arxiv: 2512.24403 · v2 · submitted 2025-12-30 · ✦ hep-ph · hep-th

Recognition: 2 theorem links

· Lean Theorem

CoLoRFulNNLO for hadron collisions: integrating the iterated single unresolved subtraction terms

L. Fek\'esh\'azy , G. Somogyi , S. Van Thurenhout

Authors on Pith no claims yet

Pith reviewed 2026-05-16 18:39 UTC · model grok-4.3

classification ✦ hep-ph hep-th

keywords CoLoRFulNNLOsubtraction termsNNLO QCDhadron collisionsinfrared subtractionphase space mappingsparametric integralscolor-singlet production

0 comments

The pith

The integrated iterated single-unresolved subtraction terms can be written as a convolution of the Born cross section with an insertion operator.

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

This paper integrates the iterated single-unresolved subtraction terms in the CoLoRFulNNLO scheme extended to color-singlet production at hadron colliders. Momentum mappings are used to create exact phase-space convolutions for real emissions, which convert the subtraction terms into parametric integrals that can be evaluated with standard numerical tools. The central result follows: the full integrated approximate cross section is expressed as a convolution of the Born cross section with a defined insertion operator. This integration is required to cancel infrared divergences when combining with virtual corrections and real-emission contributions in NNLO calculations.

Core claim

The integrated iterated single-unresolved approximate cross section can be written as a convolution of the Born cross section with an appropriately defined insertion operator, after the subtraction terms are expressed as parametric integrals via the exact phase-space convolutions produced by the momentum mappings.

What carries the argument

Momentum mappings that produce exact phase-space convolutions for real emissions, allowing integrated subtraction terms to be written as parametric integrals.

If this is right

The integrated terms become parametric integrals that can be evaluated using standard numerical integration methods.
The approximate cross section can be combined with virtual and real contributions to cancel infrared singularities at NNLO.
The scheme extends to full hadron-collider predictions for color-singlet final states.
The insertion operator provides a compact way to include the integrated subtractions in the calculation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same mapping-based convolution structure could be applied to other infrared subtraction schemes that rely on similar phase-space factorizations.
Numerical implementations of NNLO event generators might become simpler if the convolution reduces the dimensionality of required integrations.
Explicit evaluation of the insertion operator for benchmark processes like Drell-Yan production would allow direct numerical checks of the analytic results.

Load-bearing premise

The momentum mappings used to define the subtraction terms produce exact phase-space convolutions for real emissions.

What would settle it

A direct numerical computation of the integrated subtraction terms for a specific color-singlet process such as Higgs production that fails to reproduce the convolution result obtained from the parametric integrals would falsify the claim.

Figures

Figures reproduced from arXiv: 2512.24403 by G. Somogyi, L. Fek\'esh\'azy, S. Van Thurenhout.

read the original abstract

We present the analytic integration of the iterated single-unresolved subtraction terms in the extension of the CoLoRFulNNLO subtraction scheme to color-singlet production in hadron collisions. We exploit the fact that, in this scheme, subtraction terms are defined through momentum mappings which lead to exact phase space convolutions for real emissions. This allows us to write the integrated subtraction terms as parametric integrals, which can be evaluated using standard tools. Finally, we show that the integrated iterated single-unresolved approximate cross section can be written as a convolution of the Born cross section with an appropriately defined insertion operator.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The analytic integration of iterated single-unresolved terms completes a key part of the CoLoRFulNNLO scheme for hadron collisions.

read the letter

The punchline is that this paper provides the analytic integration of the iterated single-unresolved subtraction terms in the CoLoRFulNNLO scheme extended to color-singlet production in hadron collisions. They leverage the momentum mappings that define the subtraction terms to achieve exact phase space convolutions, turning the integrated terms into parametric integrals that standard tools can handle. The result is that the integrated approximate cross section becomes a convolution of the Born cross section with a defined insertion operator. What stands out as new is the specific handling of these iterated terms for the hadron collision case. Earlier work on CoLoRFulNNLO likely covered other aspects or different processes, so this fills in the gap for the full NNLO subtraction in this framework. The paper does well by building directly on the scheme's definitions without introducing new assumptions or parameters. The approach is straightforward and consistent with standard QCD integration techniques. On the soft spots, the central claim rests on the momentum mappings producing exact convolutions, which seems to hold based on the description. However, without explicit forms of the integrated terms or numerical cross-checks visible in the summary, the practical usability depends on whether the full paper shows clean expressions and tests. This is a minor concern given the established nature of the method. This kind of paper is aimed at experts in higher-order QCD calculations, particularly those implementing subtraction schemes for LHC phenomenology. A reader focused on NNLO predictions for processes like Higgs or vector boson production would find the technical details useful for extending their own calculations. It is not revolutionary but provides a necessary building block. I think it deserves peer review. The work is grounded enough in the existing framework to warrant a referee's look, even if revisions might be needed for more explicit results.

Referee Report

1 major / 2 minor

Summary. The manuscript extends the CoLoRFulNNLO subtraction scheme to color-singlet production in hadron collisions by analytically integrating the iterated single-unresolved subtraction terms. Momentum mappings are used to obtain exact phase-space convolutions, allowing the integrated terms to be written as parametric integrals that are evaluated with standard tools; the final result expresses the integrated iterated single-unresolved approximate cross section as a convolution of the Born cross section with an appropriately defined insertion operator.

Significance. If the central result holds, the work supplies a missing analytic ingredient for NNLO calculations within the CoLoRFulNNLO framework at hadron colliders. It enables the construction of fully differential NNLO predictions for processes such as Drell-Yan or Higgs production without introducing new free parameters, building directly on prior CoLoRFulNNLO definitions and standard QCD integration techniques. This step improves the practicality of higher-order QCD phenomenology at the LHC.

major comments (1)

[Section 3] The reduction of the integrated iterated single-unresolved terms to a convolution with an insertion operator (final result) rests on the assertion that the chosen momentum mappings produce exact phase-space convolutions for real emissions in hadron kinematics. An explicit verification of this property for the partonic initial-state case, including the handling of the parton distribution functions, is required to confirm that no residual finite terms are omitted.

minor comments (2)

[Abstract] The abstract refers to evaluation 'using standard tools' for the parametric integrals; the manuscript should name the specific numerical quadrature routines or libraries employed and report the achieved numerical precision.
[Section 4] Notation for the insertion operator should be introduced with an explicit equation number at first appearance and cross-referenced consistently in the text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the constructive comment. We address the single major point below.

read point-by-point responses

Referee: [Section 3] The reduction of the integrated iterated single-unresolved terms to a convolution with an insertion operator (final result) rests on the assertion that the chosen momentum mappings produce exact phase-space convolutions for real emissions in hadron kinematics. An explicit verification of this property for the partonic initial-state case, including the handling of the parton distribution functions, is required to confirm that no residual finite terms are omitted.

Authors: We agree that an explicit verification strengthens the presentation. In the revised manuscript we will insert a new subsection (3.3) that performs this verification for the partonic initial-state case. Starting from the momentum mappings defined in Eqs. (3.4)–(3.7), we explicitly compute the Jacobian of the transformation and show that the phase-space measure for the unresolved parton factors exactly into a convolution with the Born kinematics. The parton distribution functions are evaluated at the mapped momentum fractions x' = x(1−y), where y is the unresolved momentum fraction; the resulting delta functions and integration limits reproduce the standard convolution structure without additional finite remainders. The calculation is performed both analytically (for the leading singular pieces) and numerically (for a representative set of phase-space points) to confirm the absence of omitted terms. This addition does not alter the final insertion operator but makes the underlying assumption fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives the integrated iterated single-unresolved subtraction terms by exploiting momentum mappings that yield exact phase-space convolutions, recasting the terms as parametric integrals evaluable by standard methods, and expressing the result as a convolution of the Born cross section with an insertion operator. This follows directly from the prior CoLoRFulNNLO definition and established QCD integration techniques without any reduction of the central claim to fitted parameters, self-definitional loops, or load-bearing self-citations that collapse the result to its own inputs. The logical steps remain independent and externally verifiable against standard subtraction-scheme benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the established CoLoRFulNNLO subtraction scheme from prior publications and standard assumptions of perturbative QCD phase-space integration; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Momentum mappings lead to exact phase space convolutions for real emissions
Invoked to convert subtraction terms into parametric integrals

pith-pipeline@v0.9.0 · 5404 in / 1083 out tokens · 33004 ms · 2026-05-16T18:39:20.095935+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We exploit the fact that, in this scheme, subtraction terms are defined through momentum mappings which lead to exact phase space convolutions for real emissions. This allows us to write the integrated subtraction terms as parametric integrals...
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the integrated iterated single-unresolved approximate cross section can be written as a convolution of the Born cross section with an appropriately defined insertion operator

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages · 26 internal anchors

[1]

Aoyama et al.,The anomalous magnetic moment of the muon in the Standard Model,Phys

T. Aoyama et al.,The anomalous magnetic moment of the muon in the Standard Model,Phys. Rept. 887(2020) 1 [arXiv:2006.04822]. [2]Muon g-2collaboration, G.W. Bennett et al.,Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL,Phys. Rev. D73(2006) 072003 [hep-ex/0602035]. [3]Muon g-2collaboration, B. Abi et al.,Measurement of the Positi...

work page arXiv 2020
[2]

The anomalous magnetic moment of the muon in the Standard Model: an update

R. Aliberti et al.,The anomalous magnetic moment of the muon in the Standard Model: an update, Phys. Rept.1143(2025) 1 [arXiv:2505.21476]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[3]

An NNLO subtraction formalism in hadron collisions and its application to Higgs boson production at the LHC

S. Catani and M. Grazzini,An NNLO subtraction formalism in hadron collisions and its application to Higgs boson production at the LHC,Phys. Rev. Lett.98(2007) 222002 [hep-ph/0703012]

work page internal anchor Pith review Pith/arXiv arXiv 2007
[4]

N-jettiness Subtractions for NNLO QCD Calculations

J. Gaunt, M. Stahlhofen, F.J. Tackmann and J.R. Walsh,N-jettiness Subtractions for NNLO QCD Calculations,JHEP09(2015) 058 [arXiv:1505.04794]

work page internal anchor Pith review Pith/arXiv arXiv 2015
[5]

Antenna Subtraction at NNLO

A. Gehrmann-De Ridder, T. Gehrmann and E.W.N. Glover,Antenna subtraction at NNLO,JHEP 09(2005) 056 [hep-ph/0505111]

work page internal anchor Pith review Pith/arXiv arXiv 2005
[6]

Four-dimensional formulation of the sector-improved residue subtraction scheme

M. Czakon and D. Heymes,Four-dimensional formulation of the sector-improved residue subtraction scheme,Nucl. Phys. B890(2014) 152 [arXiv:1408.2500]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[7]

Local Analytic Sector Subtraction at NNLO

L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati,Local analytic sector subtraction at NNLO,JHEP12(2018) 107 [arXiv:1806.09570]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[8]

Nested soft-collinear subtractions in NNLO QCD computations

F. Caola, K. Melnikov and R. R¨ ontsch,Nested soft-collinear subtractions in NNLO QCD computations,Eur. Phys. J. C77(2017) 248 [arXiv:1702.01352]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[9]

Fully differential VBF Higgs production at NNLO

M. Cacciari, F.A. Dreyer, A. Karlberg, G.P. Salam and G. Zanderighi,Fully Differential Vector-Boson-Fusion Higgs Production at Next-to-Next-to-Leading Order,Phys. Rev. Lett.115 (2015) 082002 [arXiv:1506.02660]. 88

work page internal anchor Pith review Pith/arXiv arXiv 2015
[10]

Geometric IR subtraction for real radiation

F. Herzog,Geometric IR subtraction for final state real radiation,JHEP08(2018) 006 [arXiv:1804.07949]. [19]NNLOJETcollaboration, A. Huss et al.,NNLOJET: a parton-level event generator for jet cross sections at NNLO QCD accuracy,arXiv:2503.22804

work page internal anchor Pith review Pith/arXiv arXiv 2018
[11]

A subtraction scheme for computing QCD jet cross sections at NNLO: regularization of doubly-real emissions

G. Somogyi, Z. Trocsanyi and V. Del Duca,A Subtraction scheme for computing QCD jet cross sections at NNLO: Regularization of doubly-real emissions,JHEP01(2007) 070 [hep-ph/0609042]

work page internal anchor Pith review Pith/arXiv arXiv 2007
[12]

Matching of Singly- and Doubly-Unresolved Limits of Tree-level QCD Squared Matrix Elements

G. Somogyi, Z. Trocsanyi and V. Del Duca,Matching of singly- and doubly-unresolved limits of tree-level QCD squared matrix elements,JHEP06(2005) 024 [hep-ph/0502226]

work page internal anchor Pith review Pith/arXiv arXiv 2005
[13]

A subtraction scheme for computing QCD jet cross sections at NNLO: regularization of real-virtual emission

G. Somogyi and Z. Trocsanyi,A Subtraction scheme for computing QCD jet cross sections at NNLO: Regularization of real-virtual emission,JHEP01(2007) 052 [hep-ph/0609043]

work page internal anchor Pith review Pith/arXiv arXiv 2007
[14]

Higgs boson decay into b-quarks at NNLO accuracy

V. Del Duca, C. Duhr, G. Somogyi, F. Tramontano and Z. Tr´ ocs´ anyi,Higgs boson decay into b-quarks at NNLO accuracy,JHEP04(2015) 036 [arXiv:1501.07226]

work page internal anchor Pith review Pith/arXiv arXiv 2015
[15]

Three-jet production in electron-positron collisions using the CoLoRFulNNLO method

V. Del Duca, C. Duhr, A. Kardos, G. Somogyi and Z. Tr´ ocs´ anyi,Three-Jet Production in Electron-Positron Collisions at Next-to-Next-to-Leading Order Accuracy,Phys. Rev. Lett.117(2016) 152004 [arXiv:1603.08927]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[16]

Jet production in the CoLoRFulNNLO method: event shapes in electron-positron collisions

V. Del Duca, C. Duhr, A. Kardos, G. Somogyi, Z. Sz˝ or, Z. Tr´ ocs´ anyi et al.,Jet production in the CoLoRFulNNLO method: event shapes in electron-positron collisions,Phys. Rev. D94(2016) 074019 [arXiv:1606.03453]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[17]

Somogyi and F

G. Somogyi and F. Tramontano,Fully exclusive heavy quark-antiquark pair production from a colourless initial state at NNLO in QCD,JHEP11(2020) 142 [arXiv:2007.15015]

work page arXiv 2020
[18]

Del Duca, C

V. Del Duca, C. Duhr, L. Fek´ esh´ azy, F. Guadagni, P. Mukherjee, G. Somogyi et al.,NNLOCAL: completely local subtractions for color-singlet production in hadron collisions,JHEP05(2025) 151 [arXiv:2412.21028]

work page arXiv 2025
[19]

Del Duca, G

V. Del Duca, G. Somogyi and F. Tramontano,CoLoRFulNNLO for hadron collisions: regularizing initial-state double real emissions,arXiv:2512.05192

work page arXiv
[20]

Higgs boson production at hadron colliders in NNLO QCD

C. Anastasiou and K. Melnikov,Higgs boson production at hadron colliders in NNLO QCD,Nucl. Phys. B646(2002) 220 [hep-ph/0207004]

work page internal anchor Pith review Pith/arXiv arXiv 2002
[21]

Chetyrkin and F.V

K.G. Chetyrkin and F.V. Tkachov,Integration by parts: The algorithm to calculateβ-functions in 4 loops,Nucl. Phys. B192(1981) 159

work page 1981
[22]

Multiloop integrals in dimensional regularization made simple

J.M. Henn,Multiloop integrals in dimensional regularization made simple,Phys. Rev. Lett.110 (2013) 251601 [arXiv:1304.1806]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[23]

Angular integrals in d dimensions

G. Somogyi,Angular integrals in d dimensions,J. Math. Phys.52(2011) 083501 [arXiv:1101.3557]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[24]

Sector Decomposition

G. Heinrich,Sector Decomposition,Int. J. Mod. Phys. A23(2008) 1457 [arXiv:0803.4177]

work page internal anchor Pith review Pith/arXiv arXiv 2008
[25]

Multiple polylogarithms, cyclotomy and modular complexes

A.B. Goncharov,Multiple polylogarithms, cyclotomy and modular complexes,Math. Res. Lett.5 (1998) 497 [arXiv:1105.2076]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[26]

Duhr and F

C. Duhr and F. Dulat,PolyLogTools — polylogs for the masses,JHEP08(2019) 135 [arXiv:1904.07279]

work page arXiv 2019
[27]

Besier, P

M. Besier, P. Wasser and S. Weinzierl,RationalizeRoots: Software Package for the Rationalization of Square Roots,Comput. Phys. Commun.253(2020) 107197 [arXiv:1910.13251]. 89

work page arXiv 2020
[28]

Asymptotic expansion of Feynman integrals near threshold

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

work page internal anchor Pith review Pith/arXiv arXiv 1998
[29]

Geometric approach to asymptotic expansion of Feynman integrals

A. Pak and A. Smirnov,Geometric approach to asymptotic expansion of Feynman integrals,Eur. Phys. J. C71(2011) 1626 [arXiv:1011.4863]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[30]

Expansion by regions: revealing potential and Glauber regions automatically

B. Jantzen, A.V. Smirnov and V.A. Smirnov,Expansion by regions: revealing potential and Glauber regions automatically,Eur. Phys. J. C72(2012) 2139 [arXiv:1206.0546]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[31]

Chargeishvili, L

B. Chargeishvili, L. Fek´ esh´ azy, G. Somogyi and S. Van Thurenhout,LinApart: Optimizing the univariate partial fraction decomposition,Comput. Phys. Commun.307(2025) 109395 [arXiv:2405.20130]

work page arXiv 2025
[32]

Gradshteyn and I.M

I.S. Gradshteyn and I.M. Ryzhik,Table of integrals, series, and products, seventh ed., Elsevier/Academic Press, Amsterdam (2007)

work page 2007
[33]

A subtraction scheme for computing QCD jet cross sections at NNLO: integrating the iterated singly-unresolved subtraction terms

P. Bolzoni, G. Somogyi and Z. Trocsanyi,A subtraction scheme for computing QCD jet cross sections at NNLO: integrating the iterated singly-unresolved subtraction terms,JHEP01(2011) 059 [arXiv:1011.1909]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[34]

Altarelli and G

G. Altarelli and G. Parisi,Asymptotic Freedom in Parton Language,Nucl. Phys. B126(1977) 298

work page 1977
[35]

Del Duca, C

V. Del Duca, C. Duhr, R. Haindl, A. Lazopoulos and M. Michel,Tree-level splitting amplitudes for a quark into four collinear partons,JHEP02(2020) 189 [arXiv:1912.06425]

work page arXiv 2020
[36]

Cuba - a library for multidimensional numerical integration

T. Hahn,CUBA: A Library for multidimensional numerical integration,Comput. Phys. Commun. 168(2005) 78 [hep-ph/0404043]

work page internal anchor Pith review Pith/arXiv arXiv 2005
[37]

Lepage,A New Algorithm for Adaptive Multidimensional Integration,J

G.P. Lepage,A New Algorithm for Adaptive Multidimensional Integration,J. Comput. Phys.27 (1978) 192

work page 1978
[38]

Lepage,VEGAS - an adaptive multi-dimensional integration program, tech

G.P. Lepage,VEGAS - an adaptive multi-dimensional integration program, tech. rep., Cornell Univ. Lab. Nucl. Stud., Ithaca, NY, 1980

work page 1980
[39]

Introduction to the GiNaC Framework for Symbolic Computation within the C++ Programming Language

C.W. Bauer, A. Frink and R. Kreckel,Introduction to the GiNaC framework for symbolic computation within the C++ programming language,J. Symb. Comput.33(2002) 1 [cs/0004015]

work page internal anchor Pith review Pith/arXiv arXiv 2002
[40]

Ma,Identifying regions for asymptotic expansions of amplitudes: fundamentals and recent advances,arXiv:2505.01368

Y. Ma,Identifying regions for asymptotic expansions of amplitudes: fundamentals and recent advances,arXiv:2505.01368

work page arXiv
[41]

Barber, D.P

C.B. Barber, D.P. Dobkin and H. Huhdanpaa,The quickhull algorithm for convex hulls,ACM Trans. Math. Softw.22(Dec., 1996) 469–483

work page 1996
[42]

Ferguson, D

H. Ferguson, D. Bailey and S. Arno,Analysis of pslq, an integer relation finding algorithm, Mathematics of Computation68(12, 1998)

work page 1998
[43]

Del Duca, N

V. Del Duca, N. Deutschmann and S. Lionetti,Momentum mappings for subtractions at higher orders in QCD,JHEP12(2019) 129 [arXiv:1910.01024]. 90

work page arXiv 2019