Self-consistent tensor-force constraints in the relativistic Hartree-Fock theory: RHF + particle-vibration coupling and the global strength of the pion-exchange tensor

In the relativistic Hartree-Fock (RHF) theory, the tensor force carried by the exchange (Fock) terms — dominated by the pion pseudovector coupling — was shown to drive the evolution of the neutron 1i13/2 - 1h9/2 and proton 1h11/2 - 1g7/2 energy differences along the N = 82 isotones and Z = 50 isotopes, and moderately strengthening that coupling, preferably by weakening its density dependence, markedly improves agreement with experiment. That conclusion, however, rests on pseudodata from which particle-vibration-coupling (PVC) shifts estimated in a relativistic mean-field (RMF) framework were subtracted, on illustrative rather than globally refitted strength parameters, on an unquantified interplay between the pion tensor and the rho-meson tensor and vector-tensor couplings, and on a merely assumed connection to the renormalization persistency of the tensor force. The open problem is to put this picture on a firm footing: construct a fully self-consistent RHF + PVC framework, use it to quantify how dynamical correlations modify mean-field single-particle energies, carry out the systematic overall refit of the pion-coupling strength parameters called for by the source study and its follow-up, test the robustness of the pion-dominated tensor picture against cancellations with the rho-meson tensor and vector-tensor couplings, and anchor the preferred weak density dependence of the pion coupling to microscopic (relativistic Brueckner-Hartree-Fock and ab initio) calculations.

Background

Covariant density functional theory (CDFT) describes atomic nuclei as Dirac nucleons interacting through the exchange of mesons (sigma, omega, rho, pion) with density-dependent coupling strengths. In the relativistic mean-field (RMF) approximation only the direct (Hartree) terms of these interactions are kept. The relativistic Hartree-Fock (RHF) theory additionally keeps the exchange (Fock) terms, through which the pion pseudovector coupling (pi-PV) and the rho-meson tensor (rho-T) and vector-tensor (rho-VT) couplings enter; the pion couples to the nucleon spin and isospin, and its exchange generates a tensor force — the part of the two-nucleon interaction, familiar from the one-pion-exchange Rarita-Schwinger term, that shifts the two spin-orbit partners of a given orbital angular momentum $l$ (the states with $j = l - 1/2$ and $j = l + 1/2$) in opposite directions, so that filling one partner with nucleons of one species systematically shifts the energies of the partners of the other species. Within the monopole-based (i.e., orbit-averaged two-body) analysis of Otsuka and collaborators, this tensor-driven shift of spin-orbit partners as protons or neutrons fill orbitals is a key mechanism of shell evolution — the change of shell structure along isotopic and isotonic chains — and its empirical signature is the systematic change of spin-orbit splittings with neutron or proton excess measured along chains such as the N = 82 isotones (Schiffer et al.). In the density-dependent RHF functionals PKO1-PKO3 and PKA1, the pion coupling strength is parametrized as $f_\pi(\rho_b) = f_\pi(0)\, e^{-a_\pi \xi}$, where $\xi = \rho_b/\rho_{\mathrm{sat}}$ is the baryon density in units of the nuclear-matter saturation density; it can be strengthened either by rescaling it as a whole, $f_\pi \to \lambda f_\pi$ with $\lambda > 1$, or by weakening its density dependence, $a_\pi \to \eta a_\pi$ with $\eta < 1$. The RHF studies of Wang, Naito, and Liang found that the second route is more effective, and interpreted the preference for weak density dependence as support for the renormalization persistency of the tensor force — the observation of Otsuka, Tsunoda, and collaborators that the in-medium effective tensor force resembles the bare one rather than being strongly renormalized. The remaining obstacle is that observed single-particle energies are not mean-field quantities: a nucleon in a nucleus can virtually emit and reabsorb a collective vibration (a phonon), and this particle-vibration coupling (PVC) shifts and fragments single-particle levels. Self-consistent PVC frameworks exist for the non-relativistic energy-density functionals of Skyrme and Gogny type and for covariant functionals without Fock terms (RMF+PVC), but no RHF-based PVC framework exists, so the comparison of RHF spectra with experiment had to rely on pseudodata obtained by subtracting RMF-based quasiparticle-vibration-coupling (QPVC) shifts from measured energies — with no guarantee that PVC effects are transferable between RMF and RHF. Related open threads are the partial cancellation between the pi-PV tensor and the rho-T/rho-VT tensors (large, for instance, in the PKA1 functional), the need for a systematic overall refit of the pion parameters instead of values fitted to nuclear matter and a few nuclei, and the lack of a microscopic derivation of the fitted weak density dependence — the relativistic Brueckner-Hartree-Fock (RBHF) theory, in which a realistic two-nucleon interaction such as a meson-exchange potential is resummed in the nuclear medium into an effective G-matrix, and ab initio calculations of neutron drops (neutron matter in a harmonic-oscillator trap, where the neutron-neutron pion tensor is isolated) provide the natural benchmarks.

Problem Statement

The open problem is to put the effective tensor force of the relativistic Hartree-Fock (RHF) energy density functional on a correlation-consistent and quantitatively constrained footing, along the perspectives named by the source study and its follow-up. Its parts are: (i) construct a fully self-consistent RHF + particle-vibration-coupling (PVC) framework - the same density-dependent functional generating the Hartree-Fock ground state, the random-phase-approximation phonons, and the particle-vibration self-energy, with the Fock terms (pi-PV, rho-T, rho-VT) kept throughout - and use it to quantify how dynamical correlations modify the mean-field single-neutron energies 1i13/2 and 1h9/2 along the N = 82 isotones and single-proton energies 1h11/2 and 1g7/2 along the Z = 50 isotopes, replacing the current RMF-based pseudodata whose transferability to RHF is unverified; (ii) perform the systematic overall refit of the pion-pseudovector coupling parameters - the zero-density strength $f_\pi(0)$ and density-dependence coefficient $a_\pi$, equivalently the multipliers $\lambda$ ($f_\pi \to \lambda f_\pi$) and $\eta$ ($a_\pi \to \eta a_\pi$) - called for by the source study and its follow-up, replacing the illustrative values fitted only to nuclear matter and a few selected nuclei; (iii) test whether the pion-dominated tensor picture is robust by quantifying the cancellations and interplay between the pi-PV tensor and the rho-T and rho-VT tensor couplings (which, for instance, largely cancel each other in the PKA1 functional) for the shell-structure observables at stake; and (iv) anchor the fitted weak density dependence of $f_\pi$ microscopically - via relativistic Brueckner-Hartree-Fock calculations (for example of neutron drops, extending the existing semiquantitative match at $\lambda \approx 1.4$ or $\eta \approx 0.3$) and other ab initio many-body approaches - so that tensor renormalization persistency is tested within one consistent framework rather than assumed. The common objective unifying these parts is a tensor-constrained, correlation-consistent covariant energy density functional, together with a defensible answer to how much of the apparent tensor-force physics in shell-structure evolution survives once exchange terms and dynamical correlations are treated consistently. The four parts are jointly required: a resolution must address all of them, since each removes a distinct source of the uncertainty named above, and no proper subset of the parts, taken alone, resolves the open problem as stated.

References

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