Consistent pairing channel and retardation effects in relativistic Hartree-Fock-Bogoliubov theory with density-dependent meson-nucleon couplings

The relativistic Hartree-Fock-Bogoliubov (RHFB) framework of Long, Ring, Van Giai and Meng [Phys. Rev. C 81, 024308 (2010)] couples a fully relativistic particle-hole channel — density-dependent meson-nucleon couplings with Fock (exchange) terms from pion and rho-meson exchange — to a phenomenological, non-relativistic pairing channel (the finite-range Gogny force D1S, or a zero-range delta force with adjustable quasiparticle-energy cutoff and strength), evaluated with instantaneous meson propagators on a discretized Dirac Woods-Saxon basis. The open problem is the internal consistency of the density-dependent RHFB framework: (i) quantifying retardation — the energy-transfer dependence of meson propagators — in the Fock self-energies and, for the first time, in the pairing kernel of modern density-dependent RHFB; and (ii) a systematic assessment and possible reparameterization of finite-range pairing interactions for the fully relativistic RHFB context, with the impact measured on pairing gaps and separation-energy systematics from stability to the neutron drip line.

Background

Covariant density functional theory (CDFT, also called relativistic mean-field theory) describes an atomic nucleus as Dirac particles — the nucleons — moving in classical meson fields ($\sigma$, $\omega$, $\rho$, and the pion $\pi$) generated by an effective Lagrangian; the meson-nucleon coupling constants are made density-dependent to imitate in-medium many-body effects. In the relativistic Hartree-Fock (RHF) approach one keeps, in addition to the direct (Hartree) fields, the exchange (Fock) diagrams; the pion and $\rho$-tensor exchange terms contained in them generate the tensor force and spatial non-locality that are absent at the Hartree level. RHFB adds the Bogoliubov transformation — a mixing of particle and hole states that describes superfluid pairing correlations in open-shell nuclei — so that quasiparticles, rather than particles, are the elementary degrees of freedom. The formalism then has two channels: the particle-hole (ph) mean field and the particle-particle (pp) pairing field. Long, Ring, Van Giai and Meng (2010) formulated RHFB with density-dependent meson-nucleon couplings and solved the resulting integro-differential equations by expanding the quasiparticle spinors on a Dirac Woods-Saxon (DWS) basis — the eigenstates of a Dirac equation with Woods-Saxon-shaped potentials in a spherical box of radius $R_{\max}$, truncated to $N_F$ positive- and $N_D$ negative-energy states — with applications to Sn isotopes and $N=82$ isotones. Two phenomenological inputs sit in the pp channel: either a zero-range $\delta$ force, which requires an adjustable pairing strength $V_0$ and a cutoff on quasiparticle energies, or the finite-range Gogny force D1S, whose range provides a natural cutoff and which was taken over unchanged from non-relativistic Hartree-Fock-Bogoliubov practice. In deriving the Fock terms, the meson propagators are taken instantaneous: the energy transferred between the two nucleons at a meson-exchange vertex is set to zero (the retardation approximation), which keeps the self-energies energy-independent. The source paper acknowledges these three controlled settings — neglected retardation, the basis-discretization parameters ($R_{\max}$, $N_F$, $N_D$), and the phenomenological pairing input — without quantifying their impact. The main consumer of the framework is the physics of weakly bound nuclei near the neutron drip line, where the last neutrons are barely bound, the one- and two-neutron separation energies $S_n$ and $S_{2n}$ are small, and continuum coupling, pairing, and the tensor-rich Fock field all act simultaneously; there, unquantified methodological settings can be as consequential as the effective interaction itself.

Problem Statement

The problem is the internal-consistency assessment of RHFB theory with density-dependent meson-nucleon couplings, focused on the particle-particle channel and on retardation (the basis-convergence item, addressed by references 7, 8 and 11, is outside the scope of this task). Concretely: (1) Retardation — go beyond the instantaneous approximation by including the energy-transfer dependence of the meson propagators in the Fock self-energies, and assess retardation effects also in the pairing channel, whose kernel is presently static and phenomenological; quantify the changes with respect to standard instantaneous RHFB for single-particle spectra, pairing gaps, and one- and two-neutron separation-energy systematics along benchmark chains from stability to the neutron drip line. The only existing quantification (Götz et al., 1989) was obtained in the linear Hartree-Fock model without density-dependent couplings, without pion/rho-tensor exchange, and without pairing; it must be re-established or revised for the modern functional. (2) Pairing-channel consistency — determine systematically whether finite-range pairing interactions parameterized in the non-relativistic context (Gogny D1S and strength-adjusted variants) remain suitable inside the fully relativistic RHFB functional, reparameterizing them or replacing them by a pairing interaction better matched to the relativistic framework if not, without spoiling the balance between the ph and pp channels. Both items serve one objective: an internally consistent, approximation-controlled RHFB whose residual uncertainty for weakly bound nuclei is attributable to the effective interaction rather than to unquantified methodological settings.

References

1. W. H. Long, P. Ring, N. Van Giai, J. Meng, 'Relativistic Hartree-Fock-Bogoliubov theory with density dependent meson-nucleon couplings', Physical Review 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', Physics Letters B 226, 213-215 (1989). DOI: 10.1016/0370-2693(89)91182-9, https://doi.org/10.1016/0370-2693(89)91182-9
3. J.-P. Ebran, E. Khan, D. Peña Arteaga, D. Vretenar, 'Relativistic Hartree-Fock-Bogoliubov model for deformed nuclei', Physical Review 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
4. X. L. Lu, B. Y. Sun, W. H. Long, 'Description of carbon isotopes within relativistic Hartree-Fock-Bogoliubov theory', Physical Review C 87, 034311 (2013). DOI: 10.1103/PhysRevC.87.034311, https://doi.org/10.1103/PhysRevC.87.034311; preprint arXiv:1210.5388, https://arxiv.org/abs/1210.5388
5. J. J. Li, J. Margueron, W. H. Long, N. Van Giai, 'Pairing phase transition: A finite-temperature relativistic Hartree-Fock-Bogoliubov study', Physical Review 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. S. Shen, H. Z. Liang, W. H. Long, J. Meng, P. Ring, 'Towards an ab initio covariant density functional theory for nuclear structure', Progress in Particle and Nuclear Physics 109, 103713 (2019). DOI: 10.1016/j.ppnp.2019.103713, https://doi.org/10.1016/j.ppnp.2019.103713; preprint arXiv:1904.04977, https://arxiv.org/abs/1904.04977
7. J. Geng, W. H. Long, 'Relativistic Hartree-Fock-Bogoliubov model for axially deformed nuclei', Physical Review 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. K. Y. Zhang, C. Pan, S. Q. Zhang, 'Optimized Dirac Woods-Saxon basis for covariant density functional theory', Physical Review 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
9. W. H. Long, J. Meng, N. Van Giai, 'Relativistic Hartree-Fock-Bogoliubov theory: ground states and excitations', in Relativistic Density Functional for Nuclear Structure, International Review of Nuclear Physics Vol. 10, pp. 143-169 (2016). DOI: 10.1142/9789814733267_0004, https://doi.org/10.1142/9789814733267_0004
10. Y. Peng, J. Geng, W. H. Long, 'Extension of relativistic Hartree-Fock-Bogoliubov theory incorporating axially symmetric octupole deformation', Physical Review C 114, 024318 (2026). DOI: 10.1103/9vfy-1c91, https://doi.org/10.1103/9vfy-1c91
11. W. Gao, T. T. Sun, W. H. Long, 'Single-Particle Resonant States in Relativistic Hartree-Fock Theory: A Green's Function Approach', Chinese Physics C 50, 084109 (2026). DOI: 10.1088/1674-1137/ae6a85, https://doi.org/10.1088/1674-1137/ae6a85; preprint arXiv:2603.04769, https://arxiv.org/abs/2603.04769
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