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Ab initio lattice study of neutron-alpha scattering with chiral forces at N3LO

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Pith's one-line read This paper reports the first ab initio lattice calculation of neutron–alpha scattering with N3LO chiral forces: phase shifts match the empirical benchmark in the $^2S_{1/2}$ and $^2P_{3/2}$ channels, while the $^2P_{1/2}$ channel shows a…

desk verdict First N3LO lattice n-alpha calculation with a real 2P1/2 discrepancy, but the benchmark is a private dataset and the toy model overstates the Lüscher check. read the letter →

arxiv 2507.08495 v1 pith:5C5F7MZH submitted 2025-07-11 nucl-th hep-latnucl-ex

Serdar Elhatisari , Fabian Hildenbrand , Ulf-G. Mei{ss}ner This is my paper
classification nucl-thhep-latnucl-ex
keywords neutron-alphascatteringnuclearlatticeeffectivefieldtheorychiralLüscherfinite-volumemethodthree-nucleonforcesphaseshifts5Heresonancesabinitiocalculation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper reports the first ab initio lattice calculation of neutron–$\alpha$ scattering with chiral nuclear forces at N3LO. Using Lüscher’s finite-volume method on a periodic lattice, it extracts phase shifts in the $^2S_{1/2}$, $^2P_{3/2}$, and $^2P_{1/2}$ channels and compares them with empirical R-matrix phase shifts. The result is close agreement in the first two channels and a persistent disagreement in $^2P_{1/2}$ for neutron energies above about 5 MeV. A dedicated toy model shows the disagreement is not a flaw of the Lüscher extraction, and a re-fit of the three-nucleon force parameters with scattering data included does not remove it. The paper concludes that the present form of the lattice three-nucleon interactions is not flexible enough to describe the $^2P_{1/2}$ channel, and that improving those interactions is the necessary next step.

What carries the argument

The load-bearing machinery is the Lüscher finite-volume quantization condition, which converts finite-volume two-body energy shifts into infinite-volume phase shifts, supplemented by a topological correction factor for the composite $^4$He. Around it sits the wavefunction-matched N3LO lattice Hamiltonian from Ref. [1], where a simple non-perturbative Hamiltonian is treated exactly and the chiral Hamiltonian difference is applied in perturbation theory. To separate extraction artifacts from interaction physics, the paper builds a two-body toy model — a neutron interacting with a point-like spinless $\alpha$ through zero-range, Gaussian central, and Gaussian spin-orbit potentials — and compares Lüscher-extracted phase shifts with spherical-wall phase shifts. For the refitting analysis, eight smeared three-nucleon contact and one-pion-exchange terms are varied in ensembles of six, seven, or eight terms, and the top 150 fits are used to bound systematic uncertainty.

What would settle it

Take the same lattice Hamiltonian and energies, extract the $^2P_{1/2}$ phase shifts with an independent method that does not rely on the Lüscher quantization condition (for example, a direct spherical-wall calculation in the full NLEFT simulation), and compare both with a published, uncertainty-quantified R-matrix analysis; if the independent extraction agrees with the Lüscher result while the published benchmark disagrees with the private one, the discrepancy is in the data, not the force.

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Extended reading notes

Core claim

The central claim is that the wavefunction-matched N3LO chiral Hamiltonian, previously calibrated to nuclear binding energies, predicts neutron-$\alpha$ phase shifts that track the empirical R-matrix curves in the $^2S_{1/2}$ and $^2P_{3/2}$ channels, while the $^2P_{1/2}$ channel is systematically off above 5 MeV. The authors establish this by computing finite-volume energies of the $^5$He states on lattices from 6.6 to 15.8 fm, applying the Lüscher quantization condition with topological corrections, and then testing the extraction method with a simplified spin-orbit toy model whose phase shifts are known from spherical-wall calculations. Since Lüscher and spherical-wall results agree in the toy model, and since Markov-chain Monte Carlo refits of the eight smeared three-nucleon terms — with scattering data added to binding energies — leave the $^2P_{1/2}$ discrepancy unchanged, the paper attributes the discrepancy to the structure of the lattice three-nucleon forces themselves rather than to the finite-volume machinery or the fitting protocol.

Load-bearing premise

The load-bearing premise is that the empirical R-matrix phase shifts used as the benchmark are accurate enough to serve as ground truth; if they carry large unstated errors in the $^2P_{1/2}$ channel above 5 MeV, the persistent discrepancy would be an artifact of the comparison rather than a deficiency of the three-nucleon force.

Editorial extensions

If this is right

  • The N3LO lattice Hamiltonian can be used for scattering observables, not just ground-state properties; the $^2S_{1/2}$ and $^2P_{3/2}$ agreement validates the whole wavefunction-matching plus finite-volume pipeline.
  • The $^2P_{1/2}$ discrepancy is diagnosed as a genuine deficiency of the current lattice three-nucleon force, so any future fix must come from new three-body operator structures rather than from better fitting of the existing ones.
  • Adding neutron-alpha scattering data to the three-nucleon force fit leaves binding energies stable, so scattering observables can be included in future calibrations without degrading nuclear structure predictions.
  • The simultaneous upward or downward shifts of both P-wave resonance peaks across fits indicate the spin-orbit splitting is correlated with the three-nucleon parameter set and is not independently tuned by the existing forces.
  • The finite-volume extraction, with topological corrections and the effective-mass treatment of the composite alpha, is adequate at these box sizes, since the toy model reproduces the reference phase shifts.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: if the missing $^2P_{1/2}$ physics comes from the short-range three-body contact, the natural test is to add spin- and isospin-dependent smeared contact terms to the operator set and see whether the phase shift and the $^5$He excited-state width move together.
  • Editorial inference: the benchmark itself is a private analysis with unstated uncertainties; publishing an error box for the $^2P_{1/2}$ R-matrix phase shifts above 5 MeV would turn the persistent-discrepancy claim into a quantified statement.
  • Editorial inference: the toy-model cross-check covers only the energies shown; applying the same Lüscher-versus-spherical-wall comparison in moving frames or with larger boxes would test whether the method remains reliable for the resonance region at higher momentum.
  • Editorial inference: because the 3N LECs were calibrated largely on nuclei whose valence neutrons sit in the $1p_{3/2}$ orbital, the $^2P_{1/2}$ deficiency may reflect a channel not represented in the calibration set; a scattering-informed fit with a channel-dependent contact term would test this directly.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper presents the first ab initio nuclear lattice effective field theory (NLEFT) calculation of neutron-alpha scattering with the wavefunction-matched N3LO chiral Hamiltonian of Ref. [1]. The authors compute phase shifts in the 2S1/2, 2P3/2, and 2P1/2 channels using the Lüscher finite-volume method, finding good agreement with empirical R-matrix data in the first two channels but a persistent discrepancy in 2P1/2 above about 5 MeV. To test whether this discrepancy is an artifact of the extraction method, they build a point-alpha toy model and show that Lüscher and spherical-wall extractions agree for that model. They then revisit the three-nucleon force fitting, adding n-alpha scattering data in the fit via MCMC sampling over subsets of the eight 3N terms, and find that binding energies remain stable while the 2P1/2 discrepancy persists. The paper concludes that the current lattice 3N interaction form is insufficiently flexible to describe the 2P1/2 channel.

Significance. If the central claim holds, this is a valuable step: it is the first NLEFT calculation of n-alpha scattering at N3LO and it suggests a concrete limitation of the lattice 3N force parameterization rather than of the finite-volume method. The paper also ships a useful cross-check in the toy model, and the MCMC refit provides a systematic uncertainty estimate for binding energies over different 3N parameter sets. However, the significance is reduced by two gaps: the empirical benchmark is a private communication with no stated uncertainties, and the toy model does not exercise the actual features of the N3LO calculation that could affect the Lüscher extraction. The conclusions are therefore conditional on additional validation and on a public, quantitative benchmark.

major comments (3)
  1. [Sec. III and Figs. 1, 4-6; Ref. [72]] The central quantitative conclusion—a persistent discrepancy in 2P1/2 above 5 MeV—is established by comparison with empirical phase shifts from Ref. [72], which is a private communication. The paper gives no uncertainties, no database, and no independent verification for these R-matrix phases. Because the same data are used as ground truth in the toy-model fit (Sec. III.B) and as the target of the MCMC fits (Sec. III.C), an error or large systematic uncertainty in those phases above 5 MeV would make the reported discrepancy an artifact of the benchmark rather than a property of the lattice interaction. Please provide the full R-matrix phase-shift database and uncertainties, or replace or augment Ref. [72] with a published analysis, and re-evaluate the discrepancy accordingly.
  2. [Sec. III.B, Figs. 2-3, Eq. (30)] The toy model validates Lüscher against the spherical-wall method only for a point-like, spinless alpha interacting with a local Gaussian central and spin-orbit potential. It does not test the actual ingredients that could plausibly introduce finite-volume or lattice artifacts in the N3LO calculation: the composite 4He cluster with the moving-state correction in Eq. (30), the nonlocally smeared operators in the Hamiltonian (14), the 3N interaction terms, the Monte Carlo sampling and Euclidean-time extrapolation in Eq. (33), or the different lattice spacing (a = 1.32 fm in the real calculation versus 1.97 fm in the toy model). Consequently, the statement in Sec. IV that 'the Lüscher finite-volume approach has no inherent methodological limitations for the energies considered here' is an overstatement; the evidence supports that conclusion for the toy potential only. Please either extend the validation to a case that includes a composite cluster or nonlocal interactions, or weaken the claim and add a quantitative estimate of residual finite-volume and lattice-spacing effects in the actual extraction.
  3. [Sec. III.A, Table I, Fig. 1] The numerical N3LO phase shifts are not tabulated anywhere in the paper. Figure 1 shows only smooth effective-range fits to the lattice data, and Table I lists energies for the 2N-only Hamiltonian, not for the 2N+3N Hamiltonian used in the main comparison. The reader cannot verify the claimed 'excellent agreement' in 2S1/2 and 2P3/2, the onset of the 2P1/2 discrepancy at 5 MeV, or the systematic effect of the O(p^6) effective-range smoothing. Please provide tabulated phase shifts and uncertainties for both the 2N and 2N+3N calculations, along with the effective-range fit parameters, so that the central quantitative claims are reproducible.
minor comments (5)
  1. [Sec. II.B] The sentence 'This approach is also used in the definition of the short-range interaction in the non-perturbative Hamiltonian given in Sec. II B' should refer to Sec. II A, since the non-perturbative Hamiltonian is defined there.
  2. [Sec. II.C, Eq. (30)] The text says 'The topological phase factor given in Eq. (30) was derived for...', but the factor tau(eta) is defined in Eq. (31); the equation reference should be corrected.
  3. [Sec. III.C] The MCMC procedure is described only by reference to a forthcoming paper [75], with no priors, likelihood, or convergence diagnostics. Since the persistence of the 2P1/2 discrepancy across parameter sets is a main result, at least a short appendix with these technical details is needed.
  4. [Fig. 1 caption] The caption states that solid lines are effective-range fits up to O(p^6), but it does not give the fit parameters or the uncertainty band from the fit; specifying these would improve interpretability.
  5. [Throughout] The paper uses '2N' and '3N' both as abbreviations and as subscripts in equations; for example, 'V Q3 3N' and 'V Q4 2N' are clear in context but the notation would benefit from a short definition list or glossary.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: phase shifts are genuine outputs; the disclosed 2P3/2 fit correlation and toy-model consistency checks do not reduce the central 2P1/2 discrepancy to its inputs.

full rationale

The central derivation is not circular. The n–alpha phase shifts are genuine outputs of the N3LO Hamiltonian from Ref. [1]: the 3N LECs were fitted to binding energies, not to the n–alpha phase shifts that are then compared, and the finite-volume spectra in Table I are Monte Carlo results converted to phase shifts by Lüscher's relation. The paper explicitly discloses that the 2P3/2 agreement is partly inherited from the binding-energy fit through 6He, 9Be, and 10Be (Sec. III.A); this is a non-independence caveat but not an equivalence-by-construction, and the central claim—the persistent 2P1/2 discrepancy above 5 MeV—is not an input to the first calculation. The re-fitting procedure in Sec. III.C explicitly includes n–alpha scattering data and still finds the discrepancy unchanged, which is evidence against circularity rather than for it. The toy model of Sec. III.B checks the Lüscher extraction against the spherical-wall method on a potential fitted to R-matrix phases; this is an internal consistency test of the extraction method, not a claim that the toy model predicts the data. Self-citations to the Hamiltonian [1], the topological correction [69], and the infinite-volume binding procedure [71] are peer-reviewed, independent methodological support; no uniqueness theorem or ansatz is smuggled in via citation. The main caveats—the private-communication R-matrix benchmark [72] and the toy model not covering the composite-alpha, nonlocal, topological-correction aspects of the full calculation—are correctness and robustness risks, not circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central scattering results rest on the eight 3N LECs fitted in Ref. [1] and re-fitted here, plus the toy model parameters used for method validation. No new particles or mediators are introduced. The key domain assumptions are chiral EFT power counting, the validity of Lüscher extraction for a composite alpha, and the accuracy of the private-communication empirical benchmark.

free parameters (4)
  • cE^(0), cE^(1), cE^(2), cD^(0), cD^(1), cD^(2), cE^(l), cE^(t) (8 3N LECs) = not quoted in this paper; fixed by fits in Ref. [1] and re-fit by MCMC in Sec. III.C
    These low-energy constants define the N3LO 3N force used in the scattering calculation. They are fitted to binding energies (and later to n-alpha phase shifts). The central scattering results depend on them.
  • toy model parameters g0, gc, gso, Rc, Rso = Table II: g0=3.3786e-1, gc=-3.4227e-1, Rc=1.7825, gso=-1.0000e-1, Rso=1.6497 (lattice units)
    These five parameters are fitted to the R-matrix phase shifts in the toy model that is used to benchmark Lüscher versus spherical wall extraction.
  • smearing parameters sL, sNL = sL=0.07, sNL=0.5
    Chosen following prior work, not fitted here; they set the range of the 2N interactions and affect the results.
  • lattice spacings a, a_t = a=1.32 fm, a_t=0.20 fm for main; a=1.97 fm for toy model
    Discretization parameters; results may be sensitive to them. Not fitted in the physical sense, but chosen inputs.
assumptions (5)
  • domain assumption Chiral effective field theory power counting at N3LO with smeared N2LO 3N interactions; higher-order terms are implicitly included through smearing.
    The Hamiltonian in Eq. (14) relies on the validity of Weinberg power counting and the smearing expansion, Sec. II.B.
  • domain assumption Lüscher quantization condition is valid for the n-alpha system with a composite alpha at box sizes L=6.6-15.8 fm and energies up to about 15 MeV.
    Sec. II.C applies Eq. (26) with the topological correction Eq. (31); assumes finite-range interaction and no inelastic channels.
  • domain assumption The empirical R-matrix phase shifts from Ref. [72] are accurate and applicable.
    Private communication, no uncertainties stated; used as the benchmark throughout Sec. III.
  • domain assumption Wavefunction matching perturbative treatment of the difference H-H_S converges.
    Sec. II.A; inherited from Ref. [1].
  • domain assumption The LECs c1, c3, c4 of the 3N TPE potential are fixed from pion-nucleon scattering data, and g_A, F_pi, M_pi take their standard values.
    Eqs. (19)-(21) use c1=-1.10(3), c3=-5.54(6), c4=4.17(4) GeV^-1 from Ref. [49].

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Pith. "Pith review of Ab initio lattice study of neutron-alpha scattering with chiral forces at N3LO." pith.science (2026). https://pith.science/paper/5C5F7MZH

@misc{pith2026250708495,
  author       = {Pith},
  title        = {Pith review of: Ab initio lattice study of neutron-alpha scattering with chiral forces at N3LO},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5C5F7MZH}},
  note         = {Machine review of arXiv:2507.08495}
}
abstract

We present the first ab initio lattice calculation of neutron-alpha ($n$-$\alpha$) scattering using nuclear lattice effective field theory (NLEFT) with chiral interactions at next-to-next-to-next-to-leading order (N3LO). Building on the high-fidelity chiral Hamiltonian introduced in Ref. [1], we compute scattering phase shifts in the $S$- and $P$-wave channels using the L\"uscher finite-volume method. Our results demonstrate excellent agreement with empirical $R$-matrix phase shifts in the $^2S_{1/2}$ and $^2P_{3/2}$ channels, while revealing persistent discrepancies in the $^2P_{1/2}$ channel for neutron energies above 5 MeV. To systematically investigate these discrepancies, we construct and analyze a simplified neutron-alpha toy model, demonstrating that these discrepancies are not due to the use of the L\"uscher finite-volume method. Additionally, we revisit our three-nucleon (3N) force fitting procedure, explicitly incorporating neutron-alpha scattering data through comprehensive Markov Chain Monte Carlo (MCMC) sampling. This analysis confirms the stability of nuclear binding-energy predictions and highlights the need for further refinements in the lattice N3LO three-nucleon forces to fully describe neutron-alpha scattering in the challenging ${}^2P_{1/2}$ channel.

Figures

Figures reproduced from arXiv: 2507.08495 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Phase shifts calculated for the neutron-alpha toy model with the spin-orbit interaction included, plotted as [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Plots of the relative deviations between calculated binding energies and experimental data. To benchmark the [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Neutron-alpha phase shifts in the [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Neutron-alpha phase shifts in the [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

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Reference graph

Works this paper leans on

75 extracted references · 30 canonical work pages · cited by 3 Pith papers

  1. [1]

    Elhatisari et al., Nature 630, 59 (2024), 2210.17488

    S. Elhatisari et al., Nature 630, 59 (2024), 2210.17488

    arXiv 2024

  2. [72]

    G. M. Hale, Private communication (2023)

    work page 2023

  3. [2]

    Ab initio many-body calculations of nucleon-nucleus scattering

    S. Quaglioni and P. Navratil, Phys. Rev. C 79, 044606 (2009), 0901.0950

    work page Pith review arXiv 2009

  4. [3]

    Navratil, R

    P. Navratil, R. Roth, and S. Quaglioni, Phys. Rev. C 82, 034609 (2010), 1007.0525

    arXiv 2010

  5. [4]

    Ab initio many-body calculations of deuteron-4He scattering and 6Li states

    P. Navratil and S. Quaglioni, Phys. Rev. C 83, 044609 (2011), 1102.2042

    work page Pith review arXiv 2011

  6. [5]

    V orabbi, M

    M. V orabbi, M. Gennari, P. Finelli, C. Giusti, and P. Navr´atil (2025), 2506.15406

    work page arXiv 2025

  7. [6]

    J. E. Lynn, I. Tews, J. Carlson, S. Gandolfi, A. Gezerlis, K. E. Schmidt, and A. Schwenk, Phys. Rev. Lett. 116, 062501 (2016), 1509.03470

    arXiv 2016

  8. [7]

    Ab Initio Many-Body Calculations of n-3H, n-4He, p-{3,4}He, and n-10Be Scattering

    S. Quaglioni and P. Navratil, Phys. Rev. Lett. 101, 092501 (2008), 0804.1560

    work page Pith review arXiv 2008

Show all 75 references
  1. [8]

    Kravvaris, K

    K. Kravvaris, K. R. Quinlan, S. Quaglioni, K. A. Wendt, and P. Navratil, Phys. Rev. C 102, 024616 (2020), 2004.08474

    2020 arXiv

  2. [9]

    S. C. Pieper and V . R. Pandharipande, Phys. Rev. Lett.70, 2541 (1993)

    1993

  3. [10]

    Hebeler, Phys

    K. Hebeler, Phys. Rept. 890, 1 (2021), 2002.09548

    2021 arXiv

  4. [11]

    Lazauskas, Phys

    R. Lazauskas, Phys. Rev. C 97, 044002 (2018), 1711.04716

    2018 arXiv

  5. [12]

    Lazauskas and J

    R. Lazauskas and J. Carbonell, Front. in Phys. 7, 251 (2020), 2002.05876

    2020 arXiv

  6. [13]

    Zhang, S

    X. Zhang, S. R. Stroberg, P. Navr ´atil, C. Gwak, J. A. Melendez, R. J. Furnstahl, and J. D. Holt, Phys. Rev. Lett. 125, 112503 (2020), 2004.13575. 21

    2020 arXiv

  7. [14]

    Mercenne, K

    A. Mercenne, K. D. Launey, T. Dytrych, J. E. Escher, S. Quaglioni, G. H. Sargsyan, D. Langr, and J. P. Draayer, Comput. Phys. Commun. 280, 108476 (2022), 2111.01979

    2022 arXiv

  8. [15]

    A. M. Shirokov, A. I. Mazur, I. A. Mazur, E. A. Mazur, I. J. Shin, Y . Kim, L. D. Blokhintsev, and J. P. Vary, Phys. Rev. C 98, 044624 (2018), 1808.03394

    2018 arXiv

  9. [16]

    Carlson, K

    J. Carlson, K. E. Schmidt, and M. H. Kalos, Phys. Rev. C 36, 27 (1987)

    1987

  10. [17]

    K. M. Nollett, S. C. Pieper, R. B. Wiringa, J. Carlson, and G. M. Hale, Phys. Rev. Lett. 99, 022502 (2007), nucl-th/0612035

    2007 arXiv

  11. [18]

    Y . Yang, E. Epelbaum, J. Meng, L. Meng, and P. Zhao (2025), 2502.09961

    2025

  12. [19]

    Lee, Prog

    D. Lee, Prog. Part. Nucl. Phys. 63, 117 (2009), 0804.3501

    2009 arXiv

  13. [20]

    T. A. L ¨ahde and U.-G. Meißner, Nuclear Lattice Effective Field Theory: An introduction , vol. 957 (Springer, 2019)

    2019

  14. [21]

    Elhatisari et al., Phys

    S. Elhatisari et al., Phys. Rev. Lett. 117, 132501 (2016), 1602.04539

    2016 arXiv

  15. [22]

    Elhatisari, E

    S. Elhatisari, E. Epelbaum, H. Krebs, T. A. L ¨ahde, D. Lee, N. Li, B.-n. Lu, U.-G. Meißner, and G. Rupak, Phys. Rev. Lett. 119, 222505 (2017), 1702.05177

    2017 arXiv

  16. [23]

    Elhatisari, D

    S. Elhatisari, D. Lee, G. Rupak, E. Epelbaum, H. Krebs, T. A. L ¨ahde, T. Luu, and U.-G. Meißner, Nature 528, 111 (2015), 1506.03513

    2015 arXiv

  17. [24]

    Elhatisari, D

    S. Elhatisari, D. Lee, U.-G. Meißner, and G. Rupak, Eur. Phys. J. A 52, 174 (2016), 1603.02333

    2016 arXiv

  18. [25]

    Elhatisari, K

    S. Elhatisari, K. Katterjohn, D. Lee, U.-G. Meißner, and G. Rupak, Phys. Lett. B 768, 337 (2017), 1610.09095

    2017 arXiv

  19. [26]

    Elhatisari, T

    S. Elhatisari, T. A. L ¨ahde, D. Lee, U.-G. Meißner, and T. V onk, JHEP02, 001 (2022), 2112.09409

    2022 arXiv

  20. [27]

    B.-N. Lu, N. Li, S. Elhatisari, D. Lee, J. E. Drut, T. A. L¨ahde, E. Epelbaum, and U.-G. Meißner, Phys. Rev. Lett. 125, 192502 (2020), 1912.05105

    2020 arXiv

  21. [28]

    Z. Ren, S. Elhatisari, T. A. L ¨ahde, D. Lee, and U.-G. Meißner, Phys. Lett. B 850, 138463 (2024), 2305.15037

    2024 arXiv

  22. [29]

    S. Shen, S. Elhatisari, T. A. L ¨ahde, D. Lee, B.-N. Lu, and U.-G. Meißner, Nature Commun. 14, 2777 (2023), 2202.13596

    2023 arXiv

  23. [30]

    S. Shen, S. Elhatisari, D. Lee, U.-G. Meißner, and Z. Ren, Phys. Rev. Lett. 134, 162503 (2025), 2411.14935

    2025 arXiv

  24. [31]

    Y .-Z. Ma, Z. Lin, B.-N. Lu, S. Elhatisari, D. Lee, N. Li, U.-G. Meißner, A. W. Steiner, and Q. Wang, Phys. Rev. Lett. 132, 232502 (2024), 2306.04500

    2024 arXiv

  25. [32]

    K ¨onig et al., Phys

    K. K ¨onig et al., Phys. Rev. Lett. 132, 162502 (2024), [Erratum: Phys.Rev.Lett. 133, 059901 (2024)], 2309.02037

    2024

  26. [33]

    Elhatisari, F

    S. Elhatisari, F. Hildenbrand, and U.-G. Meißner, Phys. Lett. B 859, 139086 (2024), 2408.06670

    2024 arXiv

  27. [34]

    Zhang, S

    S. Zhang, S. Elhatisari, U.-G. Meißner, and S. Shen (2024), 2411.17462

    2024 arXiv

  28. [35]

    Y .-H. Song, M. Kim, Y . Kim, K. Cho, S. Elhatisari, D. Lee, Y .-Z. Ma, and U.-G. Meißner (2025), 2502.18722

    2025

  29. [36]

    Hildenbrand, S

    F. Hildenbrand, S. Elhatisari, Z. Ren, and U.-G. Meißner, Eur. Phys. J. A 60, 215 (2024), 2406.17638

    2024 arXiv

  30. [37]

    B.-N. Lu, N. Li, S. Elhatisari, Y .-Z. Ma, D. Lee, and U.-G. Meißner, Phys. Rev. Lett. 128, 242501 (2022), 2111.14191

    2022 arXiv

  31. [38]

    N. Li, S. Elhatisari, E. Epelbaum, D. Lee, B.-N. Lu, and U.-G. Meißner, Phys. Rev. C 98, 044002 (2018), 1806.07994. 22

    2018 arXiv

  32. [39]

    J. Liu, T. Wang, and B.-N. Lu (2025), 2502.13565

    2025 arXiv

  33. [40]

    C. Wu, T. Wang, B.-N. Lu, and N. Li (2025), 2503.18017

    2025

  34. [41]

    T. Wang, X. Feng, and B.-N. Lu (2025), 2503.23840

    2025

  35. [42]

    B.-N. Lu, N. Li, S. Elhatisari, D. Lee, E. Epelbaum, and U.-G. Meißner, Phys. Lett. B 797, 134863 (2019), 1812.10928

    2019 arXiv

  36. [43]

    Reinert, H

    P. Reinert, H. Krebs, and E. Epelbaum, Eur. Phys. J. A 54, 86 (2018), 1711.08821

    2018 arXiv

  37. [44]

    Fettes, U.-G

    N. Fettes, U.-G. Meißner, and S. Steininger, Nucl. Phys. A 640, 199 (1998), hep-ph/9803266

    1998 arXiv

  38. [45]

    Weinberg, Nucl

    S. Weinberg, Nucl. Phys. B 363, 3 (1991)

    1991

  39. [46]

    J. L. Friar, D. Huber, and U. van Kolck, Phys. Rev. C 59, 53 (1999), nucl-th/9809065

    1999 arXiv

  40. [47]

    Epelbaum, A

    E. Epelbaum, A. Nogga, W. Gloeckle, H. Kamada, U. G. Meißner, and H. Witala, Phys. Rev. C 66, 064001 (2002), nucl-th/0208023

    2002 arXiv

  41. [48]

    Epelbaum, H

    E. Epelbaum, H. Krebs, D. Lee, and U.-G. Meißner, Eur. Phys. J. A 41, 125 (2009), 0903.1666

    2009 arXiv

  42. [49]

    Hoferichter, J

    M. Hoferichter, J. Ruiz de Elvira, B. Kubis, and U.-G. Meißner, Phys. Rev. Lett. 115, 192301 (2015), 1507.07552

    2015 arXiv

  43. [50]

    L ¨uscher, Communications in Mathematical Physics 105, 153 (1986)

    M. L ¨uscher, Communications in Mathematical Physics 105, 153 (1986)

    1986

  44. [51]

    L ¨uscher, Nuclear Physics B 354, 531 (1991)

    M. L ¨uscher, Nuclear Physics B 354, 531 (1991)

    1991

  45. [52]

    Rummukainen and S

    K. Rummukainen and S. Gottlieb, Nuclear Physics B 450, 397 (1995)

    1995

  46. [53]

    Bernard, M

    V . Bernard, M. Lage, U.-G. Meißner, and A. Rusetsky, Journal of High Energy Physics 2008, 024 (2008), [arXiv:0806.4495]

    2008 arXiv

  47. [54]

    Luu and M

    T. Luu and M. J. Savage, Phys. Rev. D 83, 114508 (2011), [arXiv:1101.3347]

    2011 arXiv

  48. [55]

    Fu, Phys

    Z. Fu, Phys. Rev. D 85, 014506 (2012), [arXiv:1110.0319]

    2012 arXiv

  49. [56]

    D ¨oring, U.-G

    M. D ¨oring, U.-G. Meißner, E. Oset, and A. Rusetsky, The European Physical Journal A 48, 114 (2012), [arXiv:1205.4838]

    2012 arXiv

  50. [57]

    Leskovec and S

    L. Leskovec and S. Prelovsek, Phys. Rev. D 85, 114507 (2012), [arXiv:1202.2145]

    2012 arXiv

  51. [58]

    G ¨ockeler, R

    M. G ¨ockeler, R. Horsley, M. Lage, U.-G. Meißner, P. E. L. Rakow, A. Rusetsky, G. Schierholz, and J. M. Zanotti, Phys. Rev. D 86, 094513 (2012), [arXiv:1206.4141]

    2012 arXiv

  52. [59]

    Brice ˜no, Z

    R. Brice ˜no, Z. Davoudi, and T. Luu, Phys. Rev. D88, 034502 (2013), [arXiv:1305.4903]

    2013 arXiv

  53. [60]

    R. A. Brice ˜no and Z. Davoudi, Phys. Rev. D88, 094507 (2013), [arXiv:1204.1110]

    2013 arXiv

  54. [61]

    R. A. Brice ˜no, Z. Davoudi, T. C. Luu, and M. J. Savage, Phys. Rev. D88, 114507 (2013), [arXiv:1309.3556]

    2013 arXiv

  55. [62]

    R. A. Brice ˜no, Phys. Rev. D 89, 074507 (2014), [arXiv:1401.3312]

    2014 arXiv

  56. [63]

    Morningstar, J

    C. Morningstar, J. Bulava, B. Singha, R. Brett, J. Fallica, A. Hanlon, and B. H ¨orz, Nucl. Phys. B 924, 477 (2017), 1707.05817

    2017 arXiv

  57. [64]

    Bernard, M

    V . Bernard, M. Lage, U.-G. Meißner, and A. Rusetsky, Journal of High Energy Physics 2011, 19 (2011), [arXiv:1010:6018]

    2011

  58. [65]

    Mart ´ınez Torres, L

    A. Mart ´ınez Torres, L. R. Dai, C. Koren, D. Jido, and E. Oset, Phys. Rev. D 85, 014027 (2012), [arXiv:1109.0396]

    2012 arXiv

  59. [66]

    D ¨oring and U.-G

    M. D ¨oring and U.-G. Meißner, Journal of High Energy Physics 2012, 9 (2012), [arXiv:1111.0616]. 23

    2012 arXiv

  60. [67]

    Li and C

    N. Li and C. Liu, Phys. Rev. D 87, 014502 (2013), [arXiv:1209.2201]

    2013 arXiv

  61. [68]

    D ¨oring, M

    M. D ¨oring, M. Mai, and U.-G. Meißner, Physics Letters B 722, 185 (2013), [arXiv:1302:4065]

    2013

  62. [69]

    S. Bour, S. Koenig, D. Lee, H. W. Hammer, and U.-G. Meißner, Phys. Rev. D 84, 091503 (2011), 1107.1272

    2011 arXiv

  63. [70]

    Elhatisari and D

    S. Elhatisari and D. Lee, Phys. Rev. C 90, 064001 (2014), 1407.2784

    2014 arXiv

  64. [71]

    K ¨onig and D

    S. K ¨onig and D. Lee, Phys. Lett. B 779, 9 (2018), 1701.00279

    2018 arXiv

  65. [73]

    Carlson, V

    J. Carlson, V . R. Pandharipande, and R. B. Wiringa, Nucl. Phys. A 424, 47 (1984)

    1984

  66. [74]

    Borasoy, E

    B. Borasoy, E. Epelbaum, H. Krebs, D. Lee, and U.-G. Meißner, Eur. Phys. J. A 34, 185 (2007), 0708.1780

    2007 arXiv

  67. [75]

    Elhatisari, U.-G

    S. Elhatisari, U.-G. Meißner, and Z. Ren, Work in progress

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