattice Hamiltonian with the inelastic breakup channel B+N → A+N+N and the nearby BN and ANN thresholds treated explicitly, and replace the schematic separable s-wave potentials by realistic chiral interactions including higher partial waves; in addition, the authors speculated that refined calculations might make it possible to find a compact formula for the influence of the continuum on a weakly bound state on the lattice, analogous in role to the Lüscher formula. Any such calculation must also keep under control the finite-volume summation-versus-integration artifact the authors observed, which grows sharply for near-threshold potentials.

Background

Along the nuclear chart, weakly bound states occur in halo nuclei — nuclei whose outer nucleons orbit a compact core with binding energies of order keV rather than the few MeV typical of nuclear binding — and in isotopes near the drip lines, the limits of nuclear stability against neutron or proton emission. Such states lie extremely close to decay thresholds, and their coupling to the continuum of scattering states must be treated explicitly. In continuum nuclear frameworks this is routine: for example, the Berggren representation generalizes the shell-model basis so that bound, resonant, and continuum states are treated on the same footing, and continuum-extended ab initio methods exist for light systems. Nuclear lattice effective field theory (NLEFT) takes a different route: space(-time) is discretized on a finite periodic cubic box of side L, nucleons occupy lattice sites, and their interactions are given by chiral effective field theory (an expansion in pions and short-distance contact operators). NLEFT has produced ab initio results such as the Hoyle state of 12^{12}C — the excited state of carbon-12 that governs carbon production in stars — and microscopic alpha-alpha scattering, but all lattice energies are real-valued, a direct use of Berggren-type complex representations is obstructed, and an explicit treatment of continuum coupling is missing. The standard bridge between the finite-volume spectrum and scattering is Lüscher's formula, which maps the volume dependence of discrete two-body energy levels to infinite-volume scattering phase shifts; this mapping, however, does not by itself provide a microscopic description of how a nearby continuum or an additional unbound particle modifies the finite-volume spectrum of a weakly bound subsystem. Wu and Meißner (2020, arXiv:2007.09907) addressed this gap with a schematic three-particle model: a heavy core A with mass about ten nucleon masses, a nucleon N, and AN interactions tuned to produce a very weakly bound state B. All interactions were taken in the s-wave (the zero-angular-momentum partial wave) as separable potentials — interactions factorizing as a coupling constant times regulator functions of the incoming and outgoing momenta, regulated by a dipole form factor with cutoff Λ — and the finite-volume spectrum was obtained with the momentum-space Hamiltonian method of Wu, Lee, Thomas, and Young: momenta are restricted to the discrete set 2πn/L, the Hamiltonian matrix is diagonalized, and the levels are compared with the Lüscher equation. In the three-body sector the authors used a dimer (fixed-center) approximation: B is treated as a stable particle, reducing the ANN system to an effective BN two-body problem whose potential receives an attractive contribution from A exchange and a repulsive triangle-loop contribution; the NN coupling was tuned so that these either cancel exactly at threshold or give net repulsion. Two findings frame the open problem. First, the weak bound state B and the third particle N visibly interplay in the lattice spectrum — for instance, a level appears below the BN threshold even for a purely repulsive effective potential, because the discrete lattice momenta make the off-diagonal Hamiltonian elements effectively attractive at small volumes. Second, the Hamiltonian method has a systematic difference with the Lüscher equation — the difference between summing a regular function over discrete lattice momenta and integrating it over continuum momenta — which is proportional to exp(−Lm), with m the scale characterizing how sharply the potential varies near threshold, and which becomes large for the near-threshold BN potential; increasing the box to L = 17 fm restored consistency with the Lüscher equation. The authors closed by stating that the model requires much refinement before definite conclusions: the full three-body ANN system (with the BN and ANN thresholds very close, so that inelastic breakup B+N → A+N+N is expected to affect the spectrum), more realistic nuclear interaction potentials, and higher partial waves, and they speculated that refined calculations might make it possible to find a compact formula for the influence of the continuum on a weakly bound state on the lattice. The broader context is that exact three-body finite-volume formalisms have meanwhile been developed in the continuum-EFT Lüscher framework (three-particle quantization conditions, particle-dimer finite-volume energy shifts in powers of 1/L, and extensions to particles of unequal masses), but these use continuum momentum integrals rather than a lattice Hamiltonian and do not address the discrete-summation artifacts specific to the Hamiltonian lattice approach.

Problem Statement

Extend the Wu–Meißner three-particle model so that the influence of continuum coupling on weakly bound states is determined — and, if possible, compactly parametrized — within the finite-volume lattice-Hamiltonian framework of nuclear lattice effective field theory. The single scientific objective is a controlled, quantitatively reliable lattice-Hamiltonian description of how a weakly bound state B (a very weakly bound A–N pair, heavy core A with mass about ten nucleon masses) is modified by its coupling to the continuum, with and without a third nucleon N present. Completing this objective requires four scientifically linked results: the three refinements posed by the source paper as needed for its schematic calculation, together with an assessment of the compact formula the authors speculated such refinements might make possible — (1) a full three-body finite-volume treatment of the ANN system that abandons the dimer (fixed-center) approximation, so that the inelastic breakup channel B+N → A+N+N and the proximity of the BN and ANN thresholds are treated explicitly, quantifying how the lattice energy levels and the phase shifts extracted from them change relative to the fixed-dimer calculation; (2) replacement of the schematic s-wave separable potentials by interactions resembling realistic nuclear forces — chiral effective-field-theory potentials with pion-exchange and short-range contact operators — with an assessment of how the balance between the attractive A-exchange and repulsive triangle-loop contributions to the BN potential, and the delicate near-threshold cancellations found in the schematic model, are modified; (3) inclusion of partial waves beyond the s-wave in both two- and three-body interactions, establishing how continuum coupling depends on angular momentum and whether finite-volume spectra and extracted phase shifts for near-threshold states remain reliably describable; and (4) an assessment, building on the refined calculations of items (1)–(3), of the authors' speculation that refined calculations make it possible to find a compact formula for the influence of the continuum on a weakly bound state on the lattice, analogous in role to the Lüscher formula for scattering states; where such a formula is found, its form and range of applicability should be made explicit. Items (1)–(3) supply the physical calculations on which item (4) must rest, and their finite-volume systematics must keep under control the summation-versus-integration discrepancy observed in the source paper — the difference between summing over discrete lattice momenta 2πn/L and integrating over continuum momenta, which scales as exp(−Lm) for sharply varying near-threshold potentials.

References

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