pith. sign in

arxiv: 2605.22134 · v1 · pith:OV3O533Jnew · submitted 2026-05-21 · ⚛️ nucl-th · hep-ph

Bootstrapping Two-Nucleon Effective Field Theories

Q.N. Micha-Mba , M.S. S\'anchez , P.G. Ortega , J.A. Oller , D.R. Entem This is my paper

Pith reviewed 2026-05-22 02:28 UTC · model grok-4.3

classification ⚛️ nucl-th hep-ph
keywords chiral effective field theorytwo-nucleon scatteringbootstrap resamplingrenormalizationphase shifts1S0 partial waveNLO validity range
0
0 comments X

The pith

Next-to-leading order chiral effective field theory for two-nucleon scattering remains consistent with data over a wider energy range than leading order.

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

The paper tests how far chiral effective field theory can describe nucleon-nucleon scattering after regularization and renormalization. It applies bootstrap resampling to measure statistical consistency between the theory and either a controllable toy model or the Granada phase-shift data in the singlet S-wave. A reader would care because the results quantify the practical energy window where each order of the theory can be used reliably for nuclear-force calculations.

Core claim

Using renormalization by contact terms and the N/D method with multiple subtractions, the authors show through bootstrap resampling on a toy model and on the 1S0 wave that the NLO potential significantly extends the energy range over which the renormalized effective field theory stays statistically consistent with the full theory or Granada phase-shift analysis.

What carries the argument

Bootstrap resampling applied to residuals between renormalized EFT predictions and reference phase shifts after fixing low-energy constants in either contact-term or N/D subtraction schemes.

If this is right

  • The NLO potential can be used reliably up to higher laboratory energies than LO in the two-nucleon system.
  • Both renormalization schemes gain from the bootstrap assessment of consistency.
  • Statistical resampling provides a systematic way to determine the domain of validity for successive orders of chiral EFT.
  • The wider validity window at NLO reduces the immediate requirement for higher-order terms in low-energy applications.

Where Pith is reading between the lines

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

  • The same bootstrap procedure could be applied to additional partial waves to map order-by-order improvements across the full two-nucleon interaction.
  • If the method proves robust, it offers a route to test consistency in three-nucleon forces where data are sparser.
  • Validated energy ranges could inform cutoff choices in practical calculations of nuclear binding and reactions.

Load-bearing premise

The bootstrap resampling procedure accurately captures the statistical consistency between the renormalized EFT predictions and the full theory or Granada phase-shift data without introducing bias from the choice of energy bins or subtraction points.

What would settle it

Finding that actual deviations between NLO EFT phase shifts and Granada data exceed the bootstrap-derived uncertainty bands at energies where consistency is claimed would falsify the reported extension of the validity range.

Figures

Figures reproduced from arXiv: 2605.22134 by D.R. Entem, J.A. Oller, M.S. S\'anchez, P.G. Ortega, Q.N. Micha-Mba.

Figure 1
Figure 1. Figure 1: Panels on the left are for the singular repulsive case while on the right for the singular attractive case. In [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: χ 2 /dof for fits with data between 1 MeV and kmax. The red (blue) band shows the 1σ (2σ) confidence level. Data are generated as explained in the text with ∆δ = 1 ◦ (purple line), 0.1 ◦ (gold line) and 0.01◦ (green line). In all cases the full theory is considered. From left to right, panels correspond to the N/D11, N/D12, and N/D22 solutions. fittings up to a certain kmax, displaying the 1σ (α = 0.6827) … view at source ↗
Figure 3
Figure 3. Figure 3: Residuals for ∆δ = 0.01◦ in three cases. The purple, gold, and green dots correspond to N/D11 with kmax = 400 MeV, N/D22 with kmax = 400 MeV, and N/D22 with kmax = 200 MeV, respectively. The solid line corresponds to the N (0,1) distribution. -0.010 -0.005 0.000 0.005 0.010 0 50 100 150 200 250 300 350 400 <δfit>-δtheo k (MeV) -0.010 -0.005 0.000 0.005 0.010 0 50 100 150 200 250 300 350 400 <δfit>-δtheo k … view at source ↗
Figure 4
Figure 4. Figure 4: Difference between the mean of the 2000 fitted phase shifts and the exact result. The errors show the mean [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: χ 2 /dof for fits with data between 1 MeV and kmax. The red band shows the 1σ confidence level, while the blue band shows the 2σ region. Data are generated as explained in the text with ∆δ = 1 ◦ (purple line), 0.1 ◦ (gold line) and 0.01◦ (green line). In all cases the theory at LO is considered. The upper row of panels corresponds to the N/D solutions (N/D11, N/D12, and N/D22 from left to right). The lower… view at source ↗
Figure 6
Figure 6. Figure 6: The upper panel corresponds to the distribution of the residuals in three different cases for the N/D calcu [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Same as in Fig [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: At LO, phase shift up to 400 MeV for the bootstrap fit from N/D [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: χ 2/dof for fits with data between 1 MeV and kmax. The red band shows the 1σ confidence level, while the blue band shows the 2σ region. Data are generated as explained in the text with ∆δ = 1 ◦ (purple line), 0.1 ◦ (gold line) and 0.01◦ (green line). The left figure corresponds to the N/D22 solution at NLO, while the right figure with three contact terms at NLO. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40… view at source ↗
Figure 10
Figure 10. Figure 10: Same as in Fig [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: At NLO, phase shift up to 400 MeV extrapolated from the [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: χ 2/dof for the bootstraps to the Granada phase shift analysis at LO (green points) and NLO (red points). 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 -4 -3 -2 -1 0 1 2 3 4 frecuency χ 2 /dof LO NLO N(0,1) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 -4 -3 -2 -1 0 1 2 3 4 frecuency χ 2 /dof LO NLO N(0,1) [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Residuals for the bootstrap fits corresponding to the N/D [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Bootstrap fits corresponding to the N/D22 solution for kmax = 100 MeV (left column) and kmax = 150 MeV (right column). The first row shows in a green line the difference between the mean of the fits at LO and the Granada phase shifts, the blue shaded area is the 1σ confidence level of the Granada data, and the green shaded area is the 1σ confidence level of the fits. Same in the second row in red for the … view at source ↗
Figure 15
Figure 15. Figure 15: Phase-shift extrapolation up to 400 MeV for the bootstrap fits associated to the N/D [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
read the original abstract

Chiral EFT yields singular potentials that require regularization and renormalization when implemented in a dynamical equation such as the Lippmann--Schwinger equation. We employ two different approaches, renormalization with contact terms -- as is most commonly done in chiral EFT -- and the exact N/D method with multiple subtractions. We start with a toy model in which we can control the finite-range expansion of the potential, treating the full potential as the `exact' theory. To assess the statistical consistency of the approaches with the full theory, we use the bootstrap technique. We apply the same framework to study the consistency of chiral EFT at LO and NLO with the Granada phase-shift analysis in the $^1S_0$ two-nucleon partial wave. Our results show that the NLO potential significantly extends the energy range over which the theory remains valid.

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 examines renormalization of singular potentials in chiral EFT for two-nucleon scattering via two schemes: conventional contact-term renormalization and the exact N/D method with multiple subtractions. A toy model with controllable finite-range expansion serves as the 'exact' theory for bootstrap-based statistical consistency tests. The same framework is applied to the 1S0 channel against Granada phase-shift data, with the central result that the NLO potential extends the energy range of validity relative to LO.

Significance. If the bootstrap analysis is free of bias from binning and subtraction choices, the work supplies a statistically grounded, quantitative metric for the range of applicability of chiral EFT in the NN sector. The dual renormalization approaches plus external-data comparison constitute a useful diagnostic tool for assessing order-by-order convergence.

major comments (2)
  1. [methods section on bootstrap resampling] The bootstrap resampling procedure (described in the methods section on statistical consistency) employs fixed energy bins and specific subtraction points in the N/D implementation without reported variation or sensitivity tests. Because the claim that NLO extends the validity range rests on the consistency metric remaining stable, any dependence on these discrete choices would undermine the statistical grounding of the result.
  2. [application to Granada phase-shift analysis] When the framework is applied to the Granada 1S0 phase shifts, the manuscript does not detail how data exclusions, bin widths, or weighting enter the bootstrap; without this, it is unclear whether the reported extension of the valid energy range is robust or sensitive to analysis choices that differ from the toy-model section.
minor comments (2)
  1. [Abstract] The abstract states that NLO 'significantly extends' the range but does not quote the quantitative intervals (e.g., up to what lab energy) obtained from the bootstrap; adding these numbers would improve clarity.
  2. [N/D method description] Notation for the subtraction constants in the N/D method could be introduced with an explicit equation rather than inline description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight important aspects of the statistical robustness of our bootstrap analysis. We address each major comment below and will revise the manuscript accordingly to provide the requested details and tests.

read point-by-point responses

  1. Referee: [methods section on bootstrap resampling] The bootstrap resampling procedure (described in the methods section on statistical consistency) employs fixed energy bins and specific subtraction points in the N/D implementation without reported variation or sensitivity tests. Because the claim that NLO extends the validity range rests on the consistency metric remaining stable, any dependence on these discrete choices would undermine the statistical grounding of the result.

    Authors: We agree that the absence of explicit sensitivity tests to binning and subtraction-point choices leaves open the possibility of dependence on these discrete selections. In the revised manuscript we will add a dedicated subsection reporting bootstrap results for several alternative bin widths (spanning the range used in the Granada analysis) and varied subtraction points, demonstrating that the NLO extension of the validity range remains stable within the reported uncertainties. This addition will directly address the concern and strengthen the statistical grounding of the central claim. revision: yes

  2. Referee: [application to Granada phase-shift analysis] When the framework is applied to the Granada 1S0 phase shifts, the manuscript does not detail how data exclusions, bin widths, or weighting enter the bootstrap; without this, it is unclear whether the reported extension of the valid energy range is robust or sensitive to analysis choices that differ from the toy-model section.

    Authors: We acknowledge that the manuscript does not provide a sufficiently explicit description of how the Granada data set is prepared for the bootstrap procedure. In the revision we will expand the relevant section to specify the precise data exclusions applied, the bin widths adopted for resampling, and the weighting scheme (including any covariance information from the phase-shift analysis). These details will be presented in parallel with the toy-model implementation so that readers can directly assess consistency between the two applications and confirm the robustness of the NLO result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims grounded in external comparisons

full rationale

The derivation chain relies on treating a toy model as an exact reference and performing bootstrap consistency checks against independent Granada phase-shift data in the 1S0 wave. The NLO extension of validity range is evaluated statistically via these external benchmarks rather than by fitting parameters that are then renamed as predictions or by self-definitional reductions. No load-bearing self-citations, ansatz smuggling, or uniqueness theorems imported from prior author work are indicated in the manuscript description; the statistical procedure compares renormalized EFT outputs to outside references without reducing the consistency metric to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents exhaustive extraction; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5686 in / 1106 out tokens · 51644 ms · 2026-05-22T02:28:40.696016+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages · 23 internal anchors

  1. [1]

    Weinberg, Phys

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

    work page 1990

  2. [2]

    Weinberg, Nucl

    S. Weinberg, Nucl. Phys. B363, 3 (1991)

    work page 1991

  3. [3]

    Ordóñez and U

    C. Ordóñez and U. van Kolck, Physics Letters B291, 459 (1992)

    work page 1992

  4. [4]

    Ordóñez, L

    C. Ordóñez, L. Ray, and U. van Kolck, Phys. Rev. Lett.72, 1982 (1994)

    work page 1982

  5. [5]

    Ordóñez, L

    C. Ordóñez, L. Ray, and U. van Kolck, Phys. Rev. C53, 2086 (1996)

    work page 2086

  6. [6]

    D. B. Kaplan, M. J. Savage, and M. B. Wise, Nucl. Phys. B478, 629 (1996), arXiv:nucl-th/9605002

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  7. [7]

    D. B. Kaplan, M. J. Savage, and M. B. Wise, Phys. Lett. B424, 390 (1998), arXiv:nucl-th/9801034

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  8. [8]

    D. B. Kaplan, M. J. Savage, and M. B. Wise, Nucl. Phys. B534, 329 (1998), arXiv:nucl-th/9802075 . 21

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  9. [9]

    Nogga, R

    A. Nogga, R. G. E. Timmermans, and U. van Kolck, Phys. Rev. C72, 054006 (2005)

    work page 2005

  10. [10]

    M. C. Birse, Phys. Rev. C74, 014003 (2006)

    work page 2006

  11. [11]

    Pavón Valderrama, Phys

    M. Pavón Valderrama, Phys. Rev. C83, 024003 (2011)

    work page 2011

  12. [12]

    Pavón Valderrama, Phys

    M. Pavón Valderrama, Phys. Rev. C84, 064002 (2011)

    work page 2011

  13. [13]

    Renormalizing Chiral Nuclear Forces: A Case Study of 3P0

    B. Long and C. J. Yang, Phys. Rev. C84, 057001 (2011), arXiv:1108.0985 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  14. [14]

    Renormalizing Chiral Nuclear Forces: Triplet Channels

    B. Long and C. J. Yang, Phys. Rev. C85, 034002 (2012), arXiv:1111.3993 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  15. [15]

    Short-range nuclear forces in singlet channels

    B. Long and C. J. Yang, Phys. Rev. C86, 024001 (2012), arXiv:1202.4053 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  16. [16]

    Epelbaum and U.-G

    E. Epelbaum and U.-G. Meißner, Few-Body Systems54, 2175 (2013)

    work page 2013

  17. [17]

    Epelbaum and J

    E. Epelbaum and J. Gegelia, The European Physical Journal A41, 341 (2009)

    work page 2009

  18. [18]

    Machleidt and D

    R. Machleidt and D. R. Entem, Journal of Physics G: Nuclear and Particle Physics37, 064041 (2010)

    work page 2010

  19. [19]

    Marji, A

    E. Marji, A. Canul, Q. MacPherson, R. Winzer, C. Zeoli, D. R. Entem, and R. Machleidt, Phys. Rev. C88, 054002 (2013)

    work page 2013

  20. [20]

    Epelbaum, A

    E. Epelbaum, A. M. Gasparyan, J. Gegelia, U.-G. Meißner, and X.-L. Ren, The European Physical Journal A56, 152 (2020)

    work page 2020

  21. [21]

    Wilsonian renormalization group versus subtractive renormalization in effective field theories for nucleon--nucleon scattering

    E. Epelbaum, J. Gegelia, and U.-G. Meißner, Nucl. Phys. B925, 161 (2017), arXiv:1705.02524 [nucl- th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  22. [22]

    Wilsonian renormalization group and the Lippmann-Schwinger equation with a multitude of cutoff parameters

    E. Epelbaum, J. Gegelia, and U.-G. Meißner, Commun. Theor. Phys.69, 303 (2018), arXiv:1710.04178 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  23. [23]

    How (not) to renormalize integral equations with singular potentials in effective field theory

    E. Epelbaum, A. M. Gasparyan, J. Gegelia, and U.-G. Meißner, Eur. Phys. J. A54, 186 (2018), arXiv:1810.02646 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  24. [24]

    Reply to "Comment on "How (not) to renormalize integral equations with singular potentials in effective field theory"

    E. Epelbaum, A. M. Gasparyan, J. Gegelia, and U.-G. Meißner, Eur. Phys. J. A55, 56 (2019), arXiv:1903.01273 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  25. [25]

    van Kolck, Front

    U. van Kolck, Front. in Phys.8, 79 (2020), arXiv:2003.06721 [nucl-th]

    work page arXiv 2020

  26. [26]

    R. Peng, B. Long, and F.-R. Xu, Phys. Rev. C110, 054001 (2024)

    work page 2024

  27. [27]

    C. J. Yang, A. Ekström, C. Forssén, G. Hagen, G. Rupak, and U. van Kolck, Eur. Phys. J. A59, 233 (2023), arXiv:2109.13303 [nucl-th]

    work page arXiv 2023

  28. [28]

    V . Baru, E. Epelbaum, J. Gegelia, and X. L. Ren, Phys. Lett. B798, 134987 (2019), arXiv:1905.02116 [nucl-th]

    work page arXiv 2019

  29. [29]

    A. M. Gasparyan and E. Epelbaum, Phys. Rev. C107, 034001 (2023), arXiv:2210.16225 [nucl-th]

    work page arXiv 2023

  30. [30]

    A. M. Gasparyan and E. Epelbaum, Phys. Rev. C105, 024001 (2022), arXiv:2110.15302 [nucl-th]

    work page arXiv 2022

  31. [31]

    K. M. Case, Phys. Rev.80, 797 (1950)

    work page 1950

  32. [32]

    Frank, D

    W. Frank, D. J. Land, and R. M. Spector, Rev. Mod. Phys.43, 36 (1971)

    work page 1971

  33. [33]

    Renormalization of the Deuteron with One Pion Exchange

    M. Pavón Valderrama and E. Ruiz Arriola, Phys. Rev. C72, 054002 (2005), arXiv:nucl-th/0504067 . 22

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  34. [34]

    Renormalization of NN Interaction with Chiral Two Pion Exchange Potential. Central Phases and the Deuteron

    M. Pavón Valderrama and E. Ruiz Arriola, Phys. Rev. C74, 054001 (2006), arXiv:nucl-th/0506047

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  35. [35]

    Renormalization of NN Interaction with Chiral Two Pion Exchange Potential. Non-Central Phases

    M. Pavón Valderrama and E. Ruiz Arriola, Phys. Rev. C74, 064004 (2006), [Erratum: Phys.Rev.C 75, 059905 (2007)], arXiv:nucl-th/0507075

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  36. [36]

    Infinite-cutoff renormalization of the chiral nucleon-nucleon interaction at N3LO

    C. Zeoli, R. Machleidt, and D. R. Entem, Few Body Syst.54, 2191 (2013), arXiv:1208.2657 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  37. [37]

    Chiral effective field theory and nuclear forces

    R. Machleidt and D. R. Entem, Phys. Rept.503, 1 (2011), arXiv:1105.2919 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  38. [38]

    Modern Theory of Nuclear Forces

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

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  39. [39]

    G. F. Chew and S. Mandelstam, Phys. Rev.119, 467 (1960)

    work page 1960

  40. [40]

    D. R. Entem and J. A. Oller, Phys. Lett. B773, 498 (2017), arXiv:1610.01040 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  41. [41]

    J. A. Oller and D. R. Entem, Annals Phys.411, 167965 (2019), arXiv:1810.12242 [hep-ph]

    work page arXiv 2019

  42. [42]

    D. R. Entem and J. A. Oller, Eur. Phys. J. ST230, 1675 (2021), arXiv:2103.02069 [nucl-th]

    work page arXiv 2021

  43. [43]

    D. R. Entem, J. Nieves, and J. A. Oller, Phys. Rev. D112, 096007 (2025), arXiv:2506.01461 [nucl-th]

    work page arXiv 2025

  44. [44]

    S. R. Beane and M. J. Savage, Nucl. Phys. A694, 511 (2001), arXiv:nucl-th/0011067

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  45. [45]

    Epelbaum, A

    E. Epelbaum, A. M. Gasparyan, J. Gegelia, and U.-G. Meissner, The European Physical Journal A54 (2018)

    work page 2018

  46. [46]

    J. A. Oller, Phys. Rev. C93, 024002 (2016), arXiv:1402.2449 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  47. [47]

    R. J. Furnstahl, N. Klco, D. R. Phillips, and S. Wesolowski, Phys. Rev. C92, 024005 (2015), arXiv:1506.01343 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  48. [48]

    J. A. Meléndez, S. Wesolowski, and R. J. Furnstahl, Phys. Rev. C96, 024003 (2017), arXiv:1704.03308 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  49. [49]

    J. A. Meléndez, R. J. Furnstahl, D. R. Phillips, M. T. Pratola, and S. Wesolowski, Phys. Rev. C100, 044001 (2019), arXiv:1904.10581 [nucl-th]

    work page arXiv 2019

  50. [50]

    Efron, The Annals of Statistics7, 1 (1979)

    B. Efron, The Annals of Statistics7, 1 (1979)

    work page 1979

  51. [51]

    Bootstrapping the statistical uncertainties of NN scattering data

    R. Navarro Pérez, J. E. Amaro, and E. Ruiz Arriola, Phys. Lett. B738, 155 (2014), arXiv:1407.3937 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  52. [52]

    Private communication,

    R. Navarro Pérez, “Private communication,”

    work page

  53. [53]

    Partial Wave Analysis of Nucleon-Nucleon Scattering below pion production threshold

    R. Navarro Pérez, J. E. Amaro, and E. Ruiz Arriola, Phys. Rev. C88, 024002 (2013), [Erratum: Phys.Rev.C 88, 069902 (2013)], arXiv:1304.0895 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  54. [54]

    R. N. Pérez, J. E. Amaro, and E. R. Arriola, Phys. Rev. C88, 064002 (2013). 23

    work page 2013