The model Lamb-shift operator of Shabaev, Tupitsyn, and Yerokhin, distributed as the QEDMOD program, lets radiative corrections (self-energy and vacuum polarization) be included in any many-electron calculation based on the Dirac-Coulomb-Breit Hamiltonian, but only for energy shifts, with a few-percent typical accuracy, an uncharacterized finite-rank truncation error, and approximate Wichmann-Kroll/nuclear-size treatments. The open problem is to extend this model-operator framework — whose verified scope is limited to the self-energy and vacuum-polarization (Lamb-shift) energy corrections — to vertex-type one-loop QED contributions, enabling bound-electron g factors and hyperfine structure in arbitrary many-electron systems, while systematically characterizing and reducing the model error relative to the source-stated typical few-percent accuracy; as recently as 2022 the same group described a model operator suitable for bound-electron g factors as a not-yet-constructed next step. Later work has extended the operator to superheavy nuclei, to the recoil effect, and validated it case by case for sodium-like ions, but no transferable model operator incorporating vertex corrections exists.

Background

In relativistic atomic structure, the starting point is the Dirac-Coulomb-Breit (DCB) Hamiltonian: electrons obey the Dirac equation in the nuclear Coulomb field, interact through the Coulomb potential and the Breit interaction (the leading magnetic and retardation correction to the electron-electron interaction). Radiative quantum-electrodynamics (QED) effects — collectively the Lamb shift, i.e. the shift of bound-state energies beyond the Dirac-Coulomb-Breit picture — come mainly from the electron self-energy (the interaction of the bound electron with its own virtual-photon field) and vacuum polarization (the distortion of the Coulomb field by virtual electron-positron pairs). Vacuum polarization is dominated by the Uehling potential, with the smaller Wichmann-Kroll (WK) term as the next contribution. In rigorous bound-state QED one computes these corrections order by order in perturbation theory in the Furry picture (the exact Dirac-Coulomb field is kept in the zeroth-order Hamiltonian), which is feasible for one or few electrons but prohibitively expensive for general many-electron atoms, ions, and molecules. The model-operator approach of Shabaev, Tupitsyn, and Yerokhin closes this gap: the one-loop self-energy is represented by an effective operator — a local potential plus a finite-rank nonlocal term ∑nn′Fnn′(Zα) ∣n⟩⟨n′∣\sum_{nn'} F_{nn'}(Z\alpha)\,|n\rangle\langle n'| built from low-lying hydrogenic states ∣n⟩|n\rangle, with the coefficients Fnn′F_{nn'} fitted to ab initio matrix elements computed for hydrogenlike ions — and is added to the DCB Hamiltonian, so that configuration-interaction, Dirac-Fock, or coupled-cluster codes acquire QED corrections essentially for free. The QEDMOD Fortran program implements this operator. The operator's construction carries intrinsic limitations: the finite-rank representation is exact only in the limit of a complete basis, and the residual truncation error is not systematically characterized; the WK vacuum-polarization potential uses approximate point-nucleus formulas and the package supports a single extended-nucleus model (a two-parameter Fermi charge distribution); vertex-type one-loop contributions — radiative corrections to the electron-photon interaction vertex, whose low-energy limit is the anomalous magnetic moment of the electron — are omitted, so the operator cannot as-is treat magnetic observables such as the bound-electron gg factor (the dimensionless ratio of magnetic moment to angular momentum, measured in Penning traps at parts-per-billion precision) or hyperfine structure (the splitting from the interaction of the electron with the nuclear magnetic moment); and the typical accuracy of a few percent of the QED contribution, stated by the authors for the Lamb shift, comes with no systematic strategy for reduction. For these magnetic observables, present-day theory therefore falls back on ion-by-ion ab initio QED calculations (e.g. one-loop vertex corrections evaluated to all orders in αZ\alpha Z for lithium-like and boron-like ions), which are accurate but not transferable to general many-electron systems.

Problem Statement

Extend the model QED-operator approach of Shabaev, Tupitsyn, and Yerokhin beyond its current energy-only, few-percent form, as a single coherent objective with two linked facets. (1) Vertex extension: construct an effective operator that represents the vertex-type one-loop QED contributions (radiative corrections to the electron-photon vertex, whose low-energy limit is the anomalous magnetic moment) in the same spirit as the existing model Lamb-shift operator — i.e. as a finite-rank nonlocal (plus, where appropriate, local) operator whose parameters are fixed to ab initio matrix elements computed for hydrogenlike ions — so that radiative corrections to magnetic observables, in particular the bound-electron gg factor and hyperfine-structure intervals, can be computed within any many-electron method based on the Dirac-Coulomb-Breit Hamiltonian, with the construction handling renormalization and gauge consistency and avoiding double counting with the self-energy part. (2) Accuracy control: reduce and systematically characterize the model error of the resulting operator — the finite-rank truncation in the hydrogenic model basis, the dependence on the choice of model basis and mean field, and the vacuum-polarization/nuclear-size approximations (including improved Wichmann-Kroll treatment for extended nuclei) — relative to the source-stated typical accuracy of a few percent of the QED contribution for the existing Lamb-shift operator. In this framework, accuracy is established by comparison with ab initio one-loop QED calculations, as was done for the self-energy and vacuum-polarization parts of the existing operator. Both facets are mandatory for resolution: the deliverable is the constructed vertex-extended operator together with a systematic characterization and a demonstrated reduction of the model error named in facet (2) — the finite-rank truncation, the model-basis and mean-field dependence, and the vacuum-polarization/nuclear-size approximations — each controlled against the few-percent baseline stated by the authors for the existing Lamb-shift operator; a characterization that leaves the model error unreduced does not by itself resolve facet (2).

References

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