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The Cut-Constructible Part of QCD Amplitudes

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Ruth Britto, Bo Feng, Pierpaolo Mastrolia

High Energy Physics - Phenomenology · Submitted 20 Feb 2006 · 65 references

Pith's reading: The authors complete the cut-constructible part of the one-loop six-gluon QCD amplitude by turning unitarity-cut integrals into algebraic residue extraction.

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arXiv IDhep-ph/0602178 (version 1)
Pith IDTXE6ISF3, Pith's permanent record of this paper and its review
CitationBo Feng, Pierpaolo Mastrolia, Ruth Britto. "The Cut-Constructible Part of QCD Amplitudes." arXiv:hep-ph/0602178 (2006). https://pith.science/paper/TXE6ISF3
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Claim paper
The paper

The paper claims

The authors complete the cut-constructible part of the one-loop six-gluon QCD amplitude by turning unitarity-cut integrals into algebraic residue extraction.

The reading

At one quantum loop, a gluon scattering amplitude receives contributions from a gluon, a fermion, and a scalar circulating in the loop. The authors isolate the scalar-loop term and split it into a cut-constructible part, which can be recovered by putting two internal particles on shell, and a rational remainder. This paper deals with the cut-constructible part.

The calculation works in spinor-helicity variables, where a massless momentum is represented by two spinors. A double cut fixes the loop momentum up to a one-dimensional integral. The authors rewrite that integral so that most of it becomes a total derivative, and the result is then a sum over residues at poles. Their new technical step is handling multiple poles, which had not appeared in earlier supersymmetric calculations. They also derive compact formulas for three-mass triangle coefficients, which require one Feynman parameter integration.

The main output is a set of explicit coefficients for bubble and triangle integrals for the three non-MHV six-gluon helicity configurations. Only one of these configurations, the split helicity one, had been computed before. The paper does not provide numerical checks: the authors state that automatic implementation and cross-checks are left to future work, and the formulas are complex enough that hidden spinor identities make analytic comparison difficult.

Extended reading notes

Core claim

With the complete algorithm developed in this paper, we calculate the heretofore missing cut-constructible part of the NLO six-gluon amplitudes in QCD. If the paper is correct, the explicit bubble and triangle coefficients given in Sections 3, 4, and 5 are the correct and complete cut-constructible coefficients for the three NMHV six-gluon helicity configurations.

Load-bearing premise

The reduction to the modified basis in Eq. (2.4), which drops all one-mass and two-mass triangle integrals on the grounds that their 1/epsilon^2 divergences cancel, is assumed to leave the finite coefficients unchanged. If this cancellation is not exact for the cut-constructible parts, the extracted bubble and triangle coefficients would be shifted. This premise enters in Section 2 between Eqs. (2.3) and (2.4).

Original abstract

Unitarity cuts are widely used in analytic computation of loop amplitudes in gauge theories such as QCD. We expand upon the technique introduced in hep-ph/0503132 to carry out any finite unitarity cut integral. This technique naturally separates the contributions of bubble, triangle and box integrals in one-loop amplitudes and is not constrained to any particular helicity configurations. Loop momentum integration is reduced to a sequence of algebraic operations. We discuss the extraction of the residues at higher-order poles. Additionally, we offer concise algebraic formulas for expressing coefficients of three-mass triangle integrals. As an application, we compute all remaining coefficients of bubble and triangle integrals for nonsupersymmetric six-gluon amplitudes.

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, 5,434 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.30 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Unitarity cuts are widely used in analytic computation of loop amplitudes in gauge theories such as QCD. We expand upon the technique introduced in hep-ph/0503132 to carry out any finite unitarity cut integral. This technique naturally separates the contributions of bubble, triangle and box integrals in one-loop amplitudes and is not constrained to any particular helicity configurations. Loop momentum integration is reduced to a sequence of algebraic operations. We discuss the extraction of the residues at higher-order poles. Additionally, we offer concise algebraic formulas for expressing coefficients of three-mass triangle integrals. As an application, we compute all remaining coefficients of bubble and triangle integrals for nonsupersymmetric six-gluon amplitudes.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Unitarity cuts are widely used in analytic computation of loop amplitudes in gauge theories such as QCD. background 1.00 0.05 0.03 asserted 1.00
s1 We expand upon the technique introduced in hep-ph/0503132 to carry out any finite unitarity cut integral. contribution 0.90 0.86 0.18 asserted 0.99
s2 This technique naturally separates the contributions of bubble, triangle and box integrals in one-loop amplitudes and is not constrained to any particular helicity configurations. method 0.64 0.69 0.19 asserted 1.00
s3 Loop momentum integration is reduced to a sequence of algebraic operations. method 0.93 0.70 0.12 asserted 1.00
s4 We discuss the extraction of the residues at higher-order poles. contribution 0.65 0.81 0.06 asserted 0.94
s5 Additionally, we offer concise algebraic formulas for expressing coefficients of three-mass triangle integrals. contribution 0.90 0.94 0.08 asserted 0.99
s6 As an application, we compute all remaining coefficients of bubble and triangle integrals for nonsupersymmetric six-gluon amplitudes. result 0.77 0.96 0.23 asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from proof or derivation 0.97.

What was examined, and what was not

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not examined
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  • 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. Scalar bubble and triangle integrals. (a) One-mass triangle I 1m 3;i . (b) Two￾mass triangle I 2m 3:r;i . (c) Three-mass triangle I 3m 3:r:r ′ ;i . (d) Bubble I2:r;i. Note that the modified basis in (2.4) involves only three-mass triangles and bubbles. The justification for eliminating one-mass and two-mass triangles is based on the observation that according to (2.3), Ascalar n diverges as 1/ǫ. Since one-mass and two-mass triangle integrals are the only ones with larger, 1/ǫ2 , divergences, these leading terms must conspire to cancel. After observing further that these 1/ǫ2
Figure 2. Scalar box integrals. (a) The outgoing external momenta at each of the vertices are K1, K2, K3, K4, defined to correspond to sums of the momenta of gluons in the exact orientation shown. (b) One-mass I 1m 4;i . (c) Two-mass “easy” I 2m e 4:r;i . (d) Two-mass “hard” I 2m h 4:r;i . (e) Three-mass I 3m 4:r:r′ ;i . (f) Four-mass I 4m 4:r:r′ :r”;i . 7
Figure 3. Representation of the cut integral. Left and right tree-level amplitudes are on-shell. Internal lines represent the legs coming from the cut propagators. On the right-hand side, the scalar integrals are known functions, so their disconti￾nuities are also known explicitly. We can read out the coefficients of box integrals from quadruple cuts, while coefficients of bubbles and three-mass triangle integrals from double cuts. The procedure of [12] can be extended to box integrals as well, but since it is more efficient to use quadruple cuts, we do not explore that possibility any…

Figures reproduced from arXiv:hep-ph/0602178 by the authors.

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References
How the literature uses it

53 arXiv papers cite this in 72 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 · 31 sentences without a confident label

Builds on it 1 paper
  • In this section we describe an extension of the formalism for explicit evaluation of double cuts [ref] to the single cut case [ref] .

    Massive particles and unitarity cuts · Single cut integration · 0.97

Uses its method or tool 12 papers
Compares with it 2 papers
  • We have checked numerically that the result agrees with the corresponding result derived by the technique of [ref] .[We were unable to confirm numerically the printed value of the corresponding coefficient in [ref] , so we repeated the calculation.]

    Integral Coefficients for One-Loop Amplitudes · Bubble coefficient · 0.90

  • The cut-constructible parts of the ordered amplitudes are also known analytically ( [ref] , [ref] , [ref] ), making a direct comparison possible.

    A Numerical Unitarity Formalism for Evaluating One-Loop Amplitudes · Numerical results · 0.98

Cites it as background 20 papers
  • Some widely mentioned methods include the Integration-by-Part (IBP) method [ref] , the unitarity cut method [ref] , the intersection theory method [ref] .

    General One-loop Generating Function by IBP relations · Motivation · 0.93

  • Besides the traditional Passarino-Veltman (PV) reduction [ref] which aims to solve the one-loop reduction, there are various methods available to one-loop or higher-loop reduction at the integrand and integral levels, such as Integration-By-Parts (IBP) [ref] , Ossola-Papadopoulos-Pittau(OPP) reduction [ref] , Unitarity cut [ref] , Intersection number [ref] and etc.

    Solving arbitrary one-loop reduction via generating function · Introduction · 0.93

  • The recent development of on-shell [ref] and generalized unitarity techniques [ref] for quadruple- [ref] , triple- [ref] , double- [ref] and single- [ref] cut allowed tremendous simplifications in one-loop calculations, where the knowledge of tree-level amplitudes can be exploited in order to determine the coefficients of the known basis of integrals in which any amplitude can be decomposed, [ref] .

    BCJ Identities and $d$-Dimensional Generalized Unitarity · Introduction · 0.92

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