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arxiv: 2607.01413 · v1 · pith:BZXSSLTZnew · submitted 2026-07-01 · ✦ hep-ph

One-loop matching of QCD currents to power-suppressed two-jet operators

Martin Beneke , Aleksey V. Rusov , Michel Stillger This is my paper

Pith reviewed 2026-07-03 19:15 UTC · model grok-4.3

classification ✦ hep-ph
keywords SCETmatching coefficientspower correctionstwo-jet operatorsNLOQCD currentsendpoint factorizationthree-particle operators
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The pith

QCD quark-antiquark currents are matched at one loop to two- and three-particle SCET two-jet operators up to second power in transverse momentum over energy.

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

This paper computes the next-to-leading-order matching coefficients that relate QCD currents to a complete set of two-particle and three-particle operators in soft-collinear effective theory. The operators are classified by their suppression in powers of λ equal to transverse momentum divided by the hard scale, reaching O(λ²) for the first time. Three-particle operators, which depend on the momentum fractions of two partons collinear in the same direction, are included to capture all contributions that generate endpoint singularities. The coefficient functions for these operators obey endpoint factorization relations that permit consistent cancellation of singularities inside the full factorization formula. These matching results supply the perturbative building blocks required for power corrections to event shapes and deep-inelastic scattering structure functions near the two-jet limit.

Core claim

We compute the matching of QCD quark-antiquark currents onto the set of the two-particle and three-particle two-jet operators in soft-collinear effective theory (SCET) at next-to-leading order (NLO) in the perturbative QCD series, including for the first time operators up to second order in the power expansion in the transverse momentum over energy. These results contribute to the ongoing programme of computing power corrections and summing power-suppressed logarithmically enhanced terms for event shapes in the two-jet region and deep-inelastic scattering in the Bjorken-x→1 limit. The three-particle operators depend on the partonic momentum fractions of two partons moving into the same direc

What carries the argument

The NLO matching coefficients for the two-particle and three-particle SCET two-jet operators up to O(λ²) that relate QCD currents to the effective-theory basis.

Load-bearing premise

The SCET operator basis and power counting in λ = k⊥/Q remain valid and complete up to O(λ²), with the three-particle operators correctly capturing all contributions that produce endpoint singularities requiring factorization relations for cancellation.

What would settle it

A direct NLO QCD calculation of a specific power correction to a two-jet event shape or DIS structure function that disagrees numerically with the prediction obtained by inserting these matching coefficients into the SCET factorization formula.

Figures

Figures reproduced from arXiv: 2607.01413 by Aleksey V. Rusov, Martin Beneke, Michel Stillger.

Figure 1
Figure 1. Figure 1: Contribution to the next-to-leading factorization theorem for DIS involv￾ing hard operators with three fields. leading power (NLP) term as double insertions of the O(λ) operators in the second line. As an example, in fig. 1 a contribution to the NLP factorization theorem for large-x DIS involving subleading-power SCET operators is shown. The nomenclature “B2” refers to [∂⊥Ac⊥] χc , Ac⊥ [∂⊥χc] and will be e… view at source ↗
Figure 2
Figure 2. Figure 2: Tree and one-loop diagrams contributing to the left-hand side of eq. (2.2) when projected with the matrix element ⟨q¯(p)q(k)| . . . |0⟩. Blue (green) lines represent (anti-)-collinear (c, resp. ¯c) fields and black ones hard (h) propagators. The matrix element on the left-hand side of eq. (3.1) is calculated in full QCD and then expanded in λ. The QCD spinors are related to the SCET ones by [10] v(p) =  1… view at source ↗
Figure 3
Figure 3. Figure 3: Tree and one-loop diagrams contributing to the left-hand side of eq. (2.2) when projected with the matrix element ⟨g(p1)¯q(p2)q(k)| . . . |0⟩. The colours are the same as in fig. 2. We do not show one-loop diagrams without hard region. and to ignore n+εc(p1) contributions to the right-hand side of the matching equation. It is now straightforward to calculate the diagrams in fig. 3 and by comparing to eq. (… view at source ↗
Figure 4
Figure 4. Figure 4: Tree and one-loop diagrams contributing to the left-hand side of eq. (2.2) when projected with the matrix element ⟨g(p1)¯q(p2)q(k)| . . . |0⟩. The colours are the same as in fig. 2. Even though the purple propagator is collinear these diagrams contribute to the matching coefficients as described in the main text. To obtain the expansion of Λµ one needs to evaluate the two diagrams in fig. 2 for an off￾shel… view at source ↗
Figure 5
Figure 5. Figure 5: Diagrammatic representation of the cancellation of endpoint divergences in eq. (5.1). The circled “2” denotes the L (2) ξ insertion. The colours are the same as in fig. 2 and the the soft gluon is drawn in red. where the soft Wilson lines Y ensure soft gauge invariance. The coefficients D (B2) k can be regarded as soft-collinear splitting amplitudes that describe the splitting of an en￾ergetic, slightly of… view at source ↗
read the original abstract

We compute the matching of QCD quark-antiquark currents onto the set of the two-particle and three-particle two-jet operators in soft-collinear effective theory (SCET) at next-to-leading order (NLO) in the perturbative QCD series, including for the first time operators up to second order in the power expansion in the transverse momentum over energy. These results contribute to the ongoing programme of computing power corrections and summing power-suppressed logarithmically enhanced terms for event shapes in the two-jet region and deep-inelastic scattering in the Bjorken-$x\to 1$ limit. The three-particle operators depend on the partonic momentum fractions of two partons moving into the same direction. When one of the momentum fractions approaches zero, the coefficient functions are shown to satisfy endpoint factorization relations, which allow for a consistent cancellation of endpoint singularities among various terms in the complete factorization formula for power corrections.

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 paper computes the NLO matching of QCD quark-antiquark currents onto the complete set of two-particle and three-particle two-jet operators in SCET through O(λ²) with λ = k⊥/Q. It derives the coefficient functions explicitly and verifies that they obey endpoint factorization relations when a parton momentum fraction approaches zero, ensuring consistent cancellation of endpoint singularities in the full factorization formula for power corrections to event shapes and DIS at x→1.

Significance. If the matching holds, the results supply the missing NLO ingredients for systematic power-correction calculations in the two-jet region. The explicit demonstration of endpoint factorization relations is a concrete technical advance that strengthens the consistency of the SCET power expansion. The work directly supports ongoing programs for resumming power-suppressed logarithms.

minor comments (3)
  1. §2: the operator basis is stated to be complete up to O(λ²), but the text does not explicitly list the full set of three-particle operators with their momentum-fraction dependence; adding a compact table would improve traceability of the matching results.
  2. Eq. (3.12) and following: the endpoint factorization relations are verified numerically for a subset of coefficients; stating the analytic form of the relation used for the check would make the cancellation argument more transparent.
  3. The manuscript cites prior SCET matching papers but does not compare the new O(λ²) coefficients against any existing partial results at lower orders; a brief consistency check paragraph would strengthen the presentation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the clear summary of the NLO matching calculation, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper reports a direct one-loop perturbative matching computation equating matrix elements of QCD quark-antiquark currents to a stated basis of SCET two- and three-particle operators through O(λ²). The coefficient functions are obtained from explicit diagram evaluation in full QCD and SCET; endpoint factorization relations are verified as an output of the calculation rather than imposed by definition or prior self-citation. No fitted parameters are renamed as predictions, no uniqueness theorem is invoked from overlapping prior work, and the derivation chain remains externally anchored to QCD Feynman rules without reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The calculation rests on the established SCET power-counting framework and operator classification; no new free parameters or invented entities are introduced in the abstract. The endpoint factorization is presented as a derived property rather than an additional axiom.

axioms (2)
  • domain assumption SCET power counting with λ = k⊥/Q organizes the two-jet region and remains valid through O(λ²)
    Invoked to justify inclusion of operators up to second order in the power expansion.
  • domain assumption The complete set of two- and three-particle operators at O(λ²) captures all relevant contributions
    Basis completeness is presupposed for the matching to be exhaustive.

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Works this paper leans on

34 extracted references · 32 canonical work pages · 17 internal anchors

  1. [1]

    P. A. Movilla Fernandez, S. Bethke, O. Biebel, and S. Kluth,Tests of power corrections for event shapes in e+ e- annihilation,Eur. Phys. J. C22(2001) 1–15, [hep-ex/0105059]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  2. [2]

    Nason and G

    P. Nason and G. Zanderighi,Fits ofα s using power corrections in the three-jet region,JHEP06(2023) 058, [arXiv:2301.03607]

    work page arXiv 2023

  3. [3]

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

    work page arXiv 2025

  4. [4]

    Beneke, M

    M. Beneke, M. Garny, S. Jaskiewicz, R. Szafron, L. Vernazza, and J. Wang, Leading-logarithmic threshold resummation of Higgs production in gluon fusion at next-to-leading power,JHEP01(2020) 094, [arXiv:1910.12685]

    work page arXiv 2020

  5. [5]

    van Beekveld, E

    M. van Beekveld, E. Laenen, J. Sinninghe Damst´ e, and L. Vernazza, Next-to-leading power threshold corrections for finite order and resummed colour-singlet cross sections,JHEP05(2021) 114, [arXiv:2101.07270]

    work page arXiv 2021

  6. [6]

    A. A H, P. Mukherjee, V. Ravindran, A. Sankar, and S. Tiwari,Next-to SV resummed Drell–Yan cross section beyond leading-logarithm,Eur. Phys. J. C82 (2022), no. 3 234, [arXiv:2107.09717]

    work page arXiv 2022

  7. [7]

    Boussarieet al., (2023), arXiv:2304.03302

    R. Boussarie et al.,TMD Handbook,arXiv:2304.03302

    work page arXiv

  8. [8]

    C. W. Bauer, S. Fleming, D. Pirjol, and I. W. Stewart,An Effective field theory for collinear and soft gluons: Heavy to light decays,Phys. Rev. D63(2001) 114020, [hep-ph/0011336]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  9. [9]

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

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  10. [10]

    Soft-collinear effective theory and heavy-to-light currents beyond leading power

    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) 431–476, [hep-ph/0206152]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  11. [11]

    Multipole-expanded soft-collinear effective theory with non-abelian gauge symmetry

    M. Beneke and T. Feldmann,Multipole expanded soft collinear effective theory with non-abelian gauge symmetry,Phys. Lett. B553(2003) 267–276, [hep-ph/0211358]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  12. [12]

    Anomalous dimension of subleading-power N-jet operators

    M. Beneke, M. Garny, R. Szafron, and J. Wang,Anomalous dimension of subleading-power N-jet operators,JHEP03(2018) 001, [arXiv:1712.04416]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  13. [13]

    Anomalous dimension of subleading-power ${N}$-jet operators II

    M. Beneke, M. Garny, R. Szafron, and J. Wang,Anomalous dimension of subleading-powerN-jet operators. Part II,JHEP11(2018) 112, [arXiv:1808.04742]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  14. [14]

    C. W. Bauer, C. Lee, A. V. Manohar, and M. B. Wise,Enhanced nonperturbative effects in Z decays to hadrons,Phys. Rev. D70(2004) 034014, [hep-ph/0309278]. 30

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  15. [15]

    R. N. Lee, A. von Manteuffel, R. M. Schabinger, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser,Quark and Gluon Form Factors in Four-Loop QCD,Phys. Rev. Lett.128(2022), no. 21 212002, [arXiv:2202.04660]

    work page arXiv 2022

  16. [16]

    Loop corrections to sub-leading heavy quark currents in SCET

    M. Beneke, Y. Kiyo, and D. s. Yang,Loop corrections to subleading heavy quark currents in SCET,Nucl. Phys. B692(2004) 232–248, [hep-ph/0402241]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  17. [17]

    R. J. Hill, T. Becher, S. J. Lee, and M. Neubert,Sudakov resummation for subleading SCET currents and heavy-to-light form-factors,JHEP07(2004) 081, [hep-ph/0404217]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  18. [18]

    Strohm,Reparameterization Constraints on Renormalization and Matching of SCET at Sub-Leading Power, Master Thesis, Technische Universit¨ at M¨ unchen, 2020

    J. Strohm,Reparameterization Constraints on Renormalization and Matching of SCET at Sub-Leading Power, Master Thesis, Technische Universit¨ at M¨ unchen, 2020

    2020

  19. [19]

    Vladimirov, V

    A. Vladimirov, V. Moos, and I. Scimemi,Transverse momentum dependent operator expansion at next-to-leading power,JHEP01(2022) 110, [arXiv:2109.09771]

    work page arXiv 2022

  20. [20]

    Asymptotic expansion of Feynman integrals near threshold

    M. Beneke and V. A. Smirnov,Asymptotic expansion of Feynman integrals near threshold,Nucl. Phys. B522(1998) 321–344, [hep-ph/9711391]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  21. [21]

    Shtabovenko, R

    V. Shtabovenko, R. Mertig, and F. Orellana,FeynCalc 10: Do multiloop integrals dream of computer codes?,Comput. Phys. Commun.306(2025) 109357, [arXiv:2312.14089]

    work page arXiv 2025

  22. [22]

    H. H. Patel,Package-X: A Mathematica package for the analytic calculation of one-loop integrals,Comput. Phys. Commun.197(2015) 276–290, [arXiv:1503.01469]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  23. [23]

    A. V. Manohar, T. Mehen, D. Pirjol, and I. W. Stewart,Reparameterization invariance for collinear operators,Phys. Lett. B539(2002) 59–66, [hep-ph/0204229]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  24. [24]

    B. V. Geshkenbein and M. V. Terentev,The enhanced power corrections to the asymptotics of the pion form-factor,Phys. Lett. B117(1982) 243–246

    1982

  25. [25]

    QCD factorization for exclusive, non-leptonic B meson decays: General arguments and the case of heavy-light final states

    M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda,QCD factorization for exclusive, nonleptonic B meson decays: General arguments and the case of heavy light final states,Nucl. Phys. B591(2000) 313–418, [hep-ph/0006124]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  26. [26]

    Factorization of heavy-to-light form factors in soft-collinear effective theory

    M. Beneke and T. Feldmann,Factorization of heavy to light form-factors in soft collinear effective theory,Nucl. Phys. B685(2004) 249–296, [hep-ph/0311335]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  27. [27]

    B -> chi_cJ K decays revisited

    M. Beneke and L. Vernazza,B→χ cJ Kdecays revisited,Nucl. Phys. B811(2009) 155–181, [arXiv:0810.3575]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  28. [28]

    Beneke, A

    M. Beneke, A. Broggio, S. Jaskiewicz, and L. Vernazza,Threshold factorization of the Drell-Yan process at next-to-leading power,JHEP07(2020) 078, [arXiv:1912.01585]. 31

    work page arXiv 2020

  29. [29]

    Moult, I

    I. Moult, I. W. Stewart, G. Vita, and H. X. Zhu,The Soft Quark Sudakov,JHEP 05(2020) 089, [arXiv:1910.14038]

    work page arXiv 2020

  30. [30]

    Beneke, M

    M. Beneke, M. Garny, S. Jaskiewicz, R. Szafron, L. Vernazza, and J. Wang,Large-x resummation of off-diagonal deep-inelastic parton scattering from d-dimensional refactorization,JHEP10(2020) 196, [arXiv:2008.04943]

    work page arXiv 2020

  31. [31]

    Z. L. Liu, B. Mecaj, M. Neubert, and X. Wang,Factorization at subleading power and endpoint divergences inh→γγdecay. Part II. Renormalization and scale evolution,JHEP01(2021) 077, [arXiv:2009.06779]

    work page arXiv 2021

  32. [32]

    Beneke, M

    M. Beneke, M. Garny, S. Jaskiewicz, J. Strohm, R. Szafron, L. Vernazza, and J. Wang,Next-to-leading power endpoint factorization and resummation for off-diagonal “gluon” thrust,JHEP07(2022) 144, [arXiv:2205.04479]

    work page arXiv 2022

  33. [33]

    Leading-logarithmic threshold resummation of the Drell-Yan process at next-to-leading power

    M. Beneke, A. Broggio, M. Garny, S. Jaskiewicz, R. Szafron, L. Vernazza, and J. Wang,Leading-logarithmic threshold resummation of the Drell-Yan process at next-to-leading power,JHEP03(2019) 043, [arXiv:1809.10631]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  34. [34]

    Z. L. Liu and M. Neubert,Factorization at subleading power and endpoint-divergent convolutions inh→γγdecay,JHEP04(2020) 033, [arXiv:1912.08818]. 32

    work page arXiv 2020