One-loop radiative corrections to atomic hyperfine structure consist of vacuum polarization (VP), which is local and can be included from the start in many-body atomic calculations, and the electron self-energy (SE), which generally dominates the correction but is non-local and energy-dependent. A 2024-2025 calculation of VP in the Uehling approximation for alkali-metal atoms and singly-ionized alkaline earths showed that many-body effects - core relaxation, core polarization, and valence-core correlations - change the relative VP correction by orders of magnitude and even its sign for non-ss states, but left the SE untreated with the same many-body coupling, omitted higher-order (Wichmann-Kroll) VP, used a uniform-magnetization model for the Bohr-Weisskopf effect, and included correlations only at second order. The open problem is to compute the SE correction to hyperfine constants in heavy many-electron atoms within the same consistent many-body framework, together with the remaining one-loop VP terms, so that complete one-loop QED-corrected hyperfine constants with quantified uncertainties become available.

Background

The hyperfine structure of an atom is the small splitting of its fine-structure levels caused by the interaction of the bound electrons with the magnetic dipole moment of the nucleus; for a state of total electron angular momentum JJ it is parameterized by the hyperfine constant AA. In heavy atoms, relativistic many-body calculations of AA have reached the precision at which quantum electrodynamics (QED) radiative corrections - the modifications generated by virtual processes in the quantized electromagnetic field - must be included. At one loop these consist of two parts. Vacuum polarization (VP) is the modification of the nuclear electric and magnetic fields by virtual electron-positron pairs; its dominant electric piece is the Uehling potential, a short-range local correction to the nuclear Coulomb potential; there is also a magnetic-loop VP correction to the hyperfine interaction operator itself, and the higher-order-in-ZαZ\alpha binding contribution (with ZZ the nuclear charge number and α\alpha the fine-structure constant) is called the Wichmann-Kroll term. The electron self-energy (SE) - the emission and reabsorption of a virtual photon by the bound electron, including its correction to the hyperfine vertex - generally gives the larger contribution but is a non-local, energy-dependent operator that is much harder to represent inside an atomic-structure calculation. A further nuclear-size effect, the Bohr-Weisskopf (BW) correction, accounts for the finite spatial distribution of the nuclear magnetization and reduces the hyperfine constant; it is usually modeled by a magnetization distribution F(r)F(r), the simplest choice being a uniformly magnetized sphere. For many-electron atoms there are two standard routes to these corrections. The first is rigorous bound-state QED in a frozen local screening potential (the 'extended Furry picture', in which the external field seen by the electron includes a local approximation to the field of the other electrons): Sapirstein and Cheng computed radiative corrections to hyperfine splittings for ss, p1/2p_{1/2}, and p3/2p_{3/2} states of the neutral alkali metals in this way, and Ginges, Volotka and Fritzsche evaluated the one-loop SE and VP corrections rigorously for the ground states of Rb, Cs, Fr, Ba+^+ and Ra+^+ in core-Hartree and Kohn-Sham potentials (two standard local approximations to the screening by the electron core). These calculations treat the QED part exactly for one electron in a fixed potential but do not couple the radiative corrections to the many-body response of the remaining electrons. The second route is to model the non-local SE by a local radiative potential (the radiative potential method of Flambaum and Ginges, building on the Ginges-Flambaum review of radiative and nuclear effects in heavy atoms) or by a model QED operator (Shabaev, Tupitsyn and Yerokhin), inserted into the many-body equations from the start; this allows full account of many-body effects but approximates the SE itself. The many-body effects in question are those of the relativistic Hartree-Fock (RHF) framework: core relaxation (the self-consistent change of all orbitals when a perturbing potential is added), core polarization or RPA (the random-phase-approximation response of the closed-shell core to the hyperfine field, often the dominant correction), and valence-core correlations, included through a non-local correlation potential Σ\Sigma taken in second order, Σ=Σ(2)\Sigma = \Sigma^{(2)}, which corrects the Hartree-Fock orbitals; the corrected orbitals are called Brueckner orbitals. The source paper of this problem (Hasted, Fairhall, Smits, Roberts and Ginges, arXiv:2409.17979) computed the VP correction to hyperfine constants for alkali-metal atoms and singly-ionized alkaline earths from Z=11Z = 11 to Z=88Z = 88 in the Uehling approximation, including core relaxation, RPA, and Σ(2)\Sigma^{(2)} correlations, and separating the electric-loop (perturbed-orbital) contribution - the change of the orbitals induced to first order by inserting the Uehling potential into the self-consistent field - from the magnetic-loop contribution. It found that for ss states the relative correction FVPF_{\mathrm{VP}} is close to the hydrogenlike-ion value (deviating by about 10% near Z=90Z = 90), while for non-ss states the many-body effects - above all the core polarization - are needed even to obtain the correct sign and magnitude. However, the SE, the larger of the two one-loop contributions, was not treated with this many-body coupling, the Wichmann-Kroll term was not included, the BW effect was modeled by uniform magnetization, and correlations were kept only at second order. The conclusion of that paper explicitly frames it as 'an important step towards considering the influence of many-body effects on the full (vacuum polarization and self-energy) QED contributions to the hyperfine structure in many-electron atoms'.

Problem Statement

Obtain complete one-loop QED corrections to the hyperfine structure constants of heavy many-electron atoms within a single many-body framework that treats the radiative corrections and the atomic response consistently. Concretely: (i) compute the self-energy correction to hyperfine constants for the alkali-metal atoms and singly-ionized alkaline earths (the systems treated for VP by the source work, Z=11Z = 11 to Z=88Z = 88, from Na to Ra), in which the non-local, energy-dependent SE operator is incorporated with the same many-body effects - core relaxation, RPA core polarization, and valence-core correlations at least at the Σ(2)\Sigma^{(2)} level - including its perturbed-orbital and vertex parts, and expressed as relative correction factors FSEF_{\mathrm{SE}} per state and atom in the format of the source's FVPF_{\mathrm{VP}} tables; (ii) add the missing higher-order vacuum-polarization contribution, the Wichmann-Kroll term, treated with the same many-body coupling. The common objective of these deliverables is the complete one-loop QED-corrected hyperfine constants for heavy many-electron atoms with quantified theoretical uncertainties, benchmarked against rigorous frozen-potential SE calculations and hydrogenlike-ion limits; the deliverables are scientifically linked as the successive missing pieces of that single objective, as articulated in the concluding section of the source work.

References

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