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Two-loop electron self-energy for low nuclear charges

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Pith's one-line read All-order two-loop self-energy calculations for hydrogen-like ions, extrapolated to hydrogen, give a value that differs from the accepted one by 2.8σ and shifts the Rydberg constant by one standard deviation.

desk verdict Solid all-order numerics, but the hydrogen extrapolation is hostage to unverified analytic coefficients, so the 2.8σ discrepancy is not yet credible. read the letter →

arxiv 2411.12459 v1 pith:N6FHU46U submitted 2024-11-19 physics.atom-ph

V. A. Yerokhin , Z. Harman , C. H. Keitel This is my paper · ORCID
classification physics.atom-ph
keywords two-loopelectronself-energyLambshifthydrogenRydbergconstantall-orderZαcalculationQEDbound-statecorrectionspartial-waveexpansionsubtractionscheme
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

This paper reports a calculation of the two-loop electron self-energy correction to the 1S Lamb shift of hydrogen-like ions performed to all orders in the binding parameter Zα, with numerical accuracy improved by more than an order of magnitude and calculations extended down to Z = 5. Extrapolating these all-order results to hydrogen gives G60(Z=1) = −104.1(3.8), a value twice as precise as the previously accepted one and different from it by 2.8 standard deviations. If this result is correct, the theoretical prediction for the 1S–2S transition frequency in hydrogen shifts by −2.5 kHz, lowering the Rydberg constant by about one standard deviation and breaking the previously assumed consistency between the all-order and Zα-expansion approaches.

What carries the argument

The enabling technique is an accelerated subtraction scheme for the two-loop self-energy diagrams, generalizing the one-loop method of Ref. [18] to the overlapping and nested diagrams. By subtracting and re-adding diagrams with two explicit interactions with the nuclear binding field, and computing the subtracted contributions in closed form in momentum space without partial-wave expansion, the authors improve the partial-wave convergence by one to two orders of magnitude. This permits calculations down to Z = 5 and reduces the numerical uncertainty at, for example, Z = 10 by a factor of 30.

What would settle it

A direct calculation of the missing part of the B60 coefficient (or an analytic evaluation of B71) that gives a value near the Zα-expansion prediction would shift the extrapolated G60(1) back toward the accepted value; alternatively, a nonperturbative two-loop self-energy calculation for Z = 4 or lower, performed by an independent group, would settle whether the deviation persists.

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Extended reading notes

Core claim

The central discovery is that the higher-order remainder function G60(Zα) of the two-loop self-energy, extracted from all-order numerical calculations for nuclear charges Z = 5 to 50, lies outside the band predicted by the Zα expansion and extrapolates to −104.1(3.8) at Z = 1. This disagrees with the previously recommended value of −75.8(9.4), obtained by combining Zα-expansion and older all-order results, by 2.8σ. The authors argue that the improved numerical precision challenges the assumed consistency between the nonperturbative and Zα-expansion approaches, so the theoretical hydrogen Lamb shift must be reexamined, with the updated SESE contribution shifting the 1S–2S prediction by −2.5 kHz and decreasing the Rydberg constant by one standard deviation.

Load-bearing premise

The extrapolation from the calculated nuclear charges Z ≥ 8 to hydrogen assumes the analytic constraints on the logarithmic coefficients B71, B83, B82, B81 of the Zα expansion; the numerical data alone cannot determine these logarithms, so the hydrogen value rests on the very Zα-expansion input whose consistency the paper questions.

Editorial extensions

If this is right

  • The theoretical prediction for the hydrogen 1S–2S transition frequency shifts by −2.5 kHz, changing the value of the Rydberg constant by about one standard deviation.
  • The previously recommended SESE value for hydrogen, G60(1) = −75.8(9.4), is replaced by −104.1(3.8), indicating that the consistency between all-order and Zα-expansion approaches no longer holds.
  • Improved accuracy for the 1S state also refines other nS energy predictions through the relation n^3 E(nS) = ν_n + E(1S).
  • The method opens the way to improved calculations of other two-loop QED corrections to the Lamb shift and to the bound-electron g factor.

Reading between the lines

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

  • If the disagreement is real, one of the logarithmic coefficients constrained analytically (e.g., B71) or the partially computed coefficient B60 must be substantially modified; the paper estimates that a missing contribution δB60 ≈ −40 or a 10% adjustment in B61 would reconcile the two approaches.
  • A fully independent nonperturbative calculation at Z = 4–7, where the present data are sparse and extrapolation carries the largest weight, would provide a sharp test of the claimed hydrogen value.
  • The same accelerated subtraction scheme may reduce the numerical cost of other two-loop bound-state QED calculations, including the g-factor corrections needed for ongoing experiments.
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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

3 major / 5 minor

Summary. This paper reports new all-order-in-Zalpha numerical calculations of the two-loop electron self-energy for the 1S state of hydrogen-like ions, extending the accessible range down to Z=5 and improving numerical accuracy by more than an order of magnitude through an accelerated subtraction scheme. The authors extract the higher-order remainder function G60(Zalpha) defined in Eq. (4), extrapolate it to Z=1 using a polynomial fit supplemented by analytic constraints on logarithmic coefficients, and obtain G60(1)=-104.1(3.8). This value differs by 2.8 standard deviations from the previously accepted value of -75.8(9.4), and the paper argues that the all-order and Zalpha-expansion approaches are no longer consistent, with implications for the 1S-2S hydrogen transition and the Rydberg constant.

Significance. If the hydrogen extrapolation is robust, this is an important result: it would improve the two-loop self-energy contribution to the hydrogen Lamb shift, change the theoretical prediction for the 1S-2S transition, and shift the Rydberg constant by about one standard deviation. The numerical all-order data themselves are a substantial advance, with clear improvements in accuracy and internal cross-checks between different computational schemes. However, as detailed below, the headline hydrogen value and the claimed inconsistency depend sensitively on unverified Zalpha-expansion coefficients, so the central claim is not yet established.

major comments (3)
  1. [Main text, Eq. (4) and Eq. (7)] The extracted G60 values in Table I and the extrapolated hydrogen value G60(1)=-104.1(3.8) are obtained after subtracting the logarithmic terms B63 L^3 + B62 L^2 + B61 L, where B62 and B61 are taken from Pachucki (2001) and are described in the text as "yet to be independently verified." Since L(Z=1)=ln((Zalpha)^-2) is about 9.8 and L^2 about 96.5, a change of about 3 in B61 or about 0.3 in B62 shifts G60(1) by roughly 28, which is comparable to the entire discrepancy with the accepted value -75.8(9.4). The uncertainty quoted in Eq. (7) accounts only for the constraints in Eq. (6) and polynomial extrapolation scatter, not for this dependence. The authors' own statement that "a 10% adjustment in the B61 coefficient ... could restore the consistency" confirms that the central 2.8-sigma discrepancy is not robust to plausible variation of unverified input coefficients. The paper should propagate the uncertainty of B61 and B62 into G60(1), or provide an explicit sensitivity analysis, before claiming a discrepancy.
  2. [Main text, Eq. (5)-(6); Supplemental Material, 'Details of the extrapolation procedure'] The extrapolation to Z=1 is not a direct all-order result. It uses only data at Z=8,...,50, with Z=5,6,7 excluded, and fits a polynomial to G'_60 after subtracting the B72, B71, and B84 terms using analytic values from the Zalpha expansion. The constraints for B71, B83, B82, and B81 in Eq. (6) come from Ref. [16], the same Zalpha-expansion framework whose consistency with the all-order results is the central claim. The text acknowledges that "the current numerical data cannot constrain the logarithmic coefficients." Thus the claimed inconsistency is partly a joint test of the all-order calculation and these specific analytic inputs, not a pure all-order result. The extrapolation uncertainty should include the uncertainty of these inputs without presupposing their validity, or the conclusion should be reframed accordingly.
  3. [Supplemental Material, Eqs. (22)-(23)] The exclusion of the three lowest-Z data points (Z=5,6,7) and the choice of weights w_i^2=delta_i^2+(a Z_i^{n+1})^2, with the parameter a selected by requiring that variations from removing data points "are small and approximately equal," are heuristic. These excluded points are the ones closest to hydrogen and the only direct all-order anchor near the extrapolation region. The reported polynomial-extrapolation uncertainty is estimated as twice the maximal difference among P2, P3, and P4 fits, which does not account for model-selection uncertainty beyond these three polynomials. A more systematic assessment, such as exploring dependence on the excluded points, the weight parameter a, and higher-degree polynomials, would strengthen the reliability of the extrapolated G60(1).
minor comments (5)
  1. [Summary and Conclusion] The phrase "break the previously assumed consistency" is stronger than the evidence supports; consider "challenge" or "question" in light of the caveats discussed in the major comments.
  2. [Supplemental Material, first paragraph] The text says formulas and figures of the main text will be referred to as "Eq. (?M)" and "Fig. ?M"; these placeholders should be resolved before publication.
  3. [Main text, Eq. (11)] The ellipsis in Eq. (11) hides the right-hand side of Eq. (9); the notation is understandable from context but could be made explicit.
  4. [Main text, Fig. 4 caption] The caption does not identify the shaded region or the meaning of the triangle; the text should define these elements clearly.
  5. [Main text, after Eq. (4)] The numerical values of B62 and B61 are not given; adding them would help readers assess the sensitivity discussed in the major comments.

Circularity Check

1 steps flagged · score 4.0 of 10

Extrapolated hydrogen value G60(1) = -104.1(3.8) is defined by subtracting unverified Zα-expansion coefficients (B61, B62) from the same expansion the paper claims to falsify; a 10% change in B61 would erase the 2.8σ discrepancy.

  1. other [Eq. (4) and following paragraph; extrapolation discussion near Eqs. (5)-(7); Supplemental Material 'Details of the extrapolation procedure'.]
    "The coefficients B62 and B61 derived by Pachucki [8] have yet to be independently verified. We now use the analytical results to extract the higher-order remainder function G60(Zα) from our calculations ... Specifically, a 10% adjustment in the B61 coefficient, δB61 ≈ −5, could restore the consistency with the nonperturbative calculations."

    G60(Zα) is not a raw all-order observable: Eq. (4) defines it as F(Zα) minus Zα-expansion terms, including B61L and B62L^2 taken from the very approach whose consistency with all-order results is the central claim. Because L ≈ 9.8 at Z = 1, an unverified shift δB61 ≈ −5 changes every extracted G60(Zi) by δB61·L(Zi) and moves G60(1) by roughly −49, which the authors state would restore consistency and erase the headline 2.8σ disagreement. The quoted ±3.8 uncertainty covers polynomial-extrapolation scatter and the B71/B82/B83/B81 constraints of Eq. (6), but not this B61/B62 sensitivity.

full rationale

The all-order numerical F(Zα) data in Table I are genuinely independent of the Zα expansion: they come from a nonperturbative Dirac-Coulomb calculation with an improved subtraction scheme, and they agree with previous all-order results while improving precision by an order of magnitude. There is no self-citation chain: the extrapolation form (Eq. 5) and constraints (Eq. 6) come from Karshenboim et al. (Ref. [16]), not from the present authors, and the comparison with B60 (Ref. [15]) is not used in the fit. The circularity is therefore partial and located at the extraction step: the quantity G60 that is extrapolated to hydrogen is defined by subtracting the unverified B61 and B62 coefficients of the Zα expansion, and the paper itself concedes that a 10% B61 adjustment would restore consistency. The discrepancy and derived Rydberg shift are consequently not robust to the unverified input, but they are not forced by construction; a genuine, independent all-order dataset underlies the claim. Hence score 4 rather than 6 or 8.

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

The central claim relies on the assumed functional form of the Zα expansion for the extrapolation, on external constraints for logarithmic coefficients, and on the reliability of the complex numerical renormalization scheme. No new physical entities are introduced.

free parameters (3)
  • Polynomial coefficients P_n(Z) = Not given; P_3 yields G60(0) = -101.1370
    Coefficients of the polynomial fit in Eq. (22); the intercept determines the extrapolated G60 at Z=0.
  • Weight coefficient a = Not specified
    Effective error coefficient in Eq. (23); chosen heuristically to make the fit stable when edge points are removed.
  • B71 central value = -12
    Central value of the logarithmic coefficient B71 used in the subtraction in Eq. (21); taken from Eq. (6).
assumptions (4)
  • domain assumption The Zα-expansion of G60(Zα) has the functional form of Eq. (5) with the specified logarithmic terms.
    This form underpins the subtraction in Eq. (21) and the uncertainty analysis; if it is incomplete, the extrapolation is biased.
  • domain assumption The constraints on the logarithmic coefficients in Eq. (6) from Ref. [16] are correct and sufficient.
    The extrapolation central value and uncertainty depend on these external constraints, which the authors could not verify.
  • domain assumption The renormalization/subtraction scheme and the partial-wave expansion converge to the correct two-loop self-energy.
    The calculation builds on Refs. [22-27]; no machine-checked or independently verified code is provided, so the convergence and correctness are assumed.
  • ad hoc to paper The data at Z=8-50, with weights chosen by Eq. (23), are sufficient to determine the polynomial fit; data at Z=5-7 can be excluded.
    The exclusion and the weight selection are specific to this paper and influence the extrapolated value.

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Pith. "Pith review of Two-loop electron self-energy for low nuclear charges." pith.science (2026). https://pith.science/paper/N6FHU46U

@misc{pith2026241112459,
  author       = {Pith},
  title        = {Pith review of: Two-loop electron self-energy for low nuclear charges},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N6FHU46U}},
  note         = {Machine review of arXiv:2411.12459}
}
abstract

Calculations of the two-loop electron self-energy for the $1S$ Lamb shift are reported, performed to all orders in the nuclear binding strength parameter $Z\alpha$ (where $Z$ is the nuclear charge number and $\alpha$ is the fine structure constant). Our approach allows calculations to be extended to nuclear charges lower than previously possible and improves the numerical accuracy by more than an order of magnitude. Extrapolation of our all-order results to hydrogen yields a result twice as precise as the previously accepted value [E. Tiesinga et al. Rev. Mod. Phys. 93, 025010 (2021)], differing from it by 2.8 standard deviations. The resulting shift in the theoretical prediction for the $1S$-$2S$ transition frequency in hydrogen decreases the value of the Rydberg constant by one standard deviation.

Figures

Figures reproduced from arXiv: 2411.12459 by the authors.

Figure 1
Figure 1. FIG. 1: Feynman diagrams representing the two-loop [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Two subtraction schemes used for calculations [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Same as Fig. 2 for the nested diagram. [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Two-loop self-energy higher-order function [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. One-loop electron self-energy with accelerated partial-wave expansion in Coulomb gauge

    physics.atom-ph 2024-11 conditional novelty 6.0 of 10

    Coulomb-gauge accelerated partial-wave methods now compute one-loop electron self-energies for hydrogen-like ions from Z=1 to Z=100, including high excited states.

Reference graph

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