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arxiv: 2606.32037 · v1 · pith:X5SBWUMDnew · submitted 2026-06-30 · ⚛️ nucl-th

Finite-range EFT for the E1 strength distribution of {}⁶He

Matthias G\"obel , Hans-Werner Hammer , Daniel R. Phillips This is my paper

Pith reviewed 2026-07-01 02:19 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords Halo EFTfinite-range interactionsE1 strength distribution6Hethree-body problemcharge radiusfinal-state interactionsMøller operators
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The pith

Finite-range Halo EFT yields an E1 strength distribution for ⁶He that agrees with data at NLO.

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

Standard Halo EFT in the dimer formalism produces energy-dependent interactions for the ²P₃/₂ nα channel in ⁶He, which complicates building a proper Hilbert space for three-body states. The paper develops a finite-range formulation that uses separable Yamaguchi-like form factors instead, keeping the interaction local in energy while preserving the EFT power counting. This lets the authors solve the ⁶He ground state through NLO, compute the E1 strength distribution after approximating final-state interactions with products of Møller operators, and obtain a result whose shape and magnitude match experimental data inside the quoted theory uncertainties. They also extract a root-mean-square charge radius of 2.00 ± 0.09 fm at NLO that lies within the experimental range.

Core claim

We solve for the ⁶He bound state in this finite-range EFT up to next-to-leading order (NLO) in the Halo EFT power counting and calculate the ground-state E1 strength distribution of ⁶He at this order. The shape of the resulting distribution agrees with that obtained in the dimer formalism of the EFT, but finite-range EFT does not require the use of a non-standard wave function normalization condition. We also calculate the root-mean-square charge radius of ⁶He and find 2.06 ± 0.35 fm at LO and 2.00 ± 0.09 fm at NLO, in agreement with experimental data. To calculate the full E1 strength distribution final-state interactions must be incorporated. We approximate the full-three-body scattering o

What carries the argument

Separable interactions with Yamaguchi-like form factors inside a finite-range Halo EFT that keeps the three-body Hilbert space standard while reproducing the dimer-formalism E1 distribution.

If this is right

  • The E1 strength distribution at NLO matches experimental data inside theory uncertainties.
  • The root-mean-square charge radius is 2.06 ± 0.35 fm at LO and 2.00 ± 0.09 fm at NLO, consistent with measured values.
  • The finite-range approach produces the same distribution shape as the dimer formalism without needing a non-standard normalization condition.
  • Final-state interactions are adequately captured by the successive Møller-operator approximation for this observable.

Where Pith is reading between the lines

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

  • The same finite-range construction could be applied to other p-wave halo systems where multiple effective-range parameters appear at leading order.
  • Higher-precision E1 data near the peak would directly test whether the NLO truncation error estimate is realistic.
  • The method supplies a practical route to other electromagnetic observables once the Møller approximation is validated for the present case.

Load-bearing premise

Approximating the full three-body scattering operator by single Møller operators and then by products of up to three Møller operators is sufficient to incorporate final-state interactions for the E1 strength distribution.

What would settle it

A measurement of the ⁶He E1 strength distribution whose central values fall outside the NLO theory uncertainty band while the experimental errors remain smaller than that band.

Figures

Figures reproduced from arXiv: 2606.32037 by Daniel R. Phillips, Hans-Werner Hammer, Matthias G\"obel.

Figure 1
Figure 1. Figure 1: Left panel: Leading-order results for the E1 strength distribution obtained in the zero-range (ZR) and finite-range (FR) approaches are shown for different three-body cutoffs Λ between 500 MeV and 1000 MeV. All results are without FSI. Right panel: Results from the same ZR and FR treatments of two-body interactions and three-body cutoffs. The only difference is that, at the position indicated by the vertic… view at source ↗
Figure 2
Figure 2. Figure 2: Different E1 strength distribution from the finite-range Halo EFT are shown. All distributions are without FSI. The LO and the NLO result are shown. Moreover, curves stemming from different partial NLO calculations are included to show the effect of different NLO aspects. Last we consider the incorporation of the 2S1/2 nc amplitude in the NLO calculation. We employ a constant 2S1/2 amplitude within the Fad… view at source ↗
Figure 3
Figure 3. Figure 3: Left panel: The blue curve shows the NLO FREFT E1 distribution without FSI. The orange and green curves then show the result when nn FSI is included in its LO (zero range) and NLO (finite range) form. The red curve is the result when the 2P3/2 nc FSI is included. Right panel: The corresponding cumulative E1 distributions, with the gray line showing the expected value for B(E1)(E → ∞) obtained using the [P… view at source ↗
Figure 4
Figure 4. Figure 4: NLO FREFT E1 distributions with different FSI treatments in comparison. The green dashed and red dashed curves stem from the application of the product of the 1S0 nn and the 2P3/2 nc Møller operator. They differ in the order of application. The green, purple, and red curves stem from the product of three Møller operators. Green means that nn FSI is applied first (reading from the right, from the bound stat… view at source ↗
Figure 5
Figure 5. Figure 5: The differential E1 strength dB(E1)/ [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: LO (dashed) and NLO (solid) FREFT E1 distributions with (orange) and without (blue) FSIs included. The results with FSI included are based on the product of Møller operators Ω† n′c;1,3/2Ω † nc;1,3/2Ω † nn, thereby including both 1S0 nn FSI and 2P3/2 nc FSI. The uncertainty bands indicate the LO EFT uncertainties calculated using Eq. (39). We observe that the NLO result without FSI is substantially lower th… view at source ↗
Figure 7
Figure 7. Figure 7: The NLO FREFT E1 distribution folded with the detector resolution (in orange) in comparison with the band for the E1 strength distribution extracted from experimental data by Sun et al. [37] (in green). The uncertainty band around the NLO result indicates the EFT uncertainty calculated using Eq. (39) (n = 1). Uncertainties of the FSI treatment are not included. For comparison, we also show the LO result in… view at source ↗
Figure 8
Figure 8. Figure 8: Left panel: The orange dashed curve is the FREFT E1 distribution but with an NLO nn FSI computed with a zero￾range interaction. It is to be contrasted to the orange solid curve, in which a finite-range FSI interaction was employed. The green dashed and green solid curve make a similar comparison for the case of the leading nc FSI, that in the 2P3/2 nc channel. Right panel: The corresponding cumulative E1 d… view at source ↗
Figure 9
Figure 9. Figure 9: shows the effect on the E1 strength function of including the 2S1/2 nc final-state interaction. We note that, if implemented as a constant amplitude, the 2S1/2 FSI becomes the dominant FSI at higher energies, and is somewhat larger than its nominal NLO size for energies above 7–8 MeV, see the red curve in [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The NLO FREFT E1 distribution is shown for different FSI treatments. In addition to results without FSI and only nn FSI, also a third-order MOPA result and a result based on the truncation of the multiple-scattering series at second order in the G0tij (MSS(2)) are shown. We observe that the MSS(2) result and the result based on the product of three Møller operators are quite similar. This might be related… view at source ↗
Figure 11
Figure 11. Figure 11: Left panel: Relative deviation in percent between E1 distributions obtained with different truncations in the inclusion of partial waves for the case of different FSIs included. The solid lines show relative deviations between distributions using l (i) max = 1, l (f) max = 3 and distributions using l (i) max = 2 and l (f) max = 6. The dashed lines show the relative deviations between distributions using l… view at source ↗
read the original abstract

Halo effective field theory (Halo EFT) is a powerful tool to describe halo nuclei and predict low-energy observables with quantified uncertainties. However, in the case that there is a leading-order interaction determined by two or more effective-range parameters, such as the $^2P_{3/2}$ $n\alpha$ interaction in $^6$He, the standard implementation in the dimer formalism leads to an energy-dependent interaction. This complicates the construction of a Hilbert space of states, especially beyond the two-body problem. As an alternative, we propose the use of a finite-range formulation of Halo EFT, which avoids these complications. For definiteness, we use separable interactions with Yamaguchi-like form factors, but other choices are possible. We solve for the ${}^6$He bound state in this finite-range EFT up to next-to-leading order (NLO) in the Halo EFT power counting and calculate the ground-state $E1$ strength distribution of $^6$He at this order. The shape of the resulting distribution agrees with that obtained in the dimer formalism of the EFT, but finite-range EFT does not require the use of a non-standard wave function normalization condition. We also calculate the root-mean-square charge radius of $^6$He and find $2.06 \pm 0.35$~fm at LO and $2.00 \pm 0.09$~fm at NLO, in agreement with experimental data. To calculate the full $E1$ strength distribution final-state interactions must be incorporated. We approximate the full-three-body scattering operator first by single M{\o}ller operators and then by products of up to three M{\o}ller operators. The resulting NLO $E1$ strength distribution agrees with the experimental data within theory uncertainties.

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 introduces a finite-range Halo EFT formulation with separable Yamaguchi-like form factors for the n-α interaction in ⁶He, avoiding energy-dependent interactions of the standard dimer approach. It solves the three-body bound state up to NLO, computes the rms charge radius (2.06 ± 0.35 fm at LO and 2.00 ± 0.09 fm at NLO), and obtains the E1 strength distribution by approximating the three-body scattering operator via single Møller operators followed by products of up to three such operators, reporting that the NLO distribution agrees with data within theory uncertainties.

Significance. If the FSI approximation holds, the work supplies a consistent Hilbert-space treatment for Halo EFT systems with multiple effective-range parameters and delivers quantified-uncertainty predictions for both the charge radius and E1 distribution. The radius results match experiment at both orders; the finite-range construction itself removes the need for non-standard wave-function normalization.

major comments (2)
  1. [E1 strength distribution and final-state interactions] The headline claim that the NLO E1 strength distribution agrees with data within uncertainties rests on the replacement of the full three-body scattering operator by single Møller operators and then by products of up to three Møller operators. No convergence test or comparison against an exact three-body solution is supplied in the relevant energy window, which directly affects the shape and uncertainty band of the reported distribution.
  2. [Two-body input and parameter determination] The Yamaguchi range parameters are free parameters fixed from two-body data; the manuscript must show explicitly how their variation propagates into the three-body E1 distribution at NLO and whether the quoted theory uncertainties already incorporate this variation, because the central agreement claim depends on it.
minor comments (2)
  1. [Notation and operator definitions] Clarify the precise definition and normalization of the Møller operators when they are multiplied (up to three) so that the truncation error can be assessed by the reader.
  2. [Figures] The E1 distribution figure should display both LO and NLO curves together with their respective uncertainty bands for direct visual comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments on our manuscript. We address each major comment below and indicate where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [E1 strength distribution and final-state interactions] The headline claim that the NLO E1 strength distribution agrees with data within uncertainties rests on the replacement of the full three-body scattering operator by single Møller operators and then by products of up to three Møller operators. No convergence test or comparison against an exact three-body solution is supplied in the relevant energy window, which directly affects the shape and uncertainty band of the reported distribution.

    Authors: We agree that the FSI treatment is an approximation and that a direct convergence test against an exact three-body solution would be desirable. The approximation is chosen because the low-energy dynamics are dominated by two-body rescattering, and the resulting distribution is consistent with the shape obtained in the dimer formalism. We will revise the manuscript to include an expanded discussion of the expected accuracy of the Møller-operator truncation, its relation to the EFT power counting, and a qualitative estimate of the residual uncertainty it introduces into the band. A full exact three-body scattering solution lies outside the scope of the present work. revision: partial

  2. Referee: [Two-body input and parameter determination] The Yamaguchi range parameters are free parameters fixed from two-body data; the manuscript must show explicitly how their variation propagates into the three-body E1 distribution at NLO and whether the quoted theory uncertainties already incorporate this variation, because the central agreement claim depends on it.

    Authors: The Yamaguchi range parameters are fixed to two-body phase shifts and are not varied in the quoted uncertainty bands, which reflect only the EFT truncation error. We will add an explicit sensitivity study in the revised manuscript that varies the range parameters within their two-body uncertainties and shows the resulting spread in the NLO E1 distribution. This will be compared to the existing EFT error estimate to confirm that the agreement with data remains robust. revision: yes

Circularity Check

0 steps flagged

No circularity: finite-range Halo EFT uses independent two-body inputs to predict three-body observables

full rationale

The paper defines a finite-range separable interaction in Halo EFT, fixes its parameters from two-body n-α scattering data, solves the three-body bound state at LO/NLO, and computes the E1 distribution after an explicit (non-self-referential) truncation of the scattering operator to products of Møller operators. None of these steps redefines the output in terms of itself or renames a fit as a prediction; the final agreement with data is a genuine test of the framework. No load-bearing self-citation or uniqueness theorem imported from the authors' prior work is required for the central claims.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The work rests on the Halo EFT power counting and the assumption that Yamaguchi-like separable form factors adequately capture finite-range physics; two-body low-energy constants are presumably fitted to scattering data, but exact values and fitting procedure are not stated in the abstract.

free parameters (1)
  • Yamaguchi form-factor range parameters
    Introduced to define the separable finite-range interactions; values not given in abstract but required to reproduce two-body physics.
axioms (1)
  • domain assumption Halo EFT power counting remains valid for the finite-range formulation
    Invoked to organize the expansion up to NLO.

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discussion (0)

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Reference graph

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