Microscopic foundation of the EDF-mapped interacting boson model: origin of the effective cubic three-boson term, configuration-mixing strengths, and boson effective charges

The interacting boson model (IBM) can be given a microscopic basis by mapping a potential energy surface computed with a nuclear energy density functional (EDF) onto the expectation value of an IBM-2 Hamiltonian (built from neutron and proton $s$ and $d$ bosons) in its coherent state. This EDF-mapped IBM now describes low-energy quadrupole spectra of medium-heavy and heavy nuclei, but several of its ingredients are still fixed phenomenologically rather than derived: (i) the effective three-body (cubic) $d$ -boson interaction, introduced because two-body IBM Hamiltonians cannot generate triaxial minima, whose fermionic origin (renormalization of higher-spin fermion pairs, $g$ -boson effects, or explicit three-nucleon correlations) remains unidentified; (ii) the strengths of normal-intruder configuration mixing and the configuration offset energies in shape-coexistence regions, which are determined only approximately from static mean-field surfaces and whose present forms the program's 2016 shape-coexistence review flags as possibly too simplified; (iii) the boson effective charges and $g$ -factors of the E2 and M1 transition operators, currently fitted to data; and (iv) the residual spectroscopic failures in transitional nuclei, whose source — the IBM assumptions, the mapping procedure, or the underlying EDF — has not been identified. The open problem, originating in the source thesis and stated explicitly in the program's 2016 shape-coexistence review and 2025 status review, is to derive these remaining effective bosonic quantities from the underlying fermionic/EDF degrees of freedom so that the mapped IBM becomes a parameter-free microscopic spectroscopy tool, and to identify the source of its residual spectroscopic failures.

Background

The interacting boson model (IBM) is an algebraic description of the low-energy quadrupole collective motion of medium-heavy and heavy even-even nuclei. Its degrees of freedom are $N$ bosons of two kinds, a spin-0 $s$ boson and a spin-2 $d$ boson, where $N$ equals the number of valence nucleon pairs; in the proton-neutron version (IBM-2) neutron and proton bosons are distinguished. Depending on the relative strengths of its one- and two-boson interactions the IBM exhibits three dynamical symmetries — U(5), SU(3), and O(6) — realizing the classical pictures of the spherical vibrator, the axially deformed rotor, and the $\gamma$ -unstable rotor. Here $\gamma$ is the asymmetry angle of the quadrupole deformation: $\gamma=0^\circ$ prolate, $\gamma=60^\circ$ oblate, and $\gamma\approx 30^\circ$ maximally triaxial. Historically the IBM parameters were fitted to data.

Nuclear energy density functional (EDF) theory instead computes intrinsic nuclear properties — binding energies, radii, deformations — from self-consistent mean-field (SCMF) equations of Hartree-Fock-Bogoliubov (HFB) type with effective functionals such as the Skyrme (SkM*, SLy4), Gogny (D1S, D1M), and relativistic (DD-PC1) parametrizations, but on its own it does not yield discrete spectra with good angular momentum in the laboratory frame.

The mapping program bridges the two frameworks: one computes a constrained SCMF potential energy surface (PES) $E(\beta,\gamma)$ in the triaxial quadrupole deformation variables and equates it, point by point, to the expectation value of an IBM-2 Hamiltonian in a coherent state (a boson condensate, which serves as the IBM intrinsic state and defines its classical limit). The IBM parameters are thereby fixed without phenomenological fitting. For strongly deformed rotors, whose rotational-band moments of inertia the static mapping underestimates, an additional $\hat L\cdot\hat L$ term is introduced whose strength is fixed by matching the cranking response of the boson system at small angular frequency $\omega$ to the SCMF moment of inertia in the Inglis-Belyaev or Thouless-Valatin approximation (the latter being the exact time-dependent mean-field moment of inertia, including dynamical rearrangement of the density).

Two structural facts motivate the open issues addressed here. First, a classical-limit result (Van Isacker and Chen, 1981) shows that an IBM Hamiltonian with only one- and two-body terms cannot produce a triaxial minimum on its energy surface; triaxiality requires three-boson ('cubic') terms, known phenomenologically since the 1980s (Heyde et al., 1984). In the mapped IBM-2 an $L=3$ cubic term with strength $\theta^{(3)}$ was introduced (Nomura et al., 2012) specifically to reproduce the triaxial minima that EDF surfaces predict for $\gamma$ -soft nuclei, and it simultaneously reproduces the observed systematics of the quasi- $\gamma$ band — the band built on the first excited $2^+$ state, whose odd-even-spin staggering (alternating crowding of odd- and even-spin levels) interpolates between the $\gamma$ -unstable and rigid-triaxial rotor limits. The strength $\theta^{(3)}$ , however, is determined only by fitting the mapped surface; no fermionic mechanism that generates this effective three-boson interaction has been identified. Second, in regions of shape coexistence (the simultaneous appearance of states of different intrinsic shape near the ground state, e.g. neutron-deficient Hg-Pb-Pt and the A≈100 Zr-Sr region), low-lying excited $0^+$ states arise from proton particle-hole excitations across a major shell ('intruder' states); in the IBM these correspond to boson spaces enlarged by two bosons per 2p-2h excitation, mixed with the normal configuration (Duval and Barrett's configuration-mixing IBM, IBM-CM). The EDF-based IBM-CM derives each configuration's unperturbed Hamiltonian from its own mapped energy surface, taking the configuration offset energies $\Delta_m$ approximately from the energy differences between the corresponding HFB minima and the mixing strengths from the topology of the barrier between them (set to small values, of order 0.1–0.2 MeV, in practice); the 2016 review of the program notes that the employed unperturbed Hamiltonians and mixing interactions might be of too simplified a form and that a mean-field total energy surface alone may not suffice to constrain them. Likewise the E2 transition operator carries boson effective charges $e_\nu^{\rm B}$ , $e_\pi^{\rm B}$ (and the M1 operator boson $g$ -factors) that are fitted to one measured $B(\mathrm{E}2)$ per isotopic chain rather than derived; a microscopic derivation would have to include core polarization, i.e. the coupling of the valence bosons to non-collective excitations of the closed shell.

Kosuke Nomura's thesis 'Interacting Boson Model from Energy Density Functionals' (Springer, 2013) consolidated this program and its validation for vibrational, rotational, and $\gamma$ -soft nuclei. Subsequent work extended the mapping in several directions (configuration mixing in Pb, Hg, and A≈100 nuclei; boson-number nonconserving pairing degrees of freedom; $g$ bosons for hexadecapole, i.e. fourth-rank $L=4$ , correlations; dipole $p$ bosons), but the program's own reviews show that the cubic-term origin remains unidentified and that the configuration-mixing parameters and effective charges remain only approximately or empirically determined: the 2016 shape-coexistence review (Nomura, Otsuka, and Van Isacker) calls for an advanced mapping — possibly invoking additional quantities such as symmetry-projected energy surfaces — to extract the parameters of the unperturbed Hamiltonians, the mixing strengths, the offset energies, and the effective charges for transition operators, and the 2025 status review still introduces the cubic term as the device that produces triaxial minima, reports effective charges fitted to measured transition rates, and lists identifying the source of the residual spectroscopic deficiencies in transitional nuclei as a crucial step forward. Independent beyond-mean-field references exist for validation — the generator coordinate method (GCM), which mixes symmetry-projected mean-field configurations, and the five-dimensional collective Hamiltonian, which quantizes collective motion in the quadrupole deformation variables and orientation angles on the same EDF surfaces — and comparisons against them are part of assessing the mapped IBM's accuracy.

Problem Statement

Complete the microscopic foundation of the EDF-mapped interacting boson model: derive, from the underlying fermionic (EDF) degrees of freedom and without phenomenological adjustment, the remaining effective bosonic quantities of the mapped framework, and identify the source of its residual spectroscopic discrepancies. The common objective uniting the requested results is a parameter-free, EDF-based determination of the mapped IBM's Hamiltonian and transition operators, so that the model's predictions for spectra and transition rates carry only microscopic input. Specifically, the following items remain unresolved:

Fermionic origin of the effective cubic (three-body $d$ -boson) term. Identify the microscopic mechanism that generates the effective $L=3$ three-boson interaction (for example, renormalization of higher-spin fermion pairs into the $sd$ boson space, effects of $g$ (spin-4) bosons, explicit three-body nucleonic correlations, or the truncation to $S$ - $D$ fermion pairs itself), so that its strength $\theta^{(3)}$ can be derived from the microscopic input, in place of the present procedure of adjusting it to reproduce the triaxial minimum of the mapped energy surface.
Microscopic determination of configuration-mixing parameters. Within the configuration-mixing extension for shape-coexistence regions, derive from fermionic/EDF input the mixing strengths between normal and intruder ( $2p$ - $2h$ ) boson configurations and the configuration offset energies $\Delta_m$ , going beyond their present approximate determination from static mean-field surfaces (offset energies from the energy differences between the corresponding HFB minima, mixing strengths from the topology of the barrier between them); as anticipated by the 2016 review, this may require an advanced mapping invoking additional quantities, e.g. symmetry-projected energy surfaces.
Microscopic determination of the transition operators. Derive the boson effective charges $e_\nu^{\rm B}$ , $e_\pi^{\rm B}$ of the E2 operator (and the corresponding M1 boson $g$ -factors) from EDF-based fermionic structure, including core-polarization and beyond-mean-field corrections, in place of fitting to measured $B(\mathrm{E}2)$ values.
Diagnosis of residual discrepancies. Identify the source of the residual quantitative discrepancies in transitional nuclei (for example the overestimated non-yrast levels in Sm isotopes and the yrast levels in neutron-rich Zr near $N=56$ ): determine whether they stem from assumptions in the IBM, from the mapping procedure, or from properties of the underlying EDF used as microscopic input, individually or in combination.

Items 1-3 make the model's remaining free parameters microscopic; item 4 establishes when the resulting description can be trusted. All four items are jointly required, and the objective is resolved only when all of them have been addressed; completing a proper subset does not suffice. Together they decide whether the EDF-mapped IBM is a fully microscopic theory of low-energy nuclear collective motion or remains a phenomenological model with microscopic input.

References

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