The 2011 relativistic Hartree-Fock-Bogoliubov model for axially deformed nuclei (RHFBz) closed with an explicit outlook for covariant density functional theory with explicit exchange (Fock) terms at deformation: add the tensor rho-nucleon coupling, which in the source's words 'will complete the model and thus enable studies of the role of tensor components of the effective inter-nucleon interaction in the evolution of shell structures in deformed nuclei', and remove the oscillator-basis and exchange-integral bottleneck that confined calculations to light nuclei. Its Introduction also identifies, as a general challenge for the framework of energy density functionals, the systematic treatment of collective correlations related to restoration of broken symmetries and fluctuations in collective coordinates - a challenge that RMF-based models had recently begun to address while the RHF framework was then still limited to spherical nuclei. Later work (2020-2026) resolved the methodological items by different routes, but the physics goal that the completed model was to enable remains open: a systematic quantitative assessment of the role of Fock-carried tensor components (pion pseudovector and rho-tensor couplings) in the evolution of shell structure in deformed nuclei. Beyond-mean-field treatment - angular-momentum and particle-number restoration or collective-coordinate fluctuations - within a covariant energy density functional that retains explicit Fock terms likewise remains unaddressed; it is the direction the source raises as a general challenge, not among the items of its outlook.

Background

Covariant (relativistic) density functional theory describes nuclei as Dirac nucleons coupled to meson mean fields. In the standard relativistic mean field (RMF) approach only direct (Hartree) fields are kept, and the coupling constants effectively absorb exchange effects. The relativistic Hartree-Fock (RHF) approach instead keeps the exchange (Fock) terms of the meson interactions explicitly, computed from the nucleon densities. This matters physically because the exchange of the pion (entering through its pseudovector pi-N coupling) and the tensor coupling of the rho meson to the nucleon generate the tensor components of the effective nucleon-nucleon interaction, which are known - from spherical studies such as Long, Sagawa, Van Giai and Meng, Phys. Rev. C 76, 034314 (2007) - to shape shell gaps and the evolution of single-particle structure. Treating these terms explicitly, rather than folding them into fitted constants, is expected to improve predictions of shell structure in exotic nuclei.

For open-shell deformed nuclei one must further treat pairing correlations, which is done by Bogoliubov's quasiparticle formulation: the relativistic Hartree-Fock-Bogoliubov (RHFB) equations mix particle and hole states and are solved, in the model introduced by Ebran, Khan, Peña Arteaga and Vretenar (Phys. Rev. C 83, 064323, 2011; 'RHFBz'), by expansion in a deformed harmonic-oscillator basis. That paper, restricted to light nuclei (C, Ne, Mg), closed - in Sec. IV, announced as 'a short summary and discussion of possible future studies' - with an explicit outlook of two items: (i) 'the addition of the tensor rho-nucleon coupling will complete the model and thus enable studies of the role of tensor components of the effective inter-nucleon interaction in the evolution of shell structures in deformed nuclei'; and (ii) the Fock terms make the exchange integrals nonseparable and computationally costly, limiting the oscillator basis to six fermionic shells, with work in progress toward twelve shells so that medium-mass and heavy nuclei could be described. Its Introduction separately notes that 'an important challenge for the framework of EDF is the systematic treatment of collective correlations related to restoration of broken symmetries and fluctuations in collective coordinates', which is needed to calculate excitation spectra and electromagnetic transition rates, and that RMF-based models had recently been developed with an explicit treatment of such correlations while the RHF framework was at that time limited to spherical nuclei; this challenge is not presented there as an item of the RHFBz outlook.

Substantial progress has since been made on the methodological items. The deformed RHF (D-RHF) model of Geng, Xiang, Sun and Long (Phys. Rev. C 101, 064302, 2020) and its Bogoliubov extension D-RHFB (Geng and Long, Phys. Rev. C 105, 034329, 2022) implemented the rho-tensor coupling together with the finite-range Gogny D1S pairing force, using a spherical Dirac-Woods-Saxon basis (single-particle solutions of the Dirac equation in a Woods-Saxon potential, used as the expansion basis) whose convergence was verified up to the mid-heavy nucleus 154Sm/156Sm region; octupole-deformed extensions followed in 2024-2026. Independently, Zhao, Ren, Zhao and Yoshida (Phys. Lett. B 860, 139196, 2025) solved exact-exchange relativistic density functional theory in three-dimensional coordinate space - eliminating basis truncation and reaching triaxial nuclei. What this later literature has not delivered is the physics goal that the completed framework was meant to enable: a systematic, quantitative assessment of how the pi- and rho-exchange tensor terms modify shell structure in deformed nuclei (the 2022 D-RHFB paper explicitly defers 'systematic' studies of the pi-PV and rho-T couplings coupled to deformation). Also still missing, although the source raises it only as a general challenge, not among the items of the RHFBz outlook, is any symmetry-restored or collective-fluctuation treatment that retains explicit Fock terms - the closest existing result is the configuration-interaction RHF model of Liu, Niu and Long (2025), limited to light nuclei on a spherical basis, while all deformed beyond-mean-field covariant calculations to date use no-Fock RMF functionals.

Problem Statement

Within the deformed relativistic Hartree-Fock-Bogoliubov framework - a covariant energy density functional for axially (and octupole) deformed nuclei that keeps the exchange (Fock) terms of the density-dependent meson-nucleon interaction, including the pion pseudovector coupling and the tensor rho-nucleon coupling, with Gogny-type pairing - the outlook of Ebran et al. (2011) states that adding the tensor rho-nucleon coupling 'will complete the model and thus enable studies of the role of tensor components of the effective inter-nucleon interaction in the evolution of shell structures in deformed nuclei'. The still-undelivered result that this completion was meant to enable is:

(1) Systematic tensor assessment: quantify, across deformed nuclei from the light region already accessible up to medium-mass systems, how the tensor components carried by the Fock terms (π pseudovector exchange and ρ-tensor coupling) modify the evolution of shell structure - single-particle level schemes, shell gaps, deformation energy surfaces, and the emergence or quenching of magic numbers - relative to calculations in which these exchange terms are omitted or absorbed into fitted Hartree couplings; and establish which observed features of deformed exotic nuclei are genuinely attributable to these tensor terms.

The other components of the 2011 outlook (the ρ-tensor coupling in the deformed functional, and basis or coordinate-space methods reaching medium-mass and triaxial nuclei) are already established in the existing D-RHFB and exact-exchange RDFT literature; what remains open is (1).

A related open direction, raised in the Introduction of the source as a general challenge for energy density functionals and not among the items of the RHFBz outlook:

(2) Beyond-mean-field correlations with exchange terms (related direction): a treatment of collective correlations - restoration of rotational and particle-number symmetry and fluctuations in collective shape coordinates, as realized in RMF-based spectroscopy through angular-momentum projection and generator-coordinate-method collective Hamiltonians (a superposition of mean-field configurations at different deformations) - within a covariant energy density functional that retains the explicit Fock terms, including their rearrangement contributions (the additional mean-field terms generated by the density dependence of the meson-nucleon couplings), so that spectroscopic observables (excitation energies, electromagnetic transition rates) of deformed nuclei could be computed with the exchange terms present; currently every such beyond-mean-field covariant calculation drops the Fock terms.

Known solving difficulties (the first two concern the related direction (2), the remaining ones the tensor assessment (1)):

  • Deriving symmetry-restored or collective-Hamiltonian energy functionals when the functional is density-dependent and contains explicit Fock terms requires handling rearrangement contributions under projection and avoiding the double-counting and symmetry-pathology issues of projected energy density functionals - formalism that has no existing template in the covariant case.
  • Nonseparable exchange integrals make projected or GCM energy surfaces (many deformation points, plus projection over Euler angles) orders of magnitude more expensive than their RMF counterparts; the computational bottleneck that the source paper flagged persists in the beyond-mean-field context.
  • Separating tensor from non-tensor effects on shell evolution requires controlled switching of individual couplings (π-PV, ρ-T) within a validated functional and careful convergence control, since these couplings were partially compensated during parameter fitting.
  • Assessing the tensor role 'systematically' demands coverage across mass and isotope chains with pairing, continuum, and deformation treated consistently, and disentangling tensor effects from basis truncation and from correlations implicitly absorbed in the fitted couplings.
  • Validation is complicated by the scarcity of clean experimental discriminants that isolate tensor-force effects in deformed nuclei, so claims must be argued from consistency across observables rather than single benchmarks.

References

  1. J.-P. Ebran, E. Khan, D. Peña Arteaga, D. Vretenar, 'Relativistic Hartree-Fock-Bogoliubov model for deformed nuclei', Phys. Rev. C 83, 064323 (2011), DOI:10.1103/PhysRevC.83.064323, https://arxiv.org/abs/1010.4720
  2. W. H. Long, H. Sagawa, N. Van Giai, J. Meng, 'Shell Structure and ρ-Tensor Correlations in Density-Dependent Relativistic Hartree-Fock theory', Phys. Rev. C 76, 034314 (2007), DOI:10.1103/PhysRevC.76.034314, https://arxiv.org/abs/0706.3497
  3. J. Geng, J. Xiang, B. Y. Sun, W. H. Long, 'Relativistic Hartree-Fock model for axially deformed nuclei', Phys. Rev. C 101, 064302 (2020), DOI:10.1103/PhysRevC.101.064302, https://doi.org/10.1103/PhysRevC.101.064302
  4. J. Geng, W. H. Long, 'Relativistic Hartree-Fock-Bogoliubov model for axially deformed nuclei', Phys. Rev. C 105, 034329 (2022), DOI:10.1103/PhysRevC.105.034329, https://arxiv.org/abs/2202.00482
  5. Y. Peng, J. Geng, W. H. Long, 'Relativistic Hartree-Fock model for axial-symmetric nuclei with quadruple and octupole deformations', Chinese Physics C 49, 074107 (2025), DOI:10.1088/1674-1137/adc11e, https://arxiv.org/abs/2406.06846
  6. Y. Peng, J. Geng, W. H. Long, 'Extension of relativistic Hartree-Fock-Bogoliubov theory incorporating axially symmetric octupole deformation', Phys. Rev. C 114, 024318 (2026), DOI:10.1103/9vfy-1c91, https://doi.org/10.1103/9vfy-1c91
  7. Q. Zhao, Z. Ren, P. W. Zhao, K. Yoshida, 'Exact-exchange relativistic density functional theory in three-dimensional coordinate space', Phys. Lett. B 860, 139196 (2025), DOI:10.1016/j.physletb.2024.139196, https://arxiv.org/abs/2412.08132
  8. Z. Wang, Q. Zhao, H. Liang, W. H. Long, 'Quantitative analysis of tensor effects in the relativistic Hartree-Fock theory', Phys. Rev. C 98, 034313 (2018), DOI:10.1103/PhysRevC.98.034313, https://arxiv.org/abs/1806.11270
  9. Z. J. Chen, B. Y. Sun, 'Impact of pion tensor force on alpha clustering in 20Ne', Phys. Lett. B 869, 139830 (2025), DOI:10.1016/j.physletb.2025.139830, https://arxiv.org/abs/2506.16947
  10. J.-L. An, Q. Lu, W. H. Long, S.-S. Zhang, 'Exploration on 1n halo nucleus 19C from D-RHFB structure to reaction observables' (2024), arXiv:2406.09766, https://arxiv.org/abs/2406.09766
  11. J. Liu, Y. F. Niu, W. H. Long, 'Configuration-interaction relativistic Hartree-Fock model', Chinese Physics C 49, 064104 (2025), DOI:10.1088/1674-1137/adbdba, https://arxiv.org/abs/2411.05370
  12. W. H. Long, J. Geng, J. Liu, Z. H. Wang, 'Relativistic Hartree-Fock model and its recent progress on the description of nuclear structure', Communications in Theoretical Physics 74, 097301 (2022), DOI:10.1088/1572-9494/ac70ae, https://doi.org/10.1088/1572-9494/ac70ae