Pith. sign in

REVIEW 2 major objections 4 minor 48 references

One-loop self-energy using a numerical Green function

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A numerical Green function built from exponential basis functions computes the one-loop self-energy of hydrogenlike atoms to about 10^-5 relative precision, extending to hydrogen with a convergence acceleration and Coulomb gauge, opening a

desk verdict A solid, honest methods paper that delivers the first Z=1 self-energy calculation with a numerical Green function; the error bars match benchmarks, though the partial-wave extrapolation remains the weakest link. read the letter →

arxiv 2607.16026 v1 pith:JQAWVG3E submitted 2026-07-17 physics.atom-ph

Hugo D. Nogueira , Maen Salman , Jean-Philippe Karr This is my paper · ORCID
classification physics.atom-ph
keywords self-energyLambshiftnumericalGreenfunctionDiracequationexponentialbasissethydrogenlikeatomsQEDconvergenceacceleration
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

The paper demonstrates that a numerical Green function, constructed by solving the radial Dirac equation in an exponential basis set, can evaluate the one-loop self-energy of hydrogenlike atoms with precision comparable to analytic Green function methods. For hydrogenlike uranium (Z=92) in the Feynman gauge, the total self-energy is obtained with about 10^-5 relative uncertainty; for hydrogen (Z=1), the Feynman gauge yields 10^-4 and the Coulomb gauge reaches 10^-5. The central trick is a subtraction and convergence-acceleration scheme that isolates the slowly converging partial-wave tail and handles it via extrapolation. If the method holds up, it opens a practical path to self-energy calculations in molecular ions, where no analytic Green function exists.

What carries the argument

The central object is the numerical Dirac radial Green function G_kappa(x1,x2,z) = sum_n phi_{kappa,n}(x1) phi_{kappa,n}(x2)/(z - E_{kappa,n}), where phi and E come from a finite exponential basis set that enforces dual kinetic balance (DKB). The many-potential term of the self-energy is extracted by the subtraction G^{2+} = G - G0 - G1, and the slow partial-wave convergence is tamed by subtracting an analytically known two-potential approximation G^{2+}_a and adding it back in momentum space. This combination makes the partial-wave sum short enough to extrapolate reliably, and the error cancellation among the subtracted terms is the reason the final result is far more accurate than any sing

What would settle it

Compute the total one-loop self-energy for the hydrogen ground state with an independent high-precision method (e.g., analytic Green function with partial waves carried to millions) and compare to the authors' extrapolated value; a discrepancy larger than their quoted 10^-5 or 10^-4 uncertainty would falsify the extrapolation procedure.

Watch

Extended reading notes

Core claim

The authors show that a finite-basis numerical Green function, built from exponential functions satisfying the dual kinetic balance condition, can reproduce the many-potential term of the one-loop self-energy to high precision. By subtracting the known zero- and one-potential terms and an approximate two-potential term, they convert the partial-wave expansion into a rapidly converging form, then extrapolate the residual tail to infinity using a polynomial in 1/|kappa|. This yields, for the hydrogen ground state, a total self-energy with 10^-4 relative uncertainty in the Feynman gauge and 10^-5 in the Coulomb gauge, and for hydrogenlike uranium about 10^-5 in the Feynman gauge. The achievemen

Load-bearing premise

The precision claim rests on the assumption that the partial-wave tail follows the fitted polynomial in 1/|kappa| beyond the computed partial waves; if the tail behaves differently, the extrapolated total self-energy and its quoted uncertainty are unreliable.

Editorial extensions

If this is right

  • The method provides a feasible way to compute one-loop self-energies in systems with no analytic Green function, notably the molecular hydrogen ions H2+ and HD+.
  • The DKB exponential basis set performs on par with Gaussian basis sets for high-Z atoms, and despite slower convergence it can be extended to low Z with appropriate acceleration.
  • The Coulomb gauge offers a significant precision advantage over the Feynman gauge for low-Z self-energy calculations, even with a numerical Green function.
  • The observed strong cancellation of errors between the zero-, one-, and many-potential terms suggests that the final uncertainty is dominated by the partial-wave extrapolation, not by the basis set itself.
  • Optimizing the nonlinear basis parameters for each partial wave, and pushing the extrapolation to higher |kappa|, could further reduce the uncertainty below 10^-5.

Reading between the lines

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

  • If the exponential basis set can be extended to two-center problems (as the authors imply), the same numerical Green function approach could bring all-order self-energy calculations within reach for molecular ions, where currently only nonrelativistic Z-alpha expansions are used.
  • The extrapolation method might be sharpened by combining information from both gauges, or by using a more physical model for the tail of the partial-wave series instead of a generic polynomial fit.
  • The success of this approach hints that many bound-state QED corrections, not just the self-energy, could be computed with numerical Green functions in a basis-set framework, provided the extrapolation tail is handled carefully.
  • A systematic comparison of DKB exponential basis sets versus Gaussian and B-spline bases, with optimized parameters, could quantify which family is most efficient for high-precision QED calculations in non-hydrogenic systems.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper develops a finite-basis numerical Green function approach to the one-loop self-energy in hydrogenlike atoms, using exponential basis functions and performing all radial integrations numerically. The many-potential term is computed in both Feynman and Coulomb gauges, while the zero- and one-potential terms are taken from analytical calculations. The method is benchmarked for hydrogenlike uranium (Z=92) and hydrogen (Z=1). For Z=92 in Feynman gauge the total F(Zα) is 1.49090(2), to be compared with the analytical Green function result 1.490916(3); for Z=1 the paper reports F=10.318(1) in Feynman gauge and F=10.3169(1) in Coulomb gauge, compared with 10.31679365(1). Additional contributions include the demonstration that DKB exponential basis sets improve precision substantially over NKB, the implementation of a convergence acceleration scheme, and a complexity reduction for ΔE^1_κ.

Significance. If the precision claims hold, this is a meaningful step toward extending numerical-Green-function self-energy calculations to non-hydrogenic and molecular systems, where analytical Dirac-Coulomb Green functions are unavailable. The benchmarks against independent analytical Green function calculations are a genuine strength, and no parameter is fitted to reproduce the final self-energy. The DKB-basis improvement and the O(max(n_G,n_q)^4) scaling for ΔE^1_κ are concrete, useful results. However, the advertised 10^-5-level uncertainties rest on a partial-wave tail extrapolation whose systematic uncertainty is not fully controlled; this is the main barrier to accepting the precision claims as stated.

major comments (2)
  1. [Sec. V A, Table II] The Z=92 total F=1.49090(2) depends on a polynomial-in-1/|κ| extrapolation of the |κ|=16-20 partial-wave contributions. The paper itself notes that the sum |κ|=16-35 differs from the analytical Green function result by about -2.1e-5 and attributes this to basis-set limitations that continue at higher |κ|. The uncertainty is estimated by varying the maximum fitted |κ| by 20%, but this procedure does not bound a systematic trend that grows with |κ| and could be absorbed into the fitted tail. Since the extrapolated tail is the dominant source of uncertainty, please provide a direct cross-check of the tail against analytical high-κ results or otherwise quantify how the observed P35 discrepancy is reflected in the quoted uncertainty.
  2. [Sec. V B, Table IV] For Z=1 in Coulomb gauge, the final F=10.3169(1) relies on the extrapolated tail 0.0002(1) for |κ|≥11. The fit window |κ|=11-15 is exactly where deviations from the analytical Green function grow from about 3e-6 to 1.2e-5 and are described as dominated by basis-set limitations. Varying the upper fitted |κ| by 20% tests the sensitivity to the number of fitted points, not the systematic bias from basis-set errors that increase with |κ|. Because the tail contributes the dominant uncertainty, a separate estimate of this systematic bias should be given before the 10^-5 relative-uncertainty claim can be considered established.
minor comments (4)
  1. [Table IV] The Diff. entries for the extrapolated rows with 10≤|κ|≤15 appear to be off by a factor of ten. For example, for κ=10, 0.0000426 - 0.0000472 = -4.6e-6, not -4.6e-5; similarly for κ=11 the extrapolated row gives -3.8e-6, not -3.8e-5. Please correct the notation or clarify the convention.
  2. [Sec. IV B, Eq. (28)] The expression for G^1_κ is written under the assumption x2 < x1, but this is stated rather briefly. It would help to explicitly specify the domain of the (r,y) variables after the transformation (26) so that the quadrature ranges are unambiguous.
  3. [Sec. V A, Table II] The text states that the total error on the sum of the first 20 terms is smaller than 4e-6, but the comparison shown is for the sum up to |κ|=35. It would be useful to state explicitly how the error on the first 20 terms is estimated, especially because the P35 discrepancy is attributed to a growing basis-set trend.
  4. [References] The reference values labeled 'Anal. [43]' in Table I come from a private communication. Providing a public source or an ancillary file with these numerical values would improve verifiability.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the self-energy claim is benchmarked against independent analytical Green-function results; self-citations are methodological and not load-bearing.

full rationale

The central result — the one-loop self-energy from a numerical exponential-basis Green function — is validated term-by-term against independent analytical Green-function calculations (Refs. [23], [28], and the private-communication values in Ref. [43]), not against the paper's own fitted parameters. The partial-wave tail is extrapolated by a polynomial fit in 1/|κ| to the authors' numerical data, but this is a standard convergence-acceleration procedure following Refs. [24] and [28], and the extrapolated value is not used to define the target quantity; the quoted uncertainty is estimated by varying the fitting window, which is a heuristic systematic-error estimate. A biased extrapolation would be a numerical accuracy problem, not a circular derivation. The self-citations (Refs. [32] and [35]) concern basis-set methodology and earlier numerical Green-function implementations; they do not supply the physical self-energy benchmark. The paper also explicitly borrows zero- and one-potential terms from independent calculations (Refs. [23] and [28]) when assembling the total correction, which is borrowing external input rather than assuming the conclusion. No equation reduces to its own input, no fitted parameter is renamed as a prediction, and no load-bearing uniqueness claim is imported from the authors' prior work.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The calculation rests on established QED bound-state formalism and on numerical convergence assumptions (basis-set completeness, tail extrapolation). No new physical entities or fitted physical constants are introduced.

free parameters (3)
  • Exponential basis exponents α_μ = Not specified (pseudorandomly generated over intervals)
    These nonlinear parameters define the basis functions (Eq. 24) and determine the accuracy of the numerical Green function; values are chosen by hand/heuristics rather than fitted to the self-energy.
  • Partial-wave tail fit coefficients (polynomial in 1/|κ|) = Not given
    Coefficients of the least-squares fit to partial-wave contributions for |κ|≥16 (Z=92) and |κ|≥21 (Z=1 Feynman) or |κ|≥11 (Coulomb), used to extrapolate the sum to infinity.
  • Infinite-basis extrapolation parameters (a, b, c) = Not given
    Power-law fit a + b/n_G^c to results at n_G=240, 360, 480, 600 for |κ|=10–15 at Z=1; used to reduce basis-size error.
assumptions (5)
  • domain assumption The potential expansion G = G0 + G0 V G0 + ... (Eq. 3) is convergent and the separation into zero-, one-, and many-potential terms is valid
    Standard bound-state QED perturbation theory; underpins the entire calculation.
  • domain assumption The finite-basis numerical Green function (Eq. 12) converges to the exact Dirac-Coulomb Green function as basis size increases
    Completeness of the basis set is assumed; the paper tests it via energy accuracy and sum rules in prior work [35].
  • domain assumption The subtraction scheme of Eq. (14) captures the slowly converging part of the partial-wave expansion
    Taken from Sapirstein & Cheng [24]; its validity is assumed.
  • ad hoc to paper The partial-wave tail can be extrapolated by a polynomial in 1/|κ| (Sec. V)
    A numerical modeling assumption; uncertainty is estimated by varying the fit range, but the functional form is not derived.
  • domain assumption Renormalized expressions for ΔE0_SE and ΔE1_SE (Eqs. 4-5) from Refs. [22,28] are correct
    The paper does not re-derive them; they are standard and independent of the numerical Green function.

how reviews work

0 comments
Cite this review

Pith. "Pith review of One-loop self-energy using a numerical Green function." pith.science (2026). https://pith.science/paper/JQAWVG3E

@misc{pith2026260716026,
  author       = {Pith},
  title        = {Pith review of: One-loop self-energy using a numerical Green function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JQAWVG3E}},
  note         = {Machine review of arXiv:2607.16026}
}
abstract

We calculate the one-loop self-energy in hydrogenlike atoms using a numerical Green function obtained by solving the radial Dirac equation in an exponential basis set. The self-energy correction in the ground state of hydrogenlike uranium is obtained with about $10^{-5}$ relative uncertainty in the Feynman gauge. Using a convergence acceleration scheme, we extend our calculations to the region of low nuclear charges. Our results allow calculating the self-energy correction for the hydrogen atom with $10^{-4}$ relative uncertainty. Calculations in the Coulomb gauge are also presented, improving the precision to $10^{-5}$. Present limitations and possible improvements of our method are discussed.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references

  1. [1]

    (20) leads to the following matrix representation of the Dirac equation: VLL ΠLS ΠSL VSS −2c2SSS A B = SLL 0 0 S SS A B ,(A1) 9 ∆E0 SE (Ref

    NKB Injecting the expressions (21) in Eq. (20) leads to the following matrix representation of the Dirac equation: VLL ΠLS ΠSL VSS −2c2SSS A B = SLL 0 0 S SS A B ,(A1) 9 ∆E0 SE (Ref. [28]) −168 176.156 251 ∆E1 SE (Ref. [28]) 148 579.466 946 Without subtr. With subtr. ∆E2+ substr 9 370.581 414 |κ|= 1 19 507.536 257 10 234.905 709 2 58.961 982 1.345 695 3 1...

  2. [2]

    RKB The RKB relationship (22) leads to the matrix equa- tion: VLL TLL TLL 1 4c2 WLL −TLL A B = SLL 0 0 1 2c2 TLL A B , (A3) Term This work Anal.[28] Diff. ∆E0 SE 13.849 474 1 ∆E1 SE −2.879 681 6 ∆E2+ substr −1.127 787 5 −1.127 787 5 0 |κ|= 1 0.475 589 3 0.475 625 7 −3.6[−5] 2 −0.004 692 5 −0.004 723 6 3.1[−5] 3 0.001 965 1 0.001 962 3 2.8[−6] 4 0.000 790 ...

  3. [3]

    dπS µ dx − κ x πS µ # V(x) dπS ν dx − κ x πS ν dx, (A6b) [WLS a ]µν = Z ∞ 0 πL µ V(x) dπS ν dx − κ x πS ν dx + Z ∞ 0

    DKB With the DKB expansion (23), the matrix form of the Dirac equation is TLL +V LL + 1 4c2 WLL 1 2c WLS a + WLS b 1 2c WSL a + WSL b VSS −2TSS + 1 4c2 WSS −2c2SSS × A B = SLL + 1 2c2 TLL 0 0 S SS + 1 2c2 TSS A B , (A5) with [TSS ]µν =− 1 2 Z ∞ 0 πS µ (x) d2 dx2 − κ(κ−1) x2 πS ν (x)dx, (A6a) [WSS ]µν = Z ∞ 0 " dπS µ dx − κ x πS µ # V(x) dπS ν dx − κ x πS ...

  4. [4]

    H. A. Bethe, Phys. Rev.72, 339 (1947)

    1947

  5. [5]

    W. E. Lamb and R. C. Retherford, Phys. Rev.72, 241 (1947)

    1947

  6. [6]

    M. I. Eides, H. Grotch, and V. A. Shelyuto,Theory of Light Hydrogenic Bound States, vol. 222 (Springer- Verlag, Berlin, 2007)

    2007

  7. [7]

    V. A. Yerokhin, K. Pachucki, and V. Patkos, Annalen Phys.531, 1800324 (2019)

    2019

  8. [8]

    P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, Rev. Mod. Phys.97, 025002 (2025)

    2025

Show all 48 references
  1. [9]

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

    1998

  2. [10]

    Indelicato, J

    P. Indelicato, J. Phys. B52, 232001 (2019)

    2019

  3. [11]

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

    1999

  4. [12]

    U. D. Jentschura, P. J. Mohr, and G. Soff, Phys. Rev. A 63, 042512 (2001)

    2001

  5. [13]

    U. D. Jentschura, A. Czarnecki, and K. Pachucki, Phys. Rev. A72, 062102 (2005)

    2005

  6. [14]

    S. G. Karshenboim, Z. Phys. D39, 109 (1997), hep- ph/9608462

    1997

  7. [15]

    Patra, M

    S. Patra, M. Germann, J.-Ph. Karr, M. Haidar, L. Hilico, V. I. Korobov, F. M. J. Cozijn, K. S. E. Eikema, W. Ubachs, and J. C. J. Koelemeij, Science369, 1238 (2020)

    2020

  8. [16]

    Alighanbari, I

    S. Alighanbari, I. V. Kortunov, G. S. Giri, and S. Schiller, Nat. Phys.19, 1263 (2023)

    2023

  9. [17]

    V. I. Korobov, L. Hilico, and J.-Ph. Karr, Phys. Rev. Lett.118, 233001 (2017)

    2017

  10. [18]

    V. I. Korobov and J.-Ph. Karr, Phys. Rev. A104, 032806 (2021)

    2021

  11. [19]

    V. I. Korobov, L. Hilico, and J.-Ph. Karr, Phys. Rev. Lett.112, 103003 (2014)

    2014

  12. [20]

    V. I. Korobov, L. Hilico, and J.-Ph. Karr, Phys. Rev. A 89, 032511 (2014)

    2014

  13. [21]

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

    1974

  14. [22]

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

    1974

  15. [23]

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

    1991

  16. [24]

    S. A. Blundell and N. J. Snyderman, Phys. Rev. A44, R1427 (1991)

    1991

  17. [25]

    V. A. Yerokhin and V. M. Shabaev, Phys. Rev. A60, 800 (1999)

    1999

  18. [26]

    V. A. Yerokhin, K. Pachucki, and V. M. Shabaev, Phys. Rev. A72, 042502 (2005)

    2005

  19. [27]

    Sapirstein and K

    J. Sapirstein and K. T. Cheng, Phys. Rev. A108, 042804 (2023)

    2023

  20. [28]

    G. S. Adkins, Phys. Rev. D27, 1814 (1983)

    1983

  21. [29]

    G. S. Adkins, Phys. Rev. D34, 2489 (1986)

    1986

  22. [30]

    Hedendahl and J

    D. Hedendahl and J. Holmberg, Phys. Rev. A85, 012514 (2012)

    2012

  23. [31]

    V. A. Yerokhin, Z. Harman, and C. H. Keitel, Phys. Rev. A111, 012802 (2025)

    2025

  24. [32]

    V. A. Yerokhin and A. V. Maiorova, Symmetry12, 800 (2020)

    2020

  25. [33]

    K. T. Cheng, W. R. Johnson, and J. Sapirstein, Phys. Rev. A47, 1817 (1993)

    1993

  26. [34]

    V. M. Shabaev, I. I. Tupitsyn, V. A. Yerokhin, G. Plu- nien, and G. Soff, Phys. Rev. Lett.93, 130405 (2004). 11

    2004

  27. [35]

    Ferenc, M

    D. Ferenc, M. Salman, and T. Saue, Phys. Rev. A111, L040802 (2025)

    2025

  28. [36]

    A. N. Artemyev and A. Surzhykov, Phys. Rev. Lett.114, 243004 (2015)

    2015

  29. [37]

    A. N. Artemyev, A. Surzhykov, and V. A. Yerokhin, Phys. Rev. A106, 012813 (2022)

    2022

  30. [38]

    H. D. Nogueira and J.-Ph. Karr, Phys. Rev. A107, 042817 (2023)

    2023

  31. [39]

    H. D. Nogueira, Ph.D. thesis, Sorbonne Universit´ e (2024)

    2024

  32. [40]

    R. E. Stanton and S. Havriliak, The Journal of Chemical Physics81, 1910 (1984)

    1910

  33. [41]

    V. I. Korobov, Phys. Rev. A85, 042514 (2012)

    2012

  34. [42]

    D. H. Bailey,Mpfun2020: A thread-safe arbitrary preci- sion package with special functions (full documentation), https://www.davidhbailey.com/dhbpapers/mpfun2020.pdf (2024)

    2024

  35. [43]

    D. H. Bailey, K. Jeyabalan, and X. S. Li, Experimental Mathematics14, 317 (2005)

    2005

  36. [44]

    T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms(The MIT Press, 2022)

    2022

  37. [45]

    I. I. Tupitsyn and V. M. Shabaev, Optics and Spec- troscopy105, 183 (2008)

    2008

  38. [46]

    V. A. Yerokhin, (private communication) (2025)

    2025

  39. [47]

    Q. Sun, W. Liu, and W. Kutzelnigg, Theoretical Chem- istry Accounts129, 423 (2011)

    2011

  40. [48]

    Salman and T

    M. Salman and T. Saue, Symmetry12, 1121 (2020)

    2020

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.