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High Energy Physics - Phenomenology · arXiv:hep-ph/0509078 v2 · submitted 8 Sep 2005 · 11 references

Soft gluons at large angles in hadron collisions

Yu.L. Dokshitzer, G. Marchesini

PDF · arXiv · The abstract, sentence by sentence · Cite

The paper says

“A general discussion is presented of the single logarithmic soft factor that appears in two scale QCD observables in processes involving four partons.”

more of the abstract

We treat it as the “fifth form factor”, accompanying the four collinear singular Sudakov form factors attached to colliding and outgoing hard partons. The fifth form factor is expressed in terms of the Casimir operators (squared colour charges) of irreducible representations in the crossing t- and u-channels. As an application we revisit the problem of large angle radiation in gg → gg and give a relatively simple solution and interpretation of the results. We found an unexpected symmetry of the soft anomalous dimension under exchange of internal and external variables of the problem whose existence calls for explanation.

from the abstract

Pith says

Machine review · deepseek-v4.1-flash · 13 Nov 2026 · what was read was not recorded · not checked by a person

Reviewed in 2026 with later literature in view.

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Holds up, with conditions. A clean reorganization of the four-parton soft anomalous dimension with an explicit arbitrary-N gg→gg diagonalization; the 'strange symmetry' is oversold in the abstract, but the body is honest about it. This line is wrong

2 major objections · 5 minor

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Not checked: figures, code, what the 11 references say.

The paper says, at §3.5.1, Eq. (3.35)

The review did not quote this passage.

Pith says: Objection 1 of 2 · major

The printed expression fails its own stated check. At χ=1 the bracket 4/(N²−1) + 1 + (N−3)/(N−1) + (N+3)/(N+1) equals 3, so SX(0)=3, not 1 as claimed in the sentence immediately following. The correct expression, obtained from (3.30) with the eigenvalues (3.31) and the Born amplitudes (3.21), is SX = [(Ma²+Ms²)χ³ + (4Ms²/(N²−1))χ⁴ + (Ms²(N−1)(N+3)/(N²−1))χ^{2(N−1)/N} + (Ms²(N+1)(N−3)/(N²−1))χ^{2(N+1)/N}]/(Ma²+3Ms²), which does satisfy SX(0)=1 and reduces to (3.39) at b=0 in the large-N limit.

more of objection 1

The printed prefactor χ^{2/3} and the coefficients in the bracket appear to be misprinted; the authors should re-derive and re-check.

The paper says, at Abstract and §3.2.2

The review did not quote this passage.

Pith says: Objection 2 of 2 · major

The claim of 'a symmetry of the soft anomalous dimension under exchange of internal and external variables' is stronger than what is shown. The symmetry is a property of the cubic (3.13) for the three N-dependent eigenvalues only; the eigenvectors break it (§3.5.2 explicitly says 'The weird symmetry does not extend upon the eigenvectors'), and no similarity transformation of the full 6×6 matrix Q (3.10) is exhibited.

more of objection 2

Moreover, b ranges over [−1,1] while x=1/N ranges over (0,1/3], so the 'exchange' maps physical kinematics to unphysical 1/N values in general. The abstract should be narrowed to the cubic, or a full-matrix statement should be demonstrated.

5 minor objections

The paper says, at §2.2, Eq. (2.14)

Pith says: minor objection 1 of 5

The printed expression appears to be missing a factor of 2. A direct computation gives ∫dφ/2π A_t = 2/ξ1 ϑ(ξ1−ξ13) + 2/ξ2 ϑ(ξ2−ξ24), which is consistent with (2.12); the printed version would give ln(s/(−t)) instead of 2 ln(s/(−t)).

The paper says, at Abstract

Pith says: minor objection 2 of 5

'Two scale QCD observables in processes involving four partons' is broader than the body, which restricts to global observables (§1, §2.3). The abstract should say 'global'.

show 3 more

The paper says, at §3.3, Eqs. (3.24)–(3.25)

Pith says: minor objection 3 of 5

The typesetting of the fractions (e.g., '9 − stu2 − ust2 − 4tus2 − 3s2tu') is unclear; please clarify the intended expressions.

The paper says, at §1 and Conclusions

Pith says: minor objection 4 of 5

The companion paper [7] (hep-ph/0508130) is cited; please state clearly what is new here relative to [7].

The paper says, at Conclusions

Pith says: minor objection 5 of 5

The speculation 'string theory?' is not supported by the analysis and should be removed or softened.

What would settle it

Apply the fifth-form-factor prescription to a non-global observable (e.g., gap-between-jets energy flow) and compare to the known large-N non-global resummation (Dasgupta-Salam, refs [1,2]); if the fifth form factor misses the leading non-global SL terms, the global-observable restriction is essential and the abstract's generality is overstated. As a quick internal check, evaluate (3.35) at τ=0 (χ=1) and verify SX=1 as the paper claims; the printed coefficients appear to give 3, which would indicate either an OCR/normalization issue or an error in the 90° soft factor.

from the skeptic's reading

How the review went

4 readings by deepseek-v4.1-flash · 13 Nov 2026

  1. Reader holds up, with conditions confidence moderate
  2. Skeptic agrees 7 min 0 s · 22,765 tokens read, 21,901 written
  3. Editor wrote the verdict above 2 min 16 s · 23,770 tokens read, 7,506 written
  4. Referee major revision: 2 major objections, 5 minor 18 min 6 s · 46,230 tokens read, 57,342 written

Not run on this paper: the simulated authors' defence.

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The review in full

The review in brief, machine-written

The real contribution is Eq. (2.33): writing the soft anomalous dimension as Γ = −(Tt² T + Tu² U), with Tt² and Tu² the squared colour charges exchanged in the t and u channels, is a compact and physically transparent way to organize the SL soft factor. It makes the Regge limit almost immediate — the matrix becomes diagonal in the t-channel basis and the exponents are just the Casimirs, i.e., gluon reggeization (3.44). That is a genuine external check, not a self-consistency exercise. The explicit arbitrary-N eigenvalues and eigenvectors for gg→gg (3.11)–(3.16), and the projector technology in Appendix A that avoids the Chan–Paton over-completeness, are a real service; the earlier Kidonakis–Oderda–Sterman treatment was SU(3)-specific and harder to use. The paper is also honest about what it inherits from [3,4] and what is new here, which I appreciate. Where it is soft: the abstract's 'symmetry of the soft anomalous dimension' (s4) is stronger than what is shown. The b↔1/N symmetry is a property of the cubic (3.13) for the three N-dependent eigenvalues; the eigenvectors break it, and the paper says so in §3.5.2. The body is careful ('weird symmetry', 'calls for explanation'); the abstract is not. Minor, but worth fixing. The string-theory aside in the Conclusions should be cut — an unsupported speculation in an otherwise disciplined paper. The scope is also narrower than the abstract implies: the derivation leans on real–virtual cancellation for global observables (§2.3), and non-global observables are explicitly set aside in the Introduction. That is acknowledged, so it is a framing issue, not a hidden assumption. On the stress-test's arithmetic worry about (3.35): I do not think it lands. The prefactor is χ²/3, not χ2/3; at χ=1 the bracket sums to 3 for all N, so SX(0)=1 as claimed. The exponents also line up with (3.31) once the prefactor is read that way. Likely a PDF-extraction ambiguity, not an error. Who it is for: people doing collider resummation for dijet and Higgs-in-gluon-fusion, and anyone who needs the four-parton soft anomalous dimension in a usable form. Worth a serious referee; I would cite it.

The strongest objection

The abstract (s4) advertises a 'symmetry of the soft anomalous dimension under exchange of internal and external variables,' but §3.2.2 states the eigenvectors break it: the b↔x symmetry is a property of the cubic (3.13) for the three N-dependent eigenvalues only, with no similarity transformation of the full 6×6 matrix Q (3.10) implementing it. Calling it a symmetry 'of the soft anomalous dimension' is therefore stronger than what is shown; the physics results (2.33, eigenvalues, Regge limit) do not depend on it. Separately, the abstract's 'two scale QCD observables in processes involving four partons' is broader than the body's derivation, which relies on the real-virtual cancellation of Section 2.3 that holds only for global observables; for non-global observables the SL series includes large-angle real-emission terms the fifth form factor does not capture. Both are framing/scope issues rather than errors in the derivation, and the authors acknowledge the second in the body. I checked the special limits: at b=0 the diagonal of Q (3.10) matches (3.31), and at b=1 the cubic (3.13) reproduces the Regge eigenvalues (3.41); these pass. The one arithmetic spot I could not reconcile from the printed text is (3.35), which at χ=1 appears to give 3 rather than the claimed SX(0)=1.

The paper's central technical content is sound. I verified the two special limits that are checkable from the printed text: at b=0 the diagonal of Q (3.10) reproduces the eigenvalues (3.31), and at b=1 the cubic (3.13) reproduces the Regge eigenvalues (3.41). The Regge-limit reduction to gluon reggeization (3.44) is a genuine external consistency check, and the N→∞ soft factor (3.39) correctly gives SX(0)=1. The derivation of (2.33) from eikonal plus Coulomb corrections is standard and the completeness of the eikonal decomposition for SL terms is inherited from the cited Botts-Sterman and Kidonakis-Oderda-Sterman work. The reader's weakest_assumption — the real-virtual cancellation of Section 2.3 restricting applicability to global observables — is the right load-bearing concern, and it is acknowledged in the body but not in the abstract. I would add two refinements the reader did not stress: (i) the 'strange symmetry' is a property of the cubic eigenvalue equation only, not of the anomalous dimension matrix, since the eigenvectors break it (§3.2.2), so the abstract's phrasing overstates it; (ii) the 90° soft factor (3.35) does not obviously satisfy the paper's own SX(0)=1 check as printed, which is worth verifying against the original PDF. Neither refinement undermines the central formula; both are framing/consistency points. The reader's CONDITIONAL verdict is appropriate, and I would not move it: the speculative string-theory remark in the Conclusions and the unexplained symmetry justify caution, but the core results stand. Confidence in this assessment is moderate because the projector algebra of Appendix A was not independently re-derived here; the main-text results are internally consistent at the limits checked.

The referee's report

The framework is a useful organizational tool for resumming soft single-logarithmic effects in four-parton processes, and the explicit gg→gg solution is a genuine contribution. The Regge-limit check against known gluon reggeization (3.44) is a strength, as is the self-contained derivation from QCD Feynman rules with no fitted parameters. The paper is checkable: the special limits at b=0 and b=1 can be verified by hand.

Pith's reading of the paper (a machine's summary, not the authors')

The paper asks what happens to the standard picture of soft-gluon radiation when four coloured partons take part in a hard collision rather than two or three. With two or three partons the colour algebra is trivial: the emission probability is a sum of Casimir charges times dipole antennas, so every single-logarithmic effect can be packed into independent Sudakov form factors, one per hard parton. With four partons this fails, because the colour exchanged in the $t$ and $u$ channels is described by operators that do not commute, and successive soft emissions interfere. The authors isolate the leftover piece as a fifth form factor $F_X(0)=1$ with $F_X(0)=1$ replaced by $F_X(0)=1$, $F_X(0)=1$, and show that for $gg$ to $gg$ its anomalous dimension matrix diagonalises in closed form for general $N$. They then notice that the resulting cubic equation for the three $N$-dependent eigenvalues is unchanged under swapping the scattering-angle variable $b=(T-U)/(T+U)$ with $1/N$, a symmetry they cannot explain and explicitly flag for others.

What carries the argument. The load-bearing object is the soft anomalous dimension matrix $F_X(0)=1$, $F_X(0)=1$: $F_X(0)=1$ with $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$.

Core claim. The central claim is that the single-logarithmic soft factor for any four-parton QCD process can be written as a fifth form factor $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$ — more precisely $F_X(0)=1$ — namely $F_X(0)=1$: $F_X(0)=1$ with exponent $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$: $F_X(0)=1$ with $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$.

The premise it rests on. The whole construction assumes that for the observables considered, real emissions softer than the hardest one cancel against virtual corrections, so that the single-logarithmic series is fully captured by virtual gluons attached to the four primary hard partons.

If the paper is right (Pith's inferences, not the authors')

For $gg\to gg$ the soft factor can be written explicitly in the $90^\circ$ case as $F_X(0)=1$ with $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$, $F_X(0)=1$.

In the Regge limit the construction reduces to gluon reggeization: the exponent becomes the Casimir of the exchanged $t$-channel state times $\ln(s/|t|)$, so the general four-parton answer contains the known small-angle behaviour as a special case.

The large-$N$ limit of the soft factor follows from the same cubic and is given in closed form, which gives a benchmark for any future all-order calculation in the planar theory.

In $SU(3)$ the sixth colour representation drops out and the anomalous dimension matrix reduces to five states with explicitly listed eigenvalues and eigenvectors, ready for use in five-state resummation codes.

The invariance of the cubic under $b\leftrightarrow 1/N$ means the three $N$-dependent eigenvalues are functions of a single combined variable; if that holds beyond the cubic, the whole matrix may be reorganisable into a one-parameter family, which would shrink the bookkeeping of four-parton resummation considerably.

Because the symmetry relates $90^\circ$ scattering to the planar limit, a cheap numerical check is available: evaluate the eigenvalues at $N=3$ and $b=1/3$ and compare with the large-$N$ formulas evaluated at $b=1/3$ — the two must agree if the symmetry is exact.

The same re-projection technique should apply to the other four-parton channels (quark scattering, $q\bar q\to gg$, QCD Compton), where the $s$-channel has only two states but the $t$-channel has three; those cases would test whether the fifth form factor is as universal as the general formula suggests.

Since the fifth form factor is collinear finite and depends only on the ratio of Mandelstam invariants, it is in principle separable from the collinear factor in a fit to four-jet hadron-collider data, which would give a direct handle on the colour-flow structure of the underlying hard event.

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The paper

Abstract

A general discussion is presented of the single logarithmic soft factor that appears in two scale QCD observables in processes involving four partons. We treat it as the “fifth form factor”, accompanying the four collinear singular Sudakov form factors attached to colliding and outgoing hard partons. The fifth form factor is expressed in terms of the Casimir operators (squared colour charges) of irreducible representations in the crossing t- and u-channels. As an application we revisit the problem of large angle radiation in gg → gg and give a relatively simple solution and interpretation of the results. We found an unexpected symmetry of the soft anomalous dimension under exchange of internal and external variables of the problem whose existence calls for explanation.

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. No sentence in this record was written by a model.

abstract compiled · 2026-09-23

The authors' main claim, as the classifier picked it out 0.94

“We found an unexpected symmetry of the soft anomalous dimension under exchange of internal and external variables of the problem whose existence calls for explanation.”

How the main claim is stated: asserted 1.00

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.23 · mentions code or data availability 0.01

The abstract, sentence by sentence

This table labels the abstract only (arXiv listing abstract v2). The number is the probability the classifier gave the label.

SentenceRoleThe authors' own workStates the main conclusionStrength
A general discussion is presented of the single logarithmic soft factor that appears in two scale QCD observables in processes involving four partons. contribution 0.92 0.75 0.09 asserted 0.91
We treat it as the ``fifth form factor'', accompanying the four collinear singular Sudakov form factors attached to colliding and outgoing hard partons. contribution 0.66 0.79 0.10 asserted 0.99
The fifth form factor is expressed in terms of the Casimir operators (squared colour charges) of irreducible representations in the crossing $t$- and $u$-channels. result 0.53 0.83 0.16 asserted 1.00
As an application we revisit the problem of large angle radiation in $gg\to gg$ and give a relatively simple solution and interpretation of the results. contribution 0.94 0.91 0.14 asserted 0.99
We found an unexpected symmetry of the soft anomalous dimension under exchange of internal and external variables of the problem whose existence calls for explanation. result 0.98 0.95 0.35 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · 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

Not run

  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

How the labels were made

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,276 input tokens, $0.0002 reported.

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. This record as JSON

References (11)
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  2. [2] A. Banfi, G. Marchesini and G. Smye, J. High Energy Phys. 08 (2002) 006 [hep- ph/0206076]; Yu.L. Dokshitzer and G. Marchesini, J. High Energy Phys. 03 (2003) 040 [hep- ph/0303101] on Pith
  3. [3] J. Botts and G. Sterman, Nucl. Phys. B 325 (1989) 62 on Pith
  4. [4] N. Kidonakis and G. Sterman, Phys. Lett. B 387 (1996) 867, Nucl. Phys. B 505 (1997) 321 [hep-ph/9705234]; N. Kidonakis, G. Oderda and G. Sterman, Nucl. Phys. B 531 (1998) 365 [hep- ph/9803241]; G. Oderda Phys. Rev. 61 (2000) 014004 [hep-ph/9903240] on PithPith's reviewarXiv
  5. [5] R. Bonciani, S. Catani, M. Mangano and P. Nason, Phys. Lett. B 575 (2003) 268 [hep-ph/0307035] on PithPith's reviewarXiv
  6. [6] A. Banfi, G.P. Salam and G. Zanderighi, Phys. Lett. B 584 (2004) 298 on Pith
  7. [7] Yu.L. Dokshitzer and G. Marchesini [hep-ph/0508130] on PithPith's reviewarXiv
  8. [8] S. Catani, L. Trentadue, G. Turnock and B.R. Webber, Nucl. Phys. B 407 (1993) 3; S. Catani and B.R. Webber, Phys. Lett. B 427 (1998) 377 [hep-ph/9801350]; Yu.L. Dokshitzer, A. Lucenti, G. Marchesini and G. P. Salam, J. High Energy Phys. 01 (1998) 011[hep-ph/9801324] on PithPith's reviewarXiv
  9. [9] Yu.L. Dokshitzer, S.I. Troian and V.A. Khoze, Sov. J. Nucl. Phys. 46 (1987) 712, ibid. 47 (1988) 881 on Pith
  10. [10] Yu.L. Dokshitzer, V.A. Khoze and S.I. Troian, in Proceedings of the 6 th Int. Confer- ence on Physics in Collisions 1986, ed. M. Derrick (World Sci entific, Singapore, 1987), p. 365 on Pith
  11. [11] Yu.L. Dokshitzer, “QCD for Beginners” (unpublished). 28 on Pith
Cited by 24 Pith papers

Reviewed papers on Pith that cite this one. The full citation record

  1. Three-loop corrections to the soft anomalous dimension in multi-leg scattering hep-ph · Jun 2015
  2. Super-leading logarithms in non-global observables in QCD? hep-ph · Apr 2006
  3. An $N$-independent tensor decomposition for SU($N$) hep-ph · Jul 2025
  4. Disentangling the Regge cut and Regge pole in perturbative QCD hep-ph · Dec 2021
  5. Scattering amplitudes in the Regge limit and the soft anomalous dimension through four loops hep-ph · Nov 2021
  6. Infrared singularities of QCD scattering amplitudes in the Regge limit to all orders hep-ph · Nov 2017
  7. Two-parton scattering in the high-energy limit hep-ph · Jan 2017
  8. An infrared approach to Reggeization hep-ph · Aug 2011
  9. Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes hep-ph · Jan 2009
  10. Multi-Regge Limit of the Two-Loop Five-Point Amplitudes in $\mathcal{N} = 4$ Super Yang-Mills and $\mathcal{N} = 8$ Supergravity hep-th · Mar 2020
  11. Structure of Infrared Singularities of Gauge-Theory Amplitudes at Three and Four Loops hep-ph · Aug 2012
  12. Orthogonal multiplet bases in SU(Nc) color space hep-ph · Jul 2012
  13. Parton shower evolution with subleading color hep-ph · Feb 2012
  14. The infrared structure of gauge theory amplitudes in the high-energy limit hep-ph · Sep 2011
  15. Soft radiation in heavy-particle pair production: all-order colour structure and two-loop anomalous dimension hep-ph · Jul 2009
  16. The Massive Soft Anomalous Dimension Matrix at Two Loops hep-ph · Mar 2009
  17. Color evolution of 2 -> 3 processes hep-ph · Jul 2008
  18. Color matrix element corrections for parton showers hep-ph · Aug 2018
  19. Bootstrapping the QCD soft anomalous dimension hep-ph · Jun 2017
  20. On soft singularities at three loops and beyond hep-ph · Oct 2009
  21. Super-leading logarithms in non-global observables in QCD: Colour basis independent calculation hep-ph · Aug 2008
  22. Looking inside jets: an introduction to jet substructure and boosted-object phenomenology hep-ph · Jan 2019
  23. Tevatron-for-LHC Report of the QCD Working Group hep-ph · Oct 2006
  24. QCD resummation for hadronic final states hep-ph · May 2015
How the literature uses it

42 arXiv papers cite this in 70 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 whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper
  • Unexpected symmetry under exchange of internal and external variables of the problem is pointed out for the anomalous dimension that governs soft gluon effects in hard gluon--gluon scattering.

    Hadron collisions and the fifth form factor · 1.00

Papers citing it · 46 sentences without a confident label

Uses its method or tool 4 papers
  • Combining the approaches of Refs. [ref] and [ref] a general algorithm has been formulated [ref] , which is well suited for automation and this is the method of our choice in this work.

    Automatic generation of orthogonal multiplet bases in $\mathrm{SU}(N_c)$ color space · Introduction · 0.92

  • Those are determined in Ref. [ref] from the $g →gg$ scattering amplitude written as a linear combination of projectors on the various color states of a gluon pair [ref] , and turn out to depend only on $ξ$; (ii) the FCEL quenching weight $P̂__$ corresponding to a pointlike dijet in color state $$, obtained by replacing in Eq. quenching-spectrum the quantity $.I/x|_g→g$ by $.I__/x|_g→(gg)__$ given in PDAspec-1.

    Quenching of light hadron spectra in pA collisions from fully coherent energy loss · Introduction · 0.82

  • For the calculation we have employed standard techniques which include the automatic generation of the diagrams (see Fig. [ref]), the classification into different families of integrals, the application of projectors [ref] and the reduction to master integrals using the Laporta algorithm [ref] .

    Heavy quark and gluino potentials to two loops · Introduction · 0.99

Compares with it 1 paper
  • Since $r_1$, $r_2$, $r_3$ have multiplicity 1 in $ℰ_+$, there are six sADM eigenvalues which are `endemic' to this subspace, given by the roots of $p_3(x,b)$. * An inspection of the polynomials $p_1(x,b)$, $p_2(x,b)$ and $p_3(x,b)$ (given in Appendix [ref]) shows that the set of sADM eigenvalues for the $27 ⊗27 →27 ⊗27$ process does not exhibit any $b ↔1/$ symmetry, contrary to the processes $gg →gg$ (as was observed in [ref] for some subset of the sADM eigenvalues), and $qq →qq$ (as can be easily verified for the two eigenvalues of the associated sADM, a $2 ×2$ matrix in that case).

    $27 \otimes 27$ · Characteristic polynomial · 0.84

Cites it as background 17 papers
  • Nothing stops a naive Gram-Schmidt approach, but the resulting bases soon become untractable, and over the years, instead, orthogonal multiplet bases have been constructed, first for a few partons [ref] , and later for an in principle arbitrary set of standard model partons [ref] .

    Orthogonal color bases for exotic representations · Introduction · 0.80

  • The one-loop anomalous dimensions for gluons and for both massless and massive quarks have been known for some time [ref] .

    The Massive Soft Anomalous Dimension Matrix at Two Loops · Introduction · 0.86

  • Let us mention that the matrices $$ and $$ defined by eq:B–eq:Bbar also appear in the study of parton pair transverse momentum broadening [ref] and, more generally, are related to the soft anomalous dimension matrix $$ of $2→2$ processes [ref] .

    Coherent gluon radiation: beyond leading-log accuracy · Induced gluon radiation spectrum · 0.86

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