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Low-energy theory of jet processes and PDF factorization

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A three-loop Glauber contribution to low-energy matrix elements turns the double-logarithmic super-leading evolution into single-log DGLAP running, reconciling collinear factorization violation with PDF factorization.

desk verdict A careful three-loop consistency check that Glauber exchanges turn super-leading double logs into DGLAP below Q0—convincing where it computes, but the all-order decoupling of the soft-collinear mode remains an open assumption the authors openly flag. read the letter →

arxiv 2509.07082 v1 pith:ZYK5VFGR submitted 2025-09-08 hep-ph

Thomas Becher , Patrick Hager , Sebastian Jaskiewicz , Matthias Neubert , Dominik Schwienbacher This is my paper
classification hep-ph
keywords PDFfactorizationcollinearviolationsuper-leadinglogarithmsGlaubergluonsnon-globalobservablesDGLAPevolutionsoft-collineareffectivetheorythree-loopconsistency
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 tries to resolve a long-standing puzzle in hadron-collider QCD: amplitudes with space-like collinear splittings violate collinear factorization, yet parton distribution functions (PDFs) are routinely factorized in cross-section predictions. It argues that both statements are compatible, provided a perturbative Glauber contribution is included in the low-energy matrix elements. At three loops, the paper computes that contribution explicitly and shows it exactly reproduces the pole structure, hard logarithms, and momentum-fraction dependence demanded by the hard anomalous dimensions. The result converts the double-logarithmic super-leading evolution above the veto scale into ordinary single-logarithmic DGLAP evolution below it. If correct, PDF factorization for non-global jet observables is restored while the physical super-leading logarithms remain as an important high-scale effect.

What carries the argument

The central object is the perturbative Glauber contribution to the low-energy soft-collinear matrix elements, mediated by a genuine Glauber gluon mode with scaling k ~ (λ, λ^2, λ) connecting soft and collinear fields. In the consistency relation, the terms proportional to the commutator [Γ^C, V^G Γ] and the cusp term Γ_c V^G Γ cannot be produced by soft or collinear physics alone; the Glauber diagram reproduces them exactly. This mechanism replaces the color-aware collinear evolution above the veto scale with the color-diagonal DGLAP kernels below it.

What would settle it

Perform a four-loop computation of the low-energy matrix elements including the soft-collinear region: if the mode k ~ (λ, λ^2, λ^{3/2}) yields a non-vanishing contribution after the collinear phase-space integrals, the consistency relation would fail and PDF factorization would be compromised.

Watch

Extended reading notes

Core claim

The central claim is that the low-energy matrix elements of the factorization theorem for gap-between-jets cross sections contain a genuine, regulator-free Glauber contribution at three loops that couples the soft and collinear sectors. This contribution is not contained in the purely soft or purely collinear matrix elements; it breaks soft-collinear factorization at the level of the cross section. Yet when inserted into the consistency relation derived from renormalization-group invariance, its pole structure, hard logarithms, and momentum-fraction dependence exactly match the terms needed to turn the color-aware double-logarithmic running into color-diagonal DGLAP running below the veto sc

Load-bearing premise

The entire proof depends on the claim that the soft-collinear mode with scaling k ~ (λ, λ^2, λ^{3/2}) always vanishes at higher orders because the collinear emission it would couple to remains scaleless; the paper states this has not been rigorously demonstrated to all orders, even for inclusive Drell-Yan.

Editorial extensions

If this is right

  • Below the veto scale Q0, the evolution of the hard functions is ordinary DGLAP, so standard PDF evolution and the effective-theory low-energy evolution are reconciled.
  • The super-leading logarithms above Q0 are physical and arise from collinear factorization violation; they survive in the resummed cross section as an important numerical effect.
  • Below Q0 the low-energy theory matches the inclusive Drell-Yan theory, so the established Drell-Yan factorization arguments can be extended to a wider class of jet observables.
  • Beyond two loops, hard functions with incoming gluons must carry open Lorentz indices because Glauber interactions induce non-trivial spin structures; spin-averaged hard functions are no longer sufficient.
  • The higher-order splitting functions must have a more general spin dependence than the standard ones, matching new structures in the anomalous dimensions.

Reading between the lines

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

  • The paper leaves open an all-order proof that the soft-collinear mode with scaling k ~ (λ, λ^2, λ^{3/2}) always decouples; if that mode ever contributes after the collinear integrals, the consistency argument would fail.
  • The same three-loop Glauber mechanism likely applies to global observables such as N-jettiness, where it would appear as coherence-violating logarithms rather than super-leading ones; the paper identifies this as a future direction but does not establish it.
  • A testable extension would be to compute the four-loop low-energy matrix elements including the soft-collinear region: a non-vanishing contribution there would signal genuine PDF factorization breaking.
  • The new Lorentz structures suggest that high-precision resummation for gluon-initiated processes needs a color-spin operator basis beyond the dipole-based approaches currently used.
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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

3 major / 5 minor

Summary. The paper presents a low-energy effective field theory treatment of gap-between-jets cross sections and argues that PDF factorization is restored below the veto scale Q0. Starting from a factorization theorem with hard functions H_m and soft-collinear matrix elements W_m, the authors derive consistency relations from renormalization-group invariance that constrain the pole structure of the perturbative matching coefficients I_m. They verify the one-loop anomalous dimensions using massive IR regulators, and then compute the three-loop leading-pole structure of I_m from explicit SCET diagrams, including genuine Glauber contributions. The central claim is that the double-logarithmic super-leading-logarithm evolution above Q0 becomes single-logarithmic DGLAP evolution below Q0, thereby reconciling collinear factorization violation with PDF factorization. The paper also identifies new Lorentz structures in gluonic initial-state channels at subleading poles, requiring a generalized factorization formula with open Lorentz indices.

Significance. If the central claim holds, this is a substantial step toward resolving a long-standing question: how collinear factorization violation is compatible with the standard DGLAP-based PDF factorization used in precision hadron collider physics. The paper provides an explicit, detailed three-loop computation with full color and spin structures, a well-defined Glauber mode, and a nontrivial commutator structure that converts non-DGLAP color-aware evolution into standard DGLAP evolution. The identification of the Glauber mechanism and the concrete pole-level verification are valuable and go beyond earlier two-loop or heuristic arguments. However, the result is a consistency check at a fixed perturbative order, not an all-order proof, and the paper itself concedes the key all-order decoupling assumption in Section 7.1. The significance is therefore high if the result is regarded as strong evidence; the manuscript should be careful not to overstate the status of the claim.

major comments (3)
  1. [Section 7.1 and Eq. (2.20)] The central factorization W_m = I_m * f1 f2, Eq. (2.20), is verified only through the leading poles at three loops. The paper's own Section 7.1 states that a rigorous all-order demonstration that the soft-collinear mode k_sc ~ (λ, λ^2, λ^{3/2}) decouples 'has not been performed yet, even for the inclusive Drell-Yan case.' If this mode contributes, its virtuality Q0^3/Q lies below Q0^2 and would introduce factorization-breaking physics not captured by the soft, collinear, and non-perturbative Glauber modes. The three-loop pole check is thus necessary but not sufficient for the claim that PDF factorization is restored. The manuscript should either supply an all-order argument or explicitly frame the result as evidence conditional on this unproven decoupling, rather than as a demonstration of compatibility.
  2. [Section 6.4, Eqs. (6.28), (6.29), (F.1)] The comparison with the consistency relation (6.1) is made only for the leading 1/ε^3 poles. The computation in Eq. (6.29) and the analogous gluon result (F.1) show that subleading poles contain new Lorentz structures, such as n_{j,⊥}^μ n_{j,⊥}^{\bar μ}, that are not matched by the hard anomalous dimensions presented in Section 2.3. These structures are not needed for the super-leading-logarithm claim, but they mean that the three-loop verification is incomplete for the full low-energy matrix element. The paper should state this limitation explicitly, and ideally show whether these subleading structures are consistent with the generalized factorization framework or whether they require modifications of the hard anomalous dimensions beyond leading power.
  3. [Section 3, Eq. (3.10)] The predicted pole structure in Eq. (3.10) is derived from the assumed factorization theorem and from the hard anomalous dimensions. The independent SCET computation of I_m is a meaningful cross-check, but the consistency relation is internal to the framework. The paper sometimes phrases the three-loop agreement as 'demonstrating' PDF factorization, whereas it actually verifies a necessary condition under the assumed form of the factorization theorem. This distinction should be maintained throughout the abstract, introduction, and conclusion to avoid overclaiming.
minor comments (5)
  1. [Table 1] In the definition of I_m, 'sofct-collinear' appears to be a typo for 'soft-collinear'.
  2. [Appendix B] In the first sentence of Appendix B, 'we we calculate' should be 'we calculate'.
  3. [Page 1 heading] The contents line reads 'A Zfactor at three loops'; should be 'A Z factor at three loops'.
  4. [Section 2.2, Eq. (2.19)] The generalized Mellin convolution notation is introduced compactly. A short example or a reference to an equation using it in an explicit convolution would improve readability.
  5. [Section 4.2, Eq. (4.12)] The beam function definition uses δ(\bar n_i ·(k−(1−z)p)), but the argument of the delta function is ambiguous with the subsequent text; clarify whether the factor (1−z) multiplies p or the full momentum.

Circularity Check

1 steps flagged · score 4.0 of 10

Load-bearing self-citation for the genuine Glauber mode; the three-loop pole check is otherwise an independent cross-check, not a tautology.

  1. self citation load bearing [Section 6.3, p. 35 (soft-collinear matrix elements analysis)]
    "In [24] we showed, using a detailed method-of-regions analysis, that such a hidden, genuine Glauber mode with momentum k indeed appears, connecting the soft and (anti-)collinear sectors, with its components scaling as k ∼ Q(λ^2, λ, λ)."

    The central premise that a genuine Glauber mode is the missing ingredient, and that the competing soft-collinear mode k_sc ∼ (λ, λ^2, λ^{3/2}) vanishes, is imported from the authors' own previous paper [24]. Section 6.4 then computes SCET diagrams built on this self-cited region analysis; the paper does not rederive the method-of-regions identification here. The three-loop pole check therefore verifies that the low-energy theory assumed on the basis of [24] is consistent with the hard anomalous dimensions, but the completeness of the Glauber-region description rests on a self-citation rather than on an independent derivation in the present work. Section 7.1 also concedes that the all-order absence of soft-collinear modes has not been rigorously shown even for Drell-Yan, so the load-bearing

full rationale

The main consistency check is not circular in the statistical sense: Eq. (3.10) is derived from the assumed PDF factorization W = I * f1 f2 and the hard anomalous dimensions, but the low-energy matrix elements in Sections 5 and 6 are then evaluated by explicit Feynman-diagram computations using soft currents, splitting functions, and the Glauber Lagrangian. Matching the predicted poles is a genuine, nontrivial cross-check: the soft-only terms, the VG Γ terms, and the [Γ^C, V^G Γ] commutator terms have different color, spin, and momentum-fraction structures, and the computation could have failed. No parameter is fitted to the prediction, and no known result is merely renamed. The reason the score is not 0 is that one load-bearing element — the existence and uniqueness of the genuine Glauber region and the vanishing of the soft-collinear mode — is taken from the authors' own [24] rather than rederived here. That self-citation is central, but it is not the whole derivation: the explicit three-loop diagrams and the off-diagonal DGLAP-channel verification are new independent content. Section 7.1 candidly states the all-order demonstration is missing; this is a limitation and an extrapolation risk, but it is not itself a circular reduction. Weighing all passages, the central claim 'perturbative Glauber contributions restore PDF factorization below Q0' is supported by a real three-loop consistency check, with one load-bearing self-cited region analysis, so a moderate score of 4 is appropriate.

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

No free parameters or invented entities are introduced. The central new input is the identification of the hidden Glauber region and the assumed all-order decoupling of a soft-collinear mode; both are analysis results, not fitted constants. The paper's predictive content is the exact matching of the three-loop pole structure.

assumptions (5)
  • domain assumption Factorization theorem (2.1)/(2.20) for gap-between-jets cross sections, with hard functions carrying explicit open color and Lorentz indices.
    The consistency relations (3.10) are derived from the anomalous dimensions of this factorization theorem; its validity is assumed and amended relative to [20] for the new Lorentz structures.
  • domain assumption Scale-independence of the physical cross section together with the assumed form W = I (x) f1 f2 implies the pole structure (3.10).
    This is the renormalization condition used to predict the divergences of the low-energy matrix elements; it is the framework in which the verification is performed.
  • domain assumption A genuine Glauber mode exists as the only additional region, identified by method-of-regions and Landau equations; results for the pentagon analysis are taken from the authors' previous paper [24].
    All three-loop diagrams are reduced to the single Glauber diagram of Figure 12; the all-order generalization is not proven.
  • domain assumption Below Q0 the effective theory is inclusive and identical to the Drell-Yan case, so the CSS Glauber cancellation arguments [1,7] apply.
    Invoked in Section 7.1 to extend PDF factorization below the veto scale; the paper explicitly notes this is not yet rigorously established.
  • standard math Known perturbative ingredients: one-loop soft current [52], two-loop soft current [48,49], standard DGLAP splitting functions [2-4], and the SCET framework [26-29].
    Used as inputs in Sections 5 and 6; correctness is inherited from the cited literature.

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Cite this review

Pith. "Pith review of Low-energy theory of jet processes and PDF factorization." pith.science (2026). https://pith.science/paper/ZYK5VFGR

@misc{pith2026250907082,
  author       = {Pith},
  title        = {Pith review of: Low-energy theory of jet processes and PDF factorization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZYK5VFGR}},
  note         = {Machine review of arXiv:2509.07082}
}
read the original abstract

The consistency of collinear factorization violation with PDF factorization has been an outstanding challenge and subject of considerable debate. In this work we demonstrate their compatibility using a factorization theorem for non-global jet observables. Our analysis relies on consistency relations derived from renormalization conditions in effective field theory. We verify these relations through an explicit computation at three-loop order and show that the double-logarithmic evolution sourcing the super-leading logarithms reduces to single-logarithmic DGLAP running below the lowest perturbative scale. The crucial ingredient reconciling the two evolutions is a perturbative Glauber contribution to the low-energy matrix elements which breaks soft-collinear factorization at the cross section level but restores PDF factorization.

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Forward citations

Cited by 3 Pith papers

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

  1. geoSCET: Soft Theorems from Power Counting

    hep-th 2026-07 accept novelty 8.0 of 10

    geoSCET derives geometric soft theorems for scalar field theories directly from effective-field-theory power counting and proves they are exact to all orders in perturbation theory when no potential is present.

  2. Spacelike-Collinear Scattering by the Method of Regions

    hep-ph 2026-07 conditional novelty 8.0 of 10

    The kinematic factorisation-violating part of the two-loop spacelike-collinear splitting amplitude comes entirely from a single hidden region with soft and Glauber loop momenta.

  3. Collinear Factorization Violation and Reggeization

    hep-ph 2026-07 conditional novelty 7.0 of 10

    Collinear-factorization-violating contributions from a single Glauber gluon factor into collinear and soft subgraphs, and their leading rapidity logarithms exponentiate via the gluon Regge trajectory.

Reference graph

Works this paper leans on

66 extracted references · 13 canonical work pages · cited by 3 Pith papers

  1. [24]

    Becher, P

    T. Becher, P. Hager, S. Jaskiewicz, M. Neubert and D. Schwienbacher,Factorization Restoration through Glauber Gluons,Phys. Rev. Lett.134(2025) 061901 [2408.10308]

    arXiv 2025

  2. [1]

    Collins, D.E

    J.C. Collins, D.E. Soper and G.F. Sterman,Factorization for Short Distance Hadron - Hadron Scattering,Nucl. Phys. B261(1985) 104

    1985

  3. [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

    work page 1972

  4. [3]

    Dokshitzer,Calculation of the Structure Functions for Deep Inelastic Scattering and e+e− Annihilation by Perturbation Theory in Quantum Chromodynamics.,Sov

    Y.L. Dokshitzer,Calculation of the Structure Functions for Deep Inelastic Scattering and e+e− Annihilation by Perturbation Theory in Quantum Chromodynamics.,Sov. Phys. JETP 46(1977) 641

    work page 1977

  5. [4]

    Altarelli and G

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

    work page 1977

  6. [5]

    Gross and F

    D.J. Gross and F. Wilczek,Asymptotically free gauge theories. 2,Phys. Rev. D9(1974) 980

    1974

  7. [6]

    Bodwin, S.J

    G.T. Bodwin, S.J. Brodsky and G.P. Lepage,Initial State Interactions and the Drell-Yan Process,Phys. Rev. Lett.47(1981) 1799

    work page 1981

  8. [7]

    Collins, D.E

    J.C. Collins, D.E. Soper and G.F. Sterman,Soft Gluons and Factorization,Nucl. Phys. B 308(1988) 833

    work page 1988

Show all 66 references
  1. [8]

    Collins and J.-W

    J. Collins and J.-W. Qiu,k T factorization is violated in production of high-transverse-momentum particles in hadron-hadron collisions,Phys. Rev. D75(2007) 114014 [0705.2141]

    2007 arXiv

  2. [9]

    Gaunt,Glauber Gluons and Multiple Parton Interactions,JHEP07(2014) 110 [1405.2080]

    J.R. Gaunt,Glauber Gluons and Multiple Parton Interactions,JHEP07(2014) 110 [1405.2080]

    2014 arXiv

  3. [10]

    Zeng,Drell-Yan process with jet vetoes: breaking of generalized factorization,JHEP10 (2015) 189 [1507.01652]

    M. Zeng,Drell-Yan process with jet vetoes: breaking of generalized factorization,JHEP10 (2015) 189 [1507.01652]

    2015 arXiv

  4. [11]

    Catani, D

    S. Catani, D. de Florian and G. Rodrigo,Space-like (versus time-like) collinear limits in QCD: Is factorization violated?,JHEP07(2012) 026 [1112.4405]

    2012 arXiv

  5. [12]

    Forshaw, M.H

    J.R. Forshaw, M.H. Seymour and A. Siodmok,On the Breaking of Collinear Factorization in QCD,JHEP11(2012) 066 [1206.6363]

    2012 arXiv

  6. [13]

    Schwartz, K

    M.D. Schwartz, K. Yan and H.X. Zhu,Collinear factorization violation and effective field theory,Phys. Rev. D96(2017) 056005 [1703.08572]

    2017 arXiv

  7. [14]

    Cieri, P.K

    L. Cieri, P.K. Dhani and G. Rodrigo,Catani’s generalization of collinear factorization breaking,2402.14749

  8. [15]

    J. Henn, R. Ma, Y. Xu, K. Yan, Y. Zhang and H.X. Zhu,Two-Loop Spacelike Splitting Amplitude for N=4 Super-Yang-Mills Theory,2406.14604

    arXiv

  9. [16]

    X. Guan, F. Herzog, Y. Ma, B. Mistlberger and A. Suresh,Splitting amplitudes at N 3LO in QCD,JHEP01(2025) 090 [2408.03019]

    2025 arXiv

  10. [17]

    C. Duhr, A. Venkata and C. Zhang,Double spacelike collinear limits from multi-Regge kinematics,2507.05355

    arXiv

  11. [18]

    Forshaw, A

    J.R. Forshaw, A. Kyrieleis and M.H. Seymour,Super-leading logarithms in non-global observables in QCD: Colour basis independent calculation,JHEP09(2008) 128 [0808.1269]

    2008 arXiv

  12. [19]

    Becher, M

    T. Becher, M. Neubert and D.Y. Shao,Resummation of Super-Leading Logarithms,Phys. Rev. Lett.127(2021) 212002 [2107.01212]. – 56 –

    2021 arXiv

  13. [20]

    Becher, M

    T. Becher, M. Neubert, D.Y. Shao and M. Stillger,Factorization of non-global LHC observables and resummation of super-leading logarithms,JHEP12(2023) 116 [2307.06359]

    2023 arXiv

  14. [21]

    B¨ oer, P

    P. B¨ oer, P. Hager, M. Neubert, M. Stillger and X. Xu,Renormalization-group improved resummation of super-leading logarithms,JHEP08(2024) 035 [2405.05305]

    2024 arXiv

  15. [22]

    Becher, M

    T. Becher, M. Neubert, L. Rothen and D.Y. Shao,Factorization and Resummation for Jet Processes,JHEP11(2016) 019 [1605.02737]

    2016 arXiv

  16. [23]

    Balsiger, T

    M. Balsiger, T. Becher and D.Y. Shao,Non-global logarithms in jet and isolation cone cross sections,JHEP08(2018) 104 [1803.07045]

    2018 arXiv

  17. [25]

    Becher, P

    T. Becher, P. Hager, G. Martinelli, M. Neubert, D. Schwienbacher and M. Stillger, Super-leading logarithms in pp→2 jets,JHEP01(2025) 171 [2411.12742]

    2025 arXiv

  18. [26]

    Bauer, D

    C.W. Bauer, D. Pirjol and I.W. Stewart,Soft collinear factorization in effective field theory, Phys. Rev. D65(2002) 054022 [hep-ph/0109045]

    2002 arXiv

  19. [27]

    Bauer, S

    C.W. Bauer, S. Fleming, D. Pirjol, I.Z. Rothstein and I.W. Stewart,Hard scattering factorization from effective field theory,Phys. Rev. D66(2002) 014017 [hep-ph/0202088]

    2002 arXiv

  20. [28]

    Beneke, A.P

    M. Beneke, A.P. Chapovsky, M. Diehl and T. Feldmann,Soft collinear effective theory and heavy to light currents beyond leading power,Nucl. Phys. B643(2002) 431 [hep-ph/0206152]

    2002 arXiv

  21. [29]

    Beneke and T

    M. Beneke and T. Feldmann,Multipole expanded soft collinear effective theory with nonAbelian gauge symmetry,Phys. Lett. B553(2003) 267 [hep-ph/0211358]

    2003 arXiv

  22. [30]

    Becher, M

    T. Becher, M. Neubert, L. Rothen and D.Y. Shao,Effective Field Theory for Jet Processes, Phys. Rev. Lett.116(2016) 192001 [1508.06645]

    2016 arXiv

  23. [31]

    Becher, T

    T. Becher, T. Rauh and X. Xu,Two-loop anomalous dimension for the resummation of non-global observables,JHEP08(2022) 134 [2112.02108]

    2022 arXiv

  24. [32]

    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]

    1997 arXiv

  25. [33]

    Hill and M

    R.J. Hill and M. Neubert,Spectator interactions in soft collinear effective theory,Nucl. Phys. B657(2003) 229 [hep-ph/0211018]. [34]ATLAScollaboration,Measurement of dijet production with a veto on additional central jet activity inppcollisions at √s= 7TeV using the ATLAS dete...

    2003 arXiv

  26. [35]

    Becher and J

    T. Becher and J. Haag,Factorization and resummation for sequential recombination jet cross sections,JHEP01(2024) 155 [2309.17355]

    2024 arXiv

  27. [36]

    Balsiger, T

    M. Balsiger, T. Becher and A. Ferroglia,Resummation of non-global logarithms in cross sections with massive particles,JHEP09(2020) 029 [2006.00014]

    2020 arXiv

  28. [37]

    Banerjee, R

    U. Banerjee, R. Gr¨ unhofer, M. K¨ onig, Y. Li, M. Neubert and J. Scholze, in preparation

  29. [38]

    J.-y. Chiu, F. Golf, R. Kelley and A.V. Manohar,Electroweak Sudakov corrections using effective field theory,Phys. Rev. Lett.100(2008) 021802 [0709.2377]

    2008 arXiv

  30. [39]

    J.-y. Chiu, F. Golf, R. Kelley and A.V. Manohar,Electroweak Corrections in High Energy Processes using Effective Field Theory,Phys. Rev. D77(2008) 053004 [0712.0396]. – 57 –

    2008 arXiv

  31. [40]

    Becher and M

    T. Becher and M. Neubert,Drell-Yan Production at Smallq T , Transverse Parton Distributions and the Collinear Anomaly,Eur. Phys. J. C71(2011) 1665 [1007.4005]

    2011 arXiv

  32. [41]

    J.-y. Chiu, A. Jain, D. Neill and I.Z. Rothstein,The Rapidity Renormalization Group,Phys. Rev. Lett.108(2012) 151601 [1104.0881]

    2012 arXiv

  33. [42]

    J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein,A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory,JHEP05(2012) 084 [1202.0814]

    2012 arXiv

  34. [43]

    Becher and M

    T. Becher and M. Neubert,On the Structure of Infrared Singularities of Gauge-Theory Amplitudes,JHEP06(2009) 081 [0903.1126]

    2009 arXiv

  35. [44]

    Catani and M

    S. Catani and M. Grazzini,Infrared factorization of tree level QCD amplitudes at the next-to-next-to-leading order and beyond,Nucl. Phys. B570(2000) 287 [hep-ph/9908523]

    2000 arXiv

  36. [45]

    Nagy and D.E

    Z. Nagy and D.E. Soper,Parton showers with quantum interference,JHEP09(2007) 114 [0706.0017]

    2007 arXiv

  37. [46]

    Forshaw, J

    J.R. Forshaw, J. Holguin and S. Pl¨ atzer,Parton branching at amplitude level,JHEP08 (2019) 145 [1905.08686]

    2019 arXiv

  38. [47]

    Forshaw, S

    J.R. Forshaw, S. Pl¨ atzer and F.T. Gonz´ alez,Fully Differential Soft Gluon Evolution at the Amplitude Level,2505.13183

    arXiv

  39. [48]

    Duhr and T

    C. Duhr and T. Gehrmann,The two-loop soft current in dimensional regularization,Phys. Lett. B727(2013) 452 [1309.4393]

    2013 arXiv

  40. [49]

    Dixon, E

    L.J. Dixon, E. Herrmann, K. Yan and H.X. Zhu,Soft gluon emission at two loops in full color,JHEP05(2020) 135 [1912.09370]

    2020 arXiv

  41. [50]

    Rothstein and I.W

    I.Z. Rothstein and I.W. Stewart,An Effective Field Theory for Forward Scattering and Factorization Violation,JHEP08(2016) 025 [1601.04695]

    2016 arXiv

  42. [51]

    Moult, S

    I. Moult, S. Raman, G. Ridgway and I.W. Stewart,Anomalous dimensions from soft Regge constants,JHEP05(2023) 025 [2207.02859]

    2023 arXiv

  43. [52]

    Catani and M

    S. Catani and M. Grazzini,The soft gluon current at one loop order,Nucl. Phys. B591 (2000) 435 [hep-ph/0007142]

    2000 arXiv

  44. [53]

    Beneke and V.A

    M. Beneke and V.A. Smirnov,Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys.B522(1998) 321 [hep-ph/9711391]

    1998 arXiv

  45. [54]

    Bern, L.J

    Z. Bern, L.J. Dixon and D.A. Kosower,Dimensionally regulated pentagon integrals,Nucl. Phys. B412(1994) 751 [hep-ph/9306240]

    1994 arXiv

  46. [55]

    Ma,Identifying regions in wide-angle scattering via graph-theoretical approaches,JHEP 09(2024) 197 [2312.14012]

    Y. Ma,Identifying regions in wide-angle scattering via graph-theoretical approaches,JHEP 09(2024) 197 [2312.14012]

    2024 arXiv

  47. [56]

    Gardi, F

    E. Gardi, F. Herzog, S. Jones and Y. Ma,Dissecting polytopes: Landau singularities and asymptotic expansions in 2→2 scattering,JHEP08(2024) 127 [2407.13738]

    2024 arXiv

  48. [57]

    Ma,Identifying regions for asymptotic expansions of amplitudes: fundamentals and recent advances,2505.01368

    Y. Ma,Identifying regions for asymptotic expansions of amplitudes: fundamentals and recent advances,2505.01368

    arXiv

  49. [58]

    Buccioni, F

    F. Buccioni, F. Caola, F. Devoto and G. Gambuti,Investigating the universality of five-point QCD scattering amplitudes at high energy,JHEP03(2025) 129 [2411.14050]

    2025 arXiv

  50. [59]

    Nabeebaccus, J

    S. Nabeebaccus, J. Schoenleber, L. Szymanowski and S. Wallon,Breakdown of collinear factorization in the exclusive photoproduction of aπ 0γpair with large invariant mass,Phys. Rev. D111(2025) 034040 [2311.09146]. – 58 –

    2025 arXiv

  51. [60]

    Nabeebaccus, J

    S. Nabeebaccus, J. Schoenleber, L. Szymanowski and S. Wallon,Demonstration of collinear factorization breaking due to collinear-to-soft Glauber exchanges for a 2→3 exclusive process at leading twist,Phys. Rev. D111(2025) L091502 [2409.16067]

    2025 arXiv

  52. [61]

    Becher and G

    T. Becher and G. Bell,Analytic Regularization in Soft-Collinear Effective Theory,Phys. Lett. B713(2012) 41 [1112.3907]

    2012 arXiv

  53. [62]

    Becher and R.J

    T. Becher and R.J. Hill,Loop corrections to heavy-to-light form-factors and evanescent operators in SCET,JHEP10(2004) 055 [hep-ph/0408344]

    2004 arXiv

  54. [63]

    Becher and X

    T. Becher and X. Garcia i Tormo,Factorization and resummation for transverse thrust, JHEP06(2015) 071 [1502.04136]

    2015 arXiv

  55. [64]

    Becher, M

    T. Becher, M. Neubert and L. Rothen,Factorization andN 3LLp+NNLO predictions for the Higgs cross section with a jet veto,JHEP10(2013) 125 [1307.0025]

    2013 arXiv

  56. [65]

    Stewart, F.J

    I.W. Stewart, F.J. Tackmann and W.J. Waalewijn,N-Jettiness: An Inclusive Event Shape to Veto Jets,Phys. Rev. Lett.105(2010) 092002 [1004.2489]

    2010 arXiv

  57. [66]

    Banfi, G.P

    A. Banfi, G.P. Salam and G. Zanderighi,Phenomenology of event shapes at hadron colliders, JHEP06(2010) 038 [1001.4082]

    2010 arXiv

  58. [67]

    Forshaw and J

    J.R. Forshaw and J. Holguin,Coulomb gluons will generally destroy coherence,JHEP12 (2021) 084 [2109.03665]. – 59 –

    2021 arXiv

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Reviewed August 4, 2026 · model on record in the stance chip above.