Pith. sign in

REVIEW 1 minor 4 cited by

Projected Energy Correlators: Two-Loop Jet Functions and NNLL Resummation

T0 review · 0 major / 1 minor · reviewed 2026-06-28 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Two-loop jet functions enable NNLL resummation of projected N-point energy correlators up to N=6.

desk verdict The two-loop jet functions for N=4,5,6 are the actual new piece, with NNLL resummation and power corrections built on top using standard methods. read the letter →

arxiv 2606.02714 v1 pith:ERS5BQAD submitted 2026-06-01 hep-ph hep-exnucl-th

Kyle Lee , Yibei Li , Zhen Xu , Xiaoyuan Zhang This is my paper · ORCID
classification hep-phhep-exnucl-th
keywords energycorrelatorsjetfunctionsNNLLresummationcollinearfactorizationQCDnonperturbativecorrectionsalpha_sextraction
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 establishes NNLL collinear resummation for projected N-point energy correlators up to N=6 by computing the required two-loop jet functions semi-analytically. This is done for both electron-positron annihilation and Higgs decay to gluons, with matching to NLO fixed-order results and inclusion of leading non-perturbative corrections via universal soft matrix elements. A sympathetic reader would care as this brings higher-point observables under quantitative perturbative control, potentially enabling alpha_s extractions with different systematics from lower-point ones.

What carries the argument

Two-loop jet function for N=4,5,6 computed semi-analytically via Integration-by-Parts and differential equations.

What would settle it

A direct computation of the jet function at three loops or a comparison with experimental data showing significant deviation from the NNLL prediction after accounting for the included power corrections would falsify the claimed accuracy.

Watch

Extended reading notes

Core claim

The key new ingredient is the two-loop jet function for N=4,5,6 computed using Integration-by-Parts and differential equations, which permits the NNLL resummation of projected N-point energy correlators up to N=6 matched to fixed-order NLO predictions, including leading non-perturbative corrections described by two universal soft matrix elements.

Load-bearing premise

The leading non-perturbative corrections are described by two universal soft matrix elements of order Lambda_QCD whose evolution is governed by anomalous dimensions for (N-1)-point correlators.

Editorial extensions

If this is right

  • The matched NNLL distributions for ENCs up to N=6 can be compared with parton-shower simulations.
  • Sensitivity of the spectra and their ratios to alpha_s and the soft matrix elements can be analyzed.
  • Higher-point projected energy correlators achieve quantitative control at NNLL accuracy.
  • This opens the possibility for future alpha_s extractions with complementary systematics.

Reading between the lines

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

  • The anomalous dimensions for (N-1)-point correlators governing the evolution of the soft matrix elements could be used to relate corrections across different N.
  • Ratios of higher N correlators to the two-point one might reduce experimental and theoretical uncertainties in measurements.
  • Applying similar methods to other processes or observables could broaden the use of energy correlators in QCD studies.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 1 minor

Summary. The paper presents NNLL collinear resummation of projected N-point energy correlators (ENCs) up to N=6 in e+e- annihilation and Higgs decay to gluons, matched to NLO fixed order. The central technical advance is the semi-analytic computation of the two-loop jet functions for N=4,5,6 via integration-by-parts reduction and differential equations. Leading non-perturbative power corrections are modeled by two universal soft matrix elements Ω̅1q and Ω̅1g whose evolution follows (N-1)-point anomalous dimensions. Results are compared to Pythia8 and Herwig7 parton showers, with studies of sensitivity to αs and the soft parameters in both absolute spectra and ratios to the two-point correlator.

Significance. If the two-loop jet-function results hold, the work brings projected ENCs under quantitative NNLL control, providing a new set of observables for αs extractions whose systematics are complementary to event shapes. The explicit IBP+DE computation of the N=4,5,6 jet functions and the consistent matching plus power-correction framework constitute a clear technical contribution to the precision QCD literature.

minor comments (1)
  1. The abstract and introduction would benefit from a brief statement of the kinematic cuts or fiducial phase-space definitions used for the N-point correlators when comparing to parton showers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of the technical advances in the two-loop jet functions, and recommendation to accept. There are no major comments to address.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper computes two-loop jet functions for N=4,5,6 directly via Integration-by-Parts reduction and differential equations, then performs standard NNLL collinear resummation matched to NLO fixed order. Non-perturbative power corrections are introduced explicitly as an assumption using two universal soft matrix elements whose evolution follows from known anomalous dimensions; these are not derived from or fitted to the perturbative results within the paper. No quoted equation or step reduces the claimed predictions to inputs by construction, and no load-bearing self-citation chain is exhibited. The derivation chain is therefore self-contained and employs standard field methods without internal reduction.

Assumptions & free parameters 2 free parameters · 2 assumptions · 0 invented entities

The central claim rests on standard perturbative QCD factorization assumptions plus two universal non-perturbative matrix elements whose values and evolution are taken as inputs; no new entities are postulated.

free parameters (2)
  • Ω̅1q
    Universal soft matrix element for quarks of order ΛQCD used to parametrize leading non-perturbative corrections.
  • Ω̅1g
    Universal soft matrix element for gluons of order ΛQCD used to parametrize leading non-perturbative corrections.
assumptions (2)
  • domain assumption Collinear factorization and all-order resummation in perturbative QCD
    Invoked to justify the NNLL resummation of the projected correlators.
  • domain assumption Existence and universality of the two soft matrix elements governing power corrections
    Invoked to include leading non-perturbative effects via evolution with (N-1)-point anomalous dimensions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Projected Energy Correlators: Two-Loop Jet Functions and NNLL Resummation." pith.science (2026). https://pith.science/paper/ERS5BQAD

@misc{pith2026260602714,
  author       = {Pith},
  title        = {Pith review of: Projected Energy Correlators: Two-Loop Jet Functions and NNLL Resummation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ERS5BQAD}},
  note         = {Machine review of arXiv:2606.02714}
}
abstract

We present the next-to-next-to-leading logarithmic (NNLL) collinear resummation of projected $N$-point energy correlators (ENCs) up to $N=6$, matched to fixed-order predictions at NLO, in both electron-positron annihilation and Higgs decay to gluons. The key new ingredient is the two-loop jet function for $N=4,5,6$, which we compute semi-analytically using Integration-by-Parts and differential equations. We further include the leading non-perturbative corrections for ENCs, described by two universal soft matrix elements $\overline{\Omega}_{1q},\overline{\Omega}_{1g}$ of order $\Lambda_{\rm QCD}$, whose evolution is governed by anomalous dimensions for $(N-1)$-point correlators. The matched distributions are compared with parton-shower simulations from Pythia8 and Herwig7, and we study the sensitivity of both the absolute spectra and their ratios to the two-point energy correlator under variations of $\alpha_s$ and $\overline{\Omega}_{1q,1g}$. Our results show that higher-point projected energy correlators are now under quantitative control at NNLL accuracy, opening the door to future $\alpha_s$ extractions with complementary systematics.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 4 Pith papers

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

  1. Putting Jet Substructure on Track(s)

    hep-ph 2026-06 unverdicted novelty 7.0 of 10

    First complete NLL calculations of projected energy correlators (up to 4-point) on tracks via factorization theorems and RG evolution, extending prior full-jet results.

  2. Jet-by-jet energy correlators as stochastic probes of the parton-to-hadron transition

    hep-ph 2026-08 conditional novelty 6.0 of 10

    Per-jet fluctuations of binned energy-energy correlators show a smaller covariance trace and larger neighboring-shell correlations at hadron level than at parton level in Pythia and Herwig across all tested configurations.

  3. Heavy Jet Mass in Hadronic Higgs Decays

    hep-ph 2026-07 conditional novelty 6.0 of 10

    Heavy jet mass in hadronic Higgs decays is predicted at N³LL′ (dijet) plus NNLL (Sudakov shoulder) plus NNLO, with new analytic trijet-region logarithms for H→gg and H→q̄q.

  4. Dissecting Parton Showers with Multi-Point Energy Correlators

    hep-ph 2026-07 accept novelty 6.0 of 10

    Projections of four-point energy correlators cleanly separate spin from kinematic azimuthal correlations inside jets; spin effects are subdominant in accessible LHC kinematics.

Reference graph

Works this paper leans on

104 extracted references · 95 canonical work pages · cited by 4 Pith papers

  1. [1]

    G. F. Sterman,Jet Structure in e+ e- Annihilation with Massless Hadrons,ILL-TH-75-32 (1975)

    1975

  2. [2]

    N. A. Sveshnikov and F. V. Tkachov,Jets and quantum field theory,Phys. Lett. B382 (1996) 403–408, [hep-ph/9512370]

    work page Pith review arXiv 1996

  3. [3]

    F. V. Tkachov,Measuring multi - jet structure of hadronic energy flow or What is a jet?, Int. J. Mod. Phys. A12(1997) 5411–5529, [hep-ph/9601308]

    work page Pith review arXiv 1997

  4. [4]

    G. P. Korchemsky and G. F. Sterman,Power corrections to event shapes and factorization, Nucl. Phys. B555(1999) 335–351, [hep-ph/9902341]

    work page Pith review arXiv 1999

  5. [5]

    D. M. Hofman and J. Maldacena,Conformal collider physics: Energy and charge correlations,JHEP05(2008) 012, [arXiv:0803.1467]

    work page Pith review arXiv 2008

  6. [6]

    Kologlu, P

    M. Kologlu, P. Kravchuk, D. Simmons-Duffin, and A. Zhiboedov,The light-ray OPE and conformal colliders,JHEP01(2021) 128, [arXiv:1905.01311]. – 56 –

    work page arXiv 2021

  7. [7]

    Energy Correlators: A Journey From Theory to Experiment

    I. Moult and H. X. Zhu,Energy Correlators: A Journey From Theory to Experiment, arXiv:2506.09119

    work page Pith review arXiv

  8. [8]

    C. L. Basham, L. S. Brown, S. D. Ellis, and S. T. Love,Energy Correlations in electron - Positron Annihilation: Testing QCD,Phys. Rev. Lett.41(1978) 1585

    1978

Show all 104 references
  1. [9]

    C. L. Basham, L. S. Brown, S. D. Ellis, and S. T. Love,Energy Correlations in electron-Positron Annihilation in Quantum Chromodynamics: Asymptotically Free Perturbation Theory,Phys. Rev. D19(1979) 2018

    1979

  2. [10]

    C. L. Basham, L. S. Brown, S. D. Ellis, and S. T. Love,Energy Correlations in Perturbative Quantum Chromodynamics: A Conjecture for All Orders,Phys. Lett. B85(1979) 297–299

    1979

  3. [11]

    C. L. Basham, L. S. Brown, S. D. Ellis, and S. T. Love,Electron - Positron Annihilation Energy Pattern in Quantum Chromodynamics: Asymptotically Free Perturbation Theory, Phys. Rev. D17(1978) 2298

    1978

  4. [12]

    A. V. Belitsky, S. Hohenegger, G. P. Korchemsky, E. Sokatchev, and A. Zhiboedov, Energy-Energy Correlations in N=4 Supersymmetric Yang-Mills Theory,Phys. Rev. Lett. 112(2014), no. 7 071601, [arXiv:1311.6800]

    2014 arXiv

  5. [13]

    J. M. Henn, E. Sokatchev, K. Yan, and A. Zhiboedov,Energy-energy correlation inN=4 super Yang-Mills theory at next-to-next-to-leading order,Phys. Rev. D100(2019), no. 3 036010, [arXiv:1903.05314]

    2019

  6. [14]

    L. J. Dixon, M.-X. Luo, V. Shtabovenko, T.-Z. Yang, and H. X. Zhu,Analytical Computation of Energy-Energy Correlation at Next-to-Leading Order in QCD,Phys. Rev. Lett.120(2018), no. 10 102001, [arXiv:1801.03219]

    2018 arXiv

  7. [15]

    M.-X. Luo, V. Shtabovenko, T.-Z. Yang, and H. X. Zhu,Analytic Next-To-Leading Order Calculation of Energy-Energy Correlation in Gluon-Initiated Higgs Decays,JHEP06(2019) 037, [arXiv:1903.07277]

    2019 arXiv

  8. [16]

    J. Gao, V. Shtabovenko, and T.-Z. Yang,Energy-energy correlation in hadronic Higgs decays: analytic results and phenomenology at NLO,JHEP02(2021) 210, [arXiv:2012.14188]. [17]ALEPHCollaboration, D. Decamp et al.,Measurement of alpha-s from the structure of particle clusters pr...

    2021

  9. [17]

    Bossi, A

    H. Bossi, A. Baty, Y. Chen, Y.-C. Chen, G.-M. Innocenti, M. Maggi, C. McGinn, and Y.-J. Lee,Measurement of the energy-energy correlator in the back-to-back limit using the archived ALEPH e+e- data at 91.2 GeV,PoSLHCP2024(2025) 228, [arXiv:2501.01968]

    2025

  10. [18]

    Bossi, Y.-C

    H. Bossi, Y.-C. Chen, Y. Chen, J. Zhang, G. M. Innocenti, A. Badea, A. Baty, M. Maggi, C. McGinn, and Y.-J. Lee,Analysis note: measurement of energy-energy correlator ine+e− collisions at91GeV with archived ALEPH data,arXiv:2505.11828. [21]Electron-Positron AllianceCollaborati...

  11. [19]

    Zhang, T.-A

    J. Zhang, T.-A. Sheng, Y.-C. Chen, H. Bossi, A. Badea, A. Baty, C. McGinn, Y.-J. Lee, and Y. Chen,Analysis note: measurement of thrust and track energy-energy correlator in e+e- collisions at 91.2 GeV with DELPHI open data,arXiv:2510.18762. – 57 –

  12. [20]

    L. J. Dixon, I. Moult, and H. X. Zhu,Collinear limit of the energy-energy correlator,Phys. Rev. D100(2019), no. 1 014009, [arXiv:1905.01310]

    2019 arXiv

  13. [21]

    G. P. Korchemsky,Energy correlations in the end-point region,JHEP01(2020) 008, [arXiv:1905.01444]

    2020

  14. [22]

    Yan and X

    K. Yan and X. Zhang,Three-Point Energy Correlator in N=4 Supersymmetric Yang-Mills Theory,Phys. Rev. Lett.129(2022), no. 2 021602, [arXiv:2203.04349]

    2022

  15. [23]

    Chen, M.-X

    H. Chen, M.-X. Luo, I. Moult, T.-Z. Yang, X. Zhang, and H. X. Zhu,Three point energy correlators in the collinear limit: symmetries, dualities and analytic results,JHEP08 (2020) 028, [arXiv:1912.11050]

    2020

  16. [24]

    Yang and X

    T.-Z. Yang and X. Zhang,Analytic Computation of three-point energy correlator in QCD, JHEP09(2022) 006, [arXiv:2208.01051]

    2022

  17. [25]

    Yang and X

    T.-Z. Yang and X. Zhang,Three-point energy correlators in hadronic Higgs boson decays, Phys. Rev. D109(2024), no. 11 114036, [arXiv:2402.05174]

    2024

  18. [26]

    Chicherin, I

    D. Chicherin, I. Moult, E. Sokatchev, K. Yan, and Y. Zhu,Collinear limit of the four-point energy correlator in N=4 supersymmetric Yang-Mills theory,Phys. Rev. D110(2024), no. 9 L091901, [arXiv:2401.06463]

    2024

  19. [27]

    S. He, X. Jiang, Q. Yang, and Y.-Q. Zhang,From squared amplitudes to energy correlators, arXiv:2408.04222

  20. [28]

    H. Chen, I. Moult, X. Zhang, and H. X. Zhu,Rethinking jets with energy correlators: Tracks, resummation, and analytic continuation,Phys. Rev. D102(2020), no. 5 054012, [arXiv:2004.11381]

    2020

  21. [29]

    P. T. Komiske, I. Moult, J. Thaler, and H. X. Zhu,Analyzing N-Point Energy Correlators inside Jets with CMS Open Data,Phys. Rev. Lett.130(2023), no. 5 051901, [arXiv:2201.07800]

    2023

  22. [30]

    Lee and I

    K. Lee and I. W. Stewart,Dihadron Fragmentation and the Confinement Transition in Energy Correlators,Phys. Rev. Lett.136(2026), no. 8 081902, [arXiv:2507.11495]

    2026

  23. [31]

    Chang, H

    C.-H. Chang, H. Chen, X. Liu, D. Simmons-Duffin, F. Yuan, and H. X. Zhu,Quantum Scaling in Energy Correlators beyond the Confinement Transition,Phys. Rev. Lett.136 (2026), no. 8 081903, [arXiv:2507.15923]

    2026

  24. [32]

    Z.-B. Kang, A. Metz, D. Pitonyak, and C. Zhang,Dihadron Fragmentation Framework for Near-Side Energy-Energy Correlators,Phys. Rev. Lett.136(2026), no. 8 081905, [arXiv:2507.17444]

    2026

  25. [33]

    K. Lee, B. Meçaj, and I. Moult,Conformal collider physics meets LHC data,Phys. Rev. D 111(2025), no. 1 L011502, [arXiv:2205.03414]

    2025

  26. [34]

    W. Chen, J. Gao, Y. Li, Z. Xu, X. Zhang, and H. X. Zhu,NNLL resummation for projected three-point energy correlator,JHEP05(2024) 043, [arXiv:2307.07510]. [38]Flavour Lattice A veraging Group (FLAG)Collaboration, Y. Aoki et al.,FLAG review 2024,Phys. Rev. D113(2026), no. 1 0145...

    2024

  27. [35]

    Abbate, M

    R. Abbate, M. Fickinger, A. H. Hoang, V. Mateu, and I. W. Stewart,Thrust atN3LLwith Power Corrections and a Precision Global Fit forαs(mZ),Phys. Rev. D83(2011) 074021, [arXiv:1006.3080]. – 58 –

    2011 arXiv

  28. [36]

    Becher and M

    T. Becher and M. D. Schwartz,A precise determination ofαs from LEP thrust data using effective field theory,JHEP07(2008) 034, [arXiv:0803.0342]

    2008 arXiv

  29. [37]

    A. H. Hoang, D. W. Kolodrubetz, V. Mateu, and I. W. Stewart,Precise determination of αs from theC-parameter distribution,Phys. Rev. D91(2015), no. 9 094018, [arXiv:1501.04111]

    2015 arXiv

  30. [38]

    A. H. Hoang, D. W. Kolodrubetz, V. Mateu, and I. W. Stewart,C-parameter distribution at N3LL’ including power corrections,Phys. Rev. D91(2015), no. 9 094017, [arXiv:1411.6633]

    2015 arXiv

  31. [39]

    M. A. Benitez, A. H. Hoang, V. Mateu, I. W. Stewart, and G. Vita,On determining αs(mZ) from dijets in e+e− thrust,JHEP07(2025) 249, [arXiv:2412.15164]

    2025

  32. [40]

    M. A. Benitez, A. Bhattacharya, A. H. Hoang, V. Mateu, M. D. Schwartz, I. W. Stewart, and X. Zhang,A Precise Determination ofαs from the Heavy Jet Mass Distribution, arXiv:2502.12253

    arXiv

  33. [41]

    Huston, K

    J. Huston, K. Rabbertz, and G. Zanderighi,Quantum Chromodynamics, arXiv:2312.14015

  34. [42]

    d’Enterria et al.,The strong coupling constant: state of the art and the decade ahead,J

    D. d’Enterria et al.,The strong coupling constant: state of the art and the decade ahead,J. Phys. G51(2024), no. 9 090501, [arXiv:2203.08271]. [47]CMSCollaboration, A. Hayrapetyan et al.,Measurement of Energy Correlators inside Jets and Determination of the Strong CouplingαS(m...

    2024

  35. [43]

    H. Chen, P. F. Monni, Z. Xu, and H. X. Zhu,Perturbative evolution of hadronization effects in energy correlators,talk by Hao Chen at SCET 2024(2024)

    2024

  36. [44]

    K. Lee, A. Pathak, I. W. Stewart, and Z. Sun,Nonperturbative Effects in Energy Correlators: From Characterizing Confinement Transition to Improvingαs Extraction, Phys. Rev. Lett.133(2024), no. 23 231902, [arXiv:2405.19396]

    2024

  37. [45]

    H. Chen, P. F. Monni, Z. Xu, and H. X. Zhu,Scaling Violation in Power Corrections to Energy Correlators from the Light-Ray Operator Product Expansion,Phys. Rev. Lett.133 (2024), no. 23 231901, [arXiv:2406.06668]

    2024

  38. [46]

    Lee and G

    C. Lee and G. F. Sterman,Universality of nonperturbative effects in event shapes,eConf C0601121(2006) A001, [hep-ph/0603066]

    2006 arXiv

  39. [47]

    Budhraja, H

    A. Budhraja, H. Chen, and W. J. Waalewijn,ν-point energy correletors with FastEEC: Small-x physics from LHC jets,Phys. Lett. B861(2025) 139239, [arXiv:2409.12235]

    2025

  40. [48]

    Somogyi, Z

    G. Somogyi, Z. Trocsanyi, and V. Del Duca,A Subtraction scheme for computing QCD jet cross sections at NNLO: Regularization of doubly-real emissions,JHEP01(2007) 070, [hep-ph/0609042]

    2007 arXiv

  41. [49]

    Somogyi and Z

    G. Somogyi and Z. Trocsanyi,A Subtraction scheme for computing QCD jet cross sections at NNLO: Regularization of real-virtual emission,JHEP01(2007) 052, [hep-ph/0609043]. – 59 –

    2007 arXiv

  42. [50]

    Aglietti, V

    U. Aglietti, V. Del Duca, C. Duhr, G. Somogyi, and Z. Trocsanyi,Analytic integration of real-virtual counterterms in NNLO jet cross sections. I.,JHEP09(2008) 107, [arXiv:0807.0514]

    2008 arXiv

  43. [51]

    Catani and M

    S. Catani and M. H. Seymour,The Dipole formalism for the calculation of QCD jet cross-sections at next-to-leading order,Phys. Lett. B378(1996) 287–301, [hep-ph/9602277]

    1996 arXiv

  44. [52]

    Catani and M

    S. Catani and M. H. Seymour,A General algorithm for calculating jet cross-sections in NLO QCD,Nucl. Phys. B485(1997) 291–419, [hep-ph/9605323]. [Erratum: Nucl.Phys.B 510, 503–504 (1998)]

    1997 arXiv

  45. [53]

    Gehrmann-De Ridder, T

    A. Gehrmann-De Ridder, T. Gehrmann, E. W. N. Glover, and G. Heinrich,EERAD3: Event shapes and jet rates in electron-positron annihilation at orderα3 s,Comput. Phys. Commun.185(2014) 3331, [arXiv:1402.4140]

    2014 arXiv

  46. [54]

    B. C. Aveleira, A. Gehrmann-De Ridder, T. Gehrmann, N. Glover, G. Heinrich, and C. T. Preuss,EERAD3 version 2: QCD corrections in hadronic colour-singlet decays,SciPost Phys. Codeb.59(2025) 1, [arXiv:2503.20610]

    2025

  47. [55]

    P. J. Rijken and W. L. van Neerven,O (alpha-s**2) contributions to the longitudinal fragmentation function in e+ e- annihilation,Phys. Lett. B386(1996) 422–428, [hep-ph/9604436]

    1996 arXiv

  48. [56]

    P. J. Rijken and W. L. van Neerven,Higher order QCD corrections to the transverse and longitudinal fragmentation functions in electron - positron annihilation,Nucl. Phys. B487 (1997) 233–282, [hep-ph/9609377]

    1997 arXiv

  49. [57]

    P. J. Rijken and W. L. van Neerven,O (alpha-s**2) contributions to the asymmetric fragmentation function in e+ e- annihilation,Phys. Lett. B392(1997) 207–215, [hep-ph/9609379]

    1997 arXiv

  50. [58]

    Mitov and S.-O

    A. Mitov and S.-O. Moch,QCD Corrections to Semi-Inclusive Hadron Production in Electron-Positron Annihilation at Two Loops,Nucl. Phys. B751(2006) 18–52, [hep-ph/0604160]

    2006 arXiv

  51. [59]

    Mitov, S

    A. Mitov, S. Moch, and A. Vogt,Next-to-Next-to-Leading Order Evolution of Non-Singlet Fragmentation Functions,Phys. Lett. B638(2006) 61–67, [hep-ph/0604053]

    2006 arXiv

  52. [60]

    Moch and A

    S. Moch and A. Vogt,On third-order timelike splitting functions and top-mediated Higgs decay into hadrons,Phys. Lett. B659(2008) 290–296, [arXiv:0709.3899]

    2008 arXiv

  53. [61]

    A. A. Almasy, S. Moch, and A. Vogt,On the Next-to-Next-to-Leading Order Evolution of Flavour-Singlet Fragmentation Functions,Nucl. Phys. B854(2012) 133–152, [arXiv:1107.2263]

    2012 arXiv

  54. [62]

    Chen, T.-Z

    H. Chen, T.-Z. Yang, H. X. Zhu, and Y. J. Zhu,Analytic Continuation and Reciprocity Relation for Collinear Splitting in QCD,Chin. Phys. C45(2021), no. 4 043101, [arXiv:2006.10534]

    2021

  55. [63]

    Gehrmann, A

    T. Gehrmann, A. von Manteuffel, V. Sotnikov, and T.-Z. Yang,CompleteN2 f contributions to four-loop pure-singlet splitting functions,JHEP01(2024) 029, [arXiv:2308.07958]

    2024

  56. [64]

    Falcioni, F

    G. Falcioni, F. Herzog, S. Moch, and A. Vogt,Four-loop splitting functions in QCD – The quark-quark case,Phys. Lett. B842(2023) 137944, [arXiv:2302.07593]

    2023

  57. [65]

    Falcioni, F

    G. Falcioni, F. Herzog, S. Moch, and A. Vogt,Four-loop splitting functions in QCD – The gluon-to-quark case,Phys. Lett. B846(2023) 138215, [arXiv:2307.04158]. – 60 –

    2023

  58. [66]

    Falcioni, F

    G. Falcioni, F. Herzog, S. Moch, A. Pelloni, and A. Vogt,Four-loop splitting functions in QCD – The quark-to-gluon case,Phys. Lett. B856(2024) 138906, [arXiv:2404.09701]

    2024

  59. [67]

    Falcioni, F

    G. Falcioni, F. Herzog, S. Moch, A. Pelloni, and A. Vogt,Four-loop splitting functions in QCD – the gluon-gluon case –,Phys. Lett. B860(2025) 139194, [arXiv:2410.08089]

    2025

  60. [68]

    Falcioni, F

    G. Falcioni, F. Herzog, S. Moch, A. Pelloni, and A. Vogt,Additional results on the four-loop flavour-singlet splitting functions in QCD,Phys. Lett. B875(2026) 140278, [arXiv:2512.10783]

    2026

  61. [69]

    Nogueira,Automatic feynman graph generation,Journal of Computational Physics105 (1993), no

    P. Nogueira,Automatic feynman graph generation,Journal of Computational Physics105 (1993), no. 2 279–289

    1993

  62. [70]

    J. A. M. Vermaseren,New features of FORM,math-ph/0010025

    arXiv

  63. [71]

    Kuipers, T

    J. Kuipers, T. Ueda, J. A. M. Vermaseren, and J. Vollinga,FORM version 4.0,Comput. Phys. Commun.184(2013) 1453–1467, [arXiv:1203.6543]

    2013 arXiv

  64. [72]

    Ruijl, T

    B. Ruijl, T. Ueda, and J. Vermaseren,FORM version 4.2,arXiv:1707.06453

    arXiv

  65. [73]

    van Ritbergen, A

    T. van Ritbergen, A. N. Schellekens, and J. A. M. Vermaseren,Group theory factors for Feynman diagrams,Int. J. Mod. Phys. A14(1999) 41–96, [hep-ph/9802376]

    1999 arXiv

  66. [74]

    Mertig, M

    R. Mertig, M. Bohm, and A. Denner,FEYN CALC: Computer algebraic calculation of Feynman amplitudes,Comput. Phys. Commun.64(1991) 345–359

    1991

  67. [75]

    Shtabovenko, R

    V. Shtabovenko, R. Mertig, and F. Orellana,New Developments in FeynCalc 9.0,Comput. Phys. Commun.207(2016) 432–444, [arXiv:1601.01167]

    2016 arXiv

  68. [76]

    Shtabovenko, R

    V. Shtabovenko, R. Mertig, and F. Orellana,FeynCalc 9.3: New features and improvements,Comput. Phys. Commun.256(2020) 107478, [arXiv:2001.04407]

    2020 arXiv

  69. [77]

    Shtabovenko, R

    V. Shtabovenko, R. Mertig, and F. Orellana,FeynCalc 10: Do multiloop integrals dream of computer codes?,arXiv:2312.14089

  70. [78]

    R. N. Lee,Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685

    arXiv

  71. [79]

    R. N. Lee,LiteRed 1.4: a powerful tool for reduction of multiloop integrals,J. Phys. Conf. Ser.523(2014) 012059, [arXiv:1310.1145]

    2014 arXiv

  72. [80]

    A. V. Smirnov and F. S. Chukharev,FIRE6: Feynman Integral REduction with modular arithmetic,Comput. Phys. Commun.247(2020) 106877, [arXiv:1901.07808]

    2020

  73. [81]

    Meyer,Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA,Comput

    C. Meyer,Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA,Comput. Phys. Commun.222(2018) 295–312, [arXiv:1705.06252]

    2018 arXiv

  74. [82]

    R. N. Lee,Reducing differential equations for multiloop master integrals,JHEP04(2015) 108, [arXiv:1411.0911]

    2015 arXiv

  75. [83]

    R. N. Lee,Libra: A package for transformation of differential systems for multiloop integrals,Comput. Phys. Commun.267(2021) 108058, [arXiv:2012.00279]

    2021

  76. [84]

    J. M. Campbell and E. W. N. Glover,Double unresolved approximations to multiparton scattering amplitudes,Nucl. Phys. B527(1998) 264–288, [hep-ph/9710255]

    1998 arXiv

  77. [85]

    Catani and M

    S. Catani and M. Grazzini,Collinear factorization and splitting functions for next-to-next-to-leading order QCD calculations,Phys. Lett. B446(1999) 143–152, [hep-ph/9810389]. – 61 –

    1999 arXiv

  78. [86]

    Ritzmann and W

    M. Ritzmann and W. J. Waalewijn,Fragmentation in Jets at NNLO,Phys. Rev. D90 (2014), no. 5 054029, [arXiv:1407.3272]

    2014 arXiv

  79. [87]

    J. Gong, A. Pokraka, K. Yan, and X. Zhang,Toward the Analytic Bootstrap of Energy Correlators,arXiv:2509.22782

  80. [88]

    Hahn,CUBA: A Library for multidimensional numerical integration,Comput

    T. Hahn,CUBA: A Library for multidimensional numerical integration,Comput. Phys. Commun.168(2005) 78–95, [hep-ph/0404043]

    2005 arXiv

  81. [89]

    S. T. Schindler, I. W. Stewart, and Z. Sun,Renormalons in the energy-energy correlator, JHEP10(2023) 187, [arXiv:2305.19311]. [Erratum: JHEP 10, 175 (2024)]

    2023

  82. [90]

    A. Gao, K. Lee, and X. Zhang,Precision Jet Substructure of Boosted Boson Decays with Energy Correlators,arXiv:2601.20933

  83. [91]

    Holguin, I

    J. Holguin, I. Moult, A. Pathak, M. Procura, and S. Sule,High precision heavy-boson-jet substructure with energy correlators,arXiv:2601.20923

  84. [92]

    Bierlich et al.,A comprehensive guide to the physics and usage of PYTHIA 8.3,SciPost Phys

    C. Bierlich et al.,A comprehensive guide to the physics and usage of PYTHIA 8.3,SciPost Phys. Codeb.2022(2022) 8, [arXiv:2203.11601]

    2022 arXiv

  85. [93]

    Bellm et al.,Herwig 7.0/Herwig++ 3.0 release note,Eur

    J. Bellm et al.,Herwig 7.0/Herwig++ 3.0 release note,Eur. Phys. J. C76(2016), no. 4 196, [arXiv:1512.01178]

    2016 arXiv

  86. [94]

    Nachtmann,Positivity constraints for anomalous dimensions,Nucl

    O. Nachtmann,Positivity constraints for anomalous dimensions,Nucl. Phys. B63(1973) 237–247

    1973

  87. [95]

    F. J. Tackmann,Beyond scale variations: perturbative theory uncertainties from nuisance parameters,JHEP08(2025) 098, [arXiv:2411.18606]. [103]ALEPH, DELPHI, L3, OP AL, LEP ElectroweakCollaboration, S. Schael et al., Electroweak Measurements in Electron-Positron Collisions at W...

    2025

  88. [96]

    Z.-B. Kang, F. Ringer, and I. Vitev,The semi-inclusive jet function in SCET and small radius resummation for inclusive jet production,JHEP10(2016) 125, [arXiv:1606.06732]

    2016 arXiv

  89. [97]

    K. Lee, I. Moult, and X. Zhang,Revisiting single inclusive jet production: timelike factorization and reciprocity,JHEP05(2025) 129, [arXiv:2409.19045]

    2025

  90. [98]

    Generet, K

    T. Generet, K. Lee, I. Moult, R. Poncelet, and X. Zhang,Small radius inclusive jet production at the LHC through NNLO+NNLL,JHEP08(2025) 015, [arXiv:2503.21866]

    2025

  91. [99]

    Chang, M

    H.-M. Chang, M. Procura, J. Thaler, and W. J. Waalewijn,Calculating Track-Based Observables for the LHC,Phys. Rev. Lett.111(2013) 102002, [arXiv:1303.6637]

    2013 arXiv

  92. [100]

    Y. Li, I. Moult, S. S. van Velzen, W. J. Waalewijn, and H. X. Zhu,Extending Precision Perturbative QCD with Track Functions,Phys. Rev. Lett.128(2022), no. 18 182001, [arXiv:2108.01674]

    2022

  93. [101]

    H. Chen, M. Jaarsma, Y. Li, I. Moult, W. J. Waalewijn, and H. X. Zhu,Collinear parton dynamics beyond Dokshitzer-Gribov-Lipatov-Altarelli-Parisi framework,Phys. Rev. D111 (2025), no. 7 076021, [arXiv:2210.10061]

    2025

  94. [102]

    Jaarsma, Y

    M. Jaarsma, Y. Li, I. Moult, W. J. Waalewijn, and H. X. Zhu,Energy correlators on tracks: resummation and non-perturbative effects,JHEP12(2023) 087, [arXiv:2307.15739]

    2023

  95. [103]

    Lee and I

    K. Lee and I. Moult,Energy Correlators Taking Charge,arXiv:2308.00746. – 62 –

  96. [104]

    Lee and I

    K. Lee and I. Moult,Joint Track Functions: Expanding the Space of Calculable Correlations at Colliders,arXiv:2308.01332. – 63 –

Pith tools

Reviewed June 28, 2026 · model on record in the stance chip above.