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Factorisation schemes for proton PDFs

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read All the major NLO factorisation schemes for proton PDFs reduce to one general form with a few discrete coefficients, and switching schemes can shift LHC cross-section shapes by more than the usual scale-variation uncertainty does.

desk verdict A genuinely useful synthesis of NLO PDF factorisation schemes, with a solid analytic core and a conditional numerical tail that the authors themselves flag. read the letter →

arxiv 2501.18289 v2 pith:BPIXGGCR submitted 2025-01-30 hep-ph hep-th

Stéphane Delorme , Aleksander Kusina , Andrzej Siódmok , James Whitehead This is my paper · ORCID
classification hep-phhep-th PACS 12.38.-t12.38.Bx
keywords factorisationschemepartondistributionfunctionsNLOQCDMSDGLAPevolutionPDFpositivityLHCphenomenologythresholdlogarithms
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

Beyond leading order, perturbative QCD leaves a genuine choice in how the short-distance partonic cross-section and the long-distance parton distribution functions (PDFs) are separated; that choice is the factorisation scheme, with $\overline{\mathrm{MS}}$ the default. The paper assembles the main alternative schemes in the literature — the DIS scheme, the Krk scheme built for NLO parton-shower matching, the positivity-motivated Pos/Mpos family, the Aversa scheme, and the Phys scheme — into one common notation, and shows that at NLO every one of them is a special case of a single general form for the four scheme-transformation kernels, differing only in a few discrete coefficients. If true, the space of scheme choices actually explored in the literature is far smaller than it appears, and a systematic survey of the remaining choices becomes feasible. The paper also estimates the practical cost of the choice: for Z-plus-jet and Higgs-plus-jet kinematics at the LHC, the spread between schemes can exceed the conventional factorisation-scale uncertainty, so scale variation alone may understate the theoretical uncertainty of an NLO calculation.

What carries the argument

The load-bearing object is the transformation kernel $K^{\overline{\mathrm{MS}}\to\mathrm{FS}}_{ab}(z)$, which converts $\overline{\mathrm{MS}}$ PDFs into another factorisation scheme by a convolution integral performed at each scale. Every kernel is decomposed into standard pieces — the plus-distributions $D_k$, $\log(1-z)$ and $\log z$ terms, a rational polynomial $P(z)$, and a $\delta(1-z)$ term — and the paper shows that all studied schemes are recovered from one general four-kernel template (Eqs. 69–72) by choosing small discrete values for the coefficients $a$, $b$, $c$ and fixing the diagonal delta terms through the momentum sum rule. The template captures the threshold logarithms that dominate the differences: schemes with the largest $a_{qq}$ and $a_{gg}$ values absorb the double-log $D_1$ soft-gluon terms into the PDFs, leaving the Drell–Yan and Higgs coefficient functions asymptotically constant in Mellin space, while schemes with smaller coefficients leave those terms in the coefficient functions.

What would settle it

Recompute the transformed PDFs and the Z-plus-jet and Higgs-plus-jet ratios using the alternative construction the paper does not use — transform the $\overline{\mathrm{MS}}$ input at the starting scale, then evolve each PDF with that scheme's modified DGLAP kernels — and compare the between-scheme spread with the local-transformation results of Figs. 14 and 15; if the spread drops below the scale-variation band or the low-$x$ gluon enhancement disappears, the reported scheme dependence is an artifact of the transformation prescription rather than a property of the schemes themselves.

Watch

Extended reading notes

Core claim

The paper's central claim is unification: at NLO, every factorisation scheme it studies is obtained from $\overline{\mathrm{MS}}$ by convolution with kernels $K_{qq}$, $K_{qg}$, $K_{gq}$, $K_{gg}$ that share one common structure (Eqs. 69–72), and the differences between schemes reduce to the values of a few discrete coefficients ($a_{qq}, a_{gg} \in \{0,1,2\}$; $a_{qg}, a_{gq} \in \{1,2\}$; the remaining $b$ and $c$ coefficients in $\{0,1\}$), with the diagonal delta-function terms fixed by the momentum sum rule. Two exceptions are noted: the DIS-scheme gluon kernels, fixed by a local momentum-conservation convention, and the Aversa gluon kernel, which uses a single rational term. On the authors' reading, 'the domain of interest for factorisation-scheme variation within the literature is much smaller than might initially be assumed' (Sec. 5). The supporting numerical claim is that the choice matters at the LHC: the transformed low-$x$ gluon can be an order of magnitude larger at low scales, and for Z-plus-jet and Higgs-plus-jet production the scheme-to-scheme spread reaches roughly 30% and 20% respectively in some regions, exceeding the factorisation-scale variation band.

Load-bearing premise

The quantitative conclusions — the order-of-magnitude low-$x$ gluon enhancement and the 20–30% cross-section shifts — are computed by taking $\overline{\mathrm{MS}}$-fitted PDFs and transforming them locally at every scale, which is only one of three inequivalent ways to define PDFs in an alternative scheme, and the paper states that its conclusions may not apply to the other two.

Editorial extensions

If this is right

  • The space of NLO factorisation schemes in the literature is far smaller than it appears: the studied schemes differ only through a handful of discrete coefficients in one general kernel form.
  • For Z-plus-jet and Higgs-plus-jet observables, the scheme-to-scheme spread exceeds the factorisation-scale variation band in several kinematic regions, so scale variation alone can understate the theoretical uncertainty of an NLO prediction.
  • The choice of scheme modifies the valence-quark number sum rules at $\mathcal{O}(\alpha_s)$ and makes them scale-dependent in most schemes, which must be handled if PDFs are fitted in those schemes.
  • Heavy-quark PDFs turn negative just above their mass thresholds in every studied scheme when they are generated perturbatively, so positivity cannot be assumed as a universal property of alternative factorisation schemes.

Reading between the lines

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

  • Because all studied schemes sit in one small discrete parameter space, a systematic scan of that space would define a complete scheme-envelope uncertainty for LHC processes; the paper samples only the schemes that happen to exist in the literature, not the full space its template spans.
  • The reported 20–30% shifts are tied to the paper's transformation-at-each-scale prescription; transforming at the input scale and evolving in the scheme instead could redistribute the effect between the PDFs and the DGLAP evolution, and checking whether the shifts survive is the most direct test of how intrinsic to the schemes they are.
  • The heavy-quark negativity just above threshold suggests a pragmatic remedy the paper mentions only in passing: moving the flavour-number transition above the quark mass would very likely restore positivity without disturbing the unification, since the transformation kernels themselves are unchanged.
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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

2 major / 4 minor

Summary. This paper presents a systematic comparison of NLO factorisation schemes for proton PDFs, collecting the definitions of the Dis, Krk/KrkDy, Dpos/Pos/Mpos/Mposδ, A versa, and Phys schemes in a common notation and expressing each transformation kernel in a unified decomposition of distributions, logarithms, rational functions, and delta-function terms (Tables 1–4). The main analytical result is that, after neglecting the polynomial piece P(z), all considered kernels are special cases of a common parametric form (Eqs. 69–72) with a handful of discrete coefficients. The paper further derives the corresponding coefficient functions for DIS, Drell–Yan and Higgs production (Tables 5–7), studies the momentum and number sum rules, examines positivity using an adaptation of the argument of [46], and presents numerical results for PDFs transformed from CT18NLO, NNPDF40MC and MSHT20nlo PDFs. These are used to compute LO Z+jet and H+jet cross-sections, finding that scheme-to-scheme variation can exceed the conventional MS factorisation-scale uncertainty, especially at low pT.

Significance. The analytical unification, if correct, is a valuable contribution: it reduces the space of scheme choices to a small set of discrete parameters and provides a common language for future scheme proposals. The tabulated kernels and coefficient functions will serve as a reference. The numerical work is carefully validated: two independent convolution codes agree to order 10^{-5}, the momentum sum rule is reproduced to approximately 10^{-6}, and the Krk and Phys implementations are checked against original codes. The paper also gives a fair presentation of the limitations of its numerical study, noting explicitly that only method (iii) for defining scheme PDFs is used. The positivity analysis extends the argument of [46] to alternative schemes and is appropriately labelled preliminary. Taken together, the paper is a useful and reliable resource, provided the phenomenological claims are properly scoped.

major comments (2)
  1. [Sec. 3 (methods), Sec. 4, Sec. 5] The central quantitative conclusion of Sec. 4, that factorisation-scheme variation can exceed the factorisation-scale uncertainty, is obtained exclusively with PDFs defined by method (iii): MS PDFs evolved in MS and transformed locally at each scale via Eq. (5). The three methods are inequivalent (Sec. 3), and the paper explicitly acknowledges in Sec. 5 that the conclusions may not apply to methods (i) and (ii). This is not a mere technicality: Sec. 3.2.3 reports that the NLO gluon contribution is comparable to the LO term for the Krk and Mpos schemes at Q = 2 GeV, and Appendix D shows that the perturbative inversion fails at Q = 1.3 GeV. A PDF fitted or evolved in the scheme would carry these large low-scale corrections through the DGLAP evolution to LHC scales, so the 20–30% cross-section shifts in Figs. 14 and 15 may be partly artefacts of applying the transformation only at high scales. To support the Sec. 4 claim as stated, the authors should either perform a consistency test using method (ii) (transform at the input scale and evolve with the modified DGLAP kernels) for at least the most extreme schemes (Krk, Mpos), or substantially soften the abstract and Sec. 4 conclusions to state clearly that the numerical estimates apply only to method (iii). The analytical unification is unaffected, but the phenomenological claim is load-bearing and currently under-supported.
  2. [Sec. 5, Eqs. (69–72)] The unified form of Eqs. (69–72) is obtained after setting the polynomial piece P(z) to zero. For the Mpos scheme, P(z) is not a numerically suppressed detail: the qq and gg kernels in Tables 1 and 4 consist solely of the soft-function terms 350/3 z^2(1−z)^2, which are the scheme's defining momentum-conservation mechanism, and Fig. 8b shows that this choice produces an O(1) difference in the momentum-fraction distribution relative to Mposδ. Consequently, the statement that Mpos is a special case of Eq. (69) is only an approximation that omits the term that distinguishes the scheme. The conclusion that the 'domain of interest for factorisation-scheme variation within the literature is much smaller than might initially be assumed' therefore needs either a quantitative demonstration that P(z) is negligible for all considered schemes (which Fig. 8b contradicts for Mpos), or a more cautious wording that the unified form captures only the distributional and logarithmic content of the kernels.
minor comments (4)
  1. [Sec. 3.2.3] The sentence 'we show the decomposition of the transformed gluon PDFs in in Fig. 4' contains a duplicated 'in'.
  2. [Eq. (33), Tables 1–4] The sign convention for the delta-function term (the decomposition writes '- Δ δ(1−z)' but the tables list positive values under a '-δ(1−z)' heading) should be explained explicitly the first time the decomposition is used.
  3. [Throughout] The scheme name is spelled inconsistently: 'A versa' in the text and tables, 'Aversa' in figure captions and the appendix; please unify.
  4. [Sec. 3.5.1, Eq. (49)] The modified cumulant of Eq. (49) places the absolute value inside the integral, which is a strengthening of the criterion in [46]; the text notes this only in a footnote, but since the subsequent numerical comparison uses this modified criterion, its role should be highlighted in the main discussion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's scheme unification is a summary of external kernels, and its numerical spreads are computed from external PDFs without any fitted or derived quantity reducing to an input.

full rationale

The central analytical claim, Eqs. (69)-(72), is explicitly a compact parametrization of the transformation kernels tabulated in Tables 1-4, drawn from the literature; it is presented as an observation that the considered schemes fit a common form, not as a prediction derived from that form. The numerical comparisons use PDFs fitted externally in the MS scheme (CT18NLO, NNPDF40MC, MSHT20nlo) and apply the literature transformation kernels via Eq. (5); no parameter is fitted to the output and no 'predicted' quantity is constructed from the quantity being claimed. The paper's acknowledged limitation that method (iii) for obtaining scheme PDFs may differ from methods (i) and (ii) is a scope restriction, not a circular step: it does not make the reported spreads equal by construction to an input, it only means the quantitative conclusion is conditional on the chosen method. Self-citations appear where the Krk scheme is defined from earlier work by the same authors, but that is background definitional context, and the kernels themselves are independently tabulated and numerically checked against the original code. The perturbative-inversion test of Appendix D is a validation, not a load-bearing derivation. No reduction of a claimed result to its own input was found.

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

No numbers are fitted to data in this paper. The transformation kernels are taken from the prior literature; the input MS PDFs are external. The Mpos scheme's soft function f_MOM(z)=60 z^2 (1-z)^2 is a functional choice inherited from [15], not a parameter fitted here. No new physical entities (particles, forces, dimensions) are introduced; the Mposδ scheme is a new definitional variant of Mpos introduced for comparison, not a physical postulate.

assumptions (5)
  • domain assumption Collinear QCD factorisation with neglected higher-twist terms (Eqs. 1-2).
    Used throughout to define PDFs and coefficient functions; standard in the field.
  • domain assumption NLO truncation of transformation kernels and validity of perturbative inversion (Eq. 6, Eq. 9, Appendix D).
    All comparisons are at O(alpha_s); the inverse transformation is computed perturbatively and tested numerically.
  • domain assumption Variable-flavour-number scheme with sudden activation of heavy-quark kernels at threshold (Eqs. 27-28).
    Chosen to match practical PDF fitting; introduces discontinuities at thresholds that affect heavy-quark PDFs.
  • domain assumption External MS PDF sets (CT18NLO, NNPDF40MC, MSHT20nlo) are accurate representations of the proton.
    The numerical results inherit the uncertainties and positivity properties of these sets; PDF uncertainties are not propagated.
  • ad hoc to paper Modified positivity criterion with absolute value inside the cumulant integral (Eq. 49).
    The paper strengthens the criterion of [46] by taking |fin[K]| inside the integral; this modification is introduced here and not derived from first principles.

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Cite this review

Pith. "Pith review of Factorisation schemes for proton PDFs." pith.science (2026). https://pith.science/paper/BPIXGGCR

@misc{pith2026250118289,
  author       = {Pith},
  title        = {Pith review of: Factorisation schemes for proton PDFs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BPIXGGCR}},
  note         = {Machine review of arXiv:2501.18289}
}
abstract

Beyond leading-order, perturbative QCD requires a choice of factorisation scheme to define the parton distribution functions (PDFs) and hard-process cross-section. The modified minimal-subtraction ($\overline{\mathrm{MS}}$) scheme has long been adopted as the default choice due to its simplicity. Alternative schemes have been proposed with specific purposes, including, recently, PDF positivity and NLO parton-shower matching. In this paper we assemble these schemes in a common notation for the first time. We perform a detailed comparison of their features, both analytically and numerically, and estimate the resulting factorisation-scheme uncertainty for LHC phenomenology.

Figures

Figures reproduced from arXiv: 2501.18289 by the authors.

Figure 1
Figure 1. Comparison of transformed CT18NLO PDFs in different schemes for u-valence, u¯, gluon and charm, at Q = 2 GeV (left) and 100 GeV (right). The remaining flavours are presented in [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Decomposition of transformed uv-quark PDF in the Krk, Mpos, and Phys schemes at factorisation scale Q = 2 GeV, as described in Sec. 3.2 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Decomposition of transformed u-antiquark PDF in the Krk, Mpos, and Phys schemes at factorisation scale Q = 2 GeV, as described in Sec. 3.2 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (28 more)
Figure 4
Figure 4. Figure 4: Decomposition of transformed gluon PDF in the [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Decomposition of transformed charm-quark PDF in the [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Decomposition of transformed charm-quark PDF in the [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Numerical calculation of the momentum sum-rule, Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: Graphical illustration of the momentum sum-rule; carried momentum here corresponds to shaded area. [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: Number sum rule as a function of the factorisation/renormalisation scale for (a) [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 10
Figure 10. Figure 10: Cumulants, as defined in Eq. (51), adapted from the positivity argument of [46]. The dotted grey lines show 2π/αs(µ) at the labelled value of µ. 10−5 10−4 10−3 10−2 10−1 x 0 10 20 30 40 50 Q [GeV] 0.5 1.0 −100 −10−1 −10−2 −10−3 −10−4 −10−5 10−5 10−4 10−3 10−2 10−1 100…
Figure 11
Figure 11. Figure 11: Heatmap showing the sign and order-of-magnitude of heavy-quark [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]
Figure 12
Figure 12. Figure 12: Heatmap showing the sign and order-of-magnitude of charm-quark PDFs [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: The NLO contributions to coefficient functions for the DIS, Drell–Yan and Higgs production processes, in [PITH_FULL_IMAGE:figures/full_fig_p027_13.png]
Figure 14
Figure 14. Figure 14: Factorisation-scheme dependence of differential cross-sections for [PITH_FULL_IMAGE:figures/full_fig_p030_14.png]
Figure 15
Figure 15. Figure 15: Factorisation-scheme dependence of differential cross-sections for [PITH_FULL_IMAGE:figures/full_fig_p031_15.png]
Figure 16
Figure 16. Figure 16: Comparison of transformed CT18NLO PDFs in different schemes for d-valence, d, ¯d, and s, at Q = 2 GeV (left) and 100 GeV (right). Companion to [PITH_FULL_IMAGE:figures/full_fig_p034_16.png]
Figure 17
Figure 17. Figure 17: Comparison of transformed NNPDF40MC PDFs in different schemes for u-valence, u¯, gluon, and charm at Q = 2 GeV (left) and 100 GeV (right). Companion to [PITH_FULL_IMAGE:figures/full_fig_p035_17.png]
Figure 18
Figure 18. Figure 18: Comparison of transformed MSHT20nlo PDFs in different schemes for u-valence, u¯, gluon, and charm, at Q = 2 GeV (left) and 100 GeV (right). Companion to [PITH_FULL_IMAGE:figures/full_fig_p036_18.png]
Figure 19
Figure 19. Figure 19: Decomposition of transformed uv-quark PDF in the Aversa, Dis and Mposδ schemes at factorisation scale Q = 2 GeV, as described in Sec. 3.2. Companion to [PITH_FULL_IMAGE:figures/full_fig_p038_19.png]
Figure 20
Figure 20. Figure 20: Decomposition of transformed u¯-quark PDF as a representative of the light sea-quark PDFs, in the Aversa, Dis and Mposδ schemes at factorisation scale Q = 2 GeV, as described in Sec. 3.2. Companion to [PITH_FULL_IMAGE:figures/full_fig_p039_20.png]
Figure 21
Figure 21. Figure 21: Decomposition of transformed gluon PDF in the [PITH_FULL_IMAGE:figures/full_fig_p040_21.png]
Figure 22
Figure 22. Figure 22: Decomposition of transformed c-quark PDF as a representative of the heavy flavour PDFs, in the Aversa, Dis and Mposδ schemes at factorisation scale Q = 2 GeV, as described in Sec. 3.2. Companion to [PITH_FULL_IMAGE:figures/full_fig_p041_22.png]
Figure 23
Figure 23. Figure 23: Decomposition of transformed uv-quark PDF in the Aversa, Dis and Mposδ schemes at factorisation scale Q = 100 GeV, as described in Sec. 3.2. Companion to [PITH_FULL_IMAGE:figures/full_fig_p042_23.png]
Figure 24
Figure 24. Figure 24: Decomposition of transformed u¯-quark PDF as a representative of the light sea-quark PDFs, in the Aversa, Dis and Mposδ schemes at factorisation scale Q = 100 GeV, as described in Sec. 3.2. Companion to [PITH_FULL_IMAGE:figures/full_fig_p043_24.png]
Figure 25
Figure 25. Figure 25: Decomposition of transformed gluon PDF in the [PITH_FULL_IMAGE:figures/full_fig_p044_25.png]
Figure 26
Figure 26. Figure 26: Decomposition of transformed c-quark PDF as a representative of the heavy flavour PDFs, in the Aversa, Dis and Mposδ schemes at factorisation scale Q = 100 GeV, as described in Sec. 3.2. Companion to [PITH_FULL_IMAGE:figures/full_fig_p045_26.png]
Figure 27
Figure 27. Figure 27: Decomposition of transformed uv-quark PDF in the Krk, Mpos, and Phys schemes at factorisation scale Q = 2 GeV, as described in Sec. 3.2, based on MSHT20nlo MS PDFs. Companion to [PITH_FULL_IMAGE:figures/full_fig_p046_27.png]
Figure 28
Figure 28. Figure 28: Decomposition of transformed uv-quark PDF in the Krk, Mpos, and Phys schemes at factorisation scale Q = 2 GeV, as described in Sec. 3.2, based on NNPDF40MC MS PDFs. Companion to [PITH_FULL_IMAGE:figures/full_fig_p047_28.png]
Figure 29
Figure 29. Figure 29: Decomposition of transformed gluon PDF in the [PITH_FULL_IMAGE:figures/full_fig_p048_29.png]
Figure 30
Figure 30. Figure 30: Decomposition of transformed gluon PDF in the [PITH_FULL_IMAGE:figures/full_fig_p049_30.png]
Figure 31
Figure 31. Figure 31: Test of the perturbative invertibility of the transformation kernels as outlined in [PITH_FULL_IMAGE:figures/full_fig_p050_31.png]

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Forward citations

Cited by 2 Pith papers

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    hep-ph 2026-08 conditional novelty 6.0 of 10

    Independent factorization and renormalization scales are inconsistent with simultaneously preserving RG invariance, Ward identities, and PDF sum rules in generalized pole subtraction schemes.

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Reference graph

Works this paper leans on

100 extracted references · 23 canonical work pages · cited by 2 Pith papers

  1. [46]

    Candido, S

    A. Candido, S. Forte, T. Giani, and F. Hekhorn,Eur. Phys. J. C84 (2024), no. 3 335,2308.00025

    arXiv 2024

  2. [1]

    Altarelli, R

    G. Altarelli, R. K. Ellis, and G. Martinelli,Nucl. Phys. B 143 (1978) 521, [Erratum:Nucl. Phys. B146 (1978) 544]

    1978

  3. [2]

    W. A. Bardeen, A. J. Buras, D. W. Duke, and T. Muta, Phys. Rev. D18 (1978) 3998

    1978

  4. [3]

    Altarelli, R

    G. Altarelli, R. K. Ellis, and G. Martinelli,Nucl. Phys. B 157 (1979) 461–497

    1979

  5. [4]

    Curci, W

    G. Curci, W. Furmanski, and R. Petronzio,Nucl. Phys. B 175 (1980) 27–92

    1980

  6. [5]

    Furmanski and R

    W. Furmanski and R. Petronzio,Z. Phys. C11 (1982) 293

    1982

  7. [6]

    H. D. Politzer,Nucl. Phys. B194 (1982) 493–512

    1982

  8. [7]

    P. M. Stevenson and H. D. Politzer,Nucl. Phys. B277 (1986) 758–763. 29Where the input PDF used a single grid we have split the grid at the heavy-flavour thresholds to accommodate the an- ticipated discontinuities in the transformed PDFs, as outlined in Sec. 2.1.5. 38 10□4 10□3 10□2 10□1 100 x −100 −10□1 −10□2 −10□3 −10□4 0 10□4 10□3 10□2 10□1 100 101 xfuv...

    1986

Show all 100 references
  1. [8]

    Ramalho and G

    A. Ramalho and G. F. Sterman,Phys. Rev. D29 (1984) 2517

    1984

  2. [9]

    G. F. Sterman,Nucl. Phys. B281 (1987) 310–364

    1987

  3. [10]

    Aurenche, R

    P. Aurenche, R. Baier, M. Fontannaz, and D. Schiff, Nucl. Phys. B286 (1987) 509–552

    1987

  4. [11]

    Aurenche, R

    P. Aurenche, R. Baier, A. Douiri, M. Fontannaz, and D. Schiff,Nucl. Phys. B286 (1987) 553–591

    1987

  5. [12]

    Aversa, M

    F. Aversa, M. Greco, P. Chiappetta, and J. P. Guillet, Phys. Lett. B211 (1988) 465

    1988

  6. [13]

    Aversa, P

    F. Aversa, P. Chiappetta, M. Greco, and J. P. Guillet, Nucl. Phys. B327 (1989) 105

    1989

  7. [14]

    Chyla,Nucl

    J. Chyla,Nucl. Phys. B321 (1989) 374–386

    1989

  8. [15]

    Candido, S

    A. Candido, S. Forte, and F. Hekhorn,JHEP 11 (2020) 129, 2006.07377

    2020 arXiv

  9. [16]

    Jadach, W

    S. Jadach, W. Płaczek, S. Sapeta, A. Siódmok, and M. Skrzypek,JHEP 10 (2015) 052, 1503.06849

    2015 arXiv

  10. [17]

    Jadach, W

    S. Jadach, W. Płaczek, S. Sapeta, A. Siódmok, and M. Skrzypek, Eur. Phys. J. C 76 (2016), no. 12 649, 1606.00355

    2016 arXiv

  11. [18]

    Jadach, G

    S. Jadach, G. Nail, W. Płaczek, S. Sapeta, A. Siódmok, and M. Skrzypek,Eur. Phys. J. C77 (2017), no. 3 164, 1607.06799

    2017 arXiv

  12. [19]

    Jadach, A

    S. Jadach, A. Kusina, W. Płaczek, M. Skrzypek, and M. Slawinska, Phys. Rev. D 87 (2013), no. 3 034029, 1103.5015

    2013 arXiv

  13. [20]

    Sarmah, A

    P. Sarmah, A. Siódmok, and J. Whitehead,JHEP 01 (2025) 062, 2409.16417

    2025 arXiv

  14. [21]

    Nagy and D

    Z. Nagy and D. E. Soper, Phys. Rev. D 106 (2022), no. 1 014024,2204.05631

    2022 arXiv

  15. [22]

    E. G. Oliveira, A. D. Martin, and M. G. Ryskin,JHEP 11 (2013) 156, 1310.8289

    2013 arXiv

  16. [23]

    G. T. Bodwin,Phys. Rev. D31 (1985) 2616, [Erratum: Phys. Rev. D34 (1986) 3932]

    1985

  17. [24]

    J. C. Collins, D. E. Soper, and G. F. Sterman,Nucl. Phys. B 261 (1985) 104–142

    1985

  18. [25]

    J. C. Collins, Phys. Rev. D 58 (1998) 094002, hep-ph/9806259

    1998 arXiv

  19. [26]

    Buckley, J

    A. Buckley, J. Ferrando, S. Lloyd, K. Nordström, B. Page, M. Rüfenacht, M. Schönherr, and G. Watt, Eur. Phys. J. C75 (2015) 132, 1412.7420

    2015 arXiv

  20. [27]

    Altarelli and G

    G. Altarelli and G. Parisi, Nucl. Phys. B 126 (1977) 298–318

    1977

  21. [28]

    V. N. Gribov and L. N. Lipatov,Sov. J. Nucl. Phys.15 (1972) 438–450

    1972

  22. [29]

    Y. L. Dokshitzer,Sov. Phys. JETP46 (1977) 641–653

    1977

  23. [30]

    C. D. White and R. S. Thorne, Eur. Phys. J. C 45 (2006) 179–192, hep-ph/0507244

    2006 arXiv

  24. [31]

    Lappi, H

    T. Lappi, H. Mäntysaari, H. Paukkunen, and M. Tevio, Eur. Phys. J. C84 (2024), no. 1 84,2304.06998

    2024 arXiv

  25. [32]

    Lappi, H

    T. Lappi, H. Mäntysaari, H. Paukkunen, and M. Tevio, Eur. Phys. J. C85 (2025), no. 4 429,2412.09589

    2025 arXiv

  26. [33]

    J. C. Collins and D. E. Soper,Nucl. Phys. B194 (1982) 445–492

    1982

  27. [34]

    M. Buza, Y. Matiounine, J. Smith, and W. L. van Neer- ven,Eur. Phys. J. C1(1998) 301–320,hep-ph/9612398

    1998 arXiv

  28. [35]

    Stavreva, F

    T. Stavreva, F. I. Olness, I. Schienbein, T. Jezo, A. Kusina, K. Kovarik, and J. Y. Yu,Phys. Rev. D85 (2012) 114014, 1203.0282

    2012 arXiv

  29. [36]

    Bertone, A

    V. Bertone, A. Glazov, A. Mitov, A. Papanastasiou, and M. Ubiali,JHEP 04 (2018) 046, 1711.03355

    2018 arXiv

  30. [37]

    M. A. G. Aivazis, F. I. Olness, and W.-K. Tung,Phys. Rev. D 50 (1994) 3085–3101, hep-ph/9312318

    1994 arXiv

  31. [38]

    M. A. G. Aivazis, J. C. Collins, F. I. Olness, and W.-K. Tung, Phys. Rev. D 50 (1994) 3102–3118, hep-ph/9312319

    1994 arXiv

  32. [39]

    R. S. Thorne and R. G. Roberts,Phys. Rev. D57(1998) 6871–6898, hep-ph/9709442

    1998 arXiv

  33. [40]

    R. S. Thorne, Phys. Rev. D 73 (2006) 054019, hep-ph/0601245

    2006 arXiv

  34. [41]

    Cacciari, M

    M. Cacciari, M. Greco, and P. Nason,JHEP 05 (1998) 007, hep-ph/9803400

    1998 arXiv

  35. [42]

    Brocket al., Rev

    CTEQ Collaboration, R. Brocket al., Rev. Mod. Phys. 67 (1995) 157–248

    1995

  36. [43]

    A. D. Martin, R. G. Roberts, W. J. Stirling, and R. S. Thorne, Phys. Lett. B 443 (1998) 301–307, hep-ph/9808371

    1998 arXiv

  37. [44]

    Catani and M

    S. Catani and M. H. Seymour,Nucl. Phys. B485(1997) 291–419, [Erratum: Nucl. Phys. B 510 (1998) 503], hep-ph/9605323

    1997 arXiv

  38. [45]

    Collins, T

    J. Collins, T. C. Rogers, and N. Sato,Phys. Rev. D105 (2022), no. 7 076010,2111.01170

    2022 arXiv

  39. [47]

    Nagy and D

    Z. Nagy and D. E. Soper, JHEP 10 (2016) 019, 1605.05845. 52

    2016 arXiv

  40. [48]

    Nagy and D

    Z. Nagy and D. E. Soper,Phys. Rev. D98 (2018), no. 1 014034, 1705.08093

    2018 arXiv

  41. [49]

    Nagy and D

    Z. Nagy and D. E. Soper, JHEP 06 (2014) 097, 1401.6364

    2014 arXiv

  42. [50]

    Nagy and D

    Z. Nagy and D. E. Soper, JHEP 07 (2015) 119, 1501.00778

    2015 arXiv

  43. [51]

    Nagy and D

    Z. Nagy and D. E. Soper,Phys. Rev. D99 (2019), no. 5 054009, 1902.02105

    2019 arXiv

  44. [52]

    Nagy and D

    Z. Nagy and D. E. Soper, Phys. Rev. D 105 (2022), no. 5 054012,2201.08056

    2022 arXiv

  45. [53]

    H. L. Lai, J. Botts, J. Huston, J. G. Morfin, J. F. Owens, J.-W. Qiu, W. K. Tung, and H. Weerts,Phys. Rev. D 51 (1995) 4763–4782, hep-ph/9410404

    1995 arXiv

  46. [54]

    H. L. Lai, J. Huston, S. Kuhlmann, F. I. Olness, J. F. Owens, D. E. Soper, W. K. Tung, and H. Weerts,Phys. Rev. D 55 (1997) 1280–1296, hep-ph/9606399

    1997 arXiv

  47. [55]

    CTEQ Collaboration, H. L. Lai, J. Huston, S. Kuhlmann, J. Morfin, F. I. Olness, J. F. Owens, J. Pumplin, and W. K. Tung, Eur. Phys. J. C 12 (2000) 375–392, hep-ph/9903282

    2000 arXiv

  48. [56]

    Houet al., Phys

    T.-J. Houet al., Phys. Rev. D103 (2021), no. 1 014013, 1912.10053

    2021 arXiv

  49. [57]

    Cruz-Martinez, S

    J. Cruz-Martinez, S. Forte, N. Laurenti, T. R. Rabe- mananjara, and J. Rojo, JHEP 09 (2024) 088, 2406.12961

    2024 arXiv

  50. [58]

    Bailey, T

    S. Bailey, T. Cridge, L. A. Harland-Lang, A. D. Martin, and R. S. Thorne,Eur. Phys. J. C81 (2021), no. 4 341, 2012.04684

    2021 arXiv

  51. [59]

    Collins, Foundations of Perturbative QCD, vol

    J. Collins, Foundations of Perturbative QCD, vol. 32 of Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology. Cambridge University Press, 2011

    2011

  52. [60]

    Private communication with John Collins

  53. [61]

    NNPDF Collaboration, R. D. Ball, V. Bertone, F. Cerutti, L. Del Debbio, S. Forte, A. Guffanti, J. I. La- torre, J. Rojo, and M. Ubiali,Nucl. Phys. B855 (2012) 153–221, 1107.2652

    2012 arXiv

  54. [62]

    Diehl and P

    M. Diehl and P. Stienemeier,Eur. Phys. J. Plus 135 (2020), no. 2 211,1904.10722

    2020 arXiv

  55. [63]

    A. D. Martin, W. J. Stirling, R. S. Thorne, and G. Watt, Eur. Phys. J. C63 (2009) 189–285, 0901.0002

    2009 arXiv

  56. [64]

    Altarelli, S

    G. Altarelli, S. Forte, and G. Ridolfi,Nucl. Phys. B534 (1998) 277–296, hep-ph/9806345

    1998 arXiv

  57. [65]

    Abramowicz et al., Eur

    H1, ZEUS Collaboration, H. Abramowicz et al., Eur. Phys. J. C75 (2015), no. 12 580,1506.06042

    2015 arXiv

  58. [66]

    R. D. Ball, L. Del Debbio, S. Forte, A. Guffanti, J. I. La- torre, J. Rojo, and M. Ubiali,Nucl. Phys. B838 (2010) 136–206, 1002.4407

    2010 arXiv

  59. [67]

    NNPDF Collaboration, R. D. Ball, L. Del Debbio, S. Forte, A. Guffanti, J. I. Latorre, A. Piccione, J. Rojo, and M. Ubiali,Nucl. Phys. B 809 (2009) 1–63, [Erra- tum: Nucl. Phys. B 816 (2009) 293], 0808.1231

    2009 arXiv

  60. [68]

    Pumplin, D

    J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, P. M. Nadolsky, and W. K. Tung, JHEP 07 (2002) 012, hep-ph/0201195

    2002 arXiv

  61. [69]

    NNPDF Collaboration, R. D. Ballet al., Eur. Phys. J. C 82 (2022), no. 5 428,2109.02653

    2022 arXiv

  62. [70]

    Kusina, F

    A. Kusina, F. I. Olness, I. Schienbein, T. Jezo, K. Ko- varik, T. Stavreva, and J. Y. Yu, Phys. Rev. D 88 (2013), no. 7 074032,1306.6553

    2013 arXiv

  63. [71]

    Bertoneet al., Eur

    xFitter Developers Team Collaboration, V. Bertoneet al., Eur. Phys. J. C77 (2017), no. 12 837,1707.05343

    2017 arXiv

  64. [72]

    Abramowicz et al., Eur

    H1, ZEUS Collaboration, H. Abramowicz et al., Eur. Phys. J. C73 (2013), no. 2 2311,1211.1182

    2013 arXiv

  65. [73]

    A. M. Cooper-Sarkar, PoS DIS2010 (2010) 023, 1006.4471

    2010 arXiv

  66. [74]

    Gauld, A

    R. Gauld, A. Huss, and G. Stagnitto,Phys. Rev. Lett. 130 (2023), no. 16 161901, [Erratum: Phys.Rev.Lett. 132, 159901 (2024)],2208.11138

    2023 arXiv

  67. [75]

    Caletti, A

    S. Caletti, A. J. Larkoski, S. Marzani, and D. Reichelt, Eur. Phys. J. C82 (2022), no. 7 632,2205.01109

    2022 arXiv

  68. [76]

    Czakon, A

    M. Czakon, A. Mitov, and R. Poncelet,JHEP 04 (2023) 138, 2205.11879

    2023 arXiv

  69. [77]

    Caola, R

    F. Caola, R. Grabarczyk, M. L. Hutt, G. P. Salam, L. Scyboz, and J. Thaler, Phys. Rev. D 108 (2023), no. 9 094010,2306.07314

    2023 arXiv

  70. [78]

    Catani and L

    S. Catani and L. Trentadue,Nucl. Phys. B327 (1989) 323–352

    1989

  71. [79]

    Catani, M

    S. Catani, M. L. Mangano, P. Nason, and L. Trentadue, Nucl. Phys. B478 (1996) 273–310, hep-ph/9604351

    1996 arXiv

  72. [80]

    Contopanagos, E

    H. Contopanagos, E. Laenen, and G. F. Sterman,Nucl. Phys. B 484 (1997) 303–330, hep-ph/9604313

    1997 arXiv

  73. [81]

    Forte and G

    S. Forte and G. Ridolfi,Nucl. Phys. B650 (2003) 229– 270, hep-ph/0209154

    2003 arXiv

  74. [82]

    A. V. Manohar, Phys. Rev. D 68 (2003) 114019, hep-ph/0309176

    2003 arXiv

  75. [83]

    Idilbi and X.-d

    A. Idilbi and X.-d. Ji,Phys. Rev. D72 (2005) 054016, hep-ph/0501006

    2005 arXiv

  76. [84]

    Becher and M

    T. Becher and M. Neubert,Phys. Rev. Lett.97 (2006) 082001, hep-ph/0605050

    2006 arXiv

  77. [85]

    Becher, M

    T. Becher, M. Neubert, and G. Xu,JHEP 07 (2008) 030, 0710.0680

    2008 arXiv

  78. [86]

    Bonvini, S

    M. Bonvini, S. Forte, and G. Ridolfi,Nucl. Phys. B847 (2011) 93–159, 1009.5691

    2011 arXiv

  79. [87]

    Bonvini, S

    M. Bonvini, S. Forte, M. Ghezzi, and G. Ridolfi,Nucl. Phys. B 861 (2012) 337–360, 1201.6364

    2012 arXiv

  80. [88]

    Forte, G

    S. Forte, G. Ridolfi, and S. Rota,JHEP 08 (2021) 110, 2106.11321

    2021 arXiv

  81. [89]

    Curci and M

    G. Curci and M. Greco,Phys. Lett. B 92 (1980) 175– 178

    1980

  82. [90]

    Anandam and D

    P. Anandam and D. E. Soper,Phys. Rev. D61 (2000) 094003, hep-ph/9912327

    2000 arXiv

  83. [91]

    Klasen and G

    M. Klasen and G. Kramer, Phys. Lett. B 386 (1996) 384–388, hep-ph/9605210

    1996 arXiv

  84. [92]

    Bellm et al., Eur

    J. Bellm et al., Eur. Phys. J. C 76 (2016), no. 4 196, 1512.01178

    2016 arXiv

  85. [93]

    Bewicket al.,Eur

    G. Bewicket al.,Eur. Phys. J. C84(2024), no. 10 1053, 2312.05175

    2024 arXiv

  86. [94]

    Aaboudet al., Eur

    ATLAS Collaboration, M. Aaboudet al., Eur. Phys. J. C 77 (2017), no. 6 361,1702.05725

    2017 arXiv

  87. [95]

    CMS Collaboration, A. M. Sirunyanet al., Eur. Phys. J. C 78 (2018), no. 11 965,1804.05252

    2018 arXiv

  88. [96]

    M.Cacciari,G.P.Salam,andG.Soyez, JHEP 04(2008) 063, 0802.1189

    2008 arXiv

  89. [97]

    C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gom- mers, P. Virtanen, D. Cournapeau, E. Wieser, J. Tay- lor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del Río, M. Wiebe, P. Peterson, P. Gérard-Marchant, K. Shep- pa...

    2020

  90. [98]

    Virtanen, R

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber- land, T. Reddy, D. Cournapeau, E. Burovski, P. Pe- terson, W. Weckesser, J. Bright, S. J. van der Walt, M.Brett,J.Wilson,K.J.Millman,N.Mayorov,A.R.J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Po- lat, Y. Feng, E. W...

    2020

  91. [99]

    Meurer, C

    A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johans- son, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Sa...

    2017

  92. [100]

    http://mpmath.org/

    The mpmath development team,mpmath: a Python li- brary for arbitrary-precision floating-point arithmetic (version 1.3.0), 2023. http://mpmath.org/

    2023

Pith tools

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