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arxiv: 2605.19062 · v1 · pith:4U2XG2C2new · submitted 2026-05-18 · ⚛️ nucl-th

Coulomb Corrections to Three-Nucleon Moments

Ha S. Nguyen , Jared Vanasse This is my paper

Pith reviewed 2026-05-20 07:14 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords pionless effective field theoryCoulomb correctionsthree-nucleon magnetic momentGamow-Teller matrix elementtriton beta decayproton-proton fusionWigner SU(4) symmetrylow-energy constants
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The pith

In pionless EFT the O(alpha) Coulomb corrections to the 3He magnetic moment and triton Gamow-Teller matrix element are only 0.18 percent and 0.08 percent.

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

The paper computes the magnetic moment of helium-3 and the Gamow-Teller matrix element for tritium beta decay in pionless effective field theory through next-to-leading order, treating the Coulomb interaction as a perturbative correction of order alpha. It reports the next-to-leading-order 3He moment as -2.130 nuclear magnetons with a Coulomb shift of 0.00335 and the leading-order Gamow-Teller strength as 0.9806 with a shift of -0.000740. These shifts are far smaller than the naive estimate of roughly 8 percent based on the three-nucleon binding momentum. The authors fit the relevant low-energy constants to two-nucleon data and the tritium lifetime, then predict the reduced matrix element for proton-proton fusion. They investigate the smallness of the corrections through a Wigner SU(4) expansion of the observables.

Core claim

Within pionless EFT the NLO 3He magnetic moment equals -2.130 nuclear magnetons after including an O(alpha) Coulomb correction of 0.00335; the LO GT matrix element for 3H beta decay equals 0.9806 after an O(alpha) correction of -0.000740. Both corrections are only 0.18 percent and 0.08 percent of their leading-order values. Fitting the axial LEC l1,A to the 3H half-life yields the prediction Lambda(0) = 2.776(331) for the pp-fusion reduced matrix element. The authors attribute the unexpectedly small electromagnetic shifts to the structure of the Wigner-SU(4) expansion.

What carries the argument

Perturbative O(alpha) inclusion of the Coulomb potential inside pionless EFT for three-nucleon observables, together with an expansion in Wigner SU(4) symmetry to diagnose the size of the corrections.

If this is right

  • The fitted axial LEC produces a definite numerical prediction for the pp-fusion matrix element that can be compared with other calculations.
  • The three-nucleon magnetic moments and GT strength at this order are stable against the leading electromagnetic perturbation.
  • Power counting in the presence of Coulomb forces appears to hold for these particular observables in the three-nucleon system.
  • The Wigner-SU(4) analysis supplies a symmetry-based reason why electromagnetic corrections stay small rather than reaching the naive 8 percent size.

Where Pith is reading between the lines

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

  • If the pattern of small corrections persists, similar perturbative treatments may work for other electromagnetic observables in light nuclei.
  • The result suggests that three-nucleon calculations can absorb the leading Coulomb effects without needing a full non-perturbative resummation for these quantities.
  • Extending the same framework to higher orders would test whether the suppression continues or eventually produces larger shifts.

Load-bearing premise

Coulomb corrections remain small enough to be treated perturbatively at order alpha without large higher-order electromagnetic pieces shifting the fitted low-energy constants.

What would settle it

A measurement of the 3He magnetic moment or 3H beta-decay strength that differs from the reported NLO values by more than the quoted uncertainty while using the same two-nucleon input.

Figures

Figures reproduced from arXiv: 2605.19062 by Ha S. Nguyen, Jared Vanasse.

Figure 1
Figure 1. Figure 1: [40] where diagram (a) is a counterterm to absorb the logarithmic divergence from (a) (b) FIG. 1. Diagram (b) shows the O(α) correction to the pp propagator and diagram (a) its associated counterterm. Time flows from left to right in all diagram in this work. diagram (b). Including these diagrams the sum of the LO and O(α) Coulomb correction to the pp propagator is given by iD¯ pp(p0, p) = iD¯ s(p0, p) (12… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Coulomb correction diagrams to the inhomogeneous term of the Coulomb corrected three [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Diagrams for the generic LO form factor in EFT( [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The NLO correction to the generic three-nucleon form factor. The circle with a “1” [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Diagrams for the [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Plot of comparison between full physical values of [PITH_FULL_IMAGE:figures/full_fig_p035_6.png] view at source ↗
read the original abstract

The Helium-3 (${}^3\mathrm{He}$) magnetic moment and Gamow-Teller (GT) matrix element in triton (${}^3\mathrm{H}$) $\beta$-decay are calculated in pionless effective field theory ($\mathrm{EFT}(/\!\!\!\pi)$) to next-to-leading order (NLO). Coulomb corrections are included perturbatively to $\mathcal{O}(\alpha)$ in this framework and should naively be $\alpha M_n/p^*\!\!\sim\!8\%$ corrections, where $p^*\!\!\sim\!88.5$ MeV is related to the three-nucleon binding momentum. Fitting the two-nucleon iso-vector magnetic current low-energy constant (LEC), $L_1$, to the ${}^3\mathrm{H}$ magnetic moment and the two-nucleon iso-scalar magnetic current LEC, $L_2$, to the deuteron magnetic moment we find the NLO ${}^3\mathrm{He}$ magnetic moment in units of nuclear magnetons is -2.130 and the surprisingly small $\mathcal{O}(\alpha)$ correction is 0.00335, $\approx\!0.18\%$ of the LO $\mathrm{EFT}(/\!\!\!\pi)$ prediction. The leading-order (LO) GT matrix element for ${}^3\mathrm{H}$ $\beta$-decay is 0.9806 while again it has a surprisingly small $\mathcal{O}(\alpha)$ Coulomb correction of $-0.000740$, $\approx\!0.08\%$ of the LO $\mathrm{EFT}(/\!\!\!\pi)$ prediction. At NLO we calculate the GT matrix element of ${}^3\mathrm{H}$ $\beta$-decay, including the $\mathcal{O}(\alpha)$ Coulomb correction, in terms of the two-nucleon axial current LEC $l_{1,A}$. Fitting $l_{1,A}$ to the ${}^3\mathrm{H}$ half-life we make a prediction for the proton-proton fusion reduced matrix element of $\Lambda(0)=2.776(331)$. Finally, we attempt to explain the unusually small size of the $\mathcal{O}(\alpha)$ corrections by investigating the Wigner-SU(4) expansion of these 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 paper computes the 3He magnetic moment and the Gamow-Teller matrix element for 3H beta decay in pionless EFT to NLO, including perturbative O(alpha) Coulomb corrections. L1 is fit to the 3H magnetic moment, L2 to the deuteron moment, and l1,A to the triton half-life; the resulting NLO 3He moment is -2.130 with a 0.00335 O(alpha) correction (~0.18% of LO), while the LO GT matrix element is 0.9806 with a -0.000740 correction (~0.08% of LO). A prediction Lambda(0)=2.776(331) for pp fusion is extracted after fitting, and the small corrections are attributed to Wigner SU(4) symmetry.

Significance. If the O(alpha) perturbative treatment holds, the work supplies concrete numerical evidence that electromagnetic corrections to these three-nucleon observables are far smaller than the naive alpha M_N/p* ~8% scale, together with an explicit Wigner-SU(4) analysis that may explain the suppression. The explicit NLO results and the pp-fusion prediction (standard once l1,A is fixed) add to the EFT literature on few-nucleon EM effects and solar-neutrino rates. The manuscript does not, however, supply error budgets or explicit higher-order checks, limiting immediate quantitative impact.

major comments (2)
  1. [Abstract] Abstract and the section presenting the O(alpha) results: the central claim that the Coulomb corrections are only 0.18% and 0.08% rests on treating the Coulomb interaction strictly perturbatively to O(alpha), yet the manuscript provides no explicit calculation or bound on O(alpha^2) or higher electromagnetic operators, even though the naive scale estimate alpha M_N/p* ~8% (with p*~88.5 MeV) is stated; this is load-bearing for the assertion that the corrections are 'surprisingly small'.
  2. [GT matrix element section] Section on the GT matrix element and pp-fusion prediction: Lambda(0) is obtained after fitting l1,A directly to the measured triton half-life, so the quoted prediction is constrained by the same observable used to determine the LEC; while this is conventional, the reliability of the small O(alpha) correction (-0.000740) and the extrapolated value both depend on the same perturbative assumption whose justification is not quantified beyond the Wigner-SU(4) discussion.
minor comments (2)
  1. [Abstract] The abstract refers to 'surprisingly small' corrections without a forward reference to the Wigner-SU(4) analysis that is invoked to explain them.
  2. Numerical results are given without accompanying uncertainty estimates or convergence tables for the EFT expansion; adding these would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us improve the presentation and justification of our results. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the section presenting the O(alpha) results: the central claim that the Coulomb corrections are only 0.18% and 0.08% rests on treating the Coulomb interaction strictly perturbatively to O(alpha), yet the manuscript provides no explicit calculation or bound on O(alpha^2) or higher electromagnetic operators, even though the naive scale estimate alpha M_N/p* ~8% (with p*~88.5 MeV) is stated; this is load-bearing for the assertion that the corrections are 'surprisingly small'.

    Authors: We agree that an explicit bound or calculation of O(alpha^2) corrections would provide additional support for the perturbative treatment. In the pionless EFT, the leading electromagnetic corrections enter at O(alpha), with higher-order EM operators suppressed by further powers of the expansion parameter Q/M_N ~ 1/3. The Wigner SU(4) symmetry analysis already presented in the manuscript explains the suppression of the O(alpha) terms relative to the naive estimate. We have revised the abstract and the relevant results section to include a brief power-counting estimate showing that O(alpha^2) contributions are expected to be O(alpha * Q/M_N) ~ 2-3% of the computed O(alpha) correction, which does not alter the conclusion that the corrections remain small. This addition quantifies the assumption without performing a full higher-order computation. revision: partial

  2. Referee: [GT matrix element section] Section on the GT matrix element and pp-fusion prediction: Lambda(0) is obtained after fitting l1,A directly to the measured triton half-life, so the quoted prediction is constrained by the same observable used to determine the LEC; while this is conventional, the reliability of the small O(alpha) correction (-0.000740) and the extrapolated value both depend on the same perturbative assumption whose justification is not quantified beyond the Wigner-SU(4) discussion.

    Authors: Fitting l_{1,A} to the triton half-life and predicting Lambda(0) for pp fusion follows the standard procedure in the EFT literature for these observables. The O(alpha) correction to the GT matrix element is computed explicitly and is found to be small; its size is directly tied to the same Wigner SU(4) symmetry that suppresses the magnetic-moment correction. We have expanded the discussion in the GT section to better quantify how the symmetry argument justifies the perturbative treatment for both observables and to note that the quoted uncertainty on Lambda(0) incorporates the fit uncertainty from the triton data. This makes the justification more explicit while preserving the conventional extraction of the prediction. revision: yes

Circularity Check

1 steps flagged

Fitted axial LEC to triton half-life presented as independent prediction for pp fusion matrix element

specific steps
  1. fitted input called prediction [Abstract]
    "Fitting l_{1,A} to the ³H half-life we make a prediction for the proton-proton fusion reduced matrix element of Λ(0)=2.776(331)."

    The reduced matrix element Λ(0) is computed from the axial LEC l_{1,A} whose value is fixed by a direct fit to the experimental triton half-life. The quoted prediction is therefore determined by construction from that experimental datum through the EFT relation, with no additional independent theoretical input beyond the fit itself.

full rationale

The paper's central results for small O(α) Coulomb corrections to the ³He moment and GT matrix element are obtained after fitting L1 to the ³H moment, L2 to the deuteron moment, and l_{1,A} to the triton half-life within the EFT framework. The explicit step of fitting l_{1,A} to the half-life and then quoting a numerical value for the pp fusion reduced matrix element Λ(0) matches the fitted-input-called-prediction pattern. This makes that particular quoted result reduce directly to the experimental input by construction via the EFT relation. The remainder of the derivation for the correction sizes retains independent content from the perturbative EFT calculation and Wigner-SU(4) analysis, so the circularity is partial rather than total.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claims rest on fitting three low-energy constants to experimental data and on the assumption that the pionless EFT plus perturbative electromagnetism captures the relevant physics at the stated order.

free parameters (3)
  • L1
    Two-nucleon iso-vector magnetic current LEC fitted to the 3H magnetic moment
  • L2
    Two-nucleon iso-scalar magnetic current LEC fitted to the deuteron magnetic moment
  • l_{1,A}
    Two-nucleon axial current LEC fitted to the 3H half-life to predict the pp fusion matrix element
axioms (2)
  • domain assumption Pionless effective field theory is valid for three-nucleon observables at low energies
    Framework invoked throughout the abstract for the NLO calculations
  • domain assumption Coulomb corrections can be treated perturbatively to O(alpha) with binding momentum p* ~ 88.5 MeV
    Explicitly stated as the method for including electromagnetic effects

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discussion (0)

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Reference graph

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