The ground-state (2p1/2p_{1/2}) bound-electron gg factor of middle-ZZ boronlike ions is now measured in Penning traps to 0.5–1.4 parts per billion, but the best theoretical values carry uncertainties of order 10−610^{-6}, dominated by interelectronic-interaction and radiative terms that are treated only approximately (Breit approximation and effective screening potentials). The rigorous, all-orders-in-ZαZ\alpha bound-state QED evaluation of the two-photon-exchange (second order in 1/Z1/Z) and two-electron (screened) self-energy and vacuum-polarization diagrams — completed years ago for lithiumlike ions — has never been carried out for the open 2p2p shell of boronlike ions, and its evaluation in the pure Coulomb potential is known to face numerical problems. This problem asks for that rigorous evaluation across the middle-ZZ range (Z=10Z=10–2020), with controlled numerical uncertainties, together with the diagnosis of the disagreement between earlier multiconfiguration Dirac–Fock predictions, so that boronlike-ion gg-factor theory approaches experimental precision and enables many-electron QED tests and an independent determination of the fine-structure constant.

Background

The bound-electron gg factor determines the energy splitting of a level in an external magnetic field (ΔE=g μB B m\Delta E = g\,\mu_B\,B\,m). For an ion stored in a Penning trap (a combination of static electric and magnetic fields), the ratio of the spin-precession (Larmor) frequency to the ion's cyclotron frequency gives the gg factor directly in a way that cancels the magnetic-field strength, and modern measurements on highly charged ions reach sub-ppb precision. Boronlike ions have five electrons, with a single electron in the 2p1/22p_{1/2} shell outside a closed 1s2 2s21s^2\,2s^2 core; their ground-state gg factor is the simplest open-pp-shell case and a sensitive probe of many-electron effects in strong fields.

The theory is organized as bound-state QED in the Furry picture (QED of electrons in the external Coulomb field of the nucleus, with ZαZ\alpha the coupling to that field, ZZ the nuclear charge number and α\alpha the fine-structure constant), with the interelectronic interaction expanded in 1/Z1/Z. Beyond the Dirac (one-electron) value, the contributions are: one-photon exchange (the leading interelectronic term), two-photon exchange (second order in 1/Z1/Z), the one-electron self-energy and vacuum polarization (radiative corrections of order α\alpha), their screened counterparts where one photon is replaced by the interelectronic interaction, and nuclear-size, nuclear-magnetization and nuclear-recoil corrections. A 'rigorous' evaluation means computing the corresponding Feynman diagrams exactly in ZαZ\alpha rather than within the Breit approximation (the instantaneous-photon, leading-order-in-ZαZ\alpha exchange) or with effective screening potentials that mimic the other electrons.

For lithiumlike ions this program is complete: Volotka et al. (2014) evaluated the many-electron QED corrections rigorously, and Yerokhin, Keitel and Harman (2021) computed the two-photon-exchange correction to the gg factor independently. This underpins the most precise determination of the electron atomic mass and enables α\alpha determinations from specific differences of hydrogenlike and lithiumlike gg factors. For boronlike ions the state of the art (Agababaev et al., 2018) reaches a total uncertainty of order 10−610^{-6}, with the two-photon-exchange term evaluated only at the Breit level and its genuine QED part estimated — not calculated — to be at most 0.2×10−60.2\times10^{-6}, and with screened self-energy treated through model screening potentials. Separately, large-scale multiconfiguration Dirac–Fock calculations — MCDHF (multiconfiguration Dirac–Hartree–Fock, a variational configuration-interaction method) by Verdebout et al. (2014) and MCDF by Marques et al. (2016) — disagree with the QED-based predictions by 4.5×10−54.5\times10^{-5} to 2.8×10−42.8\times10^{-4} for Z=10Z=10–2020, a gap suspected to stem from incomplete treatment of negative-energy (Dirac-sea) intermediate states but never traced to its origin. Experiments have since decided the physical values: boronlike argon (Arapoglou et al., 2019) and boronlike tin (Morgner et al., 2025, g=0.6447038265(4)g=0.6447038265(4) at 0.5 ppb, versus theory 0.6447029(8)0.6447029(8)) both confirm the QED-based predictions, so experiment now outstrips theory by roughly three orders of magnitude and the missing rigorous two-electron QED terms are the dominant residual theory uncertainty.

Problem Statement

Compute, rigorously within bound-state QED (exactly in ZαZ\alpha, in the Furry picture), the higher-order interelectronic-interaction and two-electron radiative contributions to the ground-state (2p1/22p_{1/2}) bound-electron gg factor of middle-ZZ boronlike ions — the range Z=10Z=10–2020 considered in the source paper — specifically the two-photon-exchange (second-order in 1/Z1/Z) contribution including its genuine QED part beyond the Breit approximation, and the screened self-energy and screened vacuum-polarization contributions beyond effective screening potentials. The source paper reports that the evaluation of the two-photon-exchange term in the pure Coulomb nuclear potential is related to numerical problems in the boronlike case, which the rigorous evaluation must address; numerical values should be delivered with controlled uncertainty budgets for each contribution, and comparison with the established lithiumlike benchmarks (where the same diagrams are already computed) offers a natural reference point. As a linked diagnostic component, identify by controlled calculation — separating positive- and negative-energy intermediate-state contributions — whether the 4.5×10−54.5\times10^{-5}–2.8×10−42.8\times10^{-4} disagreement between the earlier MCDHF (Verdebout et al.) and MCDF (Marques et al.) predictions and the QED-based values originates in incomplete negative-energy-state treatment. All four components named here — the two-photon-exchange contribution, the screened self-energy, the screened vacuum-polarization, and the negative-energy-state diagnostic — are required, and no subset of them alone resolves the task. Together, these components aim at a boronlike-ion gg-factor theory whose higher-order interelectronic-interaction and two-electron radiative contributions are calculated with controlled uncertainties, enabling precision many-electron QED tests in an open shell and an independent determination of the fine-structure constant from specific differences of boronlike and hydrogenlike or lithiumlike gg factors.

References

  1. V. A. Agababaev, D. A. Glazov, A. V. Volotka, D. V. Zinenko, V. M. Shabaev, and G. Plunien, "Ground-state g factor of middle-Z boronlike ions", J. Phys.: Conf. Ser. 1138, 012003 (2018). DOI: 10.1088/1742-6596/1138/1/012003; arXiv:1812.06429, https://doi.org/10.1088/1742-6596/1138/1/012003
  2. H. Cakir, V. A. Yerokhin, N. S. Oreshkina, B. Sikora, I. I. Tupitsyn, C. H. Keitel, and Z. Harman, "QED corrections to the g factor of Li- and B-like ions", Phys. Rev. A 101, 062513 (2020). DOI: 10.1103/PhysRevA.101.062513, https://doi.org/10.1103/PhysRevA.101.062513
  3. D. A. Glazov, D. V. Zinenko, V. A. Agababaev, A. D. Moshkin, E. V. Tryapitsyna, A. M. Volchkova, and A. V. Volotka, "g Factor of Few-Electron Highly Charged Ions", Atoms 11, 119 (2023). DOI: 10.3390/atoms11090119, https://doi.org/10.3390/atoms11090119
  4. J. Morgner, B. Tu, M. Moretti, C. M. König, F. Heiße, T. Sailer, V. A. Yerokhin, B. Sikora, N. S. Oreshkina, Z. Harman, C. H. Keitel, S. Sturm, and K. Blaum, "g Factor of Boronlike Tin", Phys. Rev. Lett. 134, 123201 (2025). DOI: 10.1103/PhysRevLett.134.123201, https://doi.org/10.1103/PhysRevLett.134.123201
  5. A. V. Volotka, D. A. Glazov, V. M. Shabaev, I. I. Tupitsyn, and G. Plunien, "Many-Electron QED Corrections to the g Factor of Lithiumlike Ions", Phys. Rev. Lett. 112, 253004 (2014). DOI: 10.1103/PhysRevLett.112.253004, https://doi.org/10.1103/PhysRevLett.112.253004
  6. V. A. Yerokhin, C. H. Keitel, and Z. Harman, "Two-photon-exchange corrections to the g factor of Li-like ions", Phys. Rev. A 104, 022814 (2021). DOI: 10.1103/PhysRevA.104.022814, https://doi.org/10.1103/PhysRevA.104.022814
  7. I. A. Aleksandrov, D. A. Glazov, A. V. Malyshev, V. M. Shabaev, and I. I. Tupitsyn, "Relativistic nuclear-recoil effect on the g factor of highly charged boronlike ions", Phys. Rev. A 98, 062521 (2018). DOI: 10.1103/PhysRevA.98.062521, https://doi.org/10.1103/PhysRevA.98.062521
  8. D. A. Glazov, A. V. Malyshev, V. M. Shabaev, and I. I. Tupitsyn, "Interelectronic-interaction contribution to the nuclear recoil effect on the g factor of boronlike ions", Phys. Rev. A 101, 012515 (2020). DOI: 10.1103/PhysRevA.101.012515, https://doi.org/10.1103/PhysRevA.101.012515
  9. I. Arapoglou, A. Egl, M. Höcker, T. Sailer, B. Tu, A. Weigel, R. Wolf, H. Cakir, V. A. Yerokhin, N. S. Oreshkina, V. A. Agababaev, A. V. Volotka, D. V. Zinenko, D. A. Glazov, Z. Harman, C. H. Keitel, S. Sturm, and K. Blaum, "g Factor of Boronlike Argon 40Ar13+", Phys. Rev. Lett. 122, 253001 (2019). DOI: 10.1103/PhysRevLett.122.253001, https://doi.org/10.1103/PhysRevLett.122.253001
  10. J. P. Marques, P. Indelicato, F. Parente, J. M. Sampaio, and J. P. Santos, "Ground-state Landé g factors for selected ions along the boron isoelectronic sequence", Phys. Rev. A 94, 042504 (2016). DOI: 10.1103/PhysRevA.94.042504, https://doi.org/10.1103/PhysRevA.94.042504
  11. S. Verdebout, C. Nazé, P. Jönsson, P. Rynkun, M. Godefroid, and G. Gaigalas, "Hyperfine structures and Landé gJg_J-factors for n=2n=2 states in beryllium-, boron-, carbon-, and nitrogen-like ions from relativistic configuration interaction calculations", At. Data Nucl. Data Tables 100, 1111–1155 (2014). DOI: 10.1016/j.adt.2014.05.001, https://doi.org/10.1016/j.adt.2014.05.001