Quantitative impact of the instantaneous-exchange approximation, basis truncation, and pairing-parameter choices in density-dependent relativistic Hartree-Fock-Bogoliubov theory

The relativistic Hartree-Fock-Bogoliubov (RHFB) framework with density-dependent meson-nucleon couplings, as implemented by Long, Ring, Giai, and Meng [Phys. Rev. C 81, 024308 (2010)] on a Dirac-Woods-Saxon basis, rests on three controlled but unquantified choices: the instantaneous (zero-frequency) meson-exchange approximation that drops retardation effects from the single-particle Hamiltonian, the truncation of the basis in box radius and in the numbers of positive- and negative-energy states, and the residual arbitrariness of the pairing channel (quasiparticle-energy cutoff and pairing strength for zero-range pairing, and the transfer, without relativistic-specific recalibration, of nonrelativistic Gogny-type finite-range pairing into the relativistic framework). The objective is to quantify the impact of these three approximations and settings on the framework's physical predictions — Dirac self-energies, pairing gaps and pairing kernels, and separation-energy systematics, especially for weakly bound nuclei.

Background

Covariant density functional theory (CDFT) describes atomic nuclei as Dirac particles (nucleons) interacting through the exchange of mesons — the scalar-isoscalar $\sigma$ meson providing attraction, the vector-isoscalar $\omega$ meson providing repulsion, and the isovector $\rho$ meson distinguishing neutrons from protons — with coupling strengths that depend on the local baryon density ("density-dependent meson-nucleon couplings"). Most applications keep only the direct (Hartree) terms, but including the exchange (Fock) diagrams, in which the interaction is antisymmetrized between the nucleons, yields relativistic Hartree-Fock (RHF); the Fock terms carry, among others, the tensor-force and pion-exchange physics absent from the Hartree level. To describe open-shell and weakly bound nuclei, pairing correlations between time-reversed nucleon states are treated by the Bogoliubov transformation to quasiparticles, giving relativistic Hartree-Fock-Bogoliubov (RHFB) theory: a set of coupled integro-differential equations with nonlocal Fock kernels in the particle-hole channel and a pairing field in the particle-particle channel. The pairing field is generated either by a zero-range (delta) force, which requires an adjustable strength $V_0$ and a cutoff on the quasiparticle energy to avoid ultraviolet divergence, or by a finite-range Gogny force (a two-gaussian parametrization, D1S being the standard), whose range provides a natural cutoff and which is routinely taken over unchanged from nonrelativistic nuclear structure work.

Because the Fock and pairing kernels are nonlocal, Long, Ring, Giai, and Meng [Phys. Rev. C 81, 024308 (2010)] did not solve the RHFB equations directly in coordinate space; instead they expanded the Bogoliubov quasiparticle spinors in a Dirac-Woods-Saxon (DWS) basis — the bound and continuum eigenstates of a spherical Dirac equation with Woods-Saxon-shaped scalar and vector potentials, discretized in a spherical box of radius $R_{\max}$. The expansion must include both positive-energy states (the Fermi sea) and negative-energy states (the Dirac sea, whose contribution reflects the relativistic nature of the formulation), truncated at $N_F$ and $N_D$ states respectively; the paper recommended $R_{\max}=20\ \mathrm{fm}$, $N_F=28$, $N_D=12$ for general applications and $R_{\max}=24\ \mathrm{fm}$, $N_F=36$ for weakly bound systems, without a systematic convergence study. In the single-particle Hamiltonian, the exchanged mesons were taken to carry no energy (the time component of the four-momentum was set to zero), so the meson propagators reduce to static Yukawa forms; the neglected frequency dependence is called the retardation effect, an approximation the paper shares with essentially all mean-field implementations. With the Gogny D1S force in the pairing channel, the paper then obtained good systematics for Sn isotopes and $N=82$ isotones from stability to the neutron-rich side, with Fock terms (notably the tensor-force coupling of the $\rho$ meson) removing artificial shell closures.

Retardation was examined quantitatively only in the older, linear (constant-coupling) meson-exchange frameworks: Götz, Ramschütz, Weber, and Weigel [Phys. Lett. B 226, 213 (1989)] studied its influence in nuclear relativistic Hartree-Fock, and Zhang and Onley [Phys. Rev. C 44, 1915 (1991)] in relativistic mean field theory for finite nuclei. No study since then has reassessed retardation in the modern density-dependent framework, even as RHFB descendants proliferated: a deformed RHFB model in a harmonic-oscillator basis [Ebran et al., Phys. Rev. C 83, 064323 (2011)], finite-temperature RHFB in the DWS basis comparing finite-range and contact pairing [Li, Margueron, Long, Giai, Phys. Rev. C 92, 014302 (2015)], and an axially deformed RHFB model in the spherical DWS basis [Geng and Long, Phys. Rev. C 105, 034329 (2022)], all of which still neglect retardation and adopt Gogny D1S pairing unchanged. On the numerical side, a detailed convergence study of the Dirac-sea basis space and an optimized DWS basis were provided by Zhang, Pan, and S. Q. Zhang [Phys. Rev. C 106, 024302 (2022)], but within the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc), which omits the Fock terms; within RHFB proper, convergence has been checked only for representative light and mid-heavy nuclei, with heavy and superheavy systems explicitly deferred. The 2022 review by Long, Geng, Liu, and Wang [Commun. Theor. Phys. 74, 097301 (2022)] documents that a "complete RHF framework" now exists for ground states and excitations, without revisiting any of the three approximations above.

Problem Statement

Bring the three controlled approximations and settings of the density-dependent relativistic Hartree-Fock-Bogoliubov (RHFB) framework under quantitative error control, by determining — for the density-dependent relativistic Hartree-Fock Lagrangians PKA1 (which includes the $\rho$-meson tensor coupling) and PKO1, and for a nucleus set that includes at minimum the Sn isotopic chain and the $N=82$ isotonic chain studied in the source paper (the Sn isotopes with neutron numbers $N=50$–$87$ and the $N=82$ isotones with proton numbers $Z=47$–$71$, including their neutron-rich, weakly bound members), with any further Lagrangians or nuclei added at the solver's choice — the impact of each on the theory's physical predictions:

(i) Retardation: include the frequency dependence of the exchanged meson and photon propagators (the time components of the four-momentum) in the Fock contribution to the single-particle Hamiltonian, instead of the standard instantaneous zero-frequency Yukawa propagators, and quantify how the Dirac self-energies, the pairing field and pairing gaps obtained from the self-consistent solution, and the two-neutron/two-proton separation-energy systematics (including the Sn isotopic and $N=82$ isotonic chains studied in the source paper) change relative to the instantaneous approximation.

(ii) Basis truncation: provide a systematic convergence study of the Dirac-Woods-Saxon basis — box radius $R_{\max}$ and the numbers of positive- and negative-energy states ($N_F$, $N_D$, or equivalent energy cutoffs) — for the nuclei to which the framework is applied, including the weakly bound systems for which the source prescribes enlarged basis parameters, and in particular for Fock-term-carrying Lagrangians whose Dirac-sea demands are known to be stronger; where useful, provide an optimized basis that achieves converged observables at reduced cost, and state the residual numerical uncertainty on binding energies, radii, separation energies, and pairing observables.

(iii) Pairing channel: assess whether finite-range pairing interactions of the Gogny type, calibrated in nonrelativistic frameworks and used unchanged in RHFB, are adequate for the relativistic context, by quantifying the sensitivity of RHFB pairing observables to the pairing functional (range, strength, cutoff conventions) in combination with the Lagrangian choice.

The three components jointly serve one objective: establishing the systematic (non-statistical) uncertainty budget of RHFB predictions, so that the effects attributed to Fock terms and pairing in applications to exotic nuclei can be certified as physics rather than artifacts of the instantaneous-exchange approximation, the basis truncation, or the pairing-parameter choice.

References

1. W. H. Long, P. Ring, N. Van Giai, J. Meng, 'Relativistic Hartree-Fock-Bogoliubov theory with density dependent meson-nucleon couplings', Phys. Rev. C 81, 024308 (2010), DOI: 10.1103/PhysRevC.81.024308, https://doi.org/10.1103/PhysRevC.81.024308 (preprint arXiv:0812.1103, https://arxiv.org/abs/0812.1103)
2. J. Götz, J. Ramschütz, F. Weber, M. K. Weigel, 'The influence of retardation in the nuclear relativistic Hartree-Fock theory', Phys. Lett. B 226, 213 (1989), DOI: 10.1016/0370-2693(89)91182-9, https://doi.org/10.1016/0370-2693(89)91182-9
3. Jian-Kang Zhang, D. S. Onley, 'Retardation effect in finite nuclei in relativistic mean field theory', Phys. Rev. C 44, 1915 (1991), DOI: 10.1103/PhysRevC.44.1915, https://doi.org/10.1103/PhysRevC.44.1915
4. 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://doi.org/10.1103/PhysRevC.83.064323 (preprint arXiv:1010.4720, https://arxiv.org/abs/1010.4720)
5. J. J. Li, J. Margueron, W. H. Long, N. Van Giai, 'Pairing phase transition: A finite-temperature relativistic Hartree-Fock-Bogoliubov study', Phys. Rev. C 92, 014302 (2015), DOI: 10.1103/PhysRevC.92.014302, https://doi.org/10.1103/PhysRevC.92.014302 (preprint arXiv:1506.04507, https://arxiv.org/abs/1506.04507)
6. K. Y. Zhang, C. Pan, S. Q. Zhang, 'Optimized Dirac Woods-Saxon basis for covariant density functional theory', Phys. Rev. C 106, 024302 (2022), DOI: 10.1103/PhysRevC.106.024302, https://doi.org/10.1103/PhysRevC.106.024302 (preprint arXiv:2208.03636, https://arxiv.org/abs/2208.03636)
7. 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://doi.org/10.1103/PhysRevC.105.034329 (preprint arXiv:2202.00482, https://arxiv.org/abs/2202.00482)
8. W. H. Long, J. Geng, J. Liu, Z. H. Wang, 'Relativistic Hartree-Fock model and its recent progress on the description of nuclear structure', Commun. Theor. Phys. 74, 097301 (2022), DOI: 10.1088/1572-9494/ac70ae, https://doi.org/10.1088/1572-9494/ac70ae
9. J. Liu, Yi-Fei Niu, Wen-Hui Long, 'Configuration-interaction relativistic Hartree-Fock model', Chin. Phys. C 49, 064104 (2025), DOI: 10.1088/1674-1137/adbdba, https://doi.org/10.1088/1674-1137/adbdba (preprint arXiv:2411.05370, https://arxiv.org/abs/2411.05370)
10. N. Van Giai, H. Liang, H.-Q. Gu, W. H. Long, J. Meng, 'Treating Coulomb exchange contributions in relativistic mean field calculations: why and how', Phys. Scr. 89, 054008 (2014), DOI: 10.1088/0031-8949/89/5/054008, https://doi.org/10.1088/0031-8949/89/5/054008 (preprint arXiv:1405.3818, https://arxiv.org/abs/1405.3818)