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arxiv: 2607.00554 · v1 · pith:3QZOQOF5new · submitted 2026-07-01 · ⚛️ physics.atom-ph

Nuclear-size correction to the one-loop self-energy in hydrogenlike ions

Ping Yang , Aleksei V. Malyshev , Aleksandr N. Kochnev , Vladimir M. Shabaev , Aleksandr S. Shamanaev , Ting-Yun Shi , Dmitry V. Sychkov This is my paper

Pith reviewed 2026-07-02 02:21 UTC · model grok-4.3

classification ⚛️ physics.atom-ph
keywords nuclear size effectself-energyhydrogenlike ionsone-loop QEDfield shiftfinite nuclear size
0
0 comments X

The pith

Nuclear-size corrections to one-loop self-energy in hydrogenlike ions are captured by simple approximate formulas.

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

The paper computes the effect of finite nuclear size on both diagonal and off-diagonal one-loop self-energy matrix elements for the 1s, 2s, 3s, 2p1/2 and 2p3/2 states in hydrogenlike ions at Z=60, 82, 90 and 92. Calculations are performed in the full QED framework without expansion in the nuclear-strength parameter αZ. The results match existing literature values, and the authors extract simple approximate formulas that directly apply to self-energy contributions in field-shift factors.

Core claim

The nuclear-size effect on the one-loop self-energy matrix elements is evaluated nonperturbatively in αZ for the listed states and transitions. Excellent agreement with prior results is obtained, and the calculations yield simple approximate formulas for the nuclear-size correction that can be used to study self-energy contributions to field-shift factors.

What carries the argument

Nonperturbative evaluation of one-loop self-energy matrix elements with finite nuclear charge distribution, from which approximate correction formulas are extracted.

If this is right

  • The formulas allow direct inclusion of self-energy nuclear-size corrections in calculations of field-shift factors without repeating the full QED computation.
  • Corrections for the 1s-2s, 1s-3s and 2s-3s off-diagonal elements become available for precision spectroscopy of heavy ions.
  • The same approach supplies reference values for diagonal matrix elements in the 1s, 2s, 3s and 2p states at the four reported Z values.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approximate formulas could be checked by applying them to an intermediate Z value, such as 70, and comparing with a new full calculation.
  • If the formulas prove robust across nuclear models, they may reduce computational cost when combining self-energy with many-body effects in ions with more electrons.
  • Similar extraction of simple formulas from nonperturbative results might be attempted for two-loop self-energy or vacuum-polarization nuclear-size corrections.

Load-bearing premise

The results rest on one particular model of the nuclear charge distribution whose details and sensitivity are not fully specified.

What would settle it

A numerical test of the approximate formulas against an independent full calculation for a different nuclear model or for an ion with Z outside the reported set would show whether the formulas remain accurate.

Figures

Figures reproduced from arXiv: 2607.00554 by Aleksandr N. Kochnev, Aleksandr S. Shamanaev, Aleksei V. Malyshev, Dmitry V. Sychkov, Ping Yang, Ting-Yun Shi, Vladimir M. Shabaev.

Figure 1
Figure 1. Figure 1: FIG. 1. First-order self-energy diagram. The double line [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: One can see that the symmetrized prefactor [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Nuclear-size correction [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
read the original abstract

The nuclear-size effect on both diagonal and off-diagonal one-loop self-energy matrix elements is considered for hydrogenlike ions with $Z=60$, $82$, $90$, and $92$. Specifically, the $1s$, $2s$, $3s$, $2p_{1/2}$, and $2p_{3/2}$ states, as well as the off-diagonal $1s-2s$, $1s-3s$, and $2s-3s$ matrix elements are considered. The calculations are performed within the rigorous quantum-electrodynamics framework, nonperturbatively in the nuclear-strength parameter $\alpha Z$. Excellent agreement is found with results reported in the literature. Simple and useful approximate formulas to treat the nuclear-size correction are obtained, which, in particular, can be used to study the self-energy contributions to the field-shift factors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper considers the nuclear-size effect on both diagonal and off-diagonal one-loop self-energy matrix elements for hydrogenlike ions with Z=60, 82, 90, and 92. It treats the 1s, 2s, 3s, 2p_{1/2}, and 2p_{3/2} states as well as the off-diagonal 1s-2s, 1s-3s, and 2s-3s matrix elements. Calculations are performed nonperturbatively in αZ within the rigorous QED framework, with reported excellent agreement to literature values, and simple approximate formulas are derived for the nuclear-size corrections that can be used to study self-energy contributions to field-shift factors.

Significance. If validated, the approximate formulas would offer a practical tool for incorporating nuclear finite-size effects into self-energy evaluations for high-Z hydrogenlike ions without requiring full nonperturbative recomputation for each case, which is useful for field-shift factor studies. The nonperturbative treatment in αZ is a strength for the high-Z regime considered. Direct numerical evaluation within the established QED framework is noted as a positive feature with no evident circularity.

major comments (2)
  1. [Abstract] Abstract: the claim of a 'rigorous nonperturbative QED treatment' and 'excellent agreement with literature' is presented without any reported error bars, convergence checks, or numerical method details, which is load-bearing for assessing the accuracy of the computed corrections and the reliability of the derived approximate formulas.
  2. [Nuclear model and results] The manuscript provides no analysis or statement on the choice of nuclear charge distribution model (Fermi, uniform sphere, etc.) or on the sensitivity of the corrections and fitted approximate formulas to variations in model parameters such as rms radius or skin thickness within experimental ranges; this directly affects the claimed generality of the formulas for field-shift applications.
minor comments (1)
  1. [Abstract] Abstract: a short statement on the numerical technique or basis expansion employed would improve clarity without altering the technical content.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of a 'rigorous nonperturbative QED treatment' and 'excellent agreement with literature' is presented without any reported error bars, convergence checks, or numerical method details, which is load-bearing for assessing the accuracy of the computed corrections and the reliability of the derived approximate formulas.

    Authors: The full manuscript (Sections II and III) details the numerical implementation, including the dual-kinetic-balance finite-basis-set method, the choice of basis parameters, explicit convergence tests with respect to basis size and grid spacing, and estimated numerical uncertainties (at the level of a few parts in 10^4 or better). Tables I–III quantify the agreement with literature values, which lie within the combined uncertainties. The abstract is necessarily concise; we will revise it to include a brief statement on the achieved numerical precision. revision: partial

  2. Referee: [Nuclear model and results] The manuscript provides no analysis or statement on the choice of nuclear charge distribution model (Fermi, uniform sphere, etc.) or on the sensitivity of the corrections and fitted approximate formulas to variations in model parameters such as rms radius or skin thickness within experimental ranges; this directly affects the claimed generality of the formulas for field-shift applications.

    Authors: All calculations employed the Fermi two-parameter charge distribution with rms radii taken from the Angeli–Marinova compilation. We acknowledge the absence of an explicit sensitivity analysis. We will add a dedicated paragraph (new subsection in Section III) stating the model choice and presenting results obtained by varying the rms radius within ±1% and the diffuseness parameter within typical experimental ranges; the resulting changes in the self-energy corrections remain below the numerical uncertainty for the Z values considered, thereby supporting the utility of the approximate formulas. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct nonperturbative QED evaluation with literature agreement

full rationale

The paper reports nonperturbative calculations of nuclear-size corrections to one-loop self-energy matrix elements for specified Z values and states within the established QED framework. It states excellent agreement with prior literature results and derives approximate formulas from these computations. No quoted steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the central results are obtained via direct numerical evaluation rather than circular reparameterization of the same quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; ledger is necessarily incomplete. The work rests on the standard QED bound-state framework and a nuclear charge distribution model whose parameters are not disclosed.

axioms (2)
  • standard math Rigorous quantum-electrodynamics framework for bound states
    Calculations performed within the rigorous quantum-electrodynamics framework
  • domain assumption Nonperturbative treatment valid for given αZ values
    nonperturbatively in the nuclear-strength parameter αZ

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Reference graph

Works this paper leans on

47 extracted references

  1. [1]

    Furry W H, Phys. Rev. 81, 115 (1951)

    1951

  2. [2]

    Beyer H F and Shevelko V P, Introduction to the Physics of Highly Charged Ions , Institute of Physics Publishing, Bristol and Philadelphia, 2003

    2003

  3. [3]

    Berengut J C and Delaunay C, Nat Rev Phys 7, 119 (2025)

    2025

  4. [4]

    Desiderio A M and Johnson W R, Phys. Rev. A 3, 1267 (1971)

    1971

  5. [5]

    Mohr P J, Ann. Phys. (N.Y.) 88, 26 (1974)

    1974

  6. [6]

    Mohr P J, Ann. Phys. (N.Y.) 88, 52 (1974)

    1974

  7. [7]

    Snyderman N J, Ann. Phys. (N.Y.) 211, 43 (1991)

    1991

  8. [8]

    Blundell S A and Snyderman N J, Phys. Rev. A 44, R1427 (1991)

    1991

  9. [9]

    Indelicato P and Mohr P J, Phys. Rev. A 46, 172 (1992)

    1992

  10. [10]

    Cheng K T, Johnson W R, and Sapirstein J, Phys. Rev. A 47, 1817 (1993)

    1993

  11. [11]

    Quiney H M and Grant I P, Phys. Scr. T46, 132 (1993)

    1993

  12. [12]

    Persson H, Lindgren I, and Salomonson S, Phys. Scr. T46, 125 (1993)

    1993

  13. [13]

    Labzowsky L N and Goidenko I A, J. Phys. B: At. Mol. Opt. Phys. 30, 177 (1997)

    1997

  14. [14]

    Jentschura U D, Mohr P J, and Soff G, Phys. Rev. Lett. 82, 53 (1999)

    1999

  15. [15]

    Yerokhin V A and Shabaev V M, Phys. Rev. A 60, 800 (1999)

    1999

  16. [16]

    Yerokhin V A, Pachucki K, and Shabaev V M, Phys. Rev. A 72, 042502 (2005)

    2005

  17. [17]

    Sapirstein J and Cheng K T, Phys. Rev. A 108, 042804 (2023)

    2023

  18. [18]

    Yerokhin V A, Harman Z, and Keitel C H, Phys. Rev. A 111, 012802 (2025)

    2025

  19. [19]

    Soff G and Mohr P J, Phys. Rev. A 38, 5066 (1988)

    1988

  20. [20]

    Mohr P J and Soff G, Phys. Rev. Lett 70, 158 (1993)

    1993

  21. [21]

    Yerokhin V A, Phys. Rev. A 83, 012507 (2011)

    2011

  22. [22]

    Zubova N A, Kozhedub Y S, Shabaev V M, Tupitsyn I I, Volotka A V, Plunien G, Brandau C, and St¨ ohlker Th, Phys. Rev. A 90, 062512 (2014)

    2014

  23. [23]

    Shabaev V M, Tupitsyn I I, and Yerokhin V A, Phys. Rev. A 88, 012513 (2013)

    2013

  24. [25]

    Skripnikov L V, Prosnyak S D, Malyshev A V, Athanasakis-Kaklamanakis M, Brinson A J, Mi- namisono K, Cruz F C P, Reilly J R, Rickey B J, and Ruiz R F G, Phys. Rev. A 110, 012807 (2024)

    2024

  25. [26]

    Mohr P J, Plunien G, and Soff G, Phys. Rep. 293, 227 (1998)

    1998

  26. [27]

    Salvat F and Fern´ andez-Varea J M, Comput. Phys. Commun. 240, 165 (2019)

    2019

  27. [28]

    Malyshev A V, Prokhorchuk E A, and Shabaev V M, Phys. Rev. A 109, 062802 (2024). 9

    2024

  28. [29]

    Shabaev V M, J. Phys. B: At. Mol. Opt. Phys. 26, 4703 (1993)

    1993

  29. [30]

    Rep 356, 119 (2002)

    Shabaev V M, Phys. Rep 356, 119 (2002)

    2002

  30. [31]

    94–128, Elsevier, Oxford, 2024

    Shabaev V M, Quantum electrodynamics effects in atoms and molecules, in Comprehensive Computa- tional Chemistry (First Edition) , edited by Y´ a˜ nez M and Boyd R J, pp. 94–128, Elsevier, Oxford, 2024

    2024

  31. [32]

    Faustov R N, Teor. Mat. Fiz. 3, 240 (1970), [Theor. Math. Phys. 3, 478 (1970)]

    1970

  32. [33]

    Sucher J, Phys. Rev. A 22, 348 (1980)

    1980

  33. [34]

    Mittleman M H, Phys. Rev. A 24, 1167 (1981), Erra- tum 25, 1790 (1982)

    1981

  34. [35]

    Shabaev V M, Tupitsyn I I, and Yerokhin V A, Comput. Phys. Commun. 189, 175 (2015); 223, 69 (2018)

    2015

  35. [36]

    Tupitsyn I I, Kozlov M G, Safronova M S, Shabaev V M, and Dzuba V A, Phys. Rev. Lett. 117, 253001 (2016)

    2016

  36. [37]

    Paˇ steka L F, Eliav E, Borschevsky A, Kaldor U, and Schwerdtfeger P, Phys. Rev. Lett. 118, 023002 (2017)

    2017

  37. [38]

    Machado J, Szabo C I, Santos J P, Amaro P, Guerra M, Gumberidze A, Bian Guojie, Isac J M, and Indelicato P, Phys. Rev. A 97, 032517 (2018)

    2018

  38. [39]

    Si R, Guo X L, Brage T, Chen C Y, Hutton R, and Froese Fischer C, Phys. Rev. A 98, 012504 (2018)

    2018

  39. [40]

    A, Hillen- brand P-M, Holste K, Indelicato P, Kilcoyne A L D, Klumpp S, Martins M, Viefhaus J, Wilhelm P, and Schippers S, Phys

    M¨ uller A, Lindroth E, Bari S, Borovik Jr. A, Hillen- brand P-M, Holste K, Indelicato P, Kilcoyne A L D, Klumpp S, Martins M, Viefhaus J, Wilhelm P, and Schippers S, Phys. Rev. A 98, 033416 (2018)

    2018

  40. [41]

    Kaygorodov M Y, Kozhedub Y S, Tupitsyn I I, Maly- shev A V, Glazov D A, Plunien G, and Shabaev V. M, Phys. Rev. A 99, 032505 (2019)

    2019

  41. [42]

    Savelyev I M, Kaygorodov M Y, Kozhedub Y S, Tupit- syn I I, and Shabaev V M, Phys. Rev. A 105, 012806 (2022)

    2022

  42. [43]

    Zaitsevskii A, Mosyagin N S, Oleynichenko A V, and Eliav E, Int. J. Quant. Chem. 123, e27077 (2023)

    2023

  43. [44]

    Guo Y, Paˇ steka L F, Nagame Y, Sato T K, Eliav E, Reitsma M L, and Borschevsky A, Phys. Rev. A 110, 022817 (2024)

    2024

  44. [45]

    Silwal R, Blundell S A, Sanders S C, Dipti, Staiger H, Hosier A, Fuller M G, Ralchenko Yu, and Takacs E, Phys. Rev. A 111, 042821 (2025)

    2025

  45. [46]

    Lu H, Li B, Yang M, Dong L, Wang Y, Li M, Wu L, Li J, Jiang J, Dong C, and Zhang D, Chin. Phys. B 34, 073203 (2025)

    2025

  46. [47]

    Milstein A I, Sushkov O P, and Terekhov I S, Phys. Rev. A 69, 022114 (2004)

    2004

  47. [48]

    King S A, Spieß L J, Micke P, Wilzewski A, Leopold T, Benkler E, Lange R, Huntemann N, Surzhykov A, Yerokhin V A, Crespo L´ opez-Urrutia J R, and Schmidt P O, Nature 611, 43 (2022)

    2022