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Testing inter-electronic interaction in lithium-like tin

T0 review · 1 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper reports a 0.5 ppb measurement of the g factor of lithium-like tin and argues that its agreement with the ab initio and experiment-enhanced QED predictions confirms the revised interelectronic theory at $Z=50$.

desk verdict New measurement at Z=50 with a strong theory upgrade, but the 'independent confirmation' claim outstrips the experimental precision and relies on a heuristic two-loop QED error estimate. read the letter →

arxiv 2505.24286 v1 pith:ZRS42F7D submitted 2025-05-30 physics.atom-ph

Jonathan Morgner , Vladimir A. Yerokhin , Charlotte M. Konig , Fabian Hei{ss}e , Bingsheng Tu , Tim Sailer , Bastian Sikora , Zoltán Harman
show 6 more authors
Jose R. Crespo López-Urrutia Christoph H. Keitel Sven Sturm Klaus Blaum (Max-Planck-Institut für Kernphysik Heidelberg Germany)
This is my paper
classification physics.atom-ph
keywords g-factorlithium-likeionsbound-stateQEDinterelectronicinteractionPenning-trapspectroscopyhighlychargedtin-118two-loop
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Bound electrons in strong electric fields are a testbed for quantum electrodynamics, and the magnetic moment of an ion is one of the most precise observables. This paper reports a measurement of the $g$ factor of lithium-like tin ($Z=50$) at 0.5 parts per billion relative accuracy, together with ab initio QED calculations that treat the interaction between the valence electron and the two core electrons more completely than before. The measured value $g_{exp} = 1.980354799750(84)(54)(944)$ agrees with both the purely theoretical prediction $g_{theo} = 1.980354769(35)$ and the experiment-enhanced prediction $g_{theo}(enh) = 1.980354796(12)$, the latter obtained by using the hydrogen-like tin measurement to pin down the poorly known higher-order two-loop QED contribution. Because the experimental data were blinded while the theory was evaluated, the authors read the agreement as an independent confirmation of the revised interelectronic QED theory at a nuclear charge where these effects are much larger than in earlier silicon and calcium tests.

What carries the argument

The central object is the $g$ factor of the $2s$ valence electron of a lithium-like tin ion, measured by Penning-trap spectroscopy as the ratio of Larmor to cyclotron frequency. The theoretical machinery is a sum of QED corrections: Dirac binding, one-electron self-energy and vacuum polarization, finite nuclear size, nuclear recoil, and the many-electron electron-structure and QED-screening corrections. The load-bearing device for the sharpest test is the experimentally enhanced prediction: because the unknown higher-order two-loop QED terms are about eight times smaller for a $2s$ electron than for a $1s$ electron, the authors extract the two-loop contribution from the measured hydrogen-like tin $g$ factor and scale it by the calculated $1s$-to-$2s$ ratio, cutting the dominant theoretical uncertainty from about 33 parts per billion to about 6 parts per billion.

What would settle it

Compute the two-loop QED contribution of order $alpha^{2}$ (Z $\alpha$)^6 for the 2s electron at Z=50 directly to all orders in Z $\alpha$; if that computed value lies outside 0.0268(61) x $10^{{-6}}$ (the value the authors extract by scaling the hydrogen-like result), while the measured g factor stays fixed, the enhanced prediction would no longer match experiment and the central claim would need revision.

Watch

Extended reading notes

Core claim

The paper's central claim is that lithium-like tin's measured $g$ factor agrees with theory only when the interelectronic QED interaction is treated with the revised, fully non-perturbative screening corrections; the older advanced calculations that deviated by up to five standard deviations for silicon and calcium would not survive this high-$Z$ test. The measured value $g_{exp} = 1.980354799750(84)(54)(944)$ falls within the quoted uncertainties of both the ab initio prediction $g_{theo} = 1.980354769(35)$ and the experiment-enhanced prediction $g_{theo}(enh) = 1.980354796(12)$. The enhanced prediction uses the measured hydrogen-like tin $g$ factor to replace the dominant two-loop QED uncertainty, reducing that error by a factor of 5.5 and making the test sensitive mainly to the electron-electron interaction. This is the first $g$-factor measurement of a lithium-like ion at $Z = 50$, where binding QED corrections scale roughly as $Z^4$ and are substantially larger than in previous tests.

Load-bearing premise

Everything rests on the assumption that the authors' estimates of the effects they could not calculate—scaling a known one-loop term by a conservative factor of two, and taking the spread between different screening potentials—realistically bound the missing higher-order terms.

Editorial extensions

If this is right

  • If the central claim is right, the revised interelectronic QED treatment is the one to use for lithium-like $g$ factors at high nuclear charge, and the older calculations that disagreed with silicon and calcium data are not simply recoverable by re-fitting.
  • The agreement with the enhanced prediction checks the consistency of the hydrogen-like and lithium-like tin experiments and the two-loop QED theory connecting them.
  • The ion-mass uncertainty, not QED, now dominates the experimental error budget, so an improved mass measurement would sharpen the test without requiring new theory.
  • The refined methods for interelectronic QED corrections can be applied to more complex ions such as boron-like and carbon-like systems, as well as to parity-nonconserving transition amplitudes in neutral atoms.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: if the ion mass is improved by the order of magnitude the authors deem feasible, the same apparatus could discriminate between the Z-alpha-expansion value of the two-loop QED contribution and the value derived from hydrogen-like scaling, directly testing the scaling assumption.
  • Beyond the paper: the enhanced-prediction strategy could be repeated for other pairs of charge states of the same element; systematic consistency across several elements would provide a strong check on the 1s-to-2s scaling factor of 8.9 ± 1.8.
  • Beyond the paper: the procedure of estimating omitted higher-order interelectronic effects from the spread over different screening potentials is itself testable; a future all-order calculation falling outside that spread would show the uncertainty estimate was not conservative.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. This manuscript reports a Penning-trap measurement of the g-factor of lithium-like tin (118Sn47+), together with ab initio bound-state QED calculations that include an improved treatment of interelectronic effects. The measured value, gexp = 1.980 354 799 750(84)(54)(944), is compared with an ab initio prediction gtheo = 1.980 354 769(35) and an 'enhanced' prediction gtheo(enh) = 1.980 354 796(12) that uses the experimental hydrogen-like tin g-factor to infer the unknown higher-order two-loop QED contribution. The authors report agreement in both cases and interpret the ab initio agreement as an independent confirmation of the revised interelectronic QED theory at Z=50.

Significance. If the theoretical uncertainty estimates are reliable, this work is a significant milestone: it extends high-precision g-factor tests of bound-state QED to a previously unexplored regime of intermediate nuclear charge (Z=50), where the Zα expansion is poorly convergent, and it consolidates recent revisions of interelectronic QED corrections that had previously shown discrepancies at Z=14 and Z=20. The experimental work is carefully blinded, the systematic error budget is detailed, and the data and theoretical inputs are made available via the cited repositories (Refs. 52, 53).

major comments (1)
  1. [Supplementary Sec. 1.4, Eq. (33)] The dominant uncertainty of the ab initio prediction, ±33×10⁻⁹ out of ±35×10⁻⁹, is assigned to the omitted α²(Zα)^6+ two-loop QED contribution by scaling the one-loop α(Zα)^6+ term with the two-loop/one-loop ratio at (Zα)^5 and multiplying by a 'conservative factor of 2.' This is an estimate, not a calculation, and at Zα≈0.36 the expansion is known to converge slowly (the same section already ascribes a 10% uncertainty to c50 for missing light-by-light terms). If the true omitted term is larger than the factor-of-2 estimate, the observed difference gexp − gtheo ≈ 31×10⁻⁹ (about 0.9σ) would no longer provide the stated independent confirmation. The authors should either benchmark this scaling (for example, by applying the same prescription to the 1s state of hydrogen-like tin and comparing with the experimental extraction from Ref. 33) or, failing that, state explicitly that the ab initio confirmation is contingent on the reliability of this heuristic uncertainty. As written, the load-bearing claim 'independently confirms the revised theory' is stronger than the supporting evidence.
minor comments (4)
  1. [Eq. (3) vs Table 1 and Supplementary Table 5] Eq. (3) gives gtheo(enh) = 1.980 354 796 (12), while Table 1 and Supplementary Table 5 list the enhanced value as 1 980 354.797 (12); the 1×10⁻⁹ discrepancy should be resolved.
  2. [Eq. (1)] Eq. (1), the Breit formula for the point-nucleus Dirac contribution, appears garbled in the typesetting ('δgD(pnt) = 2/3 r 2 ...'); please check the formula and its rendering.
  3. [Supplementary Sec. 1.5] Supplementary Sec. 1.5 states that the two-loop QED values are '−0.107 (33)×10⁻⁹' and '−0.080 (6)×10⁻⁹', but Table 1 is in units of ×10⁻⁶; the units in the text should be corrected to ×10⁻⁶.
  4. [Supplementary Table 5] In Supplementary Table 5, the experimental g-factor is listed as '1980354.800(1)'; given that the table is in units of ×10⁻⁶, this implies an uncertainty of 1×10⁻⁶, inconsistent with the error budget in Table 2. Please clarify the units or the notation.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the ab initio prediction is self-contained, and the enhanced prediction transparently uses a same-group hydrogen-like measurement as an external input.

full rationale

The central ab initio prediction, Eq. (2), is built from independent calculations of Dirac binding, one-loop QED, electron structure, QED screening, nuclear recoil, and a separately estimated two-loop QED uncertainty. The dominant two-loop uncertainty is obtained in Sec. 1.4 by scaling the one-loop (Zα)^6 contribution by the known two-loop/one-loop ratio at (Zα)^5 and multiplying by a conservative factor of 2. This is a heuristic uncertainty estimate, not a fit to the lithium-like result, and it does not import the target quantity. The paper's headline agreement with Eq. (8) does not depend on the hydrogen-like tin measurement. The 'experimentally enhanced' prediction, Eq. (3), does use the same group's hydrogen-like tin g-factor (Ref. 33) to infer the missing two-loop QED contribution and rescale it to the 2s state. However, this is explicitly labeled as an enhancement, it is applied to a different charge state, and the resulting value is not adjusted to match the lithium-like experiment. The comparison between Eq. (8) and Eq. (3) is therefore a consistency check rather than a circular derivation. The weakest points, such as the factor-of-2 uncertainty on omitted α²(Zα)^6+ QED terms and the screening-potential spread used for interelectronic uncertainties, are conservative heuristics that affect the strength of the claim but do not make the derivation circular. The blinding statement further supports that the experimental result was not used to tune the theory. Overall, the derivation chain is self-contained for the main claim; the only near-miss is the transparent use of a same-group experimental input in the auxiliary enhanced value, which is not load-bearing for the independent confirmation claim.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard QED machinery and on several heuristic uncertainty estimates that are not derived from first principles. The most important are the scaling of the two-loop QED contribution from hydrogen-like to lithium-like tin, the assignment of a 10% uncertainty to the c50 coefficient, and the use of screening-potential spread as an error bound. No new entities are introduced.

free parameters (3)
  • 1s-to-2s scaling factor for two-loop QED = 8.88 (20% uncertainty)
    Converts the experimentally inferred 1s two-loop QED contribution in hydrogen-like tin to the 2s state in lithium-like tin. The value is derived from one-loop QED ratios, but the 20% uncertainty is an ad hoc estimate; this scaling is a potential source of bias in the enhanced prediction (Supplementary, Sec. 1.4).
  • Uncertainty for two-loop c50 coefficient = 10% relative uncertainty
    A 10% uncertainty is assigned to the (Zα)^5 two-loop coefficient c50 to account for missing light-by-light contributions; this contributes to the dominant theory error (Supplementary, Sec. 1.4).
  • Conservative factor for missing α^2(Zα)^6+ effects = 2
    Multiplier used to estimate unknown higher-order two-loop QED contributions from scaled one-loop results; chosen by hand (Supplementary, Sec. 1.4).
assumptions (4)
  • domain assumption Standard bound-state QED and renormalization theory applies to the lithium-like ion at Z=50.
    The entire calculation is built on this framework; no new QED effects are postulated.
  • domain assumption NRQED leading-order (in Zα) evaluation of three-or-more-photon exchange corrections is accurate enough.
    Used for high-order interelectronic effects (g^(3+)int and parts of g^(2)sescr); only leading-order Zα terms are computed, and omitted terms are estimated by uncertainty spread (Supplementary, Secs. 1.2-1.3).
  • domain assumption The spread of results over different screening potentials bounds the omitted higher-order interelectronic effects.
    Uncertainties are estimated from the spread across Coulomb, core-Hartree, Kohn-Sham, Dirac-Hartree, and Dirac-Slater potentials; this assumes these potentials span the likely true value.
  • domain assumption The 1s-to-2s scaling of two-loop QED contributions follows the one-loop ratio.
    Used to construct the enhanced prediction; assumes the scaling factor 8.88 derived from one-loop QED also applies to two-loop, with a 20% uncertainty (Supplementary, Sec. 1.4).

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Pith. "Pith review of Testing inter-electronic interaction in lithium-like tin." pith.science (2026). https://pith.science/paper/ZRS42F7D

@misc{pith2026250524286,
  author       = {Pith},
  title        = {Pith review of: Testing inter-electronic interaction in lithium-like tin},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZRS42F7D}},
  note         = {Machine review of arXiv:2505.24286}
}
abstract

Magnetic moments of bound-electron systems are a sensitive tool for testing fundamental interactions. $g$ factors of lithium-like ions have been rigorously studied in recent years, enabling insights into the relativistic inter-electronic effects. Here, we present the $g$-factor measurement of lithium-like tin, accurate to 0.5 parts per billion, as well as \textit{ab initio} theoretical calculations that include an advanced treatment of the inter-electronic interaction. We further improve the prediction by using the experimental result for the hydrogen-like tin $g$ factor, inferring from it the unknown higher-order QED effects. The observed agreement independently confirms the revised theory at a previously inaccessible high nuclear charge $Z$ of 50, where QED effects are significantly larger.

Figures

Figures reproduced from arXiv: 2505.24286 by the authors.

Figure 1
Figure 1. Main Feynman diagrams describing the g factor of a few-electron ion. The double lines denote the bound electrons, the wavy lines represent the virtual photon exchange, and the wavy lines terminated by a triangle indicate the interaction with the external magnetic field [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Schematic and results of the experiment. a The precision measurement takes place in two separate traps, between which the ion is shuttled by adiabatic transport. In the Preci￾sion Trap, the electric field is extremely harmonic and the magnetic field is as homogeneous as possible, which are optimal conditions for precision spectroscopy. In the Analysis Trap the center electrode is made of a ferromagnetic material tha… view at source ↗

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    Acknowledgements This work was supported by the Max Planck Society (MPG), the Inter- national Max Planck Research School for Quantum Dynamics in Physics, Chemistry and Bi- ology (IMPRS-QD), the German Research Foundation (DFG) Collaborative Research Cen- tre SFB 1225 (ISOQUANT...

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