pith. sign in

arxiv: 2606.23813 · v1 · pith:NCINYVIKnew · submitted 2026-06-22 · ✦ hep-ph

PDF evolution in alternative factorisation schemes

St\'ephane Delorme , Aleksander Kusina , Andrzej Si\'odmok , James Whitehead This is my paper

Pith reviewed 2026-06-26 07:47 UTC · model grok-4.3

classification ✦ hep-ph
keywords parton distribution functionsDGLAP evolutionfactorisation schemesNLO splitting functionsanomalous dimensionsQCD calculationsPDF fits
0
0 comments X

The pith

Alternative factorisation schemes produce modified NLO DGLAP splitting functions for parton distributions whose leading behaviour corresponds to a rescaled evolution scale.

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

The paper derives the next-to-leading-order DGLAP splitting functions that control the scale dependence of parton distribution functions once a factorisation scheme other than the conventional MS-bar choice is adopted. These functions are obtained both for specific alternative schemes and for a continuous family of schemes that parametrises a subspace of the general factorisation-scheme space. The corresponding anomalous dimensions are examined in Mellin space, with particular attention to their leading large-x and small-x limits that matter for resummation. The dominant terms in those limits are shown to admit a simple interpretation as a shift in the effective scale at which the distributions evolve. The results supply the technical ingredients required to evolve and fit PDFs consistently inside higher-order QCD calculations performed in non-standard schemes.

Core claim

Beyond leading order, parton distribution functions require a choice of factorisation scheme. Different choices lead to PDFs that satisfy modified DGLAP evolution equations. We derive the NLO DGLAP splitting functions for PDFs in alternative factorisation schemes, including for a parametrised scheme spanning a subspace of the general factorisation-scheme space. We plot their Mellin-space counterparts, the anomalous dimensions, and study the leading large- and small-x behaviour, relevant to resummation. We find that the leading behaviour admits a natural interpretation as a modified effective evolution scale.

What carries the argument

The NLO DGLAP splitting functions obtained by consistent redefinition of the factorisation procedure in alternative schemes, together with their Mellin-space anomalous-dimension counterparts.

If this is right

  • PDFs defined in the alternative schemes obey modified DGLAP evolution equations already at NLO.
  • The anomalous dimensions exhibit altered leading large-x and small-x behaviour that can be absorbed into a rescaled evolution variable.
  • The parametrised family of schemes allows continuous variation of the scheme choice while preserving the modified evolution structure.
  • The derived functions constitute the necessary input for performing PDF evolution and global fits inside QCD calculations that employ non-MS-bar schemes.

Where Pith is reading between the lines

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

  • The modified evolution could be combined with existing small-x or large-x resummation techniques to produce scheme-consistent predictions in extreme kinematic regions.
  • Extending the same redefinition procedure to NNLO would allow consistent use of alternative schemes in calculations that include higher-order corrections.
  • Global PDF fits performed directly in one of the parametrised schemes might yield different central values or uncertainties for the distributions extracted from the same data.
  • The effective-scale interpretation suggests that numerical evolution codes could be adapted with only a trivial rescaling of the evolution variable rather than a full rewrite of the splitting functions.

Load-bearing premise

The NLO splitting functions in the chosen alternative schemes can be obtained by a consistent redefinition of the factorization procedure without introducing additional scheme-dependent terms that would invalidate the modified DGLAP equations at this order.

What would settle it

An independent calculation, by a different method, of the NLO splitting function in one specific alternative scheme that yields a numerically different result from the one presented here.

Figures

Figures reproduced from arXiv: 2606.23813 by Aleksander Kusina, Andrzej Si\'odmok, James Whitehead, St\'ephane Delorme.

Figure 1
Figure 1. Figure 1: Splitting functions for the singlet sector (calculated with [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Splitting functions for the non-singlet sector (calculated with [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Anomalous dimensions for the singlet sector, with reciprocal-scaled axes below [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Anomalous dimensions for the non-singlet sector, with reciprocal-scaled axes below [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Non-singlet evolution of the prototypical up-quark valence distribution from the Les Houches PDF bench [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Splitting functions for the singlet sector, multiplied by [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
read the original abstract

Beyond leading order, parton distribution functions (PDFs) require a choice of factorisation scheme to be defined unambiguously. Different choices of factorisation scheme lead to PDFs that satisfy modified DGLAP evolution equations, relative to the conventional $\overline{\mathrm{MS}}$ scheme. In this paper we derive the NLO DGLAP splitting functions for PDFs in alternative factorisation schemes, including for a parametrised scheme spanning a subspace of the general factorisation-scheme space. We plot their Mellin-space counterparts, the anomalous dimensions, and study the leading large- and small-$x$ behaviour, relevant to resummation. We find that the leading behaviour admits a natural interpretation as a modified effective evolution scale. This is an essential step towards being able to evolve and fit PDFs in alternative schemes for use within QCD calculations.

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

0 major / 3 minor

Summary. The manuscript derives the NLO DGLAP splitting functions for parton distribution functions in alternative factorization schemes (including a parametrized scheme spanning a subspace of the general scheme space), obtains the corresponding Mellin-space anomalous dimensions, and analyzes their leading large-x and small-x behavior. The leading behavior is interpreted as corresponding to a modified effective evolution scale. The work is presented as an essential step toward evolving and fitting PDFs in schemes other than the conventional MS-bar scheme.

Significance. If the derivations are correct, the results supply explicit NLO splitting functions and anomalous dimensions outside the standard scheme, enabling controlled exploration of factorization-scheme dependence in PDF evolution and fits. The observation that leading x-behavior admits a modified-scale interpretation is a concrete, potentially useful simplification for resummation studies. The parametrized subspace provides a systematic rather than ad-hoc exploration of scheme freedom.

minor comments (3)
  1. [Abstract / §2] The abstract asserts that derivations were performed and plots produced, but the manuscript would be strengthened by an explicit statement (e.g., in §2 or §3) of the precise redefinition of the factorization procedure used to obtain the alternative-scheme splitting functions at NLO, including confirmation that no additional scheme-dependent terms arise at this order.
  2. [Figures] Figure captions and axis labels for the Mellin-space anomalous dimensions should explicitly indicate the value(s) of the scheme parameter(s) used in each curve; without this, reproduction of the plotted results is unnecessarily difficult.
  3. A short table comparing the derived NLO splitting functions (or their first few Mellin moments) against the standard MS-bar expressions for at least one concrete choice of the parametrized scheme would improve readability and allow immediate verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our manuscript deriving NLO splitting functions and anomalous dimensions in alternative factorization schemes. We note the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives NLO DGLAP splitting functions in alternative factorisation schemes by redefining the factorisation procedure starting from standard QCD inputs. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the parametrised subspace is presented as an explicit controlled choice rather than a tautological output. The leading-behaviour interpretation is an observation on the derived functions, not a prediction forced by the inputs. The work remains externally falsifiable against standard MSbar results and does not invoke uniqueness theorems or ansatze smuggled via self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; the derivation is described as following from standard factorization choices.

pith-pipeline@v0.9.1-grok · 5669 in / 1108 out tokens · 18340 ms · 2026-06-26T07:47:22.555307+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

88 extracted references · 49 linked inside Pith

  1. [1]

    Yu. L. Dokshitzer,Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e− Annihilation by Perturbation Theory in Quantum Chromodynamics,Sov. Phys. JETP46(1977) 641

    1977

  2. [2]

    Gribov and L.N

    V.N. Gribov and L.N. Lipatov,Deep inelastice pscattering in perturbation theory,Sov. J. Nucl. Phys.15(1972) 438

    1972

  3. [3]

    Altarelli and G

    G. Altarelli and G. Parisi,Asymptotic Freedom in Parton Language,Nucl. Phys. B126(1977) 298

    1977

  4. [4]

    Altarelli, R.K

    G. Altarelli, R.K. Ellis and G. Martinelli,Leptoproduction and Drell–Yan Processes Beyond the Leading Approximation in Chromodynamics,Nucl. Phys. B143(1978) 521

    1978

  5. [5]

    Bardeen, A.J

    W.A. Bardeen, A.J. Buras, D.W. Duke and T. Muta,Deep Inelastic Scattering Beyond the Leading Order in Asymptotically Free Gauge Theories,Phys. Rev. D18(1978) 3998

    1978

  6. [6]

    Sterman,Summation of Large Corrections to Short Distance Hadronic Cross-Sections,Nucl

    G.F. Sterman,Summation of Large Corrections to Short Distance Hadronic Cross-Sections,Nucl. Phys. B281(1987) 310

    1987

  7. [7]

    Catani and L

    S. Catani and L. Trentadue,Resummation of the QCD Perturbative Series for Hard Processes,Nucl. Phys. B327(1989) 323

    1989

  8. [8]

    Catani, M.L

    S. Catani, M.L. Mangano, P. Nason and L. Trentadue,The Resummation of soft gluons in hadronic collisions,Nucl. Phys. B 478(1996) 273 [hep-ph/9604351]

    Pith/arXiv arXiv 1996

  9. [9]

    Catani and F

    S. Catani and F. Hautmann,High-energy factorization and small-xdeep inelastic scattering beyond leading order,Nucl. Phys. B 427(1994) 475 [hep-ph/9405388]

    Pith/arXiv arXiv 1994

  10. [10]

    Ball and S

    R.D. Ball and S. Forte,Summation of leading logarithms at smallx,Phys. Lett. B351(1995) 313 [hep-ph/9501231]

    Pith/arXiv arXiv 1995

  11. [11]

    Ball and S

    R.D. Ball and S. Forte,Smallxbehavior of Altarelli–Parisi splitting functions,Phys. Lett. B465(1999) 271 [hep-ph/9906222]

    Pith/arXiv arXiv 1999

  12. [12]

    Altarelli, R.D

    G. Altarelli, R.D. Ball and S. Forte,Perturbatively stable resummed small x evolution kernels,Nucl. Phys. B742(2006) 1 [hep-ph/0512237]

    Pith/arXiv arXiv 2006

  13. [13]

    Bonvini, S

    M. Bonvini, S. Marzani and T. Peraro,Small-xresummation from HELL,Eur. Phys. J. C76(2016) 597 [1607.02153]

    Pith/arXiv arXiv 2016

  14. [14]

    Bonvini, S

    M. Bonvini, S. Marzani and C. Muselli,Towards parton distribution functions with small-xresummation: HELL 2.0,JHEP12 (2017) 117 [1708.07510]

    Pith/arXiv arXiv 2017

  15. [15]

    Bonvini and S

    M. Bonvini and S. Marzani,Four-loop splitting functions at smallx,JHEP06(2018) 145 [1805.06460]

    Pith/arXiv arXiv 2018

  16. [16]

    Bonvini, S

    M. Bonvini, S. Frixione and G. Stagnitto,New results on small-x resummation for splitting functions,2603.02312

    arXiv

  17. [17]

    Corcella and L

    G. Corcella and L. Magnea,Soft-gluon resummation effects on parton distributions,Phys. Rev. D72(2005) 074017 [hep-ph/0506278]

    Pith/arXiv arXiv 2005

  18. [18]

    Bonvini, S

    M. Bonvini, S. Marzani, J. Rojo, L. Rottoli et al.,Parton distributions with threshold resummation,JHEP09(2015) 191 [1507.01006]

    Pith/arXiv arXiv 2015

  19. [19]

    R.D. Ball, V. Bertone, M. Bonvini, S. Marzani et al.,Parton distributions with small-x resummation: evidence for BFKL dynamics in HERA data,Eur. Phys. J. C78(2018) 321 [1710.05935]. 20.xFitter Developers’ Teamcollaboration,Impact of low-xresummation on QCD analysis of HERA data,Eur. Phys. J. C 78(2018) 621 [1802.00064]

    Pith/arXiv arXiv 2018

  20. [20]

    Furmanski and R

    W. Furmanski and R. Petronzio,Lepton–Hadron Processes Beyond Leading Order in Quantum Chromodynamics,Z. Phys. C 11(1982) 293

    1982

  21. [21]

    Catani,Physical anomalous dimensions at small x,Z

    S. Catani,Physical anomalous dimensions at small x,Z. Phys. C75(1997) 665 [hep-ph/9609263]

    Pith/arXiv arXiv 1997

  22. [22]

    Moch and A

    S. Moch and A. Vogt,On non-singlet physical evolution kernels and large-xcoefficient functions in perturbative QCD,JHEP11 (2009) 099 [0909.2124]

    Pith/arXiv arXiv 2009

  23. [23]

    Hentschinski and M

    M. Hentschinski and M. Stratmann,On the Practical Application of Physical Anomalous Dimensions,1311.2825

    Pith/arXiv arXiv

  24. [24]

    Lappi, H

    T. Lappi, H. Mäntysaari, H. Paukkunen and M. Tevio,Evolution of structure functions in momentum space,Eur. Phys. J. C 84(2024) 84 [2304.06998]

    arXiv 2024

  25. [25]

    Lappi, H

    T. Lappi, H. Mäntysaari, H. Paukkunen and M. Tevio,Next-to-leading order evolution of structure functions without PDFs, Eur. Phys. J. C85(2025) 429 [2412.09589]

    arXiv 2025

  26. [26]

    Jadach, W

    S. Jadach, W. Płaczek, S. Sapeta, A. Siódmok et al.,Matching NLO QCD with parton shower in Monte Carlo scheme — the KrkNLO method,JHEP10(2015) 052 [1503.06849]

    Pith/arXiv arXiv 2015

  27. [27]

    Jadach, W

    S. Jadach, W. Płaczek, S. Sapeta, A. Siódmok et al.,Parton distribution functions in Monte Carlo factorisation scheme,Eur. Phys. J. C76(2016) 649 [1606.00355]

    Pith/arXiv arXiv 2016

  28. [28]

    Jadach, G

    S. Jadach, G. Nail, W. Płaczek, S. Sapeta et al.,Monte Carlo simulations of Higgs-boson production at the LHC with the KrkNLO method,Eur. Phys. J. C77(2017) 164 [1607.06799]

    Pith/arXiv arXiv 2017

  29. [29]

    Candido, S

    A. Candido, S. Forte and F. Hekhorn,CanMSparton distributions be negative?,JHEP11(2020) 129 [2006.07377]

    arXiv 2020

  30. [30]

    Candido, S

    A. Candido, S. Forte, T. Giani and F. Hekhorn,On the positivity ofMSparton distributions,Eur. Phys. J. C84(2024) 335 [2308.00025]

    Pith/arXiv arXiv 2024

  31. [31]

    Aversa, P

    F. Aversa, P. Chiappetta, M. Greco and J.P. Guillet,QCD Corrections to Parton-Parton Scattering Processes,Nucl. Phys. B 327(1989) 105

    1989

  32. [32]

    Oliveira, A.D

    E.G. Oliveira, A.D. Martin and M.G. Ryskin,Physical factorisation scheme for PDFs for non-inclusive applications,JHEP11 (2013) 156 [1310.8289]

    Pith/arXiv arXiv 2013

  33. [33]

    Ramalho and G.F

    A. Ramalho and G.F. Sterman,High Moment Corrections to the Drell–Yan Cross-section and an Alternate Parton Density, Phys. Rev. D29(1984) 2517

    1984

  34. [34]

    Nagy and D.E

    Z. Nagy and D.E. Soper,Summing threshold logs in a parton shower,JHEP10(2016) 019 [1605.05845]

    Pith/arXiv arXiv 2016

  35. [35]

    Nagy and D.E

    Z. Nagy and D.E. Soper,What is a parton shower?,Phys. Rev. D98(2018) 014034 [1705.08093]

    Pith/arXiv arXiv 2018

  36. [36]

    Nagy and D.E

    Z. Nagy and D.E. Soper,Multivariable evolution in parton showers with initial state partons,Phys. Rev. D106(2022) 014024 [2204.05631]

    arXiv 2022

  37. [37]

    Sarmah, A

    P. Sarmah, A. Siódmok and J. Whitehead,KrkNLO matching for colour-singlet processes,JHEP01(2025) 062 [2409.16417]

    arXiv 2025

  38. [38]

    Sarmah, A

    P. Sarmah, A. Siódmok and J. Whitehead,KrkNLO matching and phenomenology for vector boson processes,2511.16605

    arXiv

  39. [39]

    Lin et al.,Parton distributions and lattice QCD calculations: a community white paper,Prog

    H.-W. Lin et al.,Parton distributions and lattice QCD calculations: a community white paper,Prog. Part. Nucl. Phys.100 (2018) 107 [1711.07916]

    Pith/arXiv arXiv 2018

  40. [40]

    Cichy and M

    K. Cichy and M. Constantinou,A guide to light-cone PDFs from Lattice QCD: an overview of approaches, techniques and results,Adv. High Energy Phys.2019(2019) 3036904 [1811.07248]

    Pith/arXiv arXiv 2019

  41. [41]

    Constantinou et al.,Parton distributions and lattice-QCD calculations: Toward 3D structure,Prog

    M. Constantinou et al.,Parton distributions and lattice-QCD calculations: Toward 3D structure,Prog. Part. Nucl. Phys.121 (2021) 103908 [2006.08636]

    arXiv 2021

  42. [42]

    Martinelli, C

    G. Martinelli, C. Pittori, C.T. Sachrajda, M. Testa et al.,A general method for non-perturbative renormalization of lattice operators,Nucl. Phys. B445(1995) 81 [hep-lat/9411010]. 32

    Pith/arXiv arXiv 1995

  43. [43]

    Ma and J.-W

    Y.-Q. Ma and J.-W. Qiu,Extracting Parton Distribution Functions from Lattice QCD Calculations,Phys. Rev. D98(2018) 074021 [1404.6860]

    Pith/arXiv arXiv 2018

  44. [44]

    Ma and J.-W

    Y.-Q. Ma and J.-W. Qiu,Exploring Partonic Structure of Hadrons Using ab initio Lattice QCD Calculations,Phys. Rev. Lett. 120(2018) 022003 [1709.03018]

    Pith/arXiv arXiv 2018

  45. [45]

    Stewart and Y

    I.W. Stewart and Y. Zhao,Matching the quasiparton distribution in a momentum subtraction scheme,Phys. Rev. D97(2018) 054512 [1709.04933]

    Pith/arXiv arXiv 2018

  46. [46]

    Delorme, A

    S. Delorme, A. Kusina, A. Siódmok and J. Whitehead,Factorisation schemes for proton PDFs,Eur. Phys. J. C85(2025) 505 [2501.18289]

    arXiv 2025

  47. [47]

    Jadach, A

    S. Jadach, A. Kusina, W. Płaczek, M. Skrzypek et al.,Inclusion of the QCD next-to-leading order corrections in the quark-gluon Monte Carlo shower,Phys. Rev. D87(2013) 034029 [1103.5015]

    Pith/arXiv arXiv 2013

  48. [48]

    Ellis, W.J

    R.K. Ellis, W.J. Stirling and B.R. Webber,QCD and Collider Physics, vol. 8, Cambridge University Press (2, 2011), 10.1017/CBO9780511628788

    doi:10.1017/cbo9780511628788 2011

  49. [49]

    Gribov and L.N

    V.N. Gribov and L.N. Lipatov,e+e− pair annihilation and deep inelasticepscattering in perturbation theory,Sov. J. Nucl. Phys.15(1972) 675

    1972

  50. [50]

    Furmanski and R

    W. Furmanski and R. Petronzio,Singlet Parton Densities Beyond Leading Order,Phys. Lett. B97(1980) 437

    1980

  51. [51]

    Mathematica, Version 14.1

    Wolfram Research, Inc., “Mathematica, Version 14.1.”

  52. [52]

    Höschele, J

    M. Höschele, J. Hoff, A. Pak, M. Steinhauser et al.,MT: A Mathematica package to compute convolutions,Comput. Phys. Commun.185(2014) 528 [1307.6925]

    Pith/arXiv arXiv 2014

  53. [53]

    Maître,HPL, a Mathematica implementation of the harmonic polylogarithms,Comput

    D. Maître,HPL, a Mathematica implementation of the harmonic polylogarithms,Comput. Phys. Commun.174(2006) 222 [hep-ph/0507152]

    Pith/arXiv arXiv 2006

  54. [54]

    Maître,Extension of HPL to complex arguments,Comput

    D. Maître,Extension of HPL to complex arguments,Comput. Phys. Commun.183(2012) 846 [hep-ph/0703052]

    Pith/arXiv arXiv 2012

  55. [55]

    Moch, J.A.M

    S. Moch, J.A.M. Vermaseren and A. Vogt,The three-loop splitting functions in QCD: the nonsinglet case,Nucl. Phys. B688 (2004) 101 [hep-ph/0403192]

    Pith/arXiv arXiv 2004

  56. [56]

    A. Vogt, S. Moch and J.A.M. Vermaseren,The three-loop splitting functions in QCD: the singlet case,Nucl. Phys. B691(2004) 129 [hep-ph/0404111]

    Pith/arXiv arXiv 2004

  57. [57]

    Forte, L

    S. Forte, L. Magnea, A. Piccione and G. Ridolfi,Evolution of truncated moments of singlet parton distributions,Nucl. Phys. B 594(2001) 46 [hep-ph/0006273]

    Pith/arXiv arXiv 2001

  58. [58]

    Catani and M.H

    S. Catani and M.H. Seymour,A general algorithm for calculating jet cross-sections in NLO QCD,Nucl. Phys. B485(1997) 291 [hep-ph/9605323]

    Pith/arXiv arXiv 1997

  59. [59]

    Curci, W

    G. Curci, W. Furmanski and R. Petronzio,Evolution of Parton Densities Beyond Leading Order: The Nonsinglet Case,Nucl. Phys. B175(1980) 27

    1980

  60. [60]

    Collins,Foundations of Perturbative QCD, vol

    J. Collins,Foundations of Perturbative QCD, vol. 32 ofCambridge Monographs on Particle Physics, Nuclear Physics and Cosmology, Cambridge University Press (2011), 10.1017/9781009401845

    doi:10.1017/9781009401845 2011

  61. [61]

    Collins and D.E

    J.C. Collins and D.E. Soper,Parton Distribution and Decay Functions,Nucl. Phys. B194(1982) 445

    1982

  62. [62]

    Grozin, J

    A. Grozin, J. Henn and M. Stahlhofen,On the Casimir scaling violation in the cusp anomalous dimension at small angle,JHEP 10(2017) 052 [1708.01221]

    Pith/arXiv arXiv 2017

  63. [63]

    Henn, G.P

    J.M. Henn, G.P. Korchemsky and B. Mistlberger,The full four-loop cusp anomalous dimension inN= 4super Yang-Mills and QCD,JHEP04(2020) 018 [1911.10174]

    arXiv 2020

  64. [64]

    von Manteuffel, E

    A. von Manteuffel, E. Panzer and R.M. Schabinger,Cusp and collinear anomalous dimensions in four-loop QCD from form factors,Phys. Rev. Lett.124(2020) 162001 [2002.04617]

    arXiv 2020

  65. [65]

    Kuraev, L.N

    E.A. Kuraev, L.N. Lipatov and V.S. Fadin,Multiregge processes in the Yang-Mills theory,Sov. Phys. JETP44(1976) 443

    1976

  66. [66]

    Balitsky and L.N

    I.I. Balitsky and L.N. Lipatov,The Pomeranchuk Singularity in Quantum Chromodynamics,Sov. J. Nucl. Phys.28(1978) 822

    1978

  67. [67]

    Collins and R.K

    J.C. Collins and R.K. Ellis,Heavy Quark Production in Very High-Energy Hadron Collisions,Nucl. Phys. B360(1991) 3

    1991

  68. [68]

    Catani, M

    S. Catani, M. Ciafaloni and F. Hautmann,High-energy factorization and small x heavy flavor production,Nucl. Phys. B366 (1991) 135

    1991

  69. [69]

    Catani, M

    S. Catani, M. Ciafaloni and F. Hautmann,Gluon contributions to smallxheavy flavor production,Phys. Lett. B242(1990) 97

    1990

  70. [70]

    Kotko, K

    P. Kotko, K. Kutak, C. Marquet, E. Petreska et al.,Improved TMD factorization for forward dijet production in dilute-dense hadronic collisions,JHEP09(2015) 106 [1503.03421]

    Pith/arXiv arXiv 2015

  71. [71]

    Hentschinski, A

    M. Hentschinski, A. Kusina, K. Kutak and M. Serino,TMD splitting functions inkT factorization: the real contribution to the gluon-to-gluon splitting,Eur. Phys. J. C78(2018) 174 [1711.04587]

    Pith/arXiv arXiv 2018

  72. [72]

    van Hameren, L

    A. van Hameren, L. Motyka and G. Ziarko,Hybrid kT-factorization and impact factors at NLO,JHEP11(2022) 103 [2205.09585]

    arXiv 2022

  73. [73]

    Korchemsky,Asymptotics of the Altarelli–Parisi–Lipatov Evolution Kernels of Parton Distributions,Mod

    G.P. Korchemsky,Asymptotics of the Altarelli–Parisi–Lipatov Evolution Kernels of Parton Distributions,Mod. Phys. Lett. A4 (1989) 1257

    1989

  74. [74]

    Korchemsky and A.V

    G.P. Korchemsky and A.V. Radyushkin,Infrared factorization, Wilson lines and the heavy quark limit,Phys. Lett. B279 (1992) 359 [hep-ph/9203222]

    Pith/arXiv arXiv 1992

  75. [75]

    Albino and R.D

    S. Albino and R.D. Ball,Soft resummation of quark anomalous dimensions and coefficient functions inMSfactorization,Phys. Lett. B513(2001) 93 [hep-ph/0011133]

    Pith/arXiv arXiv 2001

  76. [76]

    Yu. L. Dokshitzer, G. Marchesini and G.P. Salam,Revisiting parton evolution and the large-xlimit,Phys. Lett. B634(2006) 504 [hep-ph/0511302]

    Pith/arXiv arXiv 2006

  77. [77]

    Vogt,Leading logarithmic large-xresummation of off-diagonal splitting functions and coefficient functions,Phys

    A. Vogt,Leading logarithmic large-xresummation of off-diagonal splitting functions and coefficient functions,Phys. Lett. B 691(2010) 77 [1005.1606]

    Pith/arXiv arXiv 2010

  78. [78]

    White and R.S

    C.D. White and R.S. Thorne,Comparison of NNLO DIS scheme splitting functions with results from exact gluon kinematics at smallx,Eur. Phys. J. C45(2006) 179 [hep-ph/0507244]

    Pith/arXiv arXiv 2006

  79. [79]

    Becher and M

    T. Becher and M. Neubert,Threshold resummation in momentum space from effective field theory,Phys. Rev. Lett.97(2006) 082001 [hep-ph/0605050]

    Pith/arXiv arXiv 2006

  80. [80]

    Becher, M

    T. Becher, M. Neubert and B.D. Pecjak,Factorization and Momentum-Space Resummation in Deep-Inelastic Scattering,JHEP 01(2007) 076 [hep-ph/0607228]

    Pith/arXiv arXiv 2007

Showing first 80 references.