Invariant Operators in Collinear Effective Theory

Christian W. Bauer; Iain W. Stewart

REVIEW

Invariant Operators in Collinear Effective Theory

Christian W. Bauer, Iain W. Stewart

High Energy Physics - Phenomenology · Submitted 2 Jul 2001 · 12 references

Pith's verdict: The label-operator machinery and the four construction rules are the real contribution here, and they hold up; the completeness claim after Eq. (13) is asserted rather than proved, and the soft-gluon scope claim leans on an unpublished companion.

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arXiv IDhep-ph/0107001 (version 2)

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CitationChristian W. Bauer, Iain W. Stewart. "Invariant Operators in Collinear Effective Theory." arXiv:hep-ph/0107001 (2001). https://pith.science/paper/XQIBCQLM

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Editor's note

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The thing worth your time here is the bookkeeping device, not the B → Dπ example. Bauer and Stewart introduce a label operator P̄ that acts on the large light-cone momenta of collinear fields, and with it they get a momentum-space Wilson line W satisfying a linear transport equation (Eq. 9). That buys them a closed-form gauge-invariant collinear quark kinetic term, Eq. (14), which Ref. [2] could only write as an infinite series — the paper says so itself in Section III, and it is a real improvement. The four rules at the end of Section II are the kind of organizing statement a subfield actually adopts, and Eq. (19), with the Wilson coefficient sandwiched between ξ̄W and W†ξ, is the payoff: it tells you the coefficient depends only on the total label and factors outside collinear loops. I checked the algebra in Eqs. (10) and (12); the unitarity step is used correctly and the covariant-derivative transformation is right. Nothing hand-wavy in the derivations themselves. The Section IV cross-check is a genuine benchmark, not a trivial identity — the α-dependence in Eq. (24) is not inherited for free from Ref. [6]. Where I'd push back is the sentence right after Eq. (13), 'the most general allowed operators must satisfy two more rules.' What is shown is that W†ξ, ξ̄W and the transverse covariant derivative are covariant, and that f(P̄ + g n̄·A) = W f(P̄) W†. That is necessity, not sufficiency. Nobody demonstrates that every invariant can be built from these blocks, and field-strength operators at order λ are not discussed at all. For B → Dπ the 'only one color-singlet operator' claim rests on that gap, though the successful reproduction of Eq. (193) of Ref. [6] is decent indirect evidence that the basis is complete at leading order for this process. Separately, Section I's 'including soft gluons does not alter any of the results presented' points at hep-ph/0107002, which was unpublished at the time; that is a scope limit, not a result. Neither gap is fatal, and the paper is honest that it treats collinear gluons only. This is for anyone building operators in SCET or working on exclusive B decays; the rules save you a lot of guesswork. Send it out. The referee should ask for the completeness argument to be stated as a conjecture rather than a conclusion, and for the soft-gluon sentence to be softened or supported.

Referee report

Significance

If the construction holds, this is a foundational paper: the label-operator and momentum-space Wilson-line technology introduced here became a standard tool for organizing operators in collinear effective theory, and the four rules give a practical recipe that has been widely used since. The Section IV comparison is a genuine external benchmark rather than a self-consistency check: a two-loop hard-collinear calculation is reproduced from one-loop effective-theory ingredients, which is a nontrivial and falsifiable test of the framework. The explicit Feynman rules of Eqs. (20) and (22) make the formalism directly usable, and the observation that the Wilson coefficient depends on the total label P̄+ rather than on individual parton momenta is a clean conceptual result. The paper is also commendably explicit about what it does and does not cover, restricting the detailed analysis to collinear gluons and deferring soft modes to a companion paper.

Major objections (1)

[Section II (after Eq. 13) and Section IV (Eq. 19)] The argument following Eq. (13) establishes that W†ξ, ξ̄W, and (P/⊥ + gA/⊥) are gauge covariant, and that f(P̄ + g n̄·A) = W f(P̄) W†. The text then concludes that 'the most general allowed operators must satisfy two more rules' (rules 3 and 4). As stated, rules 3 and 4 are necessary conditions on allowed operators, which is correct, but Section IV subsequently uses them as a sufficiency statement: 'the most general operators which satisfy the rules in section II are' Eq. (19), and 'Only one color singlet operator can be constructed which contributes to this decay.' The manuscript does not give an argument that every gauge-invariant operator can be written in terms of the stated building blocks, nor does it discuss whether operators built from the collinear field strength G_{μν} (order λ, and relevant to the 'subleading' operators advertised in the abstract) are exhausted by the rules. The successful reproduction of Eq. (193) of Ref. [6] from Eqs. (20)–(24) is strong indirect evidence that the basis is complete at leading order for this particular process, but it does not establish the general claim. I would ask the authors to state explicitly that rules 3 and 4 are necessary conditions whose sufficiency is argued constructively, to say a sentence about how field-strength operators are generated within the rules, and to soften or qualify the 'most general' language in Section IV accordingly. This is a framing and rigor issue rather than a demonstrated error, and it can be addressed with a short passage.

Minor objections (7)

[Section I versus Section IV heading] Section I refers to the decays 'B0 → D+π0 and B− → D0π−', whereas the Section IV heading and body use 'B̄0 → D+π− and B− → D0π−'. The two sets are not the same processes, and the Section I version appears to be a typographical slip; the labels should be made consistent throughout.
[Section I and Section IV, Ref. [11]] The scope statement in Section I that 'including soft gluons does not alter any of the results presented', and the deferral in Section IV of the Q_{i8} discussion and the all-orders factorization argument to Ref. [11], both rest on hep-ph/0107002, which is listed as a preprint rather than a published reference. The authors do acknowledge the restriction to collinear gluons, but a reader cannot verify the soft-gluon claim from the present manuscript; it would help to state the scope limitation more explicitly or to update the citation if the companion has since appeared.
[Eq. (21)] The argument of the logarithm in the term '(3 + 2 ln u¯u) ln z²' is ambiguous in the current typesetting; it should be written unambiguously (for example as ln(u/ū) or ln(ū u), whichever is intended) so that the comparison with Refs. [4,6] can be checked directly.
[Eq. (22)] The one-gluon Feynman rule in Eq. (22) is written as '= ig n̄μ Γh ⊗ Γℓ T^A {…}', whereas the zero-gluon rule in Eq. (20) carries an explicit overall factor of i. Please check whether the overall i is missing in Eq. (22) or is absorbed by a convention stated elsewhere.
[Eq. (17)] The intermediate step in Eq. (17), in which the matrix element is split into two terms with a relative minus sign before reducing to CΓ(μ, 2Eπ), is difficult to follow as written. One additional line showing how P̄† acting to the left on the pion state and P̄ acting to the right on the heavy-meson state combine would make the derivation transparent.
[Eqs. (24) and (26)] The notation 'iA_abc +' in Eq. (26) is not defined; presumably the '+' denotes the addition of the collinear wavefunction contribution, but this should be stated explicitly so that the relation to Eq. (24) is unambiguous.
[Section IV, below Eq. (23)] The instruction to 'drop the p⊥ and p′⊥ dependence' when reducing Eq. (23) to Eq. (24) is stated without justification. A brief remark on why these terms do not contribute to the quantity being compared with Ref. [6] would help the reader.

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The paper

The paper claims

The paper argues that gauge invariance in a theory of nearly light-like quarks and gluons collapses into four bookkeeping rules, and that a single operator built from them reproduces a two-loop factorization result for $B$ decays from one-{

The reading

The paper asks how to write down every allowed interaction operator in a theory of quarks and gluons that move nearly at the speed of light in one direction, as the pion's constituents do in $B$ decays. Gauge invariance normally forbids writing an operator with a collinear quark and a soft quark separated in space unless a Wilson line connects them, so the authors want a device that turns that constraint into a mechanical recipe. They introduce an operator $P$ (written with a bar) that reads off the sum of the large momentum labels of the fields in an operator, together with a momentum-space Wilson line $W$ built from the large-component gluon field, and show that gauge invariance reduces to four rules. If those rules are complete, the most general operator, at leading order and at suppressed orders, can be written down by inspection, with short-distance coefficients that are functions of $P$ inserted between invariant blocks. The payoff demonstrated is $B$ to $D$ plus pion: one color-singlet operator with coefficient $C_0( mu, P_+)$ reproduces a two-loop hard-collinear factorization result from one-loop effective-theory graphs, and extends the convolution to the short-distance coefficient at any order.

What carries the argument

Two objects carry the argument. The first is the label operator $P$, which acts on a product of collinear fields to give the sum of their large light-cone momentum labels minus the sum of the labels on conjugate fields, so that large phases can be pulled to the front of any operator and label sums need no longer be written explicitly. The second is the momentum-space Wilson line $W$, defined as a sum over permutations of exponentials of $-g$ times the large gluon component divided by $P$, which satisfies the linear equation $[(P + g n A) W] = 0$. That equation, plus the uniqueness of its solution, is what proves $W$ transforms as $W$ to $U$ transpose times $W$, making the combination of a collinear quark field with $W$ invariant. The identity $f(P + g n A) = W f(P) W^dagger$ then guarantees that gauge-invariant dependence on the large gluon field appears only inside $W$, which is what forces Wilson coefficients to be label-operator functions inserted between separately invariant blocks. The transverse covariant derivative built from the order-$lambda$ transverse momentum operator and the transverse gluon field is the other invariant building block, and it is what the kinetic term of Eq. (14) uses to close the previously open-ended set of collinear quark couplings.

Extended reading notes

Core claim

The central claim is that the collinear gauge symmetry of a theory of fast-moving quarks and gluons is strong enough that operator construction becomes a finite set of rules rather than a case-by-case exercise. The authors state four: label changes of variables are allowed; label momenta are conserved in Feynman rules; only operators invariant under the collinear gauge transformation of their Eq. (7) are allowed; and Wilson coefficients are functions of the label operators $P$ and $P^dagger$ inserted at every location consistent with the third rule. The load-bearing technical step is that a function of the large gluon field plus $P$ can always be rewritten as $W f(P) W^dagger$, so that gauge-invariant dependence on the large gluon component appears only through the Wilson line. Applied to $B$ to $D$ plus pion, the rules leave exactly one color-singlet operator, and expanding its $W$ factors fixes the coefficients of vertices with any number of collinear gluons in terms of the same function $C_0$. Summing the three one-loop collinear graphs built from that operator gives an expression the authors identify with the explicit two-loop hard-collinear result of an earlier calculation, and which holds with the short-distance coefficient matched at any order.

Load-bearing premise

The argument shows that the Wilson line and the transverse covariant derivative are gauge covariant and that gluon dependence can be absorbed into $W$, but it asserts rather than proves that the four rules generate every allowed operator, and it also asserts without demonstration that adding soft gluons changes none of the results.

Editorial extensions

If this is right

Coefficients for vertices with any number of collinear gluons emerging from the four-fermion operator are fixed by the same function $C_0$ through the exponentials in $W$; with tree-level matching those multi-gluon rules vanish identically because $W^dagger W = 1$.
The collinear quark kinetic term acquires a closed gauge-invariant form in which the infinite set of couplings to the large gluon component is resummed into the Wilson lines surrounding the inverse label operator.
In heavy-to-light currents the Wilson coefficient factors outside loops containing only collinear gluons, so the coefficient depends on the total large momentum $2E_pi$ rather than on any individual parton's momentum, which explains how a scale that no single parton carries enters the answer.
The one-loop effective-theory graphs yield the convolution of the short-distance coefficient with the one-loop light-cone pion wavefunction while holding the coefficient matched at arbitrary order, which the authors present as evidence for generalized factorization of $B$ to $D$ plus pion beyond two loops.

Reading between the lines

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

The label operator is best read as a device that turns momentum conservation into bookkeeping: it makes the non-commutativity of short-distance coefficients with fields explicit rather than accidental, and the same trick looks portable to other hierarchies where a large momentum is shared among many fields, such as jet or small-transverse-momentum observables.
If the completeness argument holds, counting independent operators for a given exclusive decay becomes a short combinatorial exercise, which would let operator bases for neighbouring channels be enumerated by hand rather than by exhaustive enumeration of Dirac and color structures.
The natural stress point is the treatment of soft gluons: the paper asserts in its introduction that including them alters none of the results and defers the demonstration to a companion paper, so a reader should watch whether soft emissions generate invariant structures that the Wilson line does not capture, since that would break the claimed uniqueness of the single color-singlet operator.

Original abstract

We consider processes which produce final state hadrons whose energy is much greater than their mass. In this limit interactions involving collinear fermions and gluons are constrained by a symmetry, and we give a general set of rules for constructing leading and subleading invariant operators. Wilson coefficients C(mu,P) are functions of a label operator P, and do not commute with collinear fields. The symmetry is used to reproduce a two-loop result for factorization in B -> D pi in a simple way.

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, 3,533 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.18 · compares with prior results 0.65 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We consider processes which produce final state hadrons whose energy is much greater than their mass. In this limit interactions involving collinear fermions and gluons are constrained by a symmetry, and we give a general set of rules for constructing leading and subleading invariant operators. Wilson coefficients C(mu,P) are functions of a label operator P, and do not commute with collinear fields. The symmetry is used to reproduce a two-loop result for factorization in B -> D pi in a simple way.

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	We consider processes which produce final state hadrons whose energy is much greater than their mass.	background 0.97	0.15	0.03	asserted 0.95
s1	In this limit interactions involving collinear fermions and gluons are constrained by a symmetry, and we give a general set of rules for constructing leading and subleading invariant operators.	contribution 0.95	0.89	0.51	asserted 1.00
s2	Wilson coefficients C(mu,P) are functions of a label operator P, and do not commute with collinear fields.	result 0.50	0.56	0.12	asserted 1.00
s3	The symmetry is used to reproduce a two-loop result for factorization in B -> D pi in a simple way.	result 0.86	0.86	0.33	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.83 · evidence from proof or derivation 0.99.

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

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S sections
T body sentences
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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. Typical momenta/positions for soft particles (shad

**Figure 2.** Replacing h (b) v , h (c) v ′ , and ξ (u) n,p′ ¯ξ (d) n,p with color singlet interpolating fields with the correct spin structure, the spin and color traces can be performed to give 9

Figures reproduced from arXiv:hep-ph/0107001 by the authors.

References

Works cited (12)

[2]

C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys. Rev. D 63, 114020 (2001)

View on Pith 2001
[6]

M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, N ucl. Phys. B591, 313 (2000)

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[1]

C. W. Bauer, S. Fleming and M. Luke, Phys. Rev. D63, 014006 (2001)

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[3]

See for example: J. C. Collins and D. E. Soper, Nucl. Phys. B 194, 445 (1982); J. C. Collins, D. E. Soper and G. Sterman, Nucl. Phys. B 261, 104 (1985)

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[4]

H. D. Politzer and M. B. Wise, Phys. Lett. B 257, 399 (1991)

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[5]

M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, P hys. Rev. Lett. 83, 1914 (1999)

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[7]

G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 (1980)

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[8]

H. Georgi, Phys. Lett. B240, 447 (1990)

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[9]

M. J. Dugan and B. Grinstein, Phys. Lett. B255, 583 (1991)

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[10]

J. Charles, A. Le Yaouanc, L. Oliver, O. Pene, and J. C. Ra ynal, Phys. Rev. D60, 014001 (1999)

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[11]

C. W. Bauer, D. Pirjol and I. W. Stewart, hep-ph/0107002

View on Pith Pith review arXiv
[12]

A. F. Falk, H. Georgi, B. Grinstein and M. B. Wise, Nucl. P hys. B 343, 1 (1990); A. F. Falk and B. Grinstein, Phys. Lett. B 249, 314 (1990); M. Neubert, Nucl. Phys. B 371, 149 (1992); see also M. Neubert, Phys. Rept. 245, 259 (1994). 12

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Higgs boson pair production via gluon fusion at N$^3$LO in QCD
hep-ph 2019-09 accept novelty 7.0 of 10

The N3LO QCD correction to gluon-fusion Higgs pair production raises the NNLO cross section by 3.0% at 13 TeV and cuts the scale uncertainty below 3%, in the infinite top-quark mass limit.
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.
Jet charge modification in dense QCD matter
hep-ph 2019-08 conditional novelty 7.0 of 10

The authors derive a factorized formula for the average jet charge in heavy-ion collisions from soft-collinear effective theory and predict its modification relative to proton-proton collisions.
The soft drop groomed jet radius at NLL
hep-ph 2019-08 accept novelty 7.0 of 10

A next-to-leading-logarithmic factorization and resummation for the soft drop groomed jet radius, including non-global and clustering logarithms, with predictions for the LHC and RHIC.
The Collinear Limit of the Energy-Energy Correlator
hep-ph 2019-05 conditional novelty 7.0 of 10

A factorization formula for the collinear limit of the energy-energy correlator enables next-to-next-to-leading-log resummation in QCD and N=1 SYM, and reduces to a spacelike power law in N=4 SYM.
The Transverse Energy-Energy Correlator in the Back-to-Back Limit
hep-ph 2019-01 conditional novelty 7.0 of 10

The transverse energy-energy correlator in the back-to-back limit factorizes into standard beam and jet functions times a dijet soft function that reduces to the known color-singlet TMD soft function through two loops.
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...
The Energy Distribution of Subjets and the Jet Shape
hep-ph 2017-05 accept novelty 7.0 of 10

Subjet energy fractions inside a jet are factorized and resummed at the r and R angular scales, with the standard jet axis requiring an extra soft-recoil function that earlier jet-shape calculations omitted.
Direct photon production at next-to-next-to-leading order
hep-ph 2016-12 accept novelty 7.0 of 10

Direct photon production is computed at NNLO in QCD for the first time, and the resulting NNLO plus electroweak prediction matches ATLAS 8 TeV data with substantially reduced theoretical uncertainty.
Resummation of Transverse Momentum Distributions in Distribution Space
hep-ph 2016-11 conditional novelty 7.0 of 10

A distributional scale-setting prescription allows the QCD renormalization-group evolution of the transverse-momentum spectrum to be carried out entirely in momentum space, resumming the logarithmic plus distributions...
New Angles on Energy Correlation Functions
hep-ph 2016-09 conditional novelty 7.0 of 10

Generalizing energy correlation functions to weight v of the pairwise angles yields three new jet-substructure ratios (M_i, N_i, U_i) that outperform D2 and N-subjettiness in parton-shower studies, with N_i reproducin...
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...
A Soft-Collinear Mode for Jet Cross Sections in Soft Collinear Effective Theory
hep-ph 2015-09 conditional novelty 7.0 of 10

A new soft-collinear mode in SCET factorizes the veto soft function into global soft and soft-collinear pieces at scales 2Λ and 2ΛR, enabling resummation of global R-logs in jet thrust and exclusive 2-jet cross sections.
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...
Gaining (Mutual) Information about Quark/Gluon Discrimination
hep-ph 2014-08 conditional novelty 7.0 of 10

Mutual information with the truth quantifies quark/gluon tagging power, and a two-parameter family of generalized angularities is computed to next-to-leading logarithmic accuracy, identifying which jet observables are...
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...
Combining Higher-Order Resummation with Multiple NLO Calculations and Parton Showers in GENEVA
hep-ph 2012-11 conditional novelty 7.0 of 10

A master formula multiplicatively matches N-jettiness resummation to fully differential NLO matrix elements and Pythia 8, giving a Monte Carlo formally accurate to relative O(alpha_s) at all jet resolutions.
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.
Thrust at N^3LL with Power Corrections and a Precision Global Fit for alphas(mZ)
hep-ph 2010-06 conditional novelty 7.0 of 10

Fitting all published e+e- thrust data between 35 and 207 GeV with an N3LL resummed plus O(alpha_s^3) fixed-order calculation and a fitted nonperturbative soft-function moment yields alpha_s(m_Z) = 0.1135 +/- 0.0011.

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How later papers used it

429 later papers cite it

Evidence: substantial. 354 of the citing papers share no author with it, and 77 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 595 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

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

Uses its method or tool 2 of 92 papers with this label · 1 by the same authors

Performed a field re-definition as in SCET to strip a Wilson line from the field.

On Tree Amplitudes in Gauge Theory and Gravity (2008) · 0.96 · paraphrased: the licence does not allow quoting

By the same authors

i∂_μ→P_μ+ i∂_μ , where the label operator $P$ [ref] only operates on the momentum label space, $P^i ψ_p(x) = p^i ψ_p(x)$.

Manifestly Soft Gauge Invariant Formulation of vNRQCD (2018) (same authors) · Review of NRQCD with Single Stage Matching · 0.85

Cites it as support 1 paper

Used this paper's SCET gauge invariance of the jet field to support that loop corrections to this operator and to the jet field are the same.

Decay b -> (c\bar{c}) s in the leading logarithm approximation (2007) · 0.96 · paraphrased: the licence does not allow quoting

How the literature uses it

434 arXiv papers cite this in 603 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 · 314 sentences without a confident label

Builds on it 2 papers

Here we calculate the jet charge using and extending the framework for fragmentation in jets [ref] in Soft-Collinear Effective Theory (SCET) [ref] .

Calculating the Charge of a Jet · Introduction · 0.93
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

Uses its method or tool 93 papers

[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
Then, the soft-collinear effective theory (SCET) [ref] and the heavy quark effective theory (HQET) [ref] are used to embody those dynamic modes, thereby calculating the effective amplitudes and the respective differential cross sections.

Projected transverse momentum resummation in top-antitop pair production at LHC · Introduction · 0.96

Cites it as support 1 paper

SCET gauge invariance of the jet field $[ξ̅_n,p W]$ ensures that loop corrections to this operator and to $ξ̅_n,p$ are the same [ref] .

Decay b -> (c\bar{c}) s in the leading logarithm approximation · Summing the leading logarithms · 0.96

Cites it as background 149 papers

There have been two primary ways of resumming these jet radius logarithms, one using generating functions/parton showers [ref] , and another using factorization theorem derived [ref] with effective field theory [ref] .

Revisiting Single Inclusive Jet Production: Small-$R$ Resummation at Next-to-Leading Logarithm · Introduction · 0.91
The soft collinear effective theory (SCET) [ref] , an effective field theory describing the soft and collinear limits of QCD, provides an operator and Lagrangian based formalism for deriving factorization theorems at subleading power.

Subleading Power Factorization with Radiative Functions · Introduction · 0.82
In this talk, I first review soft-collinear effective theory (SCET) [ref] , which is an effective field theory describing the dynamics of highly energetic particles moving close to the light-cone interacting with a background field of soft quanta.

Soft Collinear Effective Theory: An Overview · Introduction · 0.95

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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