Fully Differential Soft Gluon Evolution at the Amplitude Level

Fernando Torre Gonz\'alez; Jeffrey R. Forshaw; Simon Pl\"atzer

REVIEW 3 major objections 4 minor 2 cited by

Fully Differential Soft Gluon Evolution at the Amplitude Level

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Subleading colour corrections change the shape of inter-jet soft gluon radiation, not just its overall rate.

desk verdict Genuine internal check, honest Coulomb caveat, and a shape-distortion claim that is provisional but worth refereeing. read the letter →

arxiv 2505.13183 v1 pith:B6AKP5DT submitted 2025-05-19 hep-ph

Jeffrey R. Forshaw , Simon Plätzer , Fernando Torre González This is my paper · ORCID

classification hep-ph

keywords softgluonevolutionsubleadingcolourflowjetvetonon-globallogarithmsamplitude-levelresummationinter-jetradiationfullQCD

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether the subleading-colour corrections that QCD resummation must include change only the overall size of inter-jet soft gluon radiation, or also its angular shape. By evolving soft gluon emissions at the amplitude level with all colour corrections, the authors compare full-colour differential distributions with the strictly leading-colour approximation for $q\bar q\to q\bar q$ and for a colourless-initiated four-jet final state. They find that subleading colour does change shapes, not just normalisation: the $s$-channel gluon exchange contribution is badly described by leading colour, and the $t$-channel approximation fails when the jets recoil. This matters because inclusive observables such as gaps-between-jets can hide these effects after integrating over solid angle, so measurements and simulations that assume leading colour for differential radiation patterns can be wrong.

What carries the argument

The machinery is amplitude-level soft gluon evolution in a colour-flow basis: real and virtual emissions act on a density matrix whose entries are colour-flow configurations, here $|01\rangle$ and $|10\rangle$ and their interference, so that virtual gluon exchange can change the colour configuration instead of only adding a phase to a fixed dipole. The observable that carries the argument is the triple-differential cross section $\mathrm{d}^3\sigma/\mathrm{d}\Omega\,\mathrm{d}\rho$ for the highest-energy gluon emitted into the veto region, integrated into $\mathrm{d}\Sigma/\mathrm{d}(\cos\theta)$ and $\mathrm{d}\Sigma/\mathrm{d}\phi$. This is what exposes subleading-colour shape effects that vanish in fully inclusive integrals.

What would settle it

Repeat the same full-colour evolution with the phase-type exchanges included (regulating the resulting super-leading logarithms with hard-collinear physics) and compare $\mathrm{d}\Sigma/\mathrm{d}(\cos\theta)$ and $\mathrm{d}\Sigma/\mathrm{d}\phi$ for the $s$- and $t$-channel contributions; if the residuals against leading colour flatten to a constant normalisation, the claim that subleading colour changes shapes is overturned. A cheaper proxy is the $\rho=0.1$ comparison in Appendix A, where the phase-type exchanges already shift the integrated cross section by about 10%.

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Extended reading notes

Core claim

The central claim is that full-colour soft gluon evolution produces differential radiation patterns that leading-colour evolution cannot reproduce, and that the apparent success of leading colour in inclusive veto cross sections relies on cancellations across phase space. In the $q\bar q\to q\bar q$ back-to-back configuration the $s$-channel contribution shows a roughly 40% shape difference between the edges and the middle of the veto region, while the $|01\rangle\langle01|$ contribution shows 5\textendash 10% residual shape effects and the $t$-channel contribution appears flat once interference is included. In the recoiling configuration, which mimics vector-boson-fusion-like topologies, even the $t$-channel gluon exchange contribution fails to be described by the strictly leading-colour approximation, and the $|10\rangle\langle01|$ interference contribution radiates with a distinctly steeper pattern despite having no leading-colour dipoles.

Load-bearing premise

The calculation assumes the non-Abelian phase-type (Coulomb or Glauber) gluon exchange can be turned off, and Appendix A shows that including it changes the $q\bar q\to q\bar q$ integrated cross section by about 10% at $\rho=0.1$, so the differential patterns could shift once those exchanges and hard-collinear physics are added.

Editorial extensions

If this is right

Inclusive jet-veto cross sections cannot certify leading-colour accuracy; shape effects that cancel in the angular integral will survive in more differential measurements.
Event generators and resummation tools that emit only from leading-colour dipoles will mispredict the angular distribution of the hardest inter-jet gluon for $s$-channel exchange and for $t$-channel processes with recoil.
The $|10\rangle\langle01|$ interference term, although $1/N_c^2$-suppressed and absent at leading colour, contributes a numerically visible and differently shaped radiation pattern.
Colour-reconnection studies using $ZZ\to$ four jets at lepton colliders need the full-colour interference pattern as the reference, rather than a leading-colour or model-based guess.

Reading between the lines

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

I infer that observables integrating over a smaller solid-angle patch, such as azimuthal asymmetries or wedge jet shapes around a rapidity gap, will show larger subleading-colour distortions than the 5\textendash 40% residuals reported for the full veto region.
The recoiling-configuration failure suggests vector-boson-fusion-like topologies at hadron colliders are the most promising place to look for subleading-colour shape effects experimentally, e.g. in the angular distribution of the third jet.
If the neglected non-Abelian phase (Coulomb-type) exchanges are included, the picture may change: the appendix's roughly 10% shift at $\rho=0.1$ is a lower bound on the possible distortion of the differential patterns, and the true shape residuals could be larger once super-leading logarithms are regulated.
A natural next step would be to feed the same amplitude-level evolution into a parton shower for collinear-sensitive observables; that would test whether subleading-colour shape effects extend into jet substructure.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper extends CVolver from integrated jet-veto cross sections to fully differential distributions of the highest-energy inter-jet gluon, and it passes a genuine internal check: the new event-generator mode reproduces the dedicated-mode results for the |10><10| and interference contributions exactly, with different multiplicity structures. The main qualitative finding—subleading color can change the shape, not just the normalization, of radiation patterns—is plausible and consistently reproduced across back-to-back, boosted, and recoiling kinematics. The t-channel cancellation story, where the |01><01| enhancement is cancelled by interference and subleading |10><10| contributions, is a nice illustration of how integrated observables can hide differential effects.

The paper is honest about its limits. It states that Coulomb/Glauber exchanges are turned off because, absent hard-collinear physics, they generate super-leading logarithms regulated by the collinear cutoff. Appendix A shows that including Coulomb changes the qbar-q veto cross section by about 10% at rho=0.1, with some residual cutoff dependence. That means the headline claim—that s-channel exchange "completely fails" at leading color—is not yet established for full QCD. It could survive, but a 10% shift in the underlying cross section could easily alter 40% shape residuals. This is a completeness concern, not an internal inconsistency. The authors acknowledge they plan to return to Coulomb in a later study.

Other soft spots are minor in comparison. The plots carry no uncertainty bands; the authors mention fluctuations but do not quantify them. The restriction to r=0.3 is explained by the six-emission cap, and is fine for a proof-of-concept. The event-generator code is not released, though the internal check partly compensates.

This is a solid paper for a specialized audience. It deserves a serious referee. A referee should ask for uncertainty estimates and a discussion of Coulomb effects on the differential shapes, but the core check and the new differential results justify publication after revision.

Referee Report

3 major / 4 minor

Summary. The paper uses the CVolver amplitude-level evolution code in event-generator mode to compute fully differential soft-gluon observables for several 2-to-2 QCD processes and a four-parton e+e- final state. It compares full-colour and leading-colour evolution for the differential cross section of the highest-energy gluon emitted outside the jet/veto region, presenting dSigma/dOmega, dSigma/dcos(theta), dSigma/dphi and residual ratios. The main claims are that subleading colour corrections can change the shape of these distributions, that the s-channel gluon-exchange contribution is poorly described by leading-colour evolution, and that t-channel gluon exchange, which is well described in back-to-back and boosted configurations after cancellations, fails in the recoiling configuration. Coulomb/Glauber exchanges are deliberately omitted, and Appendix A quantifies their effect on the integrated jet-veto cross section at about 10% in one configuration.

Significance. If the results are correct, they provide a qualitative challenge to the common assumption that leading-colour evolution describes inter-jet radiation patterns up to an overall normalization, and they identify differential observables for which subleading colour matters. The event-generator versus dedicated-mode check in Fig. 1 is a strong internal consistency test, and the study is genuinely exploratory: no parameter is fitted to produce the claimed shapes, and the work is a step toward full-colour Monte Carlo event generation. The significance is, however, tempered by the omission of Coulomb/Glauber exchanges and by the absence of statistical uncertainties in the differential plots, both of which bear directly on the quantitative statements.

major comments (3)

[Section 2 and Appendix A] The central claims are presented as full-colour results, but the evolution is computed with Coulomb (Glauber) exchanges switched off. Section 2 states that this is necessary because, without hard-collinear physics, Coulomb exchanges generate super-leading logarithms regulated by the collinear cutoff. Appendix A then shows that including Coulomb exchanges changes the q qbar to q qbar jet-veto cross section by about 10% at rho=0.1 (Fig. 14) and that this residual is collinear-cutoff independent at high rho. Because Coulomb exchanges act on the colour-density matrix and can swap colour flows, they can in principle modify the differential shape residuals on which the paper's claims rest, e.g. the 40% s-channel effect in Section 4.1 and the claimed failure of leading colour for t-channel exchange in Section 4.3. As written, the Abstract and Conclusions assert without qualification that 'subleading colour does affect the shapes of distributions' and that the approximations 'fail'; these statements are established only in the no-Coulomb approximation. The authors should either include Coulomb exchanges in the differential analysis, estimate their differential impact, or explicitly and consistently qualify all central claims as conditional on neglecting Coulomb/Glauber contributions.
[Figures 5-12] None of the differential plots carry uncertainty bands or error bars, and the text repeatedly refers to 'fluctuations' without quantifying them (Sections 4.1 and 4.2). The quantitative claims—5-10% shape residuals in the |01><01| contribution, ~10% in the s-channel dSigma/dphi, 40% between the edges and middle of the veto region, and ~10% enhancement in the recoiling configuration—require a statement of the statistical precision of the event-generator mode, especially because the residuals are ratios of full-colour to leading-colour results. The authors should provide confidence bands or at least per-bin uncertainties for the residual panels so that the reader can distinguish genuine shape distortions from Monte Carlo noise.
[Section 2 and Fig. 1] The paper chooses r=0.3 throughout to avoid missing contributions from >=7 gluon emissions, which are visible at large rho in Fig. 1. However, no convergence test is shown for the differential observables at r=0.3: the figures break down the results by multiplicity, but they do not compare a five-emission-truncated result with the six-emission result to demonstrate that the sixth emission is negligible in each kinematic configuration and colour channel. This matters because the strongest claims, such as the failure of leading colour for s-channel exchange in Section 4.2, are made for the full six-emission evolution. The authors should add a convergence check, e.g. the relative difference between truncation at n and n-1 emissions for the integrated and differential quantities, or explicitly quantify the residual truncation uncertainty.

minor comments (4)

[Section 2] The numerical values of the collinear cutoff lambda are not given for the main results; the text says it is 'sufficiently small' and refers to [1]. For reproducibility, please state the value or values used for each figure.
[Section 2, Eq. (2.2)] The notation dSigma/dOmega used throughout the paper suppresses the r-dependence of d^2Sigma(r)/dOmega defined in Eq. (2.2). Please make the r-dependence explicit or state once that r=0.3 for all results.
[Appendix A] The sentence 'In both cases we consider the back-to-back kinematic configuration considered in the main text' is confusing, since Z->q qbar is not analysed in the main text; please clarify the geometry used for this process.
[Sections 4.1 and 4.2] The abbreviation L1, LCH is used without definition in this paper. Since the comparison depends on this leading-colour choice, please define it or give a precise reference to the partner paper [1].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the differential subleading-colour results are computed outputs of an established amplitude-level evolution code, with no fitted parameter that is then renamed as a prediction.

full rationale

The derivation chain is self-contained numerical resummation. The central claim that subleading colour changes differential shapes is obtained by evolving colour-density matrices with CVolver and comparing full-colour with strictly leading-colour evolution; the shapes are outputs, not inputs. No parameter is fitted to reproduce the claimed residuals. The check in Fig. 1 (event-generator mode vs dedicated mode) is an internal consistency test, and the agreement is non-trivial because the two modes build the multiplicity contributions differently. Self-citations to the partner paper [1] and to CVolver [2,3] supply the framework and notation, but the load-bearing numerical results do not reduce to those citations: the partner paper studied inclusive jet-veto cross sections, whereas this paper presents fully differential distributions, and the only direct comparison to the partner paper is an independent cross-check. The exclusion of Coulomb/Glauber exchanges is a physical assumption, not a circular step; Appendix A quantifies a ~10% effect at rho = 0.1 and the paper explicitly flags that a more comprehensive study including their effects will follow. Missing uncertainty bands and possible completeness concerns are correctness risks, not circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central computation rests on the soft-evolution framework from the authors' prior papers, on the choice of collinear-safe observables, and on two explicit approximations: no Coulomb exchange and a maximum of six gluon emissions. No new entities are introduced, and no parameters are fitted to data; the only adjustable numbers are technical cutoffs and a truncation order.

free parameters (3)

Energy cutoff mu = 0.1 in units of maximum gluon energy
Chosen by hand as the lowest gluon energy included in the evolution; the paper focuses on rho > 0.1 and notes lower values are possible but not explored.
Collinear cutoff lambda = 0.01, 0.005, 0.001, 0.0005 in different runs
Regularization for collinear singularities; the paper argues the main observables are independent of it, but Appendix A shows Coulomb results depend on it.
Maximum number of soft gluon emissions = 6
Computational truncation; the paper restricts r = 0.3 to mitigate missing contributions from 7 or more emissions in the |01><01| channel.

assumptions (4)

domain assumption Amplitude-level soft gluon evolution with a 1/Nc expansion correctly resums wide-angle soft radiation.
Section 2 references partner paper [1] and refs [3-5] for the framework; it is not re-derived here.
domain assumption The observables considered are collinear safe and inclusive in soft emissions below a scale, so the results are independent of the collinear cutoff.
Section 2 restricts to observables sensitive only to wide-angle soft gluons; hard-collinear contributions are deferred to future work.
ad hoc to paper Coulomb, or Glauber, exchange can be omitted without changing the qualitative subleading-color conclusions.
Section 2 turns off Coulomb exchange because hard-collinear physics is absent; Appendix A shows Coulomb effects up to 10% for qbar q to qbar q at rho = 0.1.
ad hoc to paper Truncating the evolution at six gluon emissions is adequate for r = 0.3.
Section 2 notes missing contributions from 7 or more emissions cause disagreement in the |01><01| channel at large rho, so r = 0.3 is chosen as the veto energy.

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Cite this review

Pith. "Pith review of Fully Differential Soft Gluon Evolution at the Amplitude Level." pith.science (2026). https://pith.science/paper/B6AKP5DT

@misc{pith2026250513183,
  author       = {Pith},
  title        = {Pith review of: Fully Differential Soft Gluon Evolution at the Amplitude Level},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B6AKP5DT}},
  note         = {Machine review of arXiv:2505.13183}
}

abstract

We study differential intra-jet radiation patterns in jet production at full colour. We present a systematic study of several QCD $2\to 2$ processes and also multi-jet production from a colourless initial state. We examine how subleading colour corrections are distributed differentially in phase space and find that mere normalization effects due to subleading colour can be due to subtle cancellations across phase space. In general, we find that subleading colour does affect the shapes of distributions.

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Forward citations

Cited by 2 Pith papers

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Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Forshaw, S

J.R. Forshaw, S. Pl¨ atzer and F. Torre Gonz´ alez,Exact colour evolution for jet observables , 2502.12133

arXiv

[2] [2]

Pl¨ atzer,Summing Large-N Towers in Colour Flow Evolution , Eur

S. Pl¨ atzer,Summing Large-N Towers in Colour Flow Evolution , Eur. Phys. J. C74 (2014) 2907 [1312.2448]

arXiv 2014

[3] [3]

De Angelis, J.R

M. De Angelis, J.R. Forshaw and S. Pl¨ atzer, Resummation and Simulation of Soft Gluon Effects beyond Leading Color, Phys. Rev. Lett. 126 (2021) 112001 [ 2007.09648]

arXiv 2021

[4] [4]

´Angeles Mart´ ınez, M

R. ´Angeles Mart´ ınez, M. De Angelis, J.R. Forshaw, S. Pl¨ atzer and M.H. Seymour,Soft gluon evolution and non-global logarithms , 1802.08531

arXiv

[5] [5]

Pl¨ atzer,Colour evolution and infrared physics , JHEP 07 (2023) 126 [ 2204.06956]

S. Pl¨ atzer,Colour evolution and infrared physics , JHEP 07 (2023) 126 [ 2204.06956]

arXiv 2023

[6] [6]

Forshaw, J

J.R. Forshaw, J. Holguin and S. Pl¨ atzer,Parton branching at amplitude level , JHEP 08 (2019) 145 [ 1905.08686]

arXiv 2019

[7] [7]

L3 collaboration, Search for color reconnection effects in e+e−→W +W −→ hadrons through particle flow studies at LEP , Phys. Lett. B 561 (2003) 202 [ hep-ex/0303042]

work page Pith review arXiv 2003

[8] [8]

DELPHI collaboration, Investigation of colour reconnection in WW events with the DELPHI detector at LEP-2 , Eur. Phys. J. C 51 (2007) 249 [ 0704.0597]

work page Pith review arXiv 2007