A new method for real radiation at NNLO

C. Anastasiou; Frank Petriello; Kirill Melnikov

REVIEW

A new method for real radiation at NNLO

C. Anastasiou, Kirill Melnikov, Frank Petriello

High Energy Physics - Phenomenology · Submitted 24 Nov 2003 · 18 references

Pith's verdict: A real, working method for double real emission, with the hardest sector validated only inclusively rather than differentially.

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arXiv IDhep-ph/0311311 (version 2)

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CitationC. Anastasiou, Frank Petriello, Kirill Melnikov. "A new method for real radiation at NNLO." arXiv:hep-ph/0311311 (2003). https://pith.science/paper/ZIZY6BKS

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Editor's note

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This one is worth your time if you care about NNLO infrastructure. The idea is simple to state and, as far as I can tell, correct: take the exclusive four-particle phase space, parametrize it in variables running from 0 to 1 (Eq. 21, Eq. 33), push interior singularities to the boundary, then use sector decomposition to disentangle the overlapping soft/collinear limits and expand in plus distributions. No analytic integration of counterterms, no process-specific dipole derivation, and jet functions drop in at the end as a weight on the integrand. That last point is the practical payoff, and the authors make it look easy in Section V, which is exactly what you want from a methods paper. The lineage to Binoth-Heinrich is properly credited; the new part is the exclusive phase space and the double-emission bookkeeping, not sector decomposition itself. The validation is better than I expected. The planar and non-planar real-real cuts (Eq. 40 vs Eq. 42, Eq. 47 vs Eq. 49) are checked against optical-theorem results built from MINCER and Gonsalves, and the Nf total cross section lands on the textbook number. Those are external benchmarks, so the circularity worry is low. Where I would push back is on the stress-test note, which I think mostly lands. The differential demonstration in Section V uses denominators of the form 1/(s34 s234 s134) and so exercises the lambda2-lambda3 sector decomposition, but it contains no s13 and no double-soft configuration. The lambda5 remapping and the lambda4 splitting at lambda4^s are needed precisely for the s13-containing, double-soft topologies, and those are checked only through inclusive integrals in Section IV. An inclusive check pins the epsilon expansion but cannot distinguish a correct local subtraction from one that is wrong pointwise and integrates to the right total. So the headline claim, exact kinematics preserved in all regions, is supported by integrated evidence in the hardest sector. That is a validation gap, not a demonstrated error, and the authors are upfront about deferring the non-Nf terms. The finite-piece numerical errors (a few per mille to a percent) are honest but modest, and no code ships, so reimplementation is on you. None of this sinks the paper. It is a genuine methodological step, and the fact that this line of work became standard practice is not an accident. Send it out. A referee should ask for a differential check in an s13-containing sector, but the method itself deserves to be on record.

Referee report

Significance

If the method holds, it is a substantial contribution: it removes the two bottlenecks the authors identify in the introduction, namely the construction of double-real counterterms and their analytic integration, and it delivers a fully differential, automatable framework suitable for NNLO Monte Carlo event generators. The validation strategy is genuinely strong where it is applied. The unitarity checks in Section IV are independent of the method: the analytic real-real cuts follow from the optical theorem applied to propagator integrals evaluated with MINCER [15] and the virtual corrections of Gonsalves [16], and the agreement at the 0.1% level in Eqs. 42 and 49 is a meaningful test of the sector-decomposition machinery, including the s_13 and λ_5 remapping. The N_f total cross section in Eq. 55 agrees with the textbook value in Eq. 56, and the pole cancellation in Eq. 54 is shown separately for the 2- and 3-jet cross sections, as required. I also verified the sector-decomposition algebra of Eq. 31 by hand, including the Jacobians of λ_2 → λ_2λ_3 and λ_3 → λ_3λ_2; the two expressions D_a and D_b are correct. The explicit note added, disclosing the concurrent work of Ref. [18], is a

Major objections (1)

[Section IV vs. Section V; Eqs. (32)–(36), (41), (48), (51)] The two lines of evidence are disjoint in a way that bears on the abstract's claim [abstract:s2] that exact kinematics are preserved in all regions of phase space. The differential demonstration in Section V, Eq. (51), uses the N_f topology whose denominators are 1/(s_34 s_234 s_134) (Section III, below Eq. 25); this exercises the λ_2–λ_3 sector decomposition of Eq. (31) but contains no s_13 and no double-soft configuration. The λ_5 remapping of Eqs. (32)–(33) and the λ_4 splitting at λ_4^s of Eqs. (34)–(36) are needed precisely for the s_13-containing, double-soft topologies, and those are validated only through the inclusive integrals of Eqs. (41) and (48), whose results appear in Eqs. (42) and (49). An inclusive check constrains the integrated ε-expansion but cannot distinguish a correct local subtraction from one that is wrong pointwise yet integrates to the right total. This matters here because the paper itself notes (Section V, first paragraph) that the invariant masses take different forms in terms of the λ_i in each sector, so the cross term — a jet function or differential observable applied inside an s_13 sector — is exactly where a bookkeeping error would hide. I would like either (a) one differential or jet-function-weighted check in an s_13-containing sector, for example by applying a simple jet function or a bin in an invariant-mass distribution to the integrals of Eq. (41) or Eq. (48) and comparing against the correspondingly weighted optical-theorem result, or (b) an explicit statement in the abstract and conclusions that the differential demonstration covers the non-s_13 sectors and that the s_13 sectors are validated inclusively. Relatedly, the assertion below Eq. (33) that 'sector decomposition of the λ̂_5 integration is never required' is stated rather than demonstrated; a sentence explaining why the λ̂_5 endpoints produce only integrable (λ̂_5)^{-1/2} behaviour, with no 1/ε pole requiring a plus distribution, would close this.

Minor objections (12)

[Eq. (18)] The delta function is written δ(Q_{234} − s_34), whereas the analogous constraint in Eq. (17) is δ(Q_{234}^2 − s_234). The argument here should presumably be Q_{34}^2 − s_34; as written the equation is dimensionally inconsistent with Eq. (17).
[Eqs. (21) and (33)] The quantities λ'_i appearing in the delta functions δ(λ_i − λ'_i) are never defined. Presumably they are the roots of the constraint equations obtained from the delta functions of Eq. (20), but the reader cannot reconstruct the parametrization without them. A short definition, or a pointer to where they are obtained, would make the construction reproducible.
[Section IV, below Eq. (44)] The Mandelstam assignment reads 's = s_{q q̄} = λ_2, t = s_{q̄g} = (1−λ_2)λ_1, u = s_{q̄g} = (1−λ_2)(1−λ_1)', which duplicates the label s_{q̄g} for both t and u and conflicts with Eq. (5), where s_{q̄g} = λ_2 and s_{q q̄} = (1−λ_1)(1−λ_2). The labels should be made consistent with Section II.
[Eq. (33)] The exponent in the first line is typeset as '1 − 2e' where '1 − 2ε' is meant.
[Eq. (44)] The hypergeometric function is typeset as 'F21' rather than ₂F₁ throughout, including in Eq. (45).
[Eqs. (48) and (49)] The left-hand side is called R^{num}_{np} in Eq. (48) and then simply 'R =' in Eq. (49); the labelling should be made uniform so the reader can match the two.
[Eq. (10)] The final term in braces, '1 − λ_1 (1 − λ_2/2)', is ambiguously grouped; presumably (1−λ_1)(1−λ_2/2) is intended. Parentheses would remove the ambiguity.
[Section V, first paragraph] The paper states that jet-algorithm constraints are simple to implement, but does not discuss whether the boundary of the jet region (for example the surface where a pair invariant mass crosses y_cut) requires additional sector decomposition, since the plus-distribution extraction assumes the singular factors are separated before the jet function is applied. One sentence on this point would strengthen the claim.
[Conclusions, Section VI] The phrase 'comparing with analyting results' should read 'analytic results'.
[Notation throughout] The coupling is written α_S in the abstract and introduction and α_s elsewhere, and the flavour number as Nf rather than N_f; the abstract also uses an ASCII arrow ('electron-positron --> jets') where the body uses e+e−. Standardizing these would improve readability.
[Figures 1–3] The captions do not identify which lines are cut in the unitarity checks of Section IV, nor what the internal bubble in Fig. 3 represents. Brief captions stating the cut structure would help the reader follow the cut-by-cut bookkeeping in Eqs. (37)–(49).
[Eqs. (55) and (56)] The central values differ by about 0.7 of the quoted error; stating the pull explicitly, rather than only 'agrees', would be more informative.

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The paper

The paper claims

The paper proposes a way to compute the double real-emission part of NNLO QCD calculations without deriving counterterms or performing analytic integrals, by decomposing the four-particle phase space into sectors and expanding in plus-distr

The reading

Next-to-next-to-leading-order QCD calculations need the contribution where two extra partons are radiated, and that piece is the one nobody knows how to handle: its infrared and collinear singularities overlap, so the usual tricks of phase-space slicing and dipole subtraction require hand-built counterterms plus analytic integrals to turn them into $1/eps$ poles. This paper argues that you can get the poles out without integrating anything. Parametrize the four-particle phase space with variables running from 0 to 1, use sector decomposition to pull apart the overlapping singularities, and expand in plus distributions; the $1/eps$ poles then appear explicitly while every kinematic variable is still free. Because nothing has been integrated over, a jet function or any other final-state constraint can be dropped in afterwards, which is what a Monte Carlo event generator needs. The authors test the machinery on the hardest four-propagator scalar integrals against unitarity cuts, and on the $N_f$-dependent part of $e^+e^- ->$ jets at $O(alpha_s^2)$, recovering the known total hadronic cross section.

What carries the argument

Sector decomposition of the phase space, borrowed from multi-loop integral reduction, is the load-bearing device. In its simplest form (Eq. 26-28): an integrand with $x^{-1-eps} y^{-1-eps} (x+y)^{-eps}$ cannot be expanded term by term because the two singularities overlap, so one splits the unit square into $y < x$ and $x < y$, rescales $y -> y/x$ in the first and $x -> x/y$ in the second, and each resulting integral has its singularities in separate variables that the standard identity $lambda^{-1+eps} = -delta(lambda)/eps + sum_n (eps^n/n!) [ln^n(lambda)/lambda]_+$ can handle. Applied to the four-particle phase space, this turns a matrix element with denominators like $1/(s_{34} s_{234} s_{134})$ into a sum of terms whose $lambda$-dependence factorizes, so the $1/eps$ poles can be read off analytically while all five $lambda_i$ remain integration variables. The second device is the set of remappings that move interior singularities to the edges: the split of the $lambda_4$ integration at $lambda_4^s = lambda_2(1-lambda_3)/(lambda_3 + lambda_2(1-lambda_3))$, and the nonlinear change of variable for $lambda_5$ that cancels the $s_{13}$ denominator against its Jacobian. Together these are what make the procedure automatable and what let a jet function be applied after the poles are gone.

Extended reading notes

Core claim

The claim is that sector decomposition applied to the exclusive four-particle phase space, followed by a plus-distribution expansion, extracts all singularities of double real-emission matrix elements before integration over any kinematic variable, with the exact event kinematics preserved everywhere in phase space. Concretely, the authors give a factorized parametrization of the $1 -> 4$ phase space in five variables $lambda_i in [0,1]$ (Eq. 21, and Eq. 33 for terms containing $s_{13}$), show how to split integrations so that singularities sitting inside the allowed region are pushed to the boundary (the $lambda_4$ split at $lambda_4^s$ in Eq. 36, and the nonlinear remapping $lambda_5 -> hat lambda_5$ in Eq. 32), and then apply sector decomposition to disentangle the overlapping limits. The demonstration is twofold: the most difficult real-real cuts of planar and non-planar three-loop propagator diagrams are computed and shown to satisfy the unitarity sum rules to better than 0.1% where statistics allow, and the $N_f$ contribution to $e^+e^- -> 2, 3, 4$ jets at $O(alpha_s^2)$ is computed with a JADE jet function at $y = 0.1$, with the $1/eps$ poles cancelling numerically sector by sector and the summed cross section, $(-0.117 pm 0.003) N_f (alpha_s/pi)^2$, matching the known analytic value $(2/3)zeta_3 - 11/12 = -0.115$.

Load-bearing premise

The load-bearing premise is that the five-variable parametrization together with the interior-singularity splittings (the $lambda_4$ split at $lambda_4^s$ and the $lambda_5$ remapping) moves every singularity onto the boundary of the unit cube, so that sector decomposition and the plus-distribution expansion really do capture all of them rather than only the ones exercised in the test integrals.

Editorial extensions

If this is right

Differential NNLO distributions become accessible, not just the total $e^+e^- ->$ hadrons rate: the invariant mass and angular information about the jets, which the older difference-of-cross-sections technique discards, is retained because the jet function is applied to an unintegrated integrand.
Any infrared-safe jet algorithm can be imposed on the double real-emission contribution by evaluating the jet function on the sector-dependent expressions for the invariant masses, with no new analytic work.
The same procedure extends to other massless $1 -> 4$ processes, and the authors suggest it can be combined with dipole subtraction, using sector decomposition to extract the dipole poles numerically instead of analytically.
Because the extraction is automated, the remaining non-$N_f$ contributions to $e^+e^- ->$ jets at $O(alpha_s^2)$ are presented as a bookkeeping exercise rather than a new conceptual problem.

Reading between the lines

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

The unitarity checks in Section IV are consistency tests on a handful of scalar integrals, not a proof that the remappings catch every singularity; a natural extension would be to run the same decomposition on the full non-$N_f$ double real-emission matrix element and confirm that the residual $1/eps$ coefficients vanish sector by sector rather than only in the sum.
Since the method never integrates, the numerical cost moves entirely into Monte Carlo evaluation of a five-dimensional integrand with plus-distribution subtractions; the quoted errors on the finite pieces (0.003 on -0.117, and 0.4 on the non-planar constant term) suggest that variance reduction, not singularity extraction, is the practical bottleneck for phenomenology.
The interior-singularity splitting depends on knowing where the singular manifold sits in the $lambda$ variables; for processes with more than four final-state particles, or with massive partons, that manifold is higher-dimensional, so whether the boundary-pushing step stays automatable is the question that decides how far the method scales.
A useful cross-check the authors do not attempt: apply the method to a case where the exact differential distribution is already known at NLO with a jet function, and compare point-by-point rather than only after integration, which would test the claim that exact kinematics survive the sector decomposition.

Original abstract

We propose a new method of computing real emission contributions to hard QCD processes. Our approach uses sector decomposition of the exclusive final-state phase space to enable extraction of all singularities of the real emission matrix elements before integration over any kinematic variable. The exact kinematics of the real emission process are preserved in all regions of phase space. Traditional approaches to extracting singularities from real emission matrix elements, such as phase space slicing and dipole subtraction, require both the determination of counterterms for double real emission amplitudes in singular kinematic limits and the integration of these contributions analytically to cancel the resulting singularities against virtual corrections. Our method addresses both of these issues. The implementation of constraints on the final-state phase space, including various jet algorithms, is simple using our approach. We illustrate our method using electron-positron --> jets at NNLO as an example.

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, 5,467 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.84 · 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 propose a new method of computing real emission contributions to hard QCD processes. Our approach uses sector decomposition of the exclusive final-state phase space to enable extraction of all singularities of the real emission matrix elements before integration over any kinematic variable. The exact kinematics of the real emission process are preserved in all regions of phase space. Traditional approaches to extracting singularities from real emission matrix elements, such as phase space slicing and dipole subtraction, require both the determination of counterterms for double real emission amplitudes in singular kinematic limits and the integration of these contributions analytically to cancel the resulting singularities against virtual corrections. Our method addresses both of these issues. The implementation of constraints on the final-state phase space, including various jet algorithms, is simple using our approach. We illustrate our method using electron-positron --> jets at NNLO as an example.

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 propose a new method of computing real emission contributions to hard QCD processes.	contribution 1.00	0.88	0.39	asserted 1.00
s1	Our approach uses sector decomposition of the exclusive final-state phase space to enable extraction of all singularities of the real emission matrix elements before integration over any kinematic variable.	method 0.97	0.90	0.19	asserted 1.00
s2	The exact kinematics of the real emission process are preserved in all regions of phase space.	method 0.64	0.79	0.09	asserted 1.00
s3	Traditional approaches to extracting singularities from real emission matrix elements, such as phase space slicing and dipole subtraction, require both the determination of counterterms for double real emission amplitudes in singular kinematic limits and the integration of these contributions analytically to cancel the resulting singularities against virtual corrections.	background 0.95	0.04	0.05	asserted 1.00
s4	Our method addresses both of these issues.	contribution 0.96	0.81	0.13	asserted 1.00
s5	The implementation of constraints on the final-state phase space, including various jet algorithms, is simple using our approach.	contribution 0.50	0.71	0.05	asserted 1.00
s6	We illustrate our method using electron-positron --> jets at NNLO as an example.	method 0.59	0.91	0.07	asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from proof or derivation 0.49.

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: A planar propagator-type diagram contributing to th

**Figure 2.** FIG. 2: A non-planar propagator-type diagram contributing

Figures reproduced from arXiv:hep-ph/0311311 by the authors.

References

Works cited (18)

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hep-ph 2010-11 accept novelty 6.0 of 10

The paper documents SecDec, a public Mathematica/perl/Fortran program that sector-decomposes parametric integrals, subtracts dimensional-regularisation poles, and Monte Carlo integrates the finite coefficients, valida...
On the factorization of overlapping singularities at NNLO
hep-ph 2010-11 conditional novelty 6.0 of 10

A non-linear variable-transformation technique factorizes overlapping singularities in NNLO QCD integrals with far fewer sectors than sector decomposition, demonstrated on double-real topologies and two crossed two-lo...
Antenna subtraction for gluon scattering at NNLO
hep-ph 2010-03 conditional novelty 6.0 of 10

A working antenna subtraction term for the six-gluon double-real contribution to NNLO dijet production, validated numerically in all single and double unresolved limits.
NNLO corrections to 3-jet observables in electron-positron annihilation
hep-ph 2008-07 conditional novelty 6.0 of 10

A missing soft-gluon subtraction term explains why two next-to-next-to-leading-order calculations of three-jet observables in electron-positron annihilation disagree in three colour factors.
A subtraction scheme for computing QCD jet cross sections at NNLO: regularization of doubly-real emissions
hep-ph 2006-09 conditional novelty 6.0 of 10

The paper builds explicit subtraction counterterms, with exact phase-space factorization, that regularize the doubly-real emission contribution to NNLO jet cross sections in electron-positron annihilation.
Matching of Singly- and Doubly-Unresolved Limits of Tree-level QCD Squared Matrix Elements
hep-ph 2005-02 conditional novelty 6.0 of 10

The authors construct subtraction terms that regularize tree-level QCD squared matrix elements in all singly- and doubly-unresolved regions without double or triple subtraction.
Two-loop splitting functions in QCD
hep-ph 2004-05 conditional novelty 6.0 of 10

New explicit two-loop QCD splitting functions for the quark-gluon and gluon-quark-antiquark collinear limits, with the gluon-to-gluon case confirming an independent prior calculation.
New CTEQ global analysis of quantum chromodynamics with high-precision data from the LHC
hep-ph 2019-12 unverdicted novelty 5.0 of 10

New CT18 PDFs at NLO and NNLO from global fit to HERA plus LHC jet, Drell-Yan, top-pair and Z data, with Hessian errors, Lagrange-multiplier studies, and alternate sets for data tensions and scale choices.
Mono-Neutrino at DUNE: New Signals From Neutrinophilic Thermal Dark Matter
hep-ph 2019-01 conditional novelty 5.0 of 10

Radiating a light neutrinophilic scalar off a neutrino beam yields a missing-transverse-momentum plus wrong-sign-muon signal that DUNE's near detectors could use to probe thermal sub-GeV dark matter.
Jet production in the CoLoRFulNNLO method: event shapes in electron-positron collisions
hep-ph 2016-06 conditional novelty 5.0 of 10

A general local-subtraction method reproduces published NNLO event-shape distributions in electron-positron annihilation and yields the first NNLO prediction for the jet cone energy fraction.
Color singlet production at NNLO in MCFM
hep-ph 2016-05 conditional novelty 5.0 of 10

MCFM v8.0 ships validated NNLO QCD predictions for pp to H, Z, W, ZH, WH and diphoton, with quantified jettiness-cut systematics and cluster scaling benchmarks.
The $q_T$ subtraction method for top quark production at hadron colliders
hep-ph 2015-08 accept novelty 5.0 of 10

The qT subtraction formalism is extended to top-quark pair production, reproducing MCFM at NLO and Top++ for the off-diagonal NNLO channels.
Higgs boson decay into b-quarks at NNLO accuracy
hep-ph 2015-01 accept novelty 5.0 of 10

The paper validates a general NNLO subtraction scheme by applying it to Higgs decay into b-quarks, showing analytically that its integrated counterterms cancel the infrared poles of the two-loop amplitude and reproduc...
W physics at the LHC with FEWZ 2.1
hep-ph 2012-01 accept novelty 5.0 of 10

The FEWZ 2.1 release extends next-to-next-to-leading-order QCD predictions to W production, adds new observables and an order-of-magnitude speedup, and shows that fiducial-region comparisons discriminate proton PDF se...
FEWZ 2.0: A code for hadronic Z production at next-to-next-to-leading order
hep-ph 2010-11 accept novelty 5.0 of 10

The upgraded FEWZ code delivers NNLO QCD predictions for LHC Z production with sub-percent numerical integration error, automatic histograms and PDF errors, and shows that detector acceptance qualitatively reshapes th...
FIESTA 2: parallelizeable multiloop numerical calculations
hep-ph 2009-12 conditional novelty 5.0 of 10

The rewritten FIESTA 2 code evaluates and automatically expands multiloop Feynman integrals, adding multi-computer parallelization, adaptive multi-precision arithmetic, and Speer-sector decomposition.
Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report
hep-ph 2018-03 accept novelty 4.0 of 10

The report documents the state of precision Standard Model calculations for the LHC in 2017, including new NNLO subtraction and grid techniques, a tuned comparison of electroweak one-loop tools, and a prioritized list...
The SM and NLO Multileg and SM MC Working Groups: Summary Report
hep-ph 2012-03 accept novelty 4.0 of 10

The report surveys the 2011 state of multi-leg NLO QCD calculations, their integration into parton-shower Monte Carlos, and Monte Carlo tuning, concluding that essentially all NLO two-to-three and two-to-four processe...
Les Houches 2019: Physics at TeV Colliders: Standard Model Working Group Report
hep-ph 2020-03 conditional novelty 3.0 of 10

The Les Houches 2019 Standard Model report consolidates the state of LHC precision theory, ranks the missing calculations, and presents several new technical studies on NNLO delivery, photon isolation, PDF fits, and M...
Les Houches 2023 -- Physics at TeV Colliders: Report on the Standard Model Precision Wishlist
hep-ph 2025-04 unverdicted novelty 2.0 of 10

The report reviews progress since 2021 in fixed-order computations for LHC applications and identifies processes requiring missing higher-order corrections to match anticipated experimental precision.
Sector Decomposition
hep-ph 2008-03 accept novelty 2.0 of 10

Sector decomposition is presented as a general, automatable recipe for isolating UV and IR divergences in multi-loop and real-radiation integrals.

How later papers used it

58 later papers cite it

Evidence: some. 48 of the citing papers share no author with it, and 11 of those carry a confident label from Pith's citation classifier.

Pith read 6 of the 62 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.

Uses its method or tool 1 of 12 papers with this label

Used a transformation from this paper to decouple the problem at x_4=0 from the line singularity x_1=x_2.

SecDec: A general program for sector decomposition (2010) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

56 arXiv papers cite this in 60 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 · 20 sentences without a confident label

Uses its method or tool 12 papers

The integrals are evaluated using the sector decomposition approach [ref] .

The NNLO QCD soft function for 1-jettiness · $1$-jettiness · 1.00
Generally, the discontinuity of $(t-z)^-a_1$ is [ref] [equation] Note that for $a_1=n-ϵ$, where $n$ is a positive integer, the expansion [ref] [equation] and its differential forms are used to calculate the discontinuities of the above integrals.

Analytic calculation of doubly heavy hadron spectral density in coordinate space · Integral representation of hypergeometric function · 0.81
At NNLO this involves up to two single-soft limits and the double-soft limit. * Introduce a partition of unity for the phase space that isolates the collinear limits, $1 = ∑_{p} w_{p}$, where the sum runs over the sets of partons that can produce collinear singularities and the functions $w_{p}$ go to zero whenever two partons which are not in $p$ become collinear, thereby regulating integrand in that limit. * In each collinear partition, choose a suitable phase-space parametrisation in terms of angles and energies of the partons that can become unresolved. * Use sector decomposition [ref] to map all singularities to the boundaries of the region of integration so that the singularities can be easily extracted upon using <ref>.

Higgs decay into massive b-quarks at NNLO QCD in the nested soft-collinear subtraction scheme · Outline of the subtraction scheme · 0.95

Cites it as background 27 papers

So far, successful applications of subtraction at NNLO to specific observables were accomplished with antenna subtraction [ref] , $q_T$-subtraction [ref] , sector decomposition [ref] , non-linear transformations [ref] , and most recently with an approach based on sector-improved residue subtraction [ref] .

Infrared Structure at NNLO Using Antenna Subtraction · Introduction · 0.87
Various methods have been proposed [ref] (for a recent review see Ref. [ref] ), but achieving universality and efficiency at NNLO is still a challenge.

Advances in the nested soft-collinear subtraction scheme · The problem, the idea · 0.96
Several approaches have been developed to treat these singularities at NNLO, including slicing methods [ref] , non-local subtraction schemes [ref] , and local subtraction methods like antenna subtraction [ref] , CoLoRFulNNLO [ref] , nested soft-collinear subtraction [ref] and the recently proposed Local Analytic Sector Subtraction (LASS) [ref] .

Analysis of $(n+1)$ and $n$-parton contributions for computing QCD jet cross sections in the local analytic subtraction scheme · Introduction · 0.98

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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