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REVIEW 4 major objections 6 minor 1 cited by

Fant\^omas Unconfined: global QCD fits with B\'ezier parameterizations

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Fantômas, a C++ toolkit in the xFitter framework, claims Bézier control-point parameterizations let fitters raise a fit's flexibility without refitting from scratch — faster, and a transparent alternative to neural networks.

desk verdict Useful toolkit paper with an unbenchmarked speed claim; deserves review after a few fixes. read the letter →

arxiv 2507.22969 v1 pith:OWTV3PLI submitted 2025-07-30 hep-ph

Lucas Kotz , Aurore Courtoy , T. J. Hobbs , Pavel Nadolsky , Fredrick Olness , Maximiliano Ponce-Chavez , Varada Purohit This is my paper
classification hep-ph
keywords partondistributionfunctionsBéziercurvesmetamorphparameterizationcontrolpointsglobalQCDanalysisepistemicuncertaintyxFitterpionPDFs
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

The paper is trying to establish that the shapes of parton distribution functions — the nonperturbative momentum-fraction dependence fitted from scattering data — can be explored systematically and cheaply with Bézier curves whose free parameters are function values at user-chosen control points, an approach the authors call metamorphs. Its concrete assertion is that Fantômas, a C++ implementation integrated into the xFitter global-analysis framework, speeds up this exploration: raising the polynomial degree by adding a control point placed exactly on the previous fit leaves the old minimum and its chi2 untouched, so each more flexible fit starts where the previous one ended. If that holds, fitters can rapidly generate many diverse functional forms and quantify the parametrization (epistemic) uncertainty transparently, in a way that directly refitting polynomial coefficients or training a neural network does not allow. The demonstration is a sequence of pion PDF fits, whose results are combined with the METAPDF method into a single ensemble that folds in experimental, parametrization, and nuclear uncertainties.

What carries the argument

The central object is the metamorph, a PDF parametrization $x f_a(x, Q_0^2) = F^{\rm car}_a(x) \times F^{\rm mod}_a(x)$ in which a carrier function $F^{\rm car}_a(x) = A_a x^{B_a}(1-x)^{C_a}$ sets the endpoint behavior and a modulator $F^{\rm mod}_a(x) = 1 + B^{(N_m)}(y)$ supplies mid-$x$ flexibility. The fitted parameters are the values $P_i = B^{(N_m)}(y(x_i))$ of the Bézier curve at control points $x_i$; the curve's Bernstein coefficients are then $C = M^{-1} T^{-1} P$, where $T$ is the Vandermonde matrix of powers of the stretched variable $y = x^{\alpha_x}$ in the data-covered region, and the condition number of $T$ is the monitored measure of numerical stability. The load-bearing algorithmic step is the NEW control-point option: inserting a control point exactly on the previously fitted modulator leaves the metamorph and its $\chi^2$ unchanged, so raising $N_m$ continues minimization from the previous minimum rather than restarting.

What would settle it

Generate pseudodata from a known 'truth' PDF, fit a metamorph of degree $N_m$, then add a NEW control point on the fitted modulator and evaluate chi2 at the first iteration of the higher-degree fit: the warm-start claim requires it to equal the previous minimum exactly. A visible jump — expected at higher $N_m$ or with clustered control points, where the condition numbers of Appendix B grow large — would falsify the claimed speed advantage. The direct quantitative check is the condition number of $T\cdot M$ in Eq. (11) for the actual control-point spacings used in the pion fits, together with the interpolation error near the control points.

Watch

Extended reading notes

Core claim

The paper's central claim is that Bézier parameterizations give global QCD fits an efficient and interpretable handle on the functional-form problem. A PDF is written as a carrier function $A_a x^{B_a}(1-x)^{C_a}$, which controls the small- and large-$x$ limits, multiplied by a modulator $1+B^{(N_m)}(y(x))$ built from Bernstein-basis polynomials; the fitted parameters are the values of the Bézier curve at $N_m+1$ control points, not the polynomial coefficients, and the coefficients are recovered by inverting the control-point matrix. Because the curve interpolates the control points exactly, inserting a new control point marked NEW on the previously fitted modulator leaves the best-fit function and its $\chi^2$ unchanged, so the degree of the parametrization can be raised as a warm start from the previous minimum — the property demonstrated in the pion fits, where $\chi^2$ decreases continuously across the sequence of fits. The paper further argues that the best-fit polynomial of fixed degree is independent of the placement of the free control points (unisolvence theorem), that existing Bernstein- and Chebyshev-based parametrizations can be cast into the metamorph form without refitting, and that combining many metamorph fits via METAPDF yields a PDF ensemble that accounts for parametrization uncertainty alongside experimental and external uncertainties.

Load-bearing premise

The whole efficiency argument rests on the warm-start claim: placing a new control point exactly on the previously fitted curve must leave the old best fit and its chi2 exactly unchanged, which is only true while the Vandermonde matrix T in Eq. (11) is well conditioned — and Appendix B shows the condition number grows steeply with polynomial degree and with uneven or closely spaced control points, so for higher-degree fits the benefit, and even the uniqueness of the interpolant, degrades.

Editorial extensions

If this is right

  • The degree of a PDF parametrization can be raised incrementally by adding control points, with each higher-degree fit starting at the previous minimum instead of being refit from scratch, shortening the time needed to explore parametrization space.
  • For fixed degree, moving the free control points does not change the eventual best-fit polynomial, so their positions can be adjusted to improve convergence without altering the solution.
  • Moving the fixed control points or the stretching power $\alpha_x$ does change the best-fit metamorph, providing transparent, low-dimensional knobs for generating diverse PDF shapes and probing parametrization uncertainty.
  • Many existing analytical PDF parametrizations, including Bernstein- and Chebyshev-based ones, can be re-expressed as metamorphs without refitting, making the format a common language for comparing functional forms.
  • Combining the individual fits with the METAPDF method produces a single Hessian ensemble (exemplified by Fanto10) that quantifies experimental, parametrization, and external uncertainties at once.

Reading between the lines

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

  • Because the warm-start property lets a search add flexibility without losing progress, an automated fitter could add control points adaptively in x-regions where the data or the chi2 gradient shows remaining freedom — the paper sketches this as 'an intelligent program' but does not implement it.
  • The practical ceiling on polynomial degree is set by the Vandermonde conditioning, so an automatic control-point placement that optimizes the condition number (Chebyshev-like spacing in the stretched variable) would extend the speed advantage to higher degrees than the $N_m = 3$ used here.
  • The metamorph machinery is flavor-agnostic and applies to any x-dependent nonperturbative correlator, so the same parametrization-uncertainty protocol could be carried over to fragmentation functions or transverse-momentum distributions, which the paper mentions but does not develop.
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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

4 major / 6 minor

Summary. Fantômas is a C++ toolkit for representing PDFs via Bézier control-point parameterizations ('metamorphs'), implemented both standalone and as an xFitter module. The paper describes the mathematical setup (carrier × modulator, Bernstein basis, Vandermonde inversion), steering-card I/O, sum rules and prior penalties, and demonstrates a sequence of pion PDF fits with increasing polynomial degree. It also outlines a METAPDF-based procedure for combining fits into an ensemble with aleatoric and epistemic uncertainties. The central advertised advantages are reduced computational cost relative to traditional polynomial parameterizations and an interpretable alternative to neural-network models.

Significance. The toolkit is concrete and reproducible: the interpolation formulas are standard, source code and steering cards are provided, and the pion example gives a realistic end-to-end demonstration. The paper's honest reporting of non-convergence (fit3), condition-number warnings, and prior dependence is a strength. If the computational-efficiency claim can be substantiated with a benchmark, the package would be a useful practical contribution to parametrization-uncertainty studies. However, the current evidence does not establish the headline speed advantage, and one central formula is inconsistent with the provided implementation.

major comments (4)
  1. [Abstract, Section 7, Appendix B.5] The claim that Fantômas reduces computational time compared with traditional polynomial parametrizations is not supported by any fit-level benchmark. The only timing data, Table B.5, report T-matrix inversion times for 10–30 points, not wall-clock or χ²-evaluation comparisons against a standard polynomial PDF fit; the demonstrated pion fits use Nm≤3, for which the inversion cost is negligible. Moreover, standard Bernstein or Chebyshev bases also admit exact degree elevation, so a warm start is not unique to this scheme. Please add a quantitative comparison (e.g., wall-clock per MINUIT iteration or total fit time for the same pion data with a traditional polynomial parametrization) or explicitly qualify the speed claim as asymptotic/architectural.
  2. [Eq. (9), Appendix C] Equation (9) as printed is inconsistent with the Mathematica implementation in Appendix C. With row index l and column index p, the definition m_lp = (-1)^{p-l} binom(l,n) binom(n-p,n-l) vanishes except for the degenerate case l=p=n, whereas the code defines m[p,l] = (-1)^{p-l} Binomial[n,l] Binomial[n-l,n-p]. This discrepancy makes Eqs. (10)–(11) non-reproducible from the paper alone. Please correct the indices/binomial arguments in Eq. (9) and verify that Eq. (11) follows with the same convention.
  3. [Section 5.1, Procedure A, step 2] The retention of a small number (eN=5) of 'distinct PDF shapes' by 'ad-hoc criteria' is not specified, yet this selection is the vehicle for the epistemic-uncertainty claim in Section 5.1 and Ref. [26]. Without a defined distance measure, a stopping rule, or a reproducibility statement, the resulting METAPDF ensemble cannot be distinguished from a subjective pick among acceptable fits. Please either specify the algorithm concretely or reframe the procedure as an exploratory heuristic and temper the 'captures the epistemic uncertainty' interpretation.
  4. [Sections 3.3, 4.1, Appendix B] The warm-start property (a NEW control point placed exactly on the previous modulator leaves χ² unchanged) is exact only in exact arithmetic. Appendix B shows that condition numbers of the interpolation matrix grow rapidly with Nm and with clustered/equally spaced control points (Table B.4), and the code warns above 10,000; for higher-degree fits the new point may no longer lie on the old metamorph after roundoff. Because this property is the stated basis of the efficiency advantage, please quantify the deviation for realistic Nm and CP spacings, or restrict the efficiency claim to the low-degree regime used here.
minor comments (6)
  1. [Section 2.2, Eq. (4)] The definition of g(x,n) is garbled in the typeset text; please check the formula and the limits stated in Eq. (5).
  2. [Section 4.2, Table 2] The notation {0,0,0}, {1,1,1}, etc. for free CPs is not defined before the table; add an explicit mapping to per-flavor counts.
  3. [Section 4.3] The invariance of the best-fit metamorph under relocation of free CPs is stated as a consequence of the unisolvence theorem; the argument should explicitly state the fixed-degree, fixed-carrier, fixed-data conditions under which it holds.
  4. [Section 5.1, Eq. (19)] The use of sqrt(2(Npts-Npar(M))) as a one-standard-deviation threshold assumes χ²_LL is approximately χ²-distributed; please justify or cite this approximation.
  5. [Appendix C] The code evaluates C via Inverse[T^T T] T^T rather than the T^{-1} in Eq. (11); for the square case this is equivalent, but the text should say so to avoid confusion.
  6. [Section 6, Don'ts] The statement that the Runge phenomenon arises from 'equally spaced interpolation points' conflates the phenomenon with its common trigger; please phrase more carefully.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the core Bezier interpolation and warm-start properties are standard mathematics, and the unbenchmarked speed claim is a support weakness rather than a circular argument.

full rationale

The paper's derivation chain is self-contained. The metamorph formalism is ordinary polynomial interpolation in the Bernstein basis: Eq. (6)-(11) express the Bezier polynomial through values at control points via a Vandermonde matrix, and the paper explicitly grounds this in the unisolvence theorem and standard approximation theory. The warm-start property used in Sections 3.3 and 4.1 is a mathematical identity, not a fitted prediction: a NEW control point is defined to lie exactly on the previous modulator, so by the uniqueness of polynomial interpolation the old polynomial is reproduced and chi2 is unchanged at the new fit's start. The paper states this as a construction rather than as an empirically derived result. The unisolvence theorem is cited partly through the authors' prior work [32], but it is a standard external mathematical fact and is not load-bearing in a way that would make the argument self-referential. Self-citations to the pion analyses [25, 26] provide context and prior application, but the central claims of the toolkit are supported by the released code, the included xFitter module, and the new pion illustrations in Section 4. The claimed reduction in computational time is asserted rather than benchmarked with wall-clock comparisons, and the epistemic-uncertainty combination of Section 5 relies on ad-hoc model selection; these are limitations of evidence or correctness risk, not circularity, because no central result is equivalent to its input by construction. Therefore no circular step is exhibited.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

No new physical entities are introduced; 'metamorph' and 'carrier'/'modulator' are naming constructs for a parameterization scheme, not independent physical objects. The parameterization choices and prior weights are the main human inputs that shape the results.

free parameters (6)
  • Carrier exponents B_a, C_a per flavor (Eq. 2)
    Fitted to pion data in Section 4; control the small-x and large-x power-law behavior of each PDF. Initial values appear in steering cards, final values only in output files not quoted.
  • Control point values P_i at FREE control points
    Primary modulator parameters varied by MINUIT; the metamorph is reconstructed from these values via Eq. (11). Final values are written to steering_fantomas_out.txt but not tabulated in the paper.
  • Normalization constants A_a (valence, sea, gluon)
    One normalization is fixed by the valence sum rule and one by the momentum sum rule, the remaining one is free (Eqs. 15-16). Whether Ag or AS is fitted changes the fit.
  • Stretching exponent alpha_x = 1 (nominal), 0.3 explored
    User-chosen hyperparameter in Eq. (4) that changes the Bernstein basis and noticeably changes the fitted PDFs (Table 3).
  • Prior penalty weights w_j (Table 1) = w = 1, 1/(Nm+1), 50, 50
    Hand-selected weights for AbsLog and ReLU penalties; the paper shows the choice changes the gluon and sea PDFs (fit2 vs fit2alt).
  • Polynomial degree Nm and control-point positions x_i = Nm = 0..3; CPs at 0.001, 0.3, 0.4, 0.6, 0.7, 0.8, 0.999 depending on fit
    Modeling choices made by the user; Section 4.3 shows moving fixed CPs and changing Nm changes the fit, so they are effective free choices.
assumptions (6)
  • standard math Stone-Weierstrass theorem guarantees Bezier polynomials can approximate any continuous function on [0,1] to desired accuracy.
    Invoked in the Introduction to justify the modulator as a universal approximator; assumes PDFs are continuous and smooth enough.
  • standard math Unisolvence theorem guarantees a unique interpolating polynomial through Nm+1 distinct control points, and moving free CPs for fixed Nm does not change the best-fit polynomial.
    Used in Section 2.2 (Eq. 10-11) and Section 4.3; requires distinct, non-degenerate control points and invertible Vandermonde matrix.
  • domain assumption DGLAP evolution and QCD factorization relate the input PDFs at Q0 to the fitted cross sections.
    Standard framework of the xFitter global analysis; the paper relies on it for the pion fit example.
  • ad hoc to paper The carrier form A x^B (1-x)^C with modulator 1+B(Nm)(y) spans the plausible functional forms of PDFs in the data region.
    Equation (1)-(4); the paper shows results depend on alpha_x and fixed CP positions, so the family is not neutral with respect to the inferred PDFs.
  • ad hoc to paper Weak goodness-of-fit selection plus ad-hoc retention of distinct PDF shapes yields a representative sample for epistemic uncertainty.
    Section 5.1 Procedure A; the authors explicitly call the shape-selection criteria ad hoc, so the ensemble's representativeness is assumed rather than demonstrated.
  • ad hoc to paper Prior penalties (AbsLog and ReLU) are sufficient to keep fits physical and integrable.
    Table 1 and Section 3.6.2; the fits depend strongly on these hand-selected penalties, so the physicality of results rests on this choice.

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Cite this review

Pith. "Pith review of Fant\^omas Unconfined: global QCD fits with B\'ezier parameterizations." pith.science (2026). https://pith.science/paper/OWTV3PLI

@misc{pith2026250722969,
  author       = {Pith},
  title        = {Pith review of: Fant\^omas Unconfined: global QCD fits with B\'ezier parameterizations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OWTV3PLI}},
  note         = {Machine review of arXiv:2507.22969}
}
read the original abstract

Fant\^omas is a C++ toolkit for exploring the parametrization dependence of parton distribution functions (PDFs) and other correlator functions in quantum chromodynamics (QCD). Fant\^omas facilitates the generation of adaptable polynomial parametrizations for PDFs, called metamorphs, to find best-fit PDF solutions and quantify the epistemic uncertainty associated with the parametrizations during their fitting. The method employs B\'ezier curves as universal approximators for a variety of PDF shapes. Integrated into the xFitter framework for the global QCD analysis, Fant\^omas provides a foundation for general models of PDFs, while reducing the computational time compared to the approaches utilizing traditional polynomial parametrizations as well as providing an interpretable alternative to neural-network-based models. This paper outlines the structure and practical usage of the Fant\^omas toolkit, including its inputs, outputs, and implementation within xFitter. It also provides a practical example of using Fant\^omas for uncertainty quantification as well as the combination of PDF fits into a single ensemble.

Figures

Figures reproduced from arXiv: 2507.22969 by the authors.

Figure 1
Figure 1. Plot of the stretching function g(x, n) in Eq. (4) for xmin = 0.1, xmax = 0.8, n = 6. The dashed curve indicates the dependence y = x. The methodology of Fantomas ˆ is based on the use of analytical forms called metamorphs to represent functions of a variable x defined over a supported region, x ∈ [0, 1]. In the context of PDFs, such functions represent the distribution, fa(x, Q 2 0 ), of a parent hadron’s fractiona… view at source ↗
Figure 2
Figure 2. An example of a Fantomas ˆ steering card, steering fantomas.txt. where we have made the dependence on the normalization explicit. Hence, in the external fitting program, a delta parameter must be assigned to every free parameter of the metamorphs: Sc[] as well as Sm[] for free CPs. While the deltas for FIXed CP’s are redundant and will be ignored by metamorph Collection even if assigned, the structure of the xFitter… view at source ↗
Figure 3
Figure 3. Activation functions listed in Table 1, [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Initial steering cards for (a) the simplest [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: Illustration of χ 2 0 , the lowest χ 2 value when an improvement is detected in the fitting process, which decreases when adding CPs. The number of iterations is accumulated for the three fits (fit1, fit2, and either fit3 or fit3 Cgfix) corresponding to the addition of…
Figure 6
Figure 6. Figure 6: Central values for the gluon PDF of the pion using [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: PDFs from fits in Table 3 illustrating the impact of metamorph settings: nominal fit (solid pink), change in all free control points [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: The aleatoric+epistemic+nuclear uncertainties of the published combination of Fanto10 n15 NLO pion PDFs [26] for xuv = xV/2, xu¯ = xS/6, and xg at QCD scale Q = 2 GeV. Light green curves and filled bands represent the average PDFs and uncertainties of the Monte Carlo r…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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Reference graph

Works this paper leans on

51 extracted references · 19 canonical work pages · cited by 1 Pith paper

  1. [26]

    L. Kotz, A. Courtoy, P. Nadolsky, M. Ponce-Chavez, Fantˆomas: epistemic and nuclear uncertainties for the parton distributions of the pion (5 2025). arXiv:2505.13594

    arXiv 2025

  2. [1]

    Amoroso, et al., Snowmass 2021 Whitepaper: Proton Structure at the Precision Frontier, Acta Phys

    S. Amoroso, et al., Snowmass 2021 Whitepaper: Proton Structure at the Precision Frontier, Acta Phys. Polon. B 53 (12) (2022) 12–A1. arXiv:2203.13923, doi:10.5506/APhysPolB.53.12-A1

    arXiv 2022

  3. [2]

    Kova ˇr´ık, P

    K. Kova ˇr´ık, P. M. Nadolsky, D. E. Soper, Hadronic structure in high-energy collisions, Rev. Mod. Phys. 92 (4) (2020) 045003. arXiv: 1905.06957, doi:10.1103/RevModPhys.92.045003

    arXiv 2020

  4. [3]

    J. J. Ethier, E. R. Nocera, Parton Distributions in Nucleons and Nuclei, Ann. Rev. Nucl. Part. Sci. 70 (2020) 43–76. arXiv:2001.07722, doi:10.1146/annurev-nucl-011720-042725

    arXiv 2020

  5. [4]

    J. G. Morfin, W.-K. Tung, Parton distributions from a global QCD analysis of deep inelastic scattering and lepton pair production, Z. Phys. C 52 (1991) 13–30. doi:10.1007/BF01412323. 31 1 Fc[x_, {a0_, a1_, a2_ }] := a0 *xˆa1 (1 - x)ˆa2; 2 3 Bezier[xPower_ : 1, k_ : 1, n_ : 1, q0_ : 1.3] := 4 Module[{m, nlist, M, t, T, p, l, x, P, C, f, b, B, pts, plotmod,...

    work page doi:10.1007/bf01412323 1991

  6. [5]

    Hou, et al., New CTEQ global analysis of quantum chromodynamics with high-precision data from the LHC, Phys

    T.-J. Hou, et al., New CTEQ global analysis of quantum chromodynamics with high-precision data from the LHC, Phys. Rev. D 103 (1) (2021) 014013. arXiv:1912.10053, doi:10.1103/PhysRevD.103.014013

    arXiv 2021

  7. [6]

    Bailey, T

    S. Bailey, T. Cridge, L. A. Harland-Lang, A. D. Martin, R. S. Thorne, Parton distributions from LHC, HERA, Tevatron and fixed target data: MSHT20 PDFs, Eur. Phys. J. C 81 (4) (2021) 341. arXiv:2012.04684, doi:10.1140/epjc/s10052-021-09057-0

    arXiv 2021

  8. [7]

    R. D. Ball, et al., Parton distributions from high-precision collider data, Eur. Phys. J. C 77 (10) (2017) 663. arXiv:1706.00428, doi:10.1140/epjc/s10052-017-5199-5

    arXiv 2017

Show all 51 references
  1. [8]

    R. D. Ball, et al., The PDF4LHC21 combination of global PDF fits for the LHC Run III, J. Phys. G 49 (8) (2022) 080501. arXiv: 2203.05506, doi:10.1088/1361-6471/ac7216

    2022 arXiv

  2. [9]

    R. D. Ball, et al., The path to proton structure at 1% accuracy, Eur. Phys. J. C 82 (5) (2022) 428. arXiv:2109.02653, doi:10.1140/ epjc/s10052-022-10328-7

    2022 arXiv

  3. [10]

    McGowan, T

    J. McGowan, T. Cridge, L. A. Harland-Lang, R. S. Thorne, Approximate N 3LO parton distribution functions with theoretical uncertainties: MSHT20aN3LO PDFs, Eur. Phys. J. C 83 (3) (2023) 185, [Erratum: Eur.Phys.J.C 83, 302 (2023)]. arXiv:2207.04739, doi:10. 1140/epjc/s10052-023-11236-0

    2023 arXiv

  4. [11]

    R. D. Ball, et al., The path to N 3LO parton distributions, Eur. Phys. J. C 84 (7) (2024) 659.arXiv:2402.18635, doi:10.1140/epjc/ s10052-024-12891-7

    2024 arXiv

  5. [12]

    Ablat, et al., New results in the CTEQ-TEA global analysis of parton distributions in the nucleon, Eur

    A. Ablat, et al., New results in the CTEQ-TEA global analysis of parton distributions in the nucleon, Eur. Phys. J. Plus 139 (12) (2024) 1063. arXiv:2408.04020, doi:10.1140/epjp/s13360-024-05865-x

    2024 arXiv

  6. [13]

    Cridge, et al., Combination of aN 3LO PDFs and implications for Higgs production cross-sections at the LHC, J

    T. Cridge, et al., Combination of aN 3LO PDFs and implications for Higgs production cross-sections at the LHC, J. Phys. G 52 (2025) 6. arXiv:2411.05373, doi:10.1088/1361-6471/adde78

    2025 arXiv

  7. [14]

    Courtoy, J

    A. Courtoy, J. Huston, P. Nadolsky, K. Xie, M. Yan, C. P. Yuan, Parton distributions need representative sampling, Phys. Rev. D 107 (3) (2023) 034008. arXiv:2205.10444, doi:10.1103/PhysRevD.107.034008

    2023 arXiv

  8. [15]

    Kriesten, T

    B. Kriesten, T. J. Hobbs, Learning PDFs through interpretable latent representations in Mellin space, Phys. Rev. D 111 (1) (2025) 014028. arXiv:2312.02278, doi:10.1103/PhysRevD.111.014028

    2025 arXiv

  9. [16]

    Kriesten, J

    B. Kriesten, J. Gomprecht, T. J. Hobbs, Explainable AI classification for parton density theory, JHEP 11 (2024) 007.arXiv:2407.03411, doi:10.1007/JHEP11(2024)007

    2024 arXiv

  10. [17]

    Kriesten, T

    B. Kriesten, T. J. Hobbs, Anomalous electroweak physics unraveled via evidential deep learning (12 2024). arXiv:2412.16286

    2024 arXiv

  11. [18]

    Kriesten, A

    B. Kriesten, A. NieMiera, W. Good, T. J. Hobbs, H.-W. Lin, Decoding the proton’s gluonic density with lattice QCD-informed machine learning (7 2025). arXiv:2507.17810

    2025 arXiv

  12. [19]

    Yan, T.-J

    M. Yan, T.-J. Hou, Z. Li, K. Mohan, C. P. Yuan, A generalized statistical model for fits to parton distributions (6 2024). arXiv:2406. 01664

    2024

  13. [20]

    M. N. Costantini, M. Madigan, L. Mantani, J. M. Moore, A critical study of the Monte Carlo replica method, JHEP 12 (2024) 064. arXiv: 2404.10056, doi:10.1007/JHEP12(2024)064

    2024 arXiv

  14. [22]

    Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of Control, Signals and Systems 2 (4) (1989) 303

    G. Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of Control, Signals and Systems 2 (4) (1989) 303. doi:10.1007/BF02551274. URL https://doi.org/10.1007/BF02551274

    1989 doi

  15. [24]

    Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks 4 (2) (1991) 251

    K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks 4 (2) (1991) 251. doi:https://doi.org/ 10.1016/0893-6080(91)90009-T . URL http://www.sciencedirect.com/science/article/pii/089360809190009T

    1991

  16. [25]

    L. Kotz, A. Courtoy, P. Nadolsky, F. Olness, M. Ponce-Chavez, Analysis of parton distributions in a pion with B´ezier parametrizations, Phys. Rev. D 109 (7) (2024) 074027. arXiv:2311.08447, doi:10.1103/PhysRevD.109.074027

    2024 arXiv

  17. [27]

    P. C. Barry, N. Sato, W. Melnitchouk, C.-R. Ji, First Monte Carlo Global QCD Analysis of Pion Parton Distributions, Phys. Rev. Lett. 121 (15) (2018) 152001. arXiv:1804.01965, doi:10.1103/PhysRevLett.121.152001

    2018 arXiv

  18. [28]

    Novikov, et al., Parton Distribution Functions of the Charged Pion Within The xFitter Framework, Phys

    I. Novikov, et al., Parton Distribution Functions of the Charged Pion Within The xFitter Framework, Phys. Rev. D 102 (1) (2020) 014040. arXiv:2002.02902, doi:10.1103/PhysRevD.102.014040

    2020 arXiv

  19. [29]

    P. C. Barry, C.-R. Ji, N. Sato, W. Melnitchouk, Global QCD Analysis of Pion Parton Distributions with Threshold Resummation, Phys. Rev. Lett. 127 (23) (2021) 232001. arXiv:2108.05822, doi:10.1103/PhysRevLett.127.232001

    2021 arXiv

  20. [30]

    xFitter, The xFitter project is an open source QCD fit framework ready to extract PDFs and assess the impact of new data., https: //www.xfitter.org/xFitter/

  21. [31]

    J. Gao, P. Nadolsky, A meta-analysis of parton distribution functions, JHEP 07 (2014) 035. arXiv:1401.0013, doi:10.1007/ JHEP07(2014)035

    2014 arXiv

  22. [32]

    Courtoy, P

    A. Courtoy, P. M. Nadolsky, Testing momentum dependence of the nonperturbative hadron structure in a global QCD analysis, Phys. Rev. D 103 (5) (2021) 054029. arXiv:2011.10078, doi:10.1103/PhysRevD.103.054029

    2021 arXiv

  23. [33]

    L. A. Harland-Lang, T. Cridge, R. S. Thorne, A Stress Test of Global PDF Fits: Closure Testing the MSHT PDFs and a First Direct Comparison to the Neural Net Approach (7 2024). arXiv:2407.07944

    2024 arXiv

  24. [34]

    W. I. Jay, E. T. Neil, Bayesian model averaging for analysis of lattice field theory results, Phys. Rev. D 103 (2021) 114502. arXiv: 2008.01069, doi:10.1103/PhysRevD.103.114502

    2021 arXiv

  25. [35]

    P. D. Dauncey, M. Kenzie, N. Wardle, G. J. Davies, Handling uncertainties in background shapes: the discrete profiling method, JINST 10 (04) (2015) P04015. arXiv:1408.6865, doi:10.1088/1748-0221/10/04/P04015

    2015 arXiv

  26. [36]

    Malinin, M

    A. Malinin, M. Gales, Predictive uncertainty estimation via prior networks (2018). arXiv:1802.10501. URL https://arxiv.org/abs/1802.10501

    2018 arXiv

  27. [37]

    Farin, Curves and surfaces for CAGD: A practical guide, Morgan Kaufmann, 2001

    G. Farin, Curves and surfaces for CAGD: A practical guide, Morgan Kaufmann, 2001

    2001

  28. [38]

    M. P. Kamermans, A primer on B ´ezier curves, https://pomax.github.io/bezierinfo (2011)

    2011

  29. [39]

    Hou, et al., Reconstruction of Monte Carlo replicas from Hessian parton distributions, JHEP 03 (2017) 099

    T.-J. Hou, et al., Reconstruction of Monte Carlo replicas from Hessian parton distributions, JHEP 03 (2017) 099. arXiv:1607.06066, doi:10.1007/JHEP03(2017)099

    2017 arXiv

  30. [40]

    De Finetti, Foresight: Its logical laws, and its subjective sources, Studies in Subjective Probability 1 (1) (1937) 1–68

    B. De Finetti, Foresight: Its logical laws, and its subjective sources, Studies in Subjective Probability 1 (1) (1937) 1–68

    1937

  31. [41]

    Kovarik, et al., nCTEQ15 - Global analysis of nuclear parton distributions with uncertainties in the CTEQ framework, Phys

    K. Kovarik, et al., nCTEQ15 - Global analysis of nuclear parton distributions with uncertainties in the CTEQ framework, Phys. Rev. D 93 (8) (2016) 085037. arXiv:1509.00792, doi:10.1103/PhysRevD.93.085037

    2016 arXiv

  32. [42]

    Gautschi, On inverses of vandermonde and confluent vandermonde matrices iii, Numerische Mathematik 29 (4) (1978) 445–450

    W. Gautschi, On inverses of vandermonde and confluent vandermonde matrices iii, Numerische Mathematik 29 (4) (1978) 445–450. doi: 10.1007/BF01432880. URL https://doi.org/10.1007/BF01432880

    1978 doi

  33. [43]

    X.-L. Meng, Statistical paradises and paradoxes in big data (I): Law of large populations, big data paradox, and the 2016 US presidential election, The Annals of Applied Statistics 12 (2) (2018) 685. doi:10.1214/18-AOAS1161SF

    2018 doi

  34. [44]

    Courtoy, P

    A. Courtoy, P. Nadolsky, From Data Defects to Response Functions: A View From Particle Physics, Harvard Data Science Review 5 (3), https://hdsr.mitpress.mit.edu/pub/qp47w8cm (sep 28 2023)

    2023

  35. [45]

    Del Debbio, Bayesian approach for inverse problems, pDF4LHC 2024 meeting (2024)

    L. Del Debbio, Bayesian approach for inverse problems, pDF4LHC 2024 meeting (2024). URL https://indico.cern.ch/event/1435677/

    2024

  36. [46]

    Alekhin, et al., HERAFitter, Eur

    S. Alekhin, et al., HERAFitter, Eur. Phys. J. C 75 (7) (2015) 304. arXiv:1410.4412, doi:10.1140/epjc/s10052-015-3480-z

    2015 arXiv

  37. [47]

    R. S. Thorne, A Variable-flavor number scheme for NNLO, Phys. Rev. D 73 (2006) 054019. arXiv:hep-ph/0601245, doi:10. 1103/PhysRevD.73.054019

    2006 arXiv

  38. [48]

    M. A. G. Aivazis, J. C. Collins, F. I. Olness, W.-K. Tung, Leptoproduction of heavy quarks. 2. A Unified QCD formulation of charged and neutral current processes from fixed target to collider energies, Phys. Rev. D 50 (1994) 3102–3118. arXiv:hep-ph/9312319, doi:10.1103/PhysRev...

    1994 arXiv

  39. [49]

    Stavreva, F

    T. Stavreva, F. I. Olness, I. Schienbein, T. Jezo, A. Kusina, K. Kovarik, J. Y . Yu, Heavy Quark Production in the ACOT Scheme at NNLO and N3LO, Phys. Rev. D 85 (2012) 114014. arXiv:1203.0282, doi:10.1103/PhysRevD.85.114014

    2012 arXiv

  40. [50]

    E. A. L ´opez Rosa, Global analysis of a twist-3 parton distribution function with theoretical constraints, Master’s Thesis (2025). URL https://tesiunam.dgb.unam.mx/

    2025

  41. [51]

    Gautschi, On inverses of vandermonde and confluent vandermonde matrices, Numerische Mathematik 4 (1) (1962) 117–123

    W. Gautschi, On inverses of vandermonde and confluent vandermonde matrices, Numerische Mathematik 4 (1) (1962) 117–123. URL https://doi.org/10.1007/BF01386302

    1962 doi

  42. [52]

    Gautschi, Norm estimates for inverses of vandermonde matrices, Numerische Mathematik 23 (4) (1974) 337–347

    W. Gautschi, Norm estimates for inverses of vandermonde matrices, Numerische Mathematik 23 (4) (1974) 337–347. doi:10.1007/ BF01438260. URL https://doi.org/10.1007/BF01438260

    1974 doi

  43. [53]

    Fant ˆomas Team, work in progress (2025). 33

    2025

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.