Complex-scaled ab initio QED theory of autoionizing states beyond non-mixing two-electron resonances: level mixing and multi-electron ions

The 2023 complex-scaled ab initio QED method of Zaytsev, Malyshev, and Shabaev gave the first fully rigorous QED treatment of autoionizing states — doubly excited, Auger-decaying levels of highly charged ions — by combining complex rotation of the coordinates with bound-state QED perturbation theory complete through second order, demonstrated on non-mixing $LL$ resonances of heliumlike argon and uranium. The problem it leaves open remains open: the formalism has not been implemented for autoionizing states with strong level mixing or for ions with more than two electrons; higher-order QED contributions beyond the complete second-order many-electron set remain uncalculated and dominate the mid-$Z$ energy uncertainty, with even 2024–2025 calculations still estimating them by a model-QED operator; in incomplete finite bases the practical rule for choosing the complex-scaling angle is not accompanied by a quantified basis-incompleteness error; and the suggested supercritical-field application has received only a first one-loop treatment. The problem is to construct a general, quantitatively controlled complex-scaled ab initio QED theory of autoionizing states of highly charged few-electron ions.

Background

Autoionizing states are doubly excited states of an atom or ion that lie above the first ionization threshold and therefore decay by ejecting an electron (Auger decay) rather than only by photon emission. In highly charged ions (HCIs) — atoms stripped of most of their electrons — such states appear as resonances in electron-ion recombination and photoionization spectra, and their energies serve as diagnostics in hot plasmas and astrophysical x-ray sources. Because the resonance wave function is not square-integrable (it is mixed with the continuum of the decay channel), a naive finite-basis calculation of an autoionizing level converges poorly, irregularly, or to incorrect values.

The most accurate theory of electronic structure in HCIs is bound-state quantum electrodynamics (QED), formulated in the Furry picture: electrons move in the exact Dirac field of the nucleus, and their interactions are treated by perturbation theory in the fine-structure constant $\alpha$, while being nonperturbative in the coupling $\alpha Z$ ($Z$ is the nuclear charge number). A standard tool is the two-time Green function method, which builds an effective Hamiltonian for the levels of interest order by order: one-photon exchange; one-loop radiative corrections (the electron self-energy, the interaction of the bound electron with virtual electron–positron pairs, i.e. vacuum polarization); the complete set of two-photon exchange diagrams (ladder and crossed, including their genuinely QED part beyond the Breit instantaneous-interaction approximation); screening QED (radiative corrections modified by spectator electrons); two-loop one-electron terms; plus nuclear recoil, nuclear polarization (virtual nuclear excitation), and the finite nuclear-size effect. For ordinary bound levels this program has produced part-per-billion predictions for hydrogen- and lithiumlike ions, but it presupposes discrete, isolated levels.

Complex scaling (complex rotation) is the standard device for resonances: the radial coordinate is transformed as $r \to r e^{i\theta_{CS}}$, which rotates the continuum spectrum into the complex plane while leaving the discrete resonance eigenvalues unchanged, thereby rendering the resonance wave functions square-integrable. The angle $\theta_{CS}$ must exceed a critical angle $\theta_{cr}$ determined by the Auger-decay channel; for a complete (infinite) basis the results do not depend on $\theta_{CS}$ within the allowed range, but practical bases are incomplete, so the choice matters. Zaytsev, Malyshev, and Shabaev (2023) combined complex scaling with the rigorous QED machinery in a B-spline dual-kinetic-balance basis (a standard finite basis for Dirac problems that treats the large and small components of the Dirac spinor symmetrically and avoids spurious states) and computed the energies of the lowest non-mixing $LL$ resonances — states with both electrons in the $n=2$ shell, namely $(2s2p)^0$, $(2p_{1/2}2p_{3/2})^1$, and $(2s2p_{3/2})^2$ — of heliumlike argon ($Z=18$) and uranium ($Z=92$), including all many-electron QED contributions through second order as well as one- and two-loop, recoil, and nuclear-polarization terms. The results are orders of magnitude more precise than the previous complex-scaled configuration-interaction values (Zaytsev, Maltsev, Tupitsyn, and Shabaev, 2019).

Three limitations of the 2023 work frame the open problem. First, accuracy: for argon the dominant uncertainty comes from uncalculated higher-order QED contributions (third and higher orders), which were only estimated with a model-QED operator — an effective local potential fitted to reproduce known QED corrections (the QEDMOD-type approach); for uranium the largest uncertainties are the nuclear-size model and the two-loop one-electron QED term. Second, scope: 'non-mixing' means each computed state is the only level of its total angular momentum $J$ and parity in its energy neighborhood; generic autoionizing states are quasidegenerate — several levels of the same $J$ and parity are strongly mixed by the interelectronic interaction and require an effective-Hamiltonian (quasidegenerate) treatment — and the formalism has not been extended to such states, nor implemented for ions with more than two electrons. Third, numerics: in incomplete bases the result depends on $\theta_{CS}$; the working rule of choosing the angle that minimizes the result-variation rate is practical but is not a quantified error-control procedure. The authors also suggested combining complex scaling with rigorous QED for external and even supercritical electromagnetic fields — fields so strong that electron levels dive into the negative-energy (Dirac-sea) continuum, making the vacuum unstable against spontaneous electron–positron pair creation — a direction since begun at one-loop order by Zaytsev, Yerokhin, Keitel, and Oreshkina.

Problem Statement

Complete the rigorous QED description of autoionizing states of highly charged ions inaugurated by the complex-scaled ab initio formalism of Zaytsev, Malyshev, and Shabaev (2023), which to date has been implemented only for the non-mixing $LL$ resonances of heliumlike ions. The common objective is a general complex-scaled ab initio QED theory of autoionizing states with quantitatively controlled uncertainties. Concretely, the objective is to extend the formalism along the directions its authors identify — the paper notes that the developed formalism 'can be readily adapted for ions with more electrons as well as for the mixing states' — to autoionizing states subject to strong interelectronic level mixing (quasidegenerate resonances of equal total angular momentum and parity) and to ions with more than two electrons. Existing quasidegenerate two-time-Green-function calculations for Be-like excited states supply energies only, without complex scaling, and do not fulfill this objective.

The source paper further observes that at mid $Z$ (e.g., the $LL$ resonances of heliumlike argon) the dominant uncertainty arises from uncalculated higher-order QED contributions, currently estimated with a model-QED operator; that at high $Z$ (uranium) the largest uncertainties are the nuclear-size effect and the two-loop one-electron QED contribution, whose resolution depends on independent programs (improved nuclear charge-radius and deformation models and complete two-loop one-electron QED at $Z \approx 92$) outside the atomic-structure toolkit proper; and that in incomplete finite bases a residual dependence on $\theta_{CS}$ remains beyond the practical angle-selection rule, without a quantified basis-incompleteness error. These accuracy and numerical-control observations, and the quantitatively controlled uncertainties named in the general objective, are context for the extension rather than additional required deliverables.

The extension of complex scaling plus rigorous QED to external and supercritical electromagnetic fields, suggested in the same paper and begun at one-loop order by Zaytsev, Yerokhin, Keitel, and Oreshkina (2024), is an adjacent suggested application rather than a required component of this problem.

References

1. V. A. Zaytsev, A. V. Malyshev, and V. M. Shabaev, 'Complex-scaled ab initio QED approach to autoionizing states', Physical Review A 107, 032801 (2023), DOI: 10.1103/PhysRevA.107.032801, https://arxiv.org/abs/2210.00773
2. V. A. Zaytsev, I. A. Maltsev, I. I. Tupitsyn, and V. M. Shabaev, 'Complex-scaled relativistic configuration-interaction study of the LL resonances in heliumlike ions: From boron to argon', Physical Review A 100, 052504 (2019), DOI: 10.1103/PhysRevA.100.052504, https://arxiv.org/abs/1909.11487
3. V. A. Zaytsev, V. A. Yerokhin, C. H. Keitel, and N. S. Oreshkina, 'QED corrections in unstable vacuum', arXiv:2409.08121 (2024); published in Physical Review D (2026), DOI: 10.1103/895d-7cpx, https://arxiv.org/abs/2409.08121
4. A. V. Malyshev, Y. S. Kozhedub, V. M. Shabaev, and I. I. Tupitsyn, 'QED calculations of intra-L-shell doubly excited states in Be-like ions', Physical Review A 112, 062811 (2025), DOI: 10.1103/97gb-k9ts, https://arxiv.org/abs/2509.04121
5. A. V. Malyshev, Y. S. Kozhedub, V. M. Shabaev, and I. I. Tupitsyn, 'QED calculations of intra-L-shell singly excited states in Be-like ions', Physical Review A 110, 062824 (2024), DOI: 10.1103/PhysRevA.110.062824, https://arxiv.org/abs/2411.01241
6. A. Müller, P.-M. Hillenbrand, S.-X. Wang, S. Schippers, E. Lindroth, F. Trinter, J. Seltmann, S. Reinwardt, M. Martins, A. S. Kheifets, and I. Bray, 'Double-K-hole resonances in single photoionization of He-like B3+ ions', Physical Review A 110, 062802 (2024), DOI: 10.1103/PhysRevA.110.062802, https://arxiv.org/abs/2501.03043
7. J. Danisch, M. Botz, C. Shah, M. Togawa, J. Goes, D. Hache, F. Grilo, P. Amaro, V. A. Yerokhin, S. Kühn, A. Mohamed, R. Totani, M. de Simone, S. Orlando, T. Pfeifer, F. Nicastro, M. Coreno, and J. R. Crespo López-Urrutia, 'Parts-per-million-accurate determination of the K-alpha photoionization resonance of Be-like oxygen with resolution of its 16O-18O isotopic shift', Physical Review A 113, 022811 (2026), DOI: 10.1103/d4h7-ykwn, https://arxiv.org/abs/2607.00996