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REVIEW 3 major objections 6 minor 70 references

Rewriting short-range three-nucleon forces in a spectroscopic basis lets nine of thirteen low-energy constants be fixed from elastic nucleon-deuteron data.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-30 15:49 UTC pith:RXHNRDTI

load-bearing objection The spectroscopic basis and emulator are genuinely useful; the “9 of 13 reliably determined” claim is softer than the abstract suggests once you look at Table V’s progressive shifts. the 3 major comments →

arxiv 2607.26958 v1 pith:RXHNRDTI submitted 2026-07-29 nucl-th hep-phnucl-ex

Spectroscopic basis for short-range three-nucleon forces

Josep Sol\`{a} Cava , Arseniy A. Filin , Sven Heihoff , Henri Paul Huesmann , Evgeny Epelbaum This is my paper
classification nucl-th hep-phnucl-ex
keywords three-nucleon forcechiral EFTspectroscopic basisnucleon-deuteron scatteringlow-energy constantscontact interactionsradial basis function emulator
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Three-nucleon short-range forces at next-to-next-to-next-to-leading order involve thirteen low-energy constants that are hard to pin down because each operator mixes many partial waves. This paper rewrites those operators as a spectroscopic basis in which each new constant controls only one total-angular-momentum and parity channel in nucleon-deuteron scattering. The reduced parameter space makes a simple radial-basis-function emulator practical, so the constants can be fit directly to elastic cross sections and analyzing powers. Exploratory fits show that nine of the thirteen constants are reliably determined by existing elastic data; two are largely redundant for elastic observables and two live only in the isospin-3/2 sector. The rewrite therefore turns an intractable multiparameter problem into a transparent, channel-by-channel determination that can guide the construction of precision three-nucleon forces.

Core claim

Once the thirteen subleading contact three-nucleon operators are linearly transformed into the spectroscopic basis S1…S13, each Si contributes to a single JP channel (or vanishes) in elastic nucleon-deuteron scattering. With that basis the authors build an RBF emulator and demonstrate that nine of the thirteen dimensionless coefficients can be extracted from elastic Nd data at 10, 70 and 135 MeV, while s2 and s4 remain essentially unconstrained by elastic observables and s12, s13 are invisible in the T=1/2 system.

What carries the argument

The spectroscopic basis: an invertible linear map from the original Ei operators onto new constants Si that each act in only one JP partial-wave sector of Nd scattering (Table II and Eqs. 3–5), collapsing the fit space enough for radial-basis-function interpolation.

Load-bearing premise

That the nine constants remain determinable and of roughly natural size after the still-missing longer-range three-nucleon pieces beyond N2LO are restored to the Hamiltonian.

What would settle it

Repeat the same elastic Nd fits after the symmetry-preserving N3LO (and N4LO long-range) three-nucleon contributions are included; if several of the nine Si then become unconstrained or wildly unnatural, the claim fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Elastic Nd data alone can fix the isospin-1/2 short-range 3NF sector once the spectroscopic rewrite is used.
  • The Ay puzzle at low energy is at least half-resolved by a natural-size s11 together with the 3/2− LECs.
  • The N2LO contact cD shifts negative once N4LO contacts are free, moving closer to the value preferred by tritium beta decay.
  • s2 and s4 must be constrained by breakup or four-body data; s12 and s13 require systems with T=3/2 components.
  • A complete N3LO Nd analysis can reuse the same spectroscopic emulator after the enhanced linear combinations of Si and Fi are restored.

Where Pith is reading between the lines

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

  • The same spectroscopic map should make eigenvector-continuation or Woodbury emulators for three-nucleon scattering dramatically cheaper, because each channel now depends on only a handful of LECs.
  • Once s12 and s13 are fixed from A=4 or neutron-matter calculations, the full contact 3NF can be frozen and used as a controlled input for medium-mass structure, testing whether the remaining discrepancies are truly long-range.
  • The opposite impact of the 3/2− LECs on nucleon versus deuteron Ay suggests a clean experimental program that measures both analyzing powers at the same energy to isolate mixing angles.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper introduces a spectroscopic basis for the 13 subleading contact three-nucleon operators of chiral EFT at N4LO, reparametrizing the standard Ei LECs into Si combinations that each act in a single JP(T) channel (or a single Nd partial-wave/mixing sector). Explicit partial-wave decompositions, triton expectation-value benchmarks, and the inverse transformation are supplied. Using this basis the authors map the sensitivity of elastic Nd analyzing powers, build a low-dimensional RBF emulator of the Nd transfer matrix, and perform progressive exploratory fits of cD, cE and the isospin-1/2 spectroscopic LECs to selected pd data at 10, 70 and 135 MeV. They conclude that nine of the thirteen LECs (cD, cE, s1, s3, s5–s11) can be constrained by elastic Nd scattering, while s2 and s4 are largely redundant for elastic observables and s12, s13 live only in T=3/2.

Significance. A transparent spectroscopic organization of the N4LO contact 3NF is a genuine and reusable contribution: it clarifies which LECs control which Nd waves, reduces parameter-space dimension enough for a simple RBF emulator, and supplies analytic partial-wave formulae and triton benchmarks that the community can adopt. The emulator validation (App. D) and held-out predictions at 200 MeV and for Cij are concrete strengths. If the determinability claim survives once longer-range 3NF pieces are restored, the work would materially ease the LEC-fitting bottleneck for precision chiral 3NFs. Even as an exploratory study the basis and methodology are immediately useful to LENPIC-style analyses.

major comments (3)
  1. [Abstract; Sec. III C; Table V] Abstract and Sec. III C / Table V: the central claim that “9 of 13 LECs can be reliably determined” is stronger than the internal evidence supports. Progressive columns of Table V show order-one and sign-flipping shifts that exceed the quoted 1σ errors (e.g. s6: −6.461±0.487 → +6.294±0.472; s5: −0.505 → +1.464; s10: −1.343 → +0.823 when the 1/2− sector is opened). The authors themselves note that changes typically exceed statistical errors and that systematics are unquantified, and the full-fit χ²/datum = 4.44 implies the covariance used for “statistical errors” is mis-scaled under an incomplete model. The language should be softened to “constrainable / determinable within the present incomplete Hamiltonian,” with an explicit caveat that values will shift once N3LO/N4LO long- and intermediate-range 3NFs are restored (already flagged in Secs. III A, III C, IV).
  2. [Sec. II; Sec. III C; Table V; Sec. IV] Sec. II and Table V full fit: the adopted natural window |si| ≲ 2 (motivated by Table I and NN experience) is violated by s6 ≈ 6 (and, in intermediate fits, |s9| ≈ 3.5–4). While the authors correctly state that naturalness cannot be judged until longer-range 3NFs are included, the abstract and summary still present the nine LECs as “reliably” fixed of natural size. Either the naturalness discussion should be moved into the main claim language, or the fit should be repeated with a soft naturalness prior / penalty so that the reader can see how much of the χ² improvement is purchased by unnaturally large si.
  3. [Sec. III C; Table III; Eq. (7)] Sec. III C data selection and Eq. (7): fits omit Coulomb, restrict θcm ∈ [60°, 160°], drop 200 MeV and Cij from the χ², and retain a database that has not been cleaned for mutual inconsistencies (the KVI vs RIKEN 135 MeV cross-section tension is discussed but not resolved by a 3σ-style rejection). With χ²/N ≈ 4.4 it is unclear how much of the remaining discrepancy is model incompleteness versus outlier data. A short robustness check—refitting after a transparent outlier cut or after inflating experimental errors to force χ²/N ≈ 1—would make the “9 of 13” count more credible even inside the exploratory setup.
minor comments (6)
  1. [Table I; Sec. II] Table I: the text notes that ⟨VE5,6,9…13⟩ differ from Ref. [26]; a one-sentence remark on whether the discrepancy is traced to antisymmetrization, regulator, or a typographical issue in [26] would help readers who rely on both papers.
  2. [Sec. III A; Figs. 1–8] Figs. 1–8 lower panels: the fixed-angle slices are useful, but the chosen angle (open square) is not always near the maximum sensitivity; a brief note on the selection criterion would avoid the impression of cherry-picking.
  3. [Sec. II; Sec. III C] Eq. (5) and the discussion of s2 redundancy: the Born-level argument that s2 is absorbable into E and s3 is clear; stating explicitly that s2 was fixed to zero only after verifying Δ(χ²/datum) ≲ 1% for |s2| = 2 would make the exclusion fully reproducible.
  4. [Appendix A] Appendix A is long and valuable; a compact machine-readable supplement (or a note that the expressions are available on request / in a repository) would increase uptake.
  5. [Sec. III] Minor typography: “APPLICA TIONS”, “SCA TTERING” (Sec. III heading) and occasional missing spaces around math in the arXiv text should be cleaned in production.
  6. [Fig. 16; Sec. III C] Fig. 16 correlation matrix: the strongest correlation (cD–s11 ≈ 0.71) is mentioned in the text; adding the numerical value in the caption would help skimmers.

Circularity Check

1 steps flagged

No load-bearing circularity: spectroscopic basis is an explicit linear reparametrization; fits constrain LECs and out-of-sample observables are genuine checks, not tautologies.

specific steps
  1. self definitional [Sec. II, Eqs. (3)–(5), Table II]
    "Accordingly, we define a new set of LECs Si via the relations [Eq. 3]... In contrast to Ei’s, the new LECs Si contribute to single JP(T)-channels as shown in Table II... the matrix elements of the short-range 3NFs contributing to elastic Nd scattering take the form [Eq. 5]."

    Si are defined as the linear combinations that isolate single JP channels; the subsequent statement that each Si affects only those partial waves is true by that definition, not an independent dynamical result. This is ordinary spectroscopic reparametrization and does not force the fit values or the ‘9 of 13’ claim.

full rationale

The paper’s central construction is a one-to-one linear map Ei ↔ Si (Eqs. 3 and C.1) chosen so each Si feeds a single JP(T) Nd channel (Table II, Eq. 5). That is definitional bookkeeping of the known contact operator set, not a claim that physics forces the map. Sensitivity plots and the RBF emulator then vary those independent parameters; nothing is predicted from a quantity that was defined to equal it. The exploratory fits (Sec. III C, Table V) determine cD, cE, s1, s3, s5–s11 from a stated elastic-Nd subset (10/70/135 MeV, θcm∈[60°,160°]), with cE fixed by the external 3H binding energy—standard practice, not self-definition. Spin correlations and 200 MeV observables (Sec. III D) are held out and compared after the fit; they are not forced by the fitted inputs. Progressive-fit LEC sign flips and χ²/datum≈4.44 undermine the rhetoric of “reliably determined,” but that is a robustness/overclaim issue, not a reduction of the result to its inputs by construction. Self-citations (SMS NN, LENPIC N2LO 3NF, prior Ei work) supply the Hamiltonian baseline and are not used as uniqueness theorems that forbid alternatives. Score 1 only for the mild definitional character of the spectroscopic labels themselves.

Axiom & Free-Parameter Ledger

6 free parameters · 7 axioms · 1 invented entities

The work sits inside standard chiral EFT power counting and Faddeev few-body theory. Load-bearing free parameters are the fitted LECs themselves and the hand-chosen naturalness/prior ranges and data cuts. Domain assumptions include the SMS NN potential, N2LO-only long-range 3NF, cutoff regulator, and neglect of Coulomb. The spectroscopic basis is a derived reparametrization, not a new dynamical entity.

free parameters (6)
  • s1, s3, s5–s11 (spectroscopic N4LO contact LECs in full fit) = Full fit Table V: s1=-0.146±0.373, s3=-1.654±0.234, s5=1.464±0.582, s6=6.294±0.472, s7=-0.810±0.472, s8=-0.263±0.130, s9
    Dimensionless contact LECs fitted by χ² minimization to selected elastic Nd observables; central empirical output of the paper.
  • cD, cE (N2LO short-range 3NF LECs) = Full fit: cD=-2.510±0.385, cE=0.329±0.052
    Fitted jointly with si; cE constrained to reproduce 3H binding energy 8.482 MeV for each grid point.
  • s2, s4 fixed to zero = 0
    Set by hand after redundancy tests showed <1% χ² change for ±2 variations; choice stabilizes the fit but is a modeling decision.
  • Naturalness / sampling ranges |si| and cD interval = si natural target |si|≲2; sampling as above
    Prior ranges si∈[-3,3] (extended to s6∈[-8,8], s9∈[-6,6]) and cD∈[-8,8] chosen by hand from NN experience and Table I; affect emulator domain and allowed minima.
  • Cutoff Λ=450 MeV and regulator form = Λ=450 MeV
    Fixed SMS regulator and cutoff for NN and 3NF; not varied in the fits.
  • Data selection cuts (energies, angles, observables) = Ndat=493 primary fit points
    10/70/135 MeV only; θcm∈[60°,160°]; Ay only at 10 MeV; Cij and 200 MeV held out; KVI 135 MeV cross section excluded from primary fit—all hand choices that define the χ².
axioms (7)
  • domain assumption Weinberg chiral power counting with contact 3NF operator basis of Girlanda et al. (13 Ei operators at N4LO)
    Sec. I–II; entire operator starting point Eq. (1) taken from prior EFT literature.
  • domain assumption SMS N4LO+ NN potential of Reinert et al. is an adequate two-body input
    Used throughout for triton WF, Faddeev solutions, and baseline observables (Ref. [10]).
  • ad hoc to paper Long- and intermediate-range 3NF beyond parameter-free N2LO 2π exchange can be omitted for an exploratory assessment of contact-LEC determinability
    Stated limitation in Sec. III A, III C, IV; load-bearing for interpreting fitted si as physically meaningful.
  • domain assumption Momentum-space Faddeev equation in partial waves (j≤5 NN, J≤25/2 3N, 3NF to J≤7/2) converges for the quoted observables
    Sec. III standard LENPIC-style truncation.
  • domain assumption Coulomb effects are negligible in the fitted mid-angle region or can be ignored for nd-equivalent comparison after 10 MeV correction of Ref. [12]
    Sec. III C; forward/backward data excluded partly for this reason.
  • domain assumption RBF interpolation on LHS grids of ~10^3–10^4 points per channel/energy reproduces Faddeev U-matrix elements to accuracy far below 3NF signal
    Sec. III B and App. D validation; underpins all fit results.
  • standard math Standard angular-momentum algebra and antisymmetrization for three-nucleon partial waves
    Appendix A derivations.
invented entities (1)
  • Spectroscopic LECs S1,...,S13 (si) independent evidence
    purpose: Linear combinations of Ei that each feed a single JP(T) channel / Nd partial-wave structure, enabling transparent sensitivity analysis and low-dimensional emulation.
    Defined in Eq. (3) and Table II; equivalent reparametrization of existing operators, not a new force. Independent checks via vanishing triton expectations for S5–S13 combinations and consistency with Ref. [28].

pith-pipeline@v1.2.0-daily-grok45 · 42506 in / 4664 out tokens · 87764 ms · 2026-07-30T15:49:26.940955+00:00 · methodology

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read the original abstract

We introduce a spectroscopic basis for the subleading contact three-nucleon forces, which allows one to classify these interactions according to the total angular momentum and parity quantum numbers in a transparent way. Using this new basis, we explore the sensitivity of nucleon-deuteron observables to the three-nucleon short-range interactions. The low dimensionality of the variable-parameter space in the spectroscopic basis allows us to build a simple nucleon-deuteron scattering emulator using radial basis function interpolation. We perform exploratory fits of the subleading contact three-nucleon interactions and demonstrate that 9 of 13 low-energy constants can be reliably determined from elastic nucleon-deuteron scattering data.

Figures

Figures reproduced from arXiv: 2607.26958 by Arseniy A. Filin, Evgeny Epelbaum, Henri Paul Huesmann, Josep Sol\`{a} Cava, Sven Heihoff.

Figure 1
Figure 1. Figure 1: FIG. 1. Sensitivity of the analyzing power [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Differential cross section [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p012_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Correlation matrix for the full fit specified in the last [PITH_FULL_IMAGE:figures/full_fig_p013_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Predictions for the differential cross section as well as [PITH_FULL_IMAGE:figures/full_fig_p014_17.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Relative errors for the emulation of the differential [PITH_FULL_IMAGE:figures/full_fig_p024_19.png] view at source ↗

discussion (0)

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Works this paper leans on

70 extracted references · 42 linked inside Pith

  1. [1]

    H. W. Hammer, A. Nogga and A. Schwenk, Rev. Mod. Phys.85(2013), 197. [arXiv:1210.4273 [nucl-th]]

    Pith/arXiv arXiv 2013

  2. [2]

    S. Endo, E. Epelbaum, P. Naidon, Y. Nishida, K. Sekiguchi and Y. Takahashi, Eur. Phys. J. A61(2025) no.1, 9. [arXiv:2405.09807 [nucl-th]]

    Pith/arXiv arXiv 2025

  3. [3]

    Kalantar-Nayestanaki, E

    N. Kalantar-Nayestanaki, E. Epelbaum, J. G. Messchen- dorp and A. Nogga, Rept. Prog. Phys.75(2012), 016301. [arXiv:1108.1227 [nucl-th]]

    Pith/arXiv arXiv 2012

  4. [4]

    Hebeler, J

    K. Hebeler, J. M. Lattimer, C. J. Pethick and A. Schwenk, Astrophys. J.773(2013), 11. [arXiv:1303.4662 [astro-ph.SR]]

    Pith/arXiv arXiv 2013

  5. [5]

    Weinberg, Phys

    S. Weinberg, Phys. Lett. B251(1990), 288-292

    1990

  6. [6]

    van Kolck, Phys

    U. van Kolck, Phys. Rev. C49(1994), 2932-2941

    1994

  7. [7]

    Epelbaum, H.-W

    E. Epelbaum, H.-W. Hammer and U.-G. Meißner, Rev. Mod. Phys.81(2009), 1773-1825. [arXiv:0811.1338 [nucl- th]]

    Pith/arXiv arXiv 2009

  8. [8]

    Reinert, H

    P. Reinert, H. Krebs and E. Epelbaum, Phys. Rev. Lett. 126(2021) no.9, 092501. [arXiv:2006.15360 [nucl-th]]

    Pith/arXiv arXiv 2021

  9. [9]

    Reinert,Precision studies in the two-nucleon sys- tem using chiral effective field theory, PhD thesis, Ruhr- Universit¨ at Bochum, 2022, doi:10.13154/294-9501

    P. Reinert,Precision studies in the two-nucleon sys- tem using chiral effective field theory, PhD thesis, Ruhr- Universit¨ at Bochum, 2022, doi:10.13154/294-9501

    doi:10.13154/294-9501 2022

  10. [10]

    Reinert, H

    P. Reinert, H. Krebs and E. Epelbaum, Eur. Phys. J. A 54(2018) no.5, 86. [arXiv:1711.08821 [nucl-th]]

    Pith/arXiv arXiv 2018

  11. [11]

    Epelbaum, H

    E. Epelbaum, H. Krebs and P. Reinert,Semi-local Nuclear Forces from Chiral EFT: State-of-the-Art and Challenges.In: Tanihata, I., Toki, H., Kajino, T. (eds) Handbook of Nuclear Physics. Springer, Singapore. [arXiv:2206.07072 [nucl-th]]

    Pith/arXiv arXiv

  12. [12]

    Epelbaum, A

    E. Epelbaum, A. Nogga, W. Gl¨ ockle, H. Kamada, U.- G. Meißner and H. Wita la, Phys. Rev. C66(2002), 064001. [arXiv:nucl-th/0208023 [nucl-th]]

    Pith/arXiv arXiv 2002

  13. [13]

    Ishikawa and M

    S. Ishikawa and M. R. Robilotta, Phys. Rev. C76(2007), 014006. [arXiv:0704.0711 [nucl-th]]

    Pith/arXiv arXiv 2007

  14. [14]

    Bernard, E

    V. Bernard, E. Epelbaum, H. Krebs and U.-G. Meißner, Phys. Rev. C77(2008), 064004. [arXiv:0712.1967 [nucl- th]]

    Pith/arXiv arXiv 2008

  15. [15]

    Bernard, E

    V. Bernard, E. Epelbaum, H. Krebs and U.-G. Meißner, Phys. Rev. C84(2011), 054001. [arXiv:1108.3816 [nucl- th]]

    Pith/arXiv arXiv 2011

  16. [16]

    Krebs, A

    H. Krebs, A. Gasparyan and E. Epelbaum, Phys. Rev. C 85(2012), 054006. [arXiv:1203.0067 [nucl-th]]

    Pith/arXiv arXiv 2012

  17. [17]

    Krebs, A

    H. Krebs, A. Gasparyan and E. Epelbaum, Phys. Rev. C 87(2013) no.5, 054007. [arXiv:1302.2872 [nucl-th]]

    Pith/arXiv arXiv 2013

  18. [18]

    Krebs, A

    H. Krebs, A. M. Gasparyan and E. Epelbaum, Phys. Rev. C98(2018) no.1, 014003. [arXiv:1803.09613 [nucl-th]]

    Pith/arXiv arXiv 2018

  19. [19]

    H. P. Huesmann, H. Krebs and E. Epelbaum, Nucl. Phys. A1075(2026), 123474. [arXiv:2602.12879 [nucl-th]]

    arXiv 2026

  20. [20]

    Girlanda, A

    L. Girlanda, A. Kievsky and M. Viviani, Phys. Rev. C84 (2011) no.1, 014001 [erratum: Phys. Rev. C102(2020) no.1, 019903]. [arXiv:1102.4799 [nucl-th]]

    arXiv 2011

  21. [21]

    Epelbaum, H

    E. Epelbaum, H. Krebs and P. Reinert, Front. in Phys. 8(2020), 98. [arXiv:1911.11875 [nucl-th]]

    Pith/arXiv arXiv 2020

  22. [22]

    Krebs, PoSCD2018, 098 (2019)

    H. Krebs, PoSCD2018, 098 (2019). [arXiv:1908.01538 [nucl-th]]

    Pith/arXiv arXiv 2019

  23. [23]

    Krebs and E

    H. Krebs and E. Epelbaum, Phys. Rev. C110(2024) no.4, 044003. [arXiv:2311.10893 [nucl-th]]

    Pith/arXiv arXiv 2024

  24. [24]

    Krebs and E

    H. Krebs and E. Epelbaum, Phys. Rev. C110(2024) no.4, 044004. [arXiv:2312.13932 [nucl-th]]

    Pith/arXiv arXiv 2024

  25. [25]

    Girlanda, A

    L. Girlanda, A. Kievsky, M. Viviani and L. E. Marcucci, Phys. Rev. C99(2019) no.5, 054003. [arXiv:1811.09398 [nucl-th]]

    Pith/arXiv arXiv 2019

  26. [26]

    Wita la, J

    H. Wita la, J. Golak and R. Skibi´ nski, Phys. Rev. C105 (2022) no.5, 054004. [arXiv:2203.08499 [nucl-th]]

    Pith/arXiv arXiv 2022

  27. [27]

    Margaryan, R

    A. Margaryan, R. P. Springer and J. Vanasse, Phys. Rev. C93(2016) no.5, 054001. [arXiv:1512.03774 [nucl-th]]

    Pith/arXiv arXiv 2016

  28. [28]

    Filandri, L

    E. Filandri, L. Girlanda and Y. Capitani, [arXiv:2601.19033 [nucl-th]]

    arXiv

  29. [29]

    Epelbaum, J

    E. Epelbaum, J. Golak, K. Hebeler, H. Kamada, H. Krebs, U.-G. Meißner, A. Nogga, P. Reinert, R. Skibi´ nski and K. Topolnicki,et al.Eur. Phys. J. A 56(2020) no.3, 92. [arXiv:1907.03608 [nucl-th]]

    Pith/arXiv arXiv 2020

  30. [30]

    R. G. Seyler, Nucl. Phys. A124(1969), 253-272

    1969

  31. [31]

    Gl¨ ockle, H

    W. Gl¨ ockle, H. Wita la, D. H¨ uber, H. Kamada and J. Go- lak, Phys. Rept.274(1996), 107-285. 26

    1996

  32. [32]

    Epelbaumet al.[LENPIC], Phys

    E. Epelbaumet al.[LENPIC], Phys. Rev. C99(2019) no.2, 024313. [arXiv:1807.02848 [nucl-th]]

    Pith/arXiv arXiv 2019

  33. [33]

    Maris, E

    P. Maris, E. Epelbaum, R. J. Furnstahl, J. Golak, K. Hebeler, T. H¨ uther, H. Kamada, H. Krebs, U.- G. Meißner and J. A. Melendez,et al.Phys. Rev. C103 (2021) no.5, 054001. [arXiv:2012.12396 [nucl-th]]

    Pith/arXiv arXiv 2021

  34. [34]

    Mariset al.[LENPIC], Phys

    P. Mariset al.[LENPIC], Phys. Rev. C106(2022) no.6, 064002. [arXiv:2206.13303 [nucl-th]]

    Pith/arXiv arXiv 2022

  35. [35]

    Sperisen, W

    F. Sperisen, W. Gr¨ uebler, V. K¨ onig, P. A. Schmelzbach, K. Elsener, B. Jenny, C. Schweizer, J. Ulbricht and P. Do- leschall, Nucl. Phys. A422(1984), 81-102

    1984

  36. [36]

    Sekiguchi, H

    K. Sekiguchi, H. Sakai, H. Wita la, W. Gl¨ ockle, J. Go- lak, M. Hatano, H. Kamada, H. Kato, Y. Maeda and J. Nishikawa,et al.Phys. Rev. C65(2002), 034003

    2002

  37. [37]

    K. Suda, H. Okamura, T. Uesaka, J. Nishikawa, H. Ku- masaka, R. Suzuki, H. Sakai, A. Tamii, T. Ohnishi, K. Sekiguchi, K. Yako, S. Sakoda, H. Kato, M. Hatano, Y. Maeda, T. Saito, T. Ishida, N. Sakamoto, Y. Satou, K. Hatanaka, T. Wakasa, J. Kamiya,et al.Nucl. Instrum. Methods Phys. Res. A572(2) (2007) 745-753

    2007

  38. [38]

    Sekiguchi, H

    K. Sekiguchi, H. Sakai, H. Wita la, W. Gl¨ ockle, J. Go- lak, K. Hatanaka, M. Hatano, K. Itoh, H. Kamada and H. Kuboki,et al.Phys. Rev. Lett.95(2005), 162301. [arXiv:nucl-ex/0510005 [nucl-ex]]

    Pith/arXiv arXiv 2005

  39. [39]

    Ermisch, H

    K. Ermisch, H. R. Amir-Ahmadi, A. M. vanden Berg, R. Castelijns, B. Davids, A. Deltuva, E. Epelbaum, W. Gl¨ ockle, J. Golak and M. N. Harakeh,et al.Phys. Rev. C71(2005), 064004

    2005

  40. [40]

    von Przewoski, H

    B. von Przewoski, H. O. Meyer, J. T. Balewski, W. W. Daehnick, J. Doskow, W. Haeberli, R. Ibald, B. Lorentz, R. E. Pollock and P. V. Pancella,et al.Phys. Rev. C74(2006), 064003. [arXiv:nucl-ex/0411019 [nucl- ex]]

    Pith/arXiv arXiv 2006

  41. [41]

    Kievsky, M

    A. Kievsky, M. Viviani and L. E. Marcucci, Few Body Syst.54(2013), 2395-2406

    2013

  42. [42]

    Ishikawa, Phys

    S. Ishikawa, Phys. Rev. C59(1999), 1247-1251. [arXiv:nucl-th/9902035 [nucl-th]]

    Pith/arXiv arXiv 1999

  43. [43]

    Ishikawa, Few Body Syst.32(2003), 229

    S. Ishikawa, Few Body Syst.32(2003), 229. [arXiv:nucl- th/0206064 [nucl-th]]

    arXiv 2003

  44. [44]

    Strate, K

    J. Strate, K. Geissd¨ orfer, R. Lin, W. Bielmeier, J. Cub, A. Ebneth, E. Finckh, H. Friess, G. Fuchs and K. Geb- hardt,et al.Nucl. Phys. A501(1989), 51-85

    1989

  45. [45]

    H. R. Setze, C. R. Howell, W. Tornow, R. T. Braun, D. E. Gonzalez Trotter, A. H. Hussein, R. S. Pedroni, C. D. Roper, F. Salinas and I. Slaus,et al.Phys. Rev. C 71(2005), 034006

    2005

  46. [46]

    data differ from each other by a factor of∼1.5, with theoretical predictions lying between the nd and pd data sets [3, 31, 47], see Fig. 9. Theoretical predictions for the SST cross section are known to be very robust, and the 3NF models considered so far were found to have almost no impact on this observable at low energies, see Ref. [47] and references ...

    2020

  47. [47]

    Rauprich, S

    G. Rauprich, S. Lema ˆ ıtre, P. Niessen, K. R. Nyga, R. Reckenfelderbaumer, L. Sydow, H. Paetz Gen. Schieck, H. Wita la and W. Gl¨ ockle, Nucl. Phys. A535(1991), 313-330

    1991

  48. [48]

    Wita la, J

    H. Wita la, J. Golak, R. Skibi´ nski, K. Topolnicki, E. Epel- baum, H. Krebs and P. Reinert, Phys. Rev. C104(2021) no.1, 014002. [arXiv:2102.09863 [nucl-th]]

    Pith/arXiv arXiv 2021

  49. [49]

    Deltuva, A

    A. Deltuva, A. C. Fonseca and P. U. Sauer, Phys. Rev. C72(2005), 054004 [erratum: Phys. Rev. C72(2005), 059903]. [arXiv:nucl-th/0509034 [nucl-th]]

    Pith/arXiv arXiv 2005

  50. [50]

    Wita la, J

    H. Wita la, J. Golak, R. Skibi´ nski and K. Topolnicki, Few Body Syst.62(2021) no.2, 23. [arXiv:2102.13408 [nucl- th]]

    Pith/arXiv arXiv 2021

  51. [51]

    Wita la, J

    H. Wita la, J. Golak and R. Skibi´ nski, Eur. Phys. J. A57 (2021) no.7, 241. [arXiv:2103.13237 [nucl-th]]

    Pith/arXiv arXiv 2021

  52. [52]

    Zhang and R

    X. Zhang and R. J. Furnstahl, Phys. Rev. C105(2022) no.6, 064004. [arXiv:2110.04269 [nucl-th]]

    Pith/arXiv arXiv 2022

  53. [53]

    Gnech, X

    A. Gnech, X. Zhang, C. Drischler, R. J. Furnstahl, A. Grassi, A. Kievsky, L. E. Marcucci and M. Viviani, [arXiv:2511.10420 [nucl-th]]

    arXiv

  54. [54]

    Heihoff, A

    S. Heihoff, A. A. Filin and E. Epelbaum, [arXiv:2604.25792 [nucl-th]]

    Pith/arXiv arXiv

  55. [55]

    Hardy Journal of Geophysical Research76(8) (1971) 1905

    R.L. Hardy Journal of Geophysical Research76(8) (1971) 1905

    1971

  56. [56]

    E. J. Kansa, Computers & Mathematics with Applica- tions19(8-9) (1990) 147

    1990

  57. [57]

    M. D. McKay, R. J. Beckman, and W. J. Conover, Tech- nometrics,21, 2 (1979) 239

    1979

  58. [58]

    Virtanenet al., Nature Methods,17(2020), 261-272

    P. Virtanenet al., Nature Methods,17(2020), 261-272

    2020

  59. [59]

    Orear, Least squares when both variables have uncer- tainties, Am

    J. Orear, Least squares when both variables have uncer- tainties, Am. J. Phys.50(10) (1982) 912–916

    1982

  60. [60]

    D. F. R. Jimenez, S. Heihoff, J. Golak, E. Epelbaum, H. Krebs, P. Reinert, R. Skibi´ nski, K. Topolnicki and H. Wita la, [arXiv:2606.26926 [nucl-th]]

    Pith/arXiv arXiv

  61. [61]

    V. G. J. Stoks, R. A. M. Klomp, M. C. M. Rentmeester and J. J. de Swart, Phys. Rev. C48(1993), 792-815

    1993

  62. [62]

    Gross and A

    F. Gross and A. Stadler, Phys. Rev. C78(2008), 014005. [arXiv:0802.1552 [nucl-th]]

    Pith/arXiv arXiv 2008

  63. [63]

    Navarro P´ erez, J

    R. Navarro P´ erez, J. E. Amaro and E. Ruiz Arriola, Phys. Rev. C88(2013) no.6, 064002 [erratum: Phys. Rev. C 91(2015) no.2, 029901]. [arXiv:1310.2536 [nucl-th]]

    Pith/arXiv arXiv 2013

  64. [64]

    Navarro P´ erez, J

    R. Navarro P´ erez, J. E. Amaro and E. Ruiz Arriola, Phys. Rev. C95(2017) no.6, 064001. [arXiv:1606.00592 [nucl- th]]

    Pith/arXiv arXiv 2017

  65. [65]

    R. B. Wiringa, V. G. J. Stoks and R. Schiavilla, Phys. Rev. C51(1995), 38-51. [arXiv:nucl-th/9408016 [nucl- th]]

    Pith/arXiv arXiv 1995

  66. [66]

    B. S. Pudliner, V. R. Pandharipande, J. Carlson, S. C. Pieper and R. B. Wiringa, Phys. Rev. C56(1997), 1720-1750. [arXiv:nucl-th/9705009 [nucl-th]]

    Pith/arXiv arXiv 1997

  67. [67]

    Saito, K

    Y. Saito, K. Sekiguchi, A. Watanabe, K. Suzuki, H. Sug- ahara, D. Takahashi, K. Tateishi, S. Otsuka, H. Sakai and N. Sakamoto,et al.Eur. Phys. J. A62(2026) no.7, 134

    2026

  68. [68]

    L. Zuo, H. Yang and B. Long, Phys. Rev. C113(2026) no.1, 014002. [arXiv:2505.12793 [nucl-th]]

    arXiv 2026

  69. [69]

    Girlanda, A

    L. Girlanda, A. Kievsky, L. E. Marcucci and M. Viviani, Phys. Rev. C102(2020), 064003. [arXiv:2007.04161 [nucl-th]]

    Pith/arXiv arXiv 2020

  70. [70]

    Dick, J.W

    F. Dick, J.W. Norbury, Eur. J. Phys.30(2009), 403; [erratum: Eur. J. Phys.32(2011), 1731]

    2009