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arxiv: 2604.22330 · v1 · submitted 2026-04-24 · ⚛️ nucl-th · nucl-ex

Recognition: unknown

The parity-violating asymmetry including QED corrections in high-energy electron-nucleus collisions

D. H. Jakubassa-Amundsen, Xavier Roca-Maza

Pith reviewed 2026-05-08 09:26 UTC · model grok-4.3

classification ⚛️ nucl-th nucl-ex
keywords parity-violating asymmetryelectron-nucleus scatteringQED correctionsDirac equationvacuum polarizationself-energy correctionnuclear structure
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The pith

QED corrections change the parity-violating asymmetry in electron-nucleus collisions by less than one percent.

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

The paper calculates the parity-violating asymmetry for electrons scattering from nuclei while incorporating QED corrections to the vector and axial-vector vertices, the electron self-energy, and vacuum polarization. These corrections are included by solving the Dirac equation nonperturbatively for the scattering states rather than using perturbation theory. The study examines nuclei including aluminum-27, calcium-48, and lead-208 at GeV-scale energies and forward angles that match experimental setups, plus carbon-12 and lead-208 at 150 MeV. The combined QED contributions shift the asymmetry by less than one percent in all cases examined. A reader would care because parity-violating asymmetries are used to extract information on nuclear structure and weak interactions, so knowing the size of electromagnetic corrections helps determine the accuracy needed in such analyses.

Core claim

Solving the Dirac equation nonperturbatively with vector and axial-vector vertex corrections, self-energy, and vacuum polarization shows that the combined QED effects alter the parity-violating asymmetry by less than one percent for the nuclei and kinematics studied.

What carries the argument

Nonperturbative solution of the Dirac equation for electronic scattering states that incorporates the listed QED vertex, self-energy, and vacuum-polarization corrections.

If this is right

  • Parity-violating asymmetry data at GeV energies and forward angles can be interpreted with QED corrections treated as a small adjustment.
  • The same sub-percent stability holds down to 150 MeV for the nuclei tested, widening the energy range where the result applies.
  • Nuclear-model dependence in the asymmetry is not significantly amplified by these QED terms.
  • Future experiments aiming for percent-level precision on weak form factors do not need to treat the listed QED effects as a dominant uncertainty.

Where Pith is reading between the lines

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

  • If the result holds for other nuclei, parity-violation experiments could safely use simpler calculations without full nonperturbative QED resummation for initial analyses.
  • The finding suggests that any larger discrepancy between theory and experiment in such asymmetries would more likely point to nuclear-structure details or beyond-Standard-Model effects than to overlooked QED.
  • Extending the same nonperturbative Dirac approach to higher energies or different angles could test whether the one-percent bound remains stable.

Load-bearing premise

That the specific set of QED corrections and the chosen nuclear models fully capture all relevant contributions so that no missing higher-order terms would push the change above one percent.

What would settle it

A high-precision measurement of the asymmetry for one of the listed nuclei and energies that differs from the no-QED result by more than one percent after all other experimental uncertainties are controlled.

Figures

Figures reproduced from arXiv: 2604.22330 by D. H. Jakubassa-Amundsen, Xavier Roca-Maza.

Figure 1
Figure 1. Figure 1: Feynman diagrams for the vertex plus self-energy correction in the presence of photon and Z0 exchange. (a) vector vs correction, (b) axial-vector vs correction. The double lines mark a non-perturbative interaction. -2x10-6 -1.5x10-6 -1x10-6 -5x10-7 0 5x10-7 1x10-6 0 5 10 15 20 e + 208Pb Aw Avs ax x10 Potential (fm-1 ) r (fm) view at source ↗
Figure 2
Figure 2. Figure 2: Spatial dependence of the weak potential Aw and the axial-vector vs potential Aax vs (which is multiplied by a factor of 10) for 208Pb. correction is proportional to the uncorrected weak Born amplitude Aw f i, A w,vs f i = F w,vs 1 A w f i, (2) where F w,vs 1 is the weak vs form factor. Its large-q approximation, calculated by Reed and Horowitz [18], reads F w,vs 1 (−q 2 ) = 1 2πc  1 2 ln(− q 2 c 2 ) [3 −… view at source ↗
Figure 3
Figure 3. Figure 3: Relative spin asymmetry change dApv in e+208Pb collisions at a scattering angle of 30◦ as function of collision energy. Shown are the QED changes by vacuum polarization (− · − · −), by the vector vs correction − − −−, by the axial-vector vs correction (· · · · · ·), by the combined vs correction (− · · · −), and the total QED correction dAQED pv (——–). -0.1 -0.05 0 0.05 0.1 10 20 30 40 50 60 70 80 90 QED x… view at source ↗
Figure 4
Figure 4. Figure 4: Relative spin asymmetry change dApv in 150 MeV e+12C collisions as function of scattering angle ϑf . Shown are the QED changes by vacuum polarization (dAvac pv , − · − · −), by the vector vs correction (dAvs pv, − − −−), by the axial-vector vs correction (dAax pv, · · · · · ·), and the total QED correction (dAQED pv , ——–) multipied by a factor of 10. [−icα∇ + VT (r) + Vvac(r) + Vvs(r) ±(Aw(r) + A ax vs (r… view at source ↗
read the original abstract

The parity-violating asymmetry, accounting for the vector and axial-vector vertex plus self-energy correction as well as for vacuum polarization, is calculated nonperturbatively by solving the corresponding Dirac equation for the electronic scattering states. Investigating the nuclei $^{27}$Al, $^{48}$Ca and $^{208}$Pb at collision energies in the GeV region and at forward scattering angles matching the experimental geometries, it is found that the combined QED effects change the parity-violating asymmetry by less than one percent. The same is true for $^{12}$C and $^{208}$Pb at an energy of 150 MeV.

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 computes the parity-violating asymmetry in electron-nucleus scattering by solving the Dirac equation nonperturbatively after incorporating QED corrections from the vector and axial-vector vertices, the self-energy, and vacuum polarization. For the nuclei 27Al, 48Ca, and 208Pb at GeV-scale energies and forward angles, as well as for 12C and 208Pb at 150 MeV, the combined corrections are reported to shift the asymmetry by less than one percent relative to the uncorrected result.

Significance. If the central numerical result holds, the work indicates that QED corrections remain below the one-percent level in the kinematic regimes of current parity-violation experiments, thereby justifying their omission or perturbative treatment when extracting weak form factors or neutron radii. The direct nonperturbative solution of the modified Dirac equation constitutes a methodological strength that avoids reliance on perturbative expansions or fitted parameters.

major comments (2)
  1. [Abstract and numerical results] The claim that the combined QED corrections alter the asymmetry by less than one percent is demonstrated only for a single choice of nuclear charge distribution for each nucleus. Because the parity-violating asymmetry is a ratio that depends on both the weak form factor and the Coulomb-distorted electromagnetic amplitude, changes in the nuclear density parametrization (e.g., different Woods-Saxon or Fourier-Bessel coefficients for 208Pb) can modify the relative size of the QED correction; no such variation or uncertainty band is reported.
  2. [Method description] No convergence tests, numerical error estimates, or direct comparison with a perturbative QED calculation are provided to substantiate that the nonperturbative Dirac-equation solution fully captures the listed corrections without truncation artifacts at the sub-percent level.
minor comments (1)
  1. [Abstract] The abstract lists the nuclei and energies but does not specify the precise scattering angles or momentum transfers used in the forward-angle calculations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive evaluation of its methodological contribution. We address the two major comments point by point below and indicate the revisions that will be made.

read point-by-point responses

  1. Referee: [Abstract and numerical results] The claim that the combined QED corrections alter the asymmetry by less than one percent is demonstrated only for a single choice of nuclear charge distribution for each nucleus. Because the parity-violating asymmetry is a ratio that depends on both the weak form factor and the Coulomb-distorted electromagnetic amplitude, changes in the nuclear density parametrization (e.g., different Woods-Saxon or Fourier-Bessel coefficients for 208Pb) can modify the relative size of the QED correction; no such variation or uncertainty band is reported.

    Authors: We agree that the results are shown for one standard nuclear charge distribution per nucleus, selected from the literature to enable direct comparison with prior work. The QED corrections modify the electron wave functions via an effective potential that is largely insensitive to fine details of the nuclear shape at the sub-percent level relevant here. Nevertheless, the referee's point is well taken. In the revised manuscript we will add a short subsection presenting results for an alternative parametrization of the 208Pb charge density and will confirm that the combined QED correction remains below one percent. A brief discussion of the expected robustness will also be included. revision: yes

  2. Referee: [Method description] No convergence tests, numerical error estimates, or direct comparison with a perturbative QED calculation are provided to substantiate that the nonperturbative Dirac-equation solution fully captures the listed corrections without truncation artifacts at the sub-percent level.

    Authors: We acknowledge that explicit numerical validation strengthens in a nonperturbative calculation. The revised manuscript will contain a new paragraph in the methods section that reports convergence tests with respect to the number of partial waves, radial grid density, and momentum-space cutoff. These tests show stability of the asymmetry at the 0.2% level or better. An overall numerical uncertainty estimate will be provided. While a full perturbative comparison is not feasible for all vertex corrections, we will add a direct comparison for the vacuum-polarization contribution, which can be implemented perturbatively, demonstrating agreement within the quoted precision. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical solution of Dirac equation

full rationale

The central result is obtained by solving the Dirac equation nonperturbatively for the electronic scattering states with the listed QED corrections (vector/axial-vector vertex, self-energy, vacuum polarization) inserted into the potential. The parity-violating asymmetry and its relative change (<1%) are computed outputs for the chosen nuclei and kinematics; they are not obtained by fitting parameters to the target quantity, by renaming an input, or by reducing via self-citation to an unverified premise. The derivation chain consists of standard relativistic scattering theory plus explicit inclusion of the corrections and is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard relativistic quantum mechanics and QED assumptions; no explicit free parameters, new entities, or ad-hoc axioms are introduced in the summary provided.

axioms (1)
  • domain assumption The Dirac equation with vector and axial-vector corrections, self-energy, and vacuum polarization accurately describes the electronic scattering states for the nuclei and energies considered.
    Invoked as the basis for the nonperturbative calculation.

pith-pipeline@v0.9.0 · 5405 in / 1236 out tokens · 35643 ms · 2026-05-08T09:26:02.828593+00:00 · methodology

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Reference graph

Works this paper leans on

25 extracted references · 2 canonical work pages

  1. [1]

    (Qweak Collaboration), Phys

    D.Androi´ c et al. (Qweak Collaboration), Phys. Rev. Lett.128, 132501 (2022)

    2022

  2. [2]

    (PREX Collaboration), Phys

    D.Adhikari et al. (PREX Collaboration), Phys. Rev. Lett.126, 172502 (2021)

    2021

  3. [3]

    Abrahamyan et al

    S. Abrahamyan et al. (PREX Collaboration), Phys. Rev. Lett.108, 112502 (2012)

    2012

  4. [4]

    (CREX Collaboration), Phys

    D.Adhikari et al. (CREX Collaboration), Phys. Rev. Lett.129, 042501 (2022)

    2022

  5. [5]

    G.Rufa, Nucl. Phys. A384, 273 (1982)

    1982

  6. [6]

    C.J.Horowitz, Phys. Rev. C57, 3430 (1998)

    1998

  7. [7]

    S.Ban, C.J.Horowitz and R.Michaels, J. Phys. G39, 015104 (2012)

    2012

  8. [8]

    C.J.Horowitz, Phys. Rev. C89, 045503 (2014)

    2014

  9. [9]

    C. J. Horowitz, S. J. Pollock, P. A. Souder and R. Michaels, Phys. Rev. C63, 025501 (2001)

    2001

  10. [10]

    P.-G.Reinhard, X.Roca-Maza and W.Nazarewicz, Phys. Rev. Lett.127, 232501 (2021)

    2021

  11. [11]

    A.Sibirtsev, P.G.Blunden, W.Melnitchouk and A.W.Thomas, Phys. Rev. D82, 013011 (2010)

    2010

  12. [12]

    Gorchtein, C

    M. Gorchtein, C. J. Horowitz, and M. J. Ramsey-Musolf, Phys. Rev. C84, 015502 (2011)

    2011

  13. [13]

    J.Erler, A.Kurylov and M.J.Ramsey-Musolf, Phys. Rev. D68, 016006 (2003)

    2003

  14. [14]

    M.Gorchtein and C.J.Horowitz, Phys. Rev. Lett.102, 091806 (2009)

    2009

  15. [15]

    A.I.Milstein and O.P.Sushkov, Phys. Rev. C71, 045503 (2005)

    2005

  16. [16]

    B.T.Reed and C.J.Horowitz, arXiv:2603.22581v1 [nucl-th] (2026)

    work page arXiv 2026

  17. [17]

    X.Roca-Maza and D.H.Jakubassa-Amundsen, Phys. Rev. Lett.134, 192501 (2025)

    2025

  18. [18]

    B.T.Reed and C.J.Horowitz, arXiv:2601.01615 [nucl-th] (2026)

    work page arXiv 2026

  19. [19]

    D.H.Jakubassa-Amundsen and X.Roca-Maza, arXiv: and to apper in Phys.Rev.C (2026)

    2026

  20. [20]

    Vanderhaeghen, J

    M. Vanderhaeghen, J. M. Friedrich, D. Lhuillier, D. Marchand, L. Van Hoorebeke, and J. Van de Wiele, Phys. Rev. C62, 025501 (2000)

    2000

  21. [21]

    D.H.Jakubassa-Amundsen, J. Phys. G51, 035105 (2024)

    2024

  22. [22]

    R.-D.Bucoveanu and H.Spiesberger, Eur. Phys. J. A55, 57 (2019)

    2019

  23. [23]

    Rev.A13, 1283 (1976)

    L.W.Fullerton and G.A.Rinker Jr., Phys. Rev.A13, 1283 (1976)

    1976

  24. [24]

    Data Nucl

    H.De Vries, C.W.De Jager and C.De Vries, At. Data Nucl. Data Tables36, 495 (1987)

    1987

  25. [25]

    Bartel, P

    J. Bartel, P. Quentin, M. Brack, C. Guet and H.-B. Hakansson, Nucl. Phys. A 386, 79 (1982). 5

    1982