Hard Scattering Factorization from Effective Field Theory

Christian W. Bauer; Dan Pirjol; Iain W. Stewart; Ira Z. Rothstein; Sean Fleming

REVIEW

Hard Scattering Factorization from Effective Field Theory

Christian W. Bauer, Sean Fleming, Dan Pirjol, Ira Z. Rothstein, Iain W. Stewart

High Energy Physics - Phenomenology · Submitted 11 Feb 2002 · 40 references

Pith's verdict: A clean, honest methods paper: the factorization formulae are old, the SCET proof technique for back-to-back jets is the contribution, and the two real gaps are both acknowledged in the text.

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CitationChristian W. Bauer, Dan Pirjol, Iain W. Stewart, Ira Z. Rothstein, Sean Fleming. "Hard Scattering Factorization from Effective Field Theory." arXiv:hep-ph/0202088 (2002). https://pith.science/paper/A2QRJMBF

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Editor's note

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You can read this one quickly and know what you're getting. The authors say up front that the factorization formulae are not new; what's new is SCET extended to back-to-back collinear directions, plus a proof technique that runs on gauge invariance in the effective theory rather than Ward identities, induction, or a chosen gauge. That is the whole paper, and it delivers it: πγ, γ*M→M, DIS, Drell-Yan, DVCS, all to all orders in αs at leading power. Appendix A is the part I'd actually keep on my desk — the auxiliary-field construction that shows soft and collinear Wilson lines appear in operators in the right gauge-invariant places is done explicitly, order by order in offshellness, and it's the load-bearing derivation for everything downstream. The Fierz and color-octet arguments in the Drell-Yan section are also tidier than the older diagrammatic versions, and the charge-conjugation and helicity constraints in II.D are a nice piece of bookkeeping that saves the reader work later. The paper is candid about its own limits, which is rarer than it should be. Two soft spots, both flagged by the authors. First, Section III.B drops the Feynman-mechanism operator in Eq. (62) without a power-counting argument, and says so in as many words. That's a real gap in the meson form factor section, not a fatal one, and the authors don't pretend otherwise. Second, the Glauber assumption in II.B. The stress-test is right that this is load-bearing beyond Drell-Yan: the n–n̄ decoupling step after Eq. (71) is exactly where Glauber exchange would enter, and it's used again in DIS and DVCS. The citation to Politzer and Collins-Soper-Sterman is explicitly restricted to Drell-Yan in the text, so the support is narrower than the set of processes it's applied to. That said, this is an acknowledged assumption with a plausible physical argument attached, not a hidden error, and the standard lore at the time was that Glauber modes cancel for these observables. I'd want a referee to press on it, but I wouldn't expect the paper to fall. The reader's scores look about right to me, maybe significance a notch low given how much later SCET work leans on this. This is for anyone doing hard-scattering factorization who wants the EFT version of the proofs, and for students who want to see the machinery worked end to end. Send it out.

Referee report

Significance

If the central claim holds, the paper provides a unified and systematic operator framework for factorization proofs across exclusive and inclusive processes, with explicit operator definitions and matching coefficients that can be checked order by order. The explicit tree-level matching in Eqs. (57), (60), and (79) reproduces known Brodsky-Lepage results, which is a nontrivial consistency check on the formalism and on the normalization conventions. The paper is a methodological contribution of lasting value to the SCET literature, and the operator definitions in Section II.C are widely used. The derivations are self-contained EFT manipulations; the Wilson coefficients are matched from QCD rather than fitted, and the nonperturbative matrix elements are defined in terms of QCD operators, so there is no circularity in the factorization argument itself.

Major objections (2)

[Section II.B, paragraph following Eq. (18)] The paper states explicitly that Glauber gluons with momenta p ~ (λ², λ², λ) are omitted and that 'we have assumed that Glauber gluons are not necessary to describe the infrared for the processes considered in this paper.' This assumption is load-bearing for every factorization step in Sections III and IV, since the n–n̄ decoupling in Eq. (18) and the text after Eq. (71) is exactly where Glauber exchange would enter. The text gives an intuitive argument (hard interaction implies short overlap) and then cites Refs. [2, 20] for quantitative arguments, but restricts that citation to Drell-Yan: 'For the Drell-Yan process more quantitative arguments can be found in Refs. [2, 20].' For DIS, DVCS, and the exclusive form factors in Section III.B, no process-specific Glauber-cancellation argument is given. Because the abstract and introduction claim 'proofs of factorization formulae' without qualification, the scope of the Glauber assumption should be stated explicitly: which of the five processes are covered by the cited arguments, and whether the intuitive overlap argument is being relied on for the others. This is a scope-of-claim issue rather than a correctness error, but it bears directly on the headline claim.
[Section III.B, Eqs. (62)–(63) and footnote 6] The authors drop the 'Feynman mechanism' operator in Eq. (62) in favor of the four-quark operator in Eq. (63) and state plainly: 'We emphasize that we do not claim to have shown that this is justified by the effective theory power counting.' This is an honest acknowledgment, and footnote 6 correctly notes that the 'general predictions' following Eq. (70) (helicity conservation, absence of P→V transitions, determination by only two hard coefficients C₀ and C₈) depend on the dominance of Eq. (63) over Eq. (62). However, the abstract and the conclusion present the meson form factor as one of the demonstrated examples without this qualification. The caveat should be propagated to the places where the claim is made, or the section should open with a clear statement that the leading-power analysis is conditional on the operator dominance assumption. As written, a reader of the abstract would not know that the meson form factor result rests on an unproven power-counting step.

Minor objections (6)

[Section IV.C, Eqs. (119)–(120)] I was unable to reproduce Eq. (120) from Eq. (119) using the definition in Eq. (117). With ω₊ = Q(2ξ−ζ)/x and ω₋ = −Qζ/x, Eq. (119) gives C₁ = e²Q_i²[1/(1+ξ/x) − 1/(1−(ξ−ζ)/x)], whereas Eq. (120) states H₁ = −e²Q_i²[1/(1−ξ/x) − 1/(1+(ξ−ζ)/x)]. The two agree at ζ = 0 but differ in the ζ-dependent denominators. This may be a convention for the iε prescription or a sign in the definition of H_j, but the authors should check and state the convention explicitly.
[Section IV.B, Eq. (97) and the argument following Eq. (98)] The claim that Eq. (97) is the minimal set of order λ⁴ operators is supported by the argument in Eqs. (98)–(104), but the treatment of operators with contracted n–n̄ Lorentz indices (mentioned in the sentence 'λ⁴ operators also exist where a B^ν_{n,ω} field is contracted with a B^μ_{n̄,ω′} field') and of the four-gluon case with mixed color representations is compressed. A short explicit demonstration that such operators reduce to the basis in Eq. (97) or vanish would strengthen the completeness argument.
[Introduction, paragraph 3, and footnote 1] The claim that 'our proofs do not rely on making use of Ward identities and induction, or on specifying a particular gauge' is qualified by footnote 1 ('our factorization proofs rely heavily on the gauge symmetry structure of SCET'). Since the gauge invariance of the EFT is itself a Slavnov-Taylor-type statement, the wording could be sharpened to say that the proofs avoid diagrammatic Ward identities in QCD and avoid gauge fixing, rather than avoiding Ward identities altogether.
[Sections III.A and III.B, Eqs. (52), (55), (59)] The argument order of the Wilson coefficient is inconsistent: Eq. (52) writes C(P̄, P̄†, μ), Eq. (55) writes C(P̄, P̄†, μ), and Eq. (59) writes C(μ, P̄, P̄†). A uniform convention would help the reader track the matching.
[Section II.A, discussion of soft and usoft modes] The distinction between soft (λ, λ, λ) and usoft (λ², λ², λ²) modes is central to the construction, and Table I in Appendix A summarizes the mode content clearly. A forward reference to Table I at the point where the modes are introduced in Section II.A would help readers who are not already familiar with the SCET power counting.
[General] The manuscript would benefit from a careful proofreading pass; several equations and inline expressions have inconsistent index placement and spacing (for example, the trace formulae in Eq. (46) and the spin structures in Eq. (69)) that make the conventions harder to follow than necessary.

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The paper

The paper claims

Gauge symmetry in an effective theory can carry the proof of hard-scattering factorization, the paper argues, replacing gauge-specific Ward-identity arguments with operator-level decoupling.

The reading

The paper sets out to show that factorization in high-energy hadronic scattering — the separation of a calculable short-distance coefficient from universal long-distance hadronic matrix elements — can be proved more directly inside a soft-collinear effective theory. Its key move is to extend that effective theory to two back-to-back collinear directions, so that jets moving along $n$ and $n$ are described by separate collinear fields and Wilson lines. Gauge invariance then dictates where the Wilson lines sit in the operators, and field redefinitions remove soft and ultrasoft gluons from the collinear Lagrangian at leading power. The payoff claimed is a proof strategy that does not rely on a particular gauge or on Ward identities and induction, applies to exclusive and inclusive processes, and holds to all orders in $n$ at leading order in the power expansion. The paper illustrates the method on the $n$-$n$ form factor, light-meson form factors, deep inelastic scattering, Drell-Yan, and deeply virtual Compton scattering.

What carries the argument

The machinery is the back-to-back version of soft-collinear effective theory. Collinear quarks and gluons moving in the $n$ direction carry large $n$ labels and small transverse and $n$ components, while fields moving in the $n$ direction carry the opposite large labels; each direction has its own collinear Wilson line, $n$ or $n$, and its own soft Wilson line, $n$ or $n$, with ultrasoft Wilson lines $n$ and $n$ for the lowest-energy gluons. Collinear gauge invariance fixes the placement of $n$ and $n$ in external operators, and the field redefinitions $n$ and $n$ remove ultrasoft gluons from the leading collinear Lagrangian. Integrating out off-shell fluctuations with virtuality of order $n$ induces the soft Wilson lines; the identities $n$ and $n$, plus color identities for octet bilinears, then make the matrix element split into separate collinear matrix elements. That split is the factorization proof.

Extended reading notes

Core claim

The central claim is that the gauge structure of the soft-collinear effective theory, once generalized to back-to-back collinear fields, is enough to establish the factorization formulae for a broad class of hard processes. In each example the hard scattering is matched onto a small set of gauge-invariant operators built from collinear fields and Wilson lines; the soft and ultrasoft Wilson lines either cancel by unitarity or can be moved into one collinear bilinear, so the matrix element factors into a hard coefficient convolved with standard nonperturbative objects. The paper does not present new factorization formulae so much as a new derivation of known ones: the $n$ form factor reduces to a convolution of a hard coefficient with the pion light-cone wavefunction, the meson form factor to two wavefunctions and two hard coefficients, DIS to parton distributions, Drell-Yan to two parton distributions, and DVCS to non-forward parton distributions. The authors emphasize that the argument is gauge independent and valid to all orders in $n$ at leading power.

Load-bearing premise

The load-bearing premise is that Glauber gluons — gluons with momenta of order $(n,n,n)$ exchanged in the $n$-channel between the two collinear directions — can be neglected for all the processes treated here; the paper states this assumption explicitly in Section II.B and cites earlier arguments for Drell-Yan rather than proving it within the effective theory.

Editorial extensions

If this is right

Factorization proofs for DIS, Drell-Yan, and DVCS can be organized as matching calculations onto a finite set of gauge-invariant SCET operators, with the nonperturbative content isolated in parton distributions or non-forward distributions.
Exclusive processes such as $n$ and $n$ reduce to convolutions of hard coefficients with light-cone wavefunctions, with the hard coefficients computable by matching.
The same Wilson-line and field-redefinition steps can be reused across processes, so a proof for one process supplies most of the operator structure for another.
Because the argument is organized at the operator level, it provides a natural starting point for studying power corrections and for renormalization-group resummation of large logarithms.
The method covers arbitrary spin and isospin meson form factors at leading power, not just the pion case.

Reading between the lines

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

A useful stress test of the framework would be a process where Glauber exchange is kinematically enhanced, such as forward scattering or a region with overlapping jets; if the SCET proof still goes through there, the Glauber assumption would look less like a limitation and more like a theorem.
The operator-level proof suggests that factorization could be checked mechanically: enumerate the leading-power gauge-invariant operators, apply the Wilson-line identities, and see whether any color-octet bilinear survives between color-singlet states.
If the back-to-back extension is as general as claimed, it should connect directly to the standard resummation of threshold logarithms in Drell-Yan and DIS, because the same soft Wilson lines control the large logarithms.
The treatment of DVCS with non-forward distributions hints that the method could be extended to other exclusive processes with unequal incoming and outgoing hadron momenta, provided the same operator basis closes.

Original abstract

In this paper we show how gauge symmetries in an effective theory can be used to simplify proofs of factorization formulae in highly energetic hadronic processes. We use the soft-collinear effective theory, generalized to deal with back-to-back jets of collinear particles. Our proofs do not depend on the choice of a particular gauge, and the formalism is applicable to both exclusive and inclusive factorization. As examples we treat the pi-gamma form factor (gamma gamma* -> pi^0), light meson form factors (gamma* M -> M), as well as deep inelastic scattering (e- p -> e- X), Drell-Yan (p pbar -> X l+ l-), and deeply virtual Compton scattering (gamma* p -> gamma(*) p).

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,726 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.08 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

In this paper we show how gauge symmetries in an effective theory can be used to simplify proofs of factorization formulae in highly energetic hadronic processes. We use the soft-collinear effective theory, generalized to deal with back-to-back jets of collinear particles. Our proofs do not depend on the choice of a particular gauge, and the formalism is applicable to both exclusive and inclusive factorization. As examples we treat the pi-gamma form factor (gamma gamma* -> pi^0), light meson form factors (gamma* M -> M), as well as deep inelastic scattering (e- p -> e- X), Drell-Yan (p pbar -> X l+ l-), and deeply virtual Compton scattering (gamma* p -> gamma(*) p).

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper we show how gauge symmetries in an effective theory can be used to simplify proofs of factorization formulae in highly energetic hadronic processes.	contribution 1.00	0.94	0.81	asserted 1.00
s1	We use the soft-collinear effective theory, generalized to deal with back-to-back jets of collinear particles.	method 0.96	0.84	0.05	asserted 0.99
s2	Our proofs do not depend on the choice of a particular gauge, and the formalism is applicable to both exclusive and inclusive factorization.	result 0.42	0.91	0.22	asserted 1.00
s3	As examples we treat the pi-gamma form factor (gamma gamma* -> pi^0), light meson form factors (gamma* M -> M), as well as deep inelastic scattering (e- p -> e- X), Drell-Yan (p pbar -> X l+ l-), and deeply virtual Compton scattering (gamma* p -> gamma(*) p).	contribution 0.93	0.83	0.05	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 0.86 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: Tree level matching onto

**Figure 2.** FIG. 2: Tree level matching onto

**Figure 3.** FIG. 3: Tree level matching onto the operator

**Figure 4.** FIG. 4: Tree level matching onto the operators in Drell-Yan.

Figures reproduced from arXiv:hep-ph/0202088 by the authors.

References

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Bootstrapping Pion Form Factors at Large $N$
hep-th 2026-06 unverdicted novelty 7.0 of 10

Bootstrap analysis of meromorphic observables in large-N QCD yields universal and SVZ-type bounds that constrain chiral Lagrangian parameters and link hadronic data to asymptotic freedom.
High precision heavy-boson-jet substructure with energy correlators
hep-ph 2026-01 conditional novelty 7.0 of 10

The EEC peak in boosted Z jets is the Sudakov back-to-back region of the Z-pole EEC seen through a Lorentz boost, determined by one Lorentz-invariant shape function and computable at N3LL'.
Energy Correlators Resolving Proton Spin
hep-ph 2025-09 unverdicted novelty 7.0 of 10

The work establishes a correspondence between spin-dependent energy correlators and polarized TMDs/NECs using SCET, yielding N3LL/N2LL predictions for correlation patterns in current and target fragmentation regions.
Dissecting Exclusive Multijet Cross Sections
hep-ph 2025-09 conditional novelty 7.0 of 10

A general leading-power factorization framework for exclusive multijet resolution variables, with a new rapidity regulator and an all-order factorizing k_T-ness variable.
The N$^3$LO Twist-2 Matching of Linearly Polarized Gluon TMDs
hep-ph 2025-09 unverdicted novelty 7.0 of 10

Computes N³LO twist-2 matching for linearly polarized gluon TMDs with NNLL small-x resummation for fragmentation functions.
Polarized Deep Inelastic Scattering as $x \to 1$ using Soft Collinear Effective Theory
hep-ph 2025-07 conditional novelty 7.0 of 10

SCET factorization and one-loop matching for polarized DIS g1(x), g2(x) as x approaches 1, including anomalous dimensions and twist-three operator relations.
Soft background fields at next-to-leading power in transverse momentum dependent SIDIS with jets
hep-ph 2025-07 conditional novelty 7.0 of 10

New next-to-leading-power factorization for SIDIS with jets from a background-field method with explicit soft modes, including operator-level definitions of twist-3 TMDs free of rapidity and endpoint divergences.
The structure of quark mass corrections in the $gg \rightarrow HH$ amplitude at high-energy
hep-ph 2024-12 conditional novelty 7.0 of 10

The leading-power mass logarithms in high-energy gg to HH are shown to originate solely from top-quark mass renormalization, enabling a resummation that sharply reduces the mass-scheme uncertainty of the virtual amplitude.
Emergence, Formation and Dynamics of Hot QCD Matter
hep-ph 2024-12 conditional novelty 7.0 of 10

A doctoral thesis deriving the QCD correlation functions that control quarkonium dissociation and recombination in the quark-gluon plasma, and showing that hydrodynamization in a simplified QCD kinetic theory matches ...
Renormalisation group evolution of the shape function $g_{17}$ in $\bar{B}\to X_s \gamma$ and $\bar{B} \to X_s \ell^+\ell^-$ at subleading power
hep-ph 2024-11 conditional novelty 7.0 of 10

The authors derive and solve the renormalization-group equation for the subleading shape function g17(ω,ω1;μ) in inclusive B decays, and simplify the RG evolution of a related exclusive soft function.
Drell-Yan Transverse-Momentum Spectra at N$^3$LL$'$ and Approximate N$^4$LL with SCETlib
hep-ph 2024-11 conditional novelty 7.0 of 10

State-of-the-art LHC predictions for W/Z transverse momentum spectra at N3LL' and approximate N4LL accuracy, with a rigorous reduction of nonperturbative TMD physics to one effective function per process.
Di-lepton Rapidity Distribution in Drell-Yan Production to Third Order in QCD
hep-ph 2021-07 accept novelty 7.0 of 10

The first N3LO QCD prediction for the Drell-Yan lepton-pair rapidity distribution is presented, and it independently reproduces the known N3LO total cross section channel by channel.
Higgs $p_T$ Spectrum and Total Cross Section with Fiducial Cuts at Third Resummed and Fixed Order in QCD
hep-ph 2021-02 conditional novelty 7.0 of 10

The gluon-fusion Higgs pT spectrum and total fiducial cross section are computed at N3LL'+N3LO with ATLAS photon cuts, giving sigma_fid/B_gammagamma = 25.41 +/- 0.68 pb.
Joint thrust and TMD resummation in electron-positron and electron-proton collisions
hep-ph 2020-09 conditional novelty 7.0 of 10

The authors derive three-region factorization theorems that jointly resum transverse momentum and thrust logarithms in e+e- collisions, and propose a crossed DIS observable to access TMD PDFs.
Drell-Yan $q_T$ Resummation of Fiducial Power Corrections at N$^3$LL
hep-ph 2020-06 accept novelty 7.0 of 10

Linear fiducial power corrections in Drell-Yan q_T spectra arise solely from the leptonic tensor, making them computable and resummable to N3LL, which fixes q_T subtractions and the lepton p_T spectrum at the Jacobian peak.
Analytic Continuation and Reciprocity Relation for Collinear Splitting in QCD
hep-ph 2020-06 conditional novelty 7.0 of 10

A new analytic continuation rule for splitting amplitudes resolves the NNLO ambiguity in the time-like quark-gluon splitting function and yields a singlet reciprocity relation verified through three loops.
Quark Transverse Parton Distribution at the Next-to-Next-to-Next-to-Leading Order
hep-ph 2019-12 accept novelty 7.0 of 10

The quark TMD beam function matching coefficients are computed analytically through N3LO in QCD, using a generalized integration-by-parts method that handles the exponential rapidity regulator inside the integrand.
Circumnavigating Collinear Superspace
hep-th 2019-08 conditional novelty 7.0 of 10

Collinear superspace can reproduce the full range of N=1 supersymmetric Lagrangians by adding auxiliary superfields that encode F and D terms.
Infrared Singularities of Scattering Amplitudes and N$^3$LL Resummation for $n$-Jet Processes
hep-ph 2019-08 accept novelty 7.0 of 10

At four-loop order, the soft anomalous dimension for massless n-particle amplitudes contains d_R^{abcd} color structures multiplied by cusp logarithms, which breaks naive Casimir scaling but preserves a generalized scaling.
The Three-loop Quark Jet Function
hep-ph 2018-04 accept novelty 7.0 of 10

The massless quark jet function is computed to three-loop order, yielding a new analytic coefficient that feeds N3LL-prime resummation and N3LO N-jettiness calculations.
First Subleading Power Resummation for Event Shapes
hep-ph 2018-04 accept novelty 7.0 of 10

Subleading-power logarithms in thrust are resummed to all orders and shown to exponentiate into a Sudakov factor governed by the cusp anomalous dimension, with a new class of theta-function jet and soft operators requ...
Bootstrapping rapidity anomalous dimension for transverse-momentum resummation
hep-ph 2016-04 conditional novelty 7.0 of 10

The three-loop rapidity anomalous dimension for transverse-momentum resummation is computed, and it is found to equal the threshold soft anomalous dimension minus beta-function and constant terms.
Factorization for groomed jet substructure beyond the next-to-leading logarithm
hep-ph 2016-03 conditional novelty 7.0 of 10

Soft-drop groomed jet observables factorize into single-scale collinear and collinear-soft functions with no non-global logarithms, enabling NNLL resummation of the groomed jet mass matched to fixed order for e+e- and...
Analytic Boosted Boson Discrimination
hep-ph 2015-07 conditional novelty 7.0 of 10

An all-orders QCD factorization theorem for the two-prong jet discriminant D2 reproduces Monte Carlo D2 spectra for boosted Z jets and massive QCD jets, and yields analytic signal-versus-background efficiency curves.
Non-Global Logarithms, Factorization, and the Soft Substructure of Jets
hep-ph 2015-01 conditional novelty 7.0 of 10

Non-global logarithms can be factorized and resummed by isolating a soft subjet region of phase space with energy correlation function measurements, and a dressed gluon expansion reproduces the leading large-Nc resumm...
The Gluon Beam Function at Two Loops
hep-ph 2014-05 accept novelty 7.0 of 10

The two-loop gluon beam function matching coefficients are computed explicitly, completing the NNLO set of virtuality-dependent beam functions and roughly halving the residual matching-scale uncertainty in the resumme...
Toward Multi-Differential Cross Sections: Measuring Two Angularities on a Single Jet
hep-ph 2014-01 conditional novelty 7.0 of 10

The double-differential cross section for two angularities on one jet factorizes at phase-space boundaries but not in the bulk, and a scale-setting interpolation connects the two boundary factorization theorems to nex...
Unsafe but Calculable: Ratios of Angularities in Perturbative QCD
hep-ph 2013-07 conditional novelty 7.0 of 10

The ratio of two jet angularities, though not infrared safe, has a finite calculable distribution once Sudakov logarithms are resummed, and its expansion begins at the square root of the strong coupling.
The Non-Abelian Exponentiation theorem for multiple Wilson lines
hep-ph 2013-04 accept novelty 7.0 of 10

The paper proves that in the exponent of a product of semi-infinite Wilson lines, every colour factor corresponds to a fully connected graph, generalising the Wilson-loop non-Abelian exponentiation theorem to arbitrar...
Factorization and NNLL Resummation for Higgs Production with a Jet Veto
hep-ph 2012-05 conditional novelty 7.0 of 10

The authors derive the first all-order factorization theorem for Higgs production with a sequential-recombination jet veto and present NNLL resummed cross sections and efficiencies for the LHC at 8 TeV.
Factorization and Resummation for Dijet Invariant Mass Spectra
hep-ph 2011-06 accept novelty 7.0 of 10

A new effective theory, SCET+, with an added collinear-soft mode factorizes and resums both ln(m/Q) and ln(m_jj/Q) logarithms in multijet cross sections, demonstrated for e+e- to 3 jets and generalized to pp to N jets.
Non-global Structure of the O({\alpha}_s^2) Dijet Soft Function
hep-ph 2011-05 accept novelty 7.0 of 10

The full order-alpha_s-squared dijet soft function is computed analytically, revealing non-global single-logarithm and non-logarithmic structures that earlier conjectures had missed.
Direct photon production with effective field theory
hep-ph 2009-11 accept novelty 7.0 of 10

A three-direction SCET factorization theorem yields an NNLL resummed direct photon pT spectrum whose theoretical uncertainty is smaller than the NLO fixed-order result at high transverse momentum.
Factorization at the LHC: From PDFs to Initial State Jets
hep-ph 2009-10 accept novelty 7.0 of 10

When the hadronic final state is restricted, the initial state of a proton collision must be described by beam functions rather than PDFs, and a factorization theorem for isolated Drell-Yan follows.
Top Jets in the Peak Region: Factorization Analysis with NLL Resummation
hep-ph 2007-11 accept novelty 7.0 of 10

Effective-theory factorization with NLL resummation gives the top-jet invariant-mass spectrum in e+e- collisions and defines a short-distance jet mass whose peak position is stable against perturbative corrections.
Jets from Massive Unstable Particles: Top-Mass Determination
hep-ph 2007-03 accept novelty 7.0 of 10

A factorization theorem shows that top-jet invariant mass peaks are shifted upward and broadened by non-perturbative soft radiation, and defines a short-distance jet mass measurable to better than the QCD scale.
The Zero-Bin and Mode Factorization in Quantum Field Theory
hep-ph 2006-05 conditional novelty 7.0 of 10

A Lagrangian-level zero-bin subtraction removes double counting between soft, ultrasoft, and collinear fields, converting spurious infrared singularities into ultraviolet divergences that renormalize cleanly.
Factorization of heavy-to-light form factors in soft-collinear effective theory
hep-ph 2003-11 conditional novelty 7.0 of 10

Heavy-to-light B meson form factors at large recoil factorize into a single universal form factor plus a hard-scattering convolution, proven to all orders in perturbation theory within soft-collinear effective theory.
Soft-collinear effective theory and heavy-to-light currents beyond leading power
hep-ph 2002-06 accept novelty 7.0 of 10

The soft-collinear effective Lagrangian and heavy-to-light currents are derived to second order in the power expansion, and the resulting form-factor relations are shown to survive at first order but not at second.
Sequential Discontinuities of Feynman Integrals and the Monodromy Group
hep-th 2020-07 conditional novelty 6.5 of 10

Sequential discontinuities of Feynman integrals in arbitrary momentum channels equal a Stirling-number-weighted sum over multiple cuts, derived via time-ordered perturbation theory and checked in bubble, triangle, and...
Hadronization effects and electroweak contributions in DIS one-jettiness
hep-ph 2026-09 accept novelty 6.0 of 10

The leading hadronization effects in DIS 1-Jettiness event shapes tau1 and tau1a are universal and described by a single shape function, with a calculable kinematic dependence in tau1 that provides an additional lever...
QED Corrections to $B^-\to\tau^-\bar{\nu}_\tau$
hep-ph 2026-07 conditional novelty 6.0 of 10

The B−→τ−ν̄τ rate with real and virtual QED corrections is derived to O(α) with resummed logarithmic corrections in a two-heavy-fermion EFT: percent-level effects, mild photon-veto dependence, and a structure-dependen...
The one-point charge correlator in deep inelastic scattering
hep-ph 2026-06 unverdicted novelty 6.0 of 10

Defines an IRC-safe one-point charge correlator in DIS, relates it to a new nucleon charge correlator in the forward limit and to TMDs in the back-to-back limit, with SCET derivations verified in QCD to O(alpha_s^2) a...
$B_c \to \eta_c$ form factors at large recoil: SCET analysis and a three-loop consistency check
hep-ph 2026-05 unverdicted novelty 6.0 of 10

SCET factorization confirms the double-logarithmic resummation for B_c to eta_c form factors up to three loops and derives the iterative structure from RG equations of light-cone distribution amplitudes with cutoff re...
Random Reshuffling-Based Distributed Nash Equilibrium Seeking
math.OC 2026-04 unverdicted novelty 6.0 of 10

Random reshuffling yields distributed Nash-seeking algorithms that, under partial decision information, converge linearly to a neighborhood (constant steps) or exactly a.s./in mean square (diminishing steps), outperfo...

See all 156 Pith citations

How later papers used it

366 later papers cite it

Evidence: substantial. 309 of the citing papers share no author with it, and 53 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 491 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 used its method or tool. 2 more papers by the same authors are listed: 2 used its method or tool.

Uses its method or tool 3 of 59 papers with this label · 2 by the same authors

Followed this paper in starting with the unpolarized Drell-Yan cross section.

Effective field theories for quantum chromo- and electrodynamics (2016) · 0.98 · paraphrased: the licence does not allow quoting

By the same authors

Constructed its basis post BPS field redefinition, which is this paper's field redefinition performed in each collinear sector.

A Subleading Power Operator Basis for the Scalar Quark Current (2017) (same authors) · 0.94 · paraphrased: the licence does not allow quoting
Constructed its basis post BPS field redefinition, which is defined by this paper and performed in each collinear sector.

A Subleading Operator Basis and Matching for $gg \to H$ (2017) (same authors) · 0.87 · paraphrased: the licence does not allow quoting

How the literature uses it

359 arXiv papers cite this in 483 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 281 sentences without a confident label

Builds on it 4 papers

Here we extend the work of Ref. [ref] by setting up a factorization framework within Soft Collinear Effective Theory (SCET) [ref] , which allows for the systematic extension beyond LL.

The soft drop momentum sharing fraction $z_g$ beyond leading-logarithmic accuracy · Introduction · 0.91
In this paper we develop such a factorization and resummation framework for generic jet hierarchies in hard-scattering processes with large momentum transfer by considering an extension of Soft-Collinear Effective Theory (SCET) [ref] referred to as .

Factorization and Resummation for Generic Hierarchies between Jets · Introduction · 0.81
By extension from SCET [ref] , two collinear sectors in $$, $n_1$ and $n_2$, are distinct if

Systematic Improvement of Parton Showers with Effective Theory · Finite RPI · 0.89

Uses its method or tool 56 papers

It requires the combination of numerically robust NNLO Drell-Yan-plus-jet production evaluated at low transverse momentum cut-off [ref] with unresolved N$^3$LO contributions including logarithmic divergent terms [ref] predicted via Soft-Collinear Effective Theory (SCET) [ref] .

Differential N$^3$LO QCD corrections to charged current production at the LHC · Implementation · 0.81
[ref] using soft-collinear effective theory (SCET) [ref] .

Impact of Jet Veto Resummation on Slepton Searches · Jet veto resummation · 0.99
We apply the TMD factorization formula [ref] derived in soft-collinear effective theory (SCET) [ref] to obtain the N$^3$LL resummation formula, and the $𝒪(α_s^2)$ fixed-order matching distribution is obtained using the NLOJET++ event generator [ref] .

N$^{\mathbf{3}}$LL + $\mathcal{O}(\alpha_s^2)$ predictions of lepton-jet azimuthal angular distribution in deep-inelastic scattering · Introduction · 0.99

Compares with it 3 papers

This differs conceptually from the standard SCET construction [ref] , in which the scaling of all momentum components follows from the on-shell condition and the associated power counting.

Color Coherence and the Soft Structure of QCD Jets in Vacuum and the QGP · Separation of vacuum and medium time scales · 0.86
The beam function definitions in B_def can be compared with those of the standard quark, antiquark, and gluon PDFs in SCET [ref] ,

Factorization at the LHC: From PDFs to Initial State Jets · The Beam Function · 0.99
Because of time-ordered product contributions from diagrams with onshell propagators in the EFT these matching calculations are more non-trivial than the SCET matching done to yield Wilson lines in Refs. [ref] .

An Effective Field Theory for Forward Scattering and Factorization Violation · Guide for the Reader · 0.85

Cites it as background 126 papers

For this purpose, soft-collinear effective theory (SCET) [ref] has been developed and widely used until now.

Dijet invariant mass distribution near threshold · Introduction · 0.81
Another option is to use the $N$-jettiness variable [ref] , first introduced in the context of Soft Collinear Effective Theory (SCET) [ref] , to distinguish processes with a different number of jets.

Beam functions for $N$-jettiness at N$^3$LO in perturbative QCD · Introduction · 0.92
For a detailed discussion of the effective Lagrangian of SCET-1, including the leading power corrections, we refer the reader to the founding papers [ref] (see [ref] for a review).

The Simplest B Decay, Precisely · Matching to Soft-Collinear Effective Theory · 0.93

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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