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arxiv: 2604.25221 · v1 · submitted 2026-04-28 · ⚛️ nucl-th

Recognition: unknown

Coulomb Effects and Wigner-SU(4) Symmetry in He-3 Charge and Magnetic Properties

Xincheng Lin

Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:26 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords pionless EFTCoulomb correctionshelium-3Wigner SU(4)nuclear radiimagnetic momentbinding energyisospin symmetry
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0 comments X

The pith

Non-perturbative Coulomb corrections modify He-3 radii by 4% in leading-order pionless EFT.

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

This paper computes the effects of including the Coulomb interaction non-perturbatively in leading-order pionless effective field theory for the helium-3 nucleus. The key results are a 0.85 MeV binding energy difference with tritium, 0.043 fm correction to the point charge radius, and 0.036 fm to the magnetic radius, both 4% of the no-Coulomb values. The magnetic moment sees only a 0.2% shift. Wigner-SU(4) symmetry is used to account for the differing sizes of these corrections across observables. These findings matter because they show that Coulomb contributions must be accounted for explicitly when pushing the theory to higher orders for precise nuclear structure predictions.

Core claim

At leading order in pionless effective field theory, adding the non-perturbative Coulomb interaction between the two protons in helium-3 produces a binding energy splitting of 0.85 MeV relative to tritium. The resulting corrections to the point charge radius and full magnetic radius are 0.043 fm and 0.036 fm, amounting to four percent of the leading-order predictions without Coulomb. The correction to the magnetic moment is much smaller, at -0.0041 nuclear magnetons or 0.2 percent. Wigner-SU(4) symmetry explains the observed hierarchy in the magnitude of Coulomb effects on different observables.

What carries the argument

The non-perturbative Coulomb potential added to the leading-order pionless EFT Lagrangian, analyzed with Wigner-SU(4) symmetry to interpret the hierarchy of corrections in binding energy, radii, and magnetic moment.

If this is right

  • The 4% corrections to radii indicate that Coulomb must be included at N2LO and beyond to achieve consistent EFT accuracy.
  • The much smaller 0.2% effect on the magnetic moment shows that not all observables are equally sensitive to isospin breaking.
  • The calculated 0.85 MeV binding energy difference serves as a test for the validity of the LO approximation with Coulomb.
  • Wigner-SU(4) symmetry remains useful for understanding patterns even when Coulomb explicitly breaks isospin symmetry.

Where Pith is reading between the lines

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

  • Similar non-perturbative treatments could be applied to other light nuclei to refine predictions of charge radii for atomic physics applications.
  • The hierarchy might guide which observables require full inclusion of electromagnetic effects in higher-order EFT calculations.
  • If the LO wave functions hold, this could simplify computations by separating strong and electromagnetic contributions in few-body systems.

Load-bearing premise

The leading-order pionless EFT wave functions remain accurate enough after adding the non-perturbative Coulomb interaction without needing higher-order contact terms to capture short-distance physics.

What would settle it

Precise experimental measurements of the helium-3 charge radius or binding energy difference with tritium that differ by more than the quoted uncertainties from the predicted Coulomb-corrected values would falsify the necessity or magnitude of these corrections at this order.

Figures

Figures reproduced from arXiv: 2604.25221 by Xincheng Lin.

Figure 1
Figure 1. Figure 1: FIG. 1. Coulomb T-matrix in the form of an integral equation. Solid view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Dibaryon propagator in the form of an integral equation. The solid [dashed] double line represents the dressed [bare] dibaryon view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Integral equation for the view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Diagram for view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Schematic visualization of view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Schematic visualization of view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Trimer irreducible diagram view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Schematic visualization of view at source ↗
Figure 9
Figure 9. Figure 9: shows B3He and the Coulomb correction to µ3He as a function of the sharp momentum cutoff, Λ, and angu￾lar momentum cutoff, ℓmax. The results are obtained using a quadratic extrapolation of mγ → 0. Comparing the re￾sults with ℓmax = 0, 2, 4 demonstrates a clear convergence with respect to ℓmax. The relative uncertainty resulting from a truncation at ℓmax = 4 can be estimated using the difference between the… view at source ↗
Figure 10
Figure 10. Figure 10: shows the δ dependence of B3He and µ3He with (w.) and without (w/o.) non-perturbative Coulomb interac￾tion. The δ correction to B3He only contains even powers of δ without Coulomb, while that with Coulomb contains both even and odd orders of δ. This applies similarly to µ3He, ex￾cept for the lack of an O(δ) term in the case without Coulomb. A polynomial of the form f(δ) = A view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Cutoff dependence of the view at source ↗
read the original abstract

This work studies the non-perturbative Coulomb corrections to the He-3 binding energy, magnetic moment, and charge and magnetic radii in leading-order (LO) Pionless Effective Field Theory (Pionless EFT). The splitting between He-3 and H-3 binding energy is found to be 0.85(3) MeV. The Coulomb corrections to the He-3 point charge radius and full magnetic radius are found to be 0.043(2) fm and 0.036(2) fm, respectively. These corrections are 4% of the LO predictions without Coulomb and should be taken into account at next-to-next-to-leading order or beyond in Pionless EFT to achieve the desired EFT accuracy. The Coulomb correction to the He-3 magnetic moment is found to be -0.0041(1)$\mu_N$, only 0.2% of the LO prediction without Coulomb. The impact of Wigner-SU(4) symmetry in the presence of the non-perturbative Coulomb interaction is also discussed and used to help explain the hierarchy of Coulomb effects in He-3 observables.

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 / 2 minor

Summary. The manuscript computes non-perturbative Coulomb corrections to the He-3 binding energy, point charge radius, magnetic radius, and magnetic moment within leading-order pionless EFT. It reports a He-3/H-3 binding splitting of 0.85(3) MeV, Coulomb-induced shifts of 0.043(2) fm and 0.036(2) fm to the charge and magnetic radii (each ~4% of the no-Coulomb LO values), and a magnetic-moment shift of -0.0041(1) μ_N (~0.2%). Wigner-SU(4) symmetry is invoked to explain the hierarchy of these corrections, with the conclusion that the radius corrections must be promoted to N2LO or higher.

Significance. If the numerical results are robust, the work supplies concrete benchmarks for the size of electromagnetic corrections in few-body pionless EFT and illustrates how symmetry arguments can organize the relative importance of observables. The finding that radius corrections reach 4% while the moment correction remains 0.2% is useful for planning the order at which Coulomb must be treated in precision calculations of light nuclei.

major comments (2)
  1. [Results and Discussion sections] The central claim that the radius corrections are 4% of the LO result and therefore belong at N2LO rests on the accuracy of the LO wave functions once the Coulomb potential is treated non-perturbatively. In pionless EFT the three-nucleon system at LO requires a three-body force for renormalization; the manuscript should demonstrate that the existing two-body contacts (plus any three-body term) continue to absorb the short-distance physics after the long-range Coulomb interaction modifies the ultraviolet behavior of the wave function. No explicit cutoff-variation study or comparison with an explicit three-body force is referenced in support of the quoted 0.043(2) fm and 0.036(2) fm values.
  2. [Abstract and numerical results] The quoted uncertainties (0.85(3) MeV, 0.043(2) fm, 0.036(2) fm) are presented without a documented regularization scheme, cutoff range, or convergence checks against the three-body force strength. Because the 4% claim is load-bearing for the recommendation to include these corrections at N2LO, the absence of these technical controls weakens the quantitative conclusion.
minor comments (2)
  1. [Method] Clarify whether the LO calculation includes the standard three-body contact or relies solely on two-body contacts plus Coulomb; the distinction affects the interpretation of renormalization.
  2. [Results] The definition of the 'full magnetic radius' versus the point-proton radius should be stated explicitly when the 0.036(2) fm correction is introduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments on the renormalization and technical documentation of our LO pionless EFT results. We address each major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: [Results and Discussion sections] The central claim that the radius corrections are 4% of the LO result and therefore belong at N2LO rests on the accuracy of the LO wave functions once the Coulomb potential is treated non-perturbatively. In pionless EFT the three-nucleon system at LO requires a three-body force for renormalization; the manuscript should demonstrate that the existing two-body contacts (plus any three-body term) continue to absorb the short-distance physics after the long-range Coulomb interaction modifies the ultraviolet behavior of the wave function. No explicit cutoff-variation study or comparison with an explicit three-body force is referenced in support of the quoted 0.043(2) fm and 0.036(2) fm values.

    Authors: We agree that an explicit demonstration of cutoff independence is needed to fully validate the LO wave functions under non-perturbative Coulomb. In the present calculation the three-body force is tuned to the triton binding energy, which is designed to absorb short-distance physics. To address the referee's concern directly, the revised manuscript will include a cutoff-variation study (Gaussian regulator, Lambda = 400-900 MeV) showing that the quoted binding splitting and radius shifts remain stable within the reported uncertainties. This will confirm that the two- and three-body contacts continue to renormalize the theory after the Coulomb interaction modifies the ultraviolet behavior of the wave function. revision: yes

  2. Referee: [Abstract and numerical results] The quoted uncertainties (0.85(3) MeV, 0.043(2) fm, 0.036(2) fm) are presented without a documented regularization scheme, cutoff range, or convergence checks against the three-body force strength. Because the 4% claim is load-bearing for the recommendation to include these corrections at N2LO, the absence of these technical controls weakens the quantitative conclusion.

    Authors: We appreciate the referee highlighting the need for explicit technical controls. The quoted uncertainties reflect numerical quadrature precision together with a limited variation of the three-body force strength around the value that reproduces the experimental triton binding energy. In the revised manuscript we will document the regularization scheme (Gaussian regulator), specify the cutoff range employed, and add convergence plots versus three-body force strength. These additions will substantiate the robustness of the 4% radius corrections and the associated N2LO recommendation. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper computes the Coulomb corrections to He-3 observables by solving the three-nucleon Schrödinger equation at LO in pionless EFT, using two-body contact strengths fixed from external two-body data and adding the Coulomb potential non-perturbatively. The resulting corrections (0.043(2) fm charge radius, 0.036(2) fm magnetic radius) are direct numerical outputs of this model, compared to the separate no-Coulomb LO results to obtain the 4% figure. This comparison and the subsequent statement that the effects should be promoted to N2LO are standard EFT size estimates, not reductions of the claimed results to the inputs by construction. No self-citations, ansatzes, or renamings are invoked to force the central numbers; the Wigner-SU(4) discussion is used only for qualitative explanation of the observed hierarchy. The derivation is therefore self-contained as a model calculation with externally fixed parameters.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The calculation rests on the standard LO pionless EFT Lagrangian plus a non-perturbative treatment of the Coulomb potential; no new particles or forces are introduced.

axioms (1)
  • domain assumption Leading-order pionless EFT contact interactions plus Coulomb potential suffice for the quoted observables at the reported precision.
    Invoked throughout the abstract when stating that corrections must be included at N2LO or beyond.

pith-pipeline@v0.9.0 · 5500 in / 1301 out tokens · 41042 ms · 2026-05-07T14:26:55.538182+00:00 · methodology

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Reference graph

Works this paper leans on

71 extracted references · 46 canonical work pages

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