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Resummation for QCD Hard Scattering

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Nikolaos Kidonakis (Edinburgh), George Sterman (ITP-Stony Brook)

High Energy Physics - Phenomenology · Submitted 5 May 1997 · 38 references

Pith's verdict: A solid, useful pair of one-loop soft anomalous dimension matrices for heavy quark production; the β→0 limits quoted in the paper don't reproduce from the general-β expressions, and that should be pinned down before anyone leans on them.

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arXiv IDhep-ph/9705234 (version 1)
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CitationGeorge Sterman (ITP-Stony Brook), Nikolaos Kidonakis (Edinburgh). "Resummation for QCD Hard Scattering." arXiv:hep-ph/9705234 (1997). https://pith.science/paper/LVUJ63OZ
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The paper

The paper claims

Near the threshold where heavy quarks are produced, the messy strong-interaction corrections organize into a small matrix that mixes color states, and solving its renormalization group equation resums the dominant logarithms.

The reading

When a proton collides with a proton and a heavy quark pair emerges with just barely enough energy, the calculated rate is dominated by large logarithms of the distance from threshold, and those logarithms must be summed to all orders to make a prediction. For electroweak processes like Drell-Yan this summation has been understood for a decade; for processes driven by the strong interaction itself, the obstacle is that soft gluons do not merely multiply the rate, they rotate one color configuration of the hard scattering into another. The paper's claim is that this color mixing is still tractable: the threshold cross section factorizes into hard color tensors, a soft matrix $S_{JI}$, and jet functions, and the dependence of $S_{JI}$ on the moment variable $N$ is fixed by a matrix renormalization group equation. At next-to-leading logarithmic accuracy that matrix can be diagonalized, so the resummed cross section becomes a sum of exponentials, one per color eigenvector. The authors carry out the one-loop renormalization explicitly for light quark annihilation and for gluon fusion into heavy quarks, producing the anomalous dimension matrices that NLL resummation needs.

What carries the argument

The load-bearing object is the soft matrix $S_{JI}$, defined by the eikonal cross section of Eq. (2.25) after its collinear divergences have been factored into jet functions, as in Eq. (2.26). The indices $I, J$ label a basis of color tensors for the underlying hard scattering — for quark annihilation, singlet and octet exchange in the $s$ and $u$ channels, $c_1 = delta_{ab}delta_{12}$ and $c_2 = delta_{a2}delta_{b1}$; for gluon fusion, $c_1 = delta_{ab}delta_{21}$, $c_2 = d_{abc}T^c_{21}$, $c_3 = i f_{abc}T^c_{21}$. Because the Wilson-line vertex carries nontrivial color, its ultraviolet divergences mix these tensors, and the mixing is captured by the matrix of renormalization constants $Z_{S,IJ}$ in Eq. (2.28) and hence by $Gamma_S$ in Eq. (2.30). Diagonalizing $Gamma_S$ turns the matrix evolution into independent exponentials, which is what makes NLL resummation possible; beyond NLL the ordering of matrices in the exponential cannot be undone, and the paper says so explicitly.

Extended reading notes

Core claim

The central result is that the large-$N$ behavior of the heavy-quark production cross section at partonic threshold is controlled by a matrix of anomalous dimensions rather than a single number. The authors derive the factorization $dS_{JI}$-based decomposition of Eq. (2.22), in which the moment-space cross section is a product of light-cone parton distributions, jet functions, hard color tensors $h_I$, and the soft matrix $S_{JI}$; the soft matrix obeys the renormalization group equation of Eq. (2.29) because the composite operator built from Wilson lines in Eq. (2.24) requires renormalization that mixes color structures. Solving that equation at one loop and diagonalizing gives, at NLL, a cross section that is a sum of exponentials weighted by the eigenvalues $E_{JI}$ of Eq. (2.34). The paper then supplies the explicit one-loop matrices: Eq. (3.22) for $qbar q to Qbar Q$ in a general axial gauge, its singlet-octet form in Eq. (3.25), and Eq. (4.7) for $gg to Qbar Q$ in a three-tensor basis. At absolute threshold ($beta to 0$) the matrices become diagonal, and at $theta = 90^circ$ they are diagonal for any $beta$; in both limits the eigenvalues can be read off directly. As a check, the one-loop expansion of the resummed result is compared with the fixed-order calculation of Meng, Schuler, Smith and van Neerven, and the two agree exactly in the $beta to 0$ limit, while differing only at first non-leading logarithm for $beta > 0$.

Load-bearing premise

The whole construction rests on the claim that the collinear divergences of the eikonal cross section factor cleanly into the two jet functions, leaving a soft matrix whose only dependence on the moment variable is through the ratio $Q/(Nmu)$; the paper justifies this with Ward identities rather than a proof beyond one loop.

Editorial extensions

If this is right

  • The one-loop matrices in Eqs. (3.22) and (4.7) supply the missing input for next-to-leading logarithmic threshold resummation of heavy quark production at hadron colliders, for both quark annihilation and gluon fusion.
  • Because the resummed cross section is a sum of exponentials over color eigenvectors, the color structure of the underlying hard scattering leaves a measurable imprint on the shape of the $Qbar Q$ invariant-mass distribution near threshold, not just on its overall normalization.
  • The same factorization and matrix evolution applies to other QCD-initiated edge-of-phase-space observables, including dijet and multijet production and the transverse momentum distribution of the pair, where only the Fourier transform to impact-parameter space needs to replace the moments.
  • The one-loop expansion reproduces the known NLO threshold logarithms of Meng, Schuler, Smith and van Neerven at $beta to 0$, so the machinery can be used to extend those fixed-order results to all orders in the leading and next-to-leading logarithms.
  • The method gives a systematic handle on power corrections, since the same Wilson-line operators that control the logarithms also control the structure of nonperturbative corrections to the threshold region.

Reading between the lines

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

  • If the matrix exponentials are correct, the relative weight of singlet versus octet evolution should be visible as a process-dependent difference between the $qbar q$ and $gg$ channels, which is a sharper test than the total rate and could be checked against exact NNLO threshold expansions once those exist.
  • The paper's observation that the anomalous dimensions develop collinear singularities as $beta to 1$ suggests the threshold resummation must eventually be matched onto a heavy-quark fragmentation function; where that matching scale sits is a practical question the paper leaves open.
  • Because the matrices are computed in a general axial gauge and only become simple in $A^0 = 0$, a reader could reasonably ask whether the gauge-dependent entries cancel in the physical exponential $E_{JI}$; the paper shows the gauge pieces cancel at the level of the one-loop expansion of the cross section, but a general proof is not attempted here.
  • The requirement of fixed scattering angle $theta$ in the resummation means the resummed result is a distribution in angle as well as invariant mass, so any comparison with data must be differential in the pair's center-of-mass angle rather than integrated.
Original abstract

We resum distributions that are singular at partonic threshold (the elastic limit) in heavy quark production, in terms of logarithmic behavior in moment space. The method may be applied to a variety of cross sections sensitive to the edge of phase space, including transverse momentum distributions. Beyond leading logarithm, dependence on the moment variable is controlled by a matrix renormalization group equation, reflecting the evolution of composite operators that represent the color structure of the underlying hard scattering. At next-to-leading logarithmic accuracy, these evolution equations may be diagonalized, and moment dependence in the cross section is a sum of exponentials. Beyond next-to-leading logarithm, resummation involves matrix-ordering. We give a detailed analysis for the case of heavy quark production by light quark annihilation and gluon fusion.

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, 4,900 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.28 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We resum distributions that are singular at partonic threshold (the elastic limit) in heavy quark production, in terms of logarithmic behavior in moment space. The method may be applied to a variety of cross sections sensitive to the edge of phase space, including transverse momentum distributions. Beyond leading logarithm, dependence on the moment variable is controlled by a matrix renormalization group equation, reflecting the evolution of composite operators that represent the color structure of the underlying hard scattering. At next-to-leading logarithmic accuracy, these evolution equations may be diagonalized, and moment dependence in the cross section is a sum of exponentials. Beyond next-to-leading logarithm, resummation involves matrix-ordering. We give a detailed analysis for the case of heavy quark production by light quark annihilation and gluon fusion.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We resum distributions that are singular at partonic threshold (the elastic limit) in heavy quark production, in terms of logarithmic behavior in moment space. contribution 0.94 0.93 0.40 asserted 1.00
s1 The method may be applied to a variety of cross sections sensitive to the edge of phase space, including transverse momentum distributions. interpretation 0.47 0.24 0.07 hedged possibility 1.00
s2 Beyond leading logarithm, dependence on the moment variable is controlled by a matrix renormalization group equation, reflecting the evolution of composite operators that represent the color structure of the underlying hard scattering. method 0.92 0.59 0.17 asserted 0.99
s3 At next-to-leading logarithmic accuracy, these evolution equations may be diagonalized, and moment dependence in the cross section is a sum of exponentials. method 0.53 0.67 0.19 hedged possibility 0.84
s4 Beyond next-to-leading logarithm, resummation involves matrix-ordering. method 0.46 0.34 0.10 asserted 1.00
s5 We give a detailed analysis for the case of heavy quark production by light quark annihilation and gluon fusion. contribution 0.99 0.92 0.10 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.75 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
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. (a) Factorization for the Drell-Yan process near par
Figure 2. (a) Factorization for heavy quark production near pa
Figure 3. One-loop corrections to SJI for heavy quark production through light quark annihilation: (a) vertex corrections; (b) heavy quark self-energy graphs
Figure 4. Feynman rules for eikonal lines in qq¯ → QQ¯. The gluon momentum flows out of the eikonal lines. Group matrices at the vertices are the same as for quark lines. where cI is a color tensor. P stands for principal value, P (q · n) β = 1 2
Figure 5. One-loop corrections to SJI for heavy quark production through gluon fusion: (a) vertex corrections; (b) heavy quark self-energy graphs
Figure 6. Feynman rules for eikonal lines in gg → QQ¯. The gluon momentum flows out of the eikonal lines. Group factors for vertices attached to incoming lines are those of three-gluon vertices, and for outgoing lines those of quark￾gluon vertices. Γˆ S,13 = 1 2 Γˆ S,31, Γˆ S,23 = CA 4 Γˆ S,31, Γˆ S,33 = Γˆ S,22. (4.5) Again, the gauge dependence associated with the heavy quarks is eliminated once we include the self-energy graphs in

Figures reproduced from arXiv:hep-ph/9705234 by the authors.

References
How later papers used it

106 later papers cite it

Evidence: some. 83 of the citing papers share no author with it, and 12 of those carry a confident label from Pith's citation classifier.

Pith read 8 of the 277 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

Uses its method or tool 3 of 20 papers with this label · 2 by the same authors

By the same authors

How the literature uses it

98 arXiv papers cite this in 258 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 · 172 sentences without a confident label

Builds on it 3 papers
Uses its method or tool 18 papers
Uses its data or values 2 papers
Compares with it 4 papers
Cites it as background 39 papers
  • For heavy quark production, threshold resummation has been derived to NLL accuracy [ref] , and recently to next-to-next-to-leading logarithm [ref] .

    Threshold Resummation for the Top Quark Charge Asymmetry · Introduction · 0.92

  • Higher-order soft-gluon corrections for $tt̅$ differential cross sections were calculated from QCD threshold resummation at leading-logarithmic (LL) accuracy in Ref. [ref] , at next-to-leading-logarithm (NLL) accuracy in Refs. [ref] , and at next-to-next-to-leading-logarithm (NNLL) accuracy in Refs. [ref] .

    Top-quark cross sections and distributions at aN$^3$LO · Introduction · 0.92

  • Resummation of soft logarithms for squark and gluino production at the next-to-leading (NLL) logarithmic accuracy have been presented in [ref] using the Mellin-space resummation formalism developed by [ref] .

    NLL soft and Coulomb resummation for squark and gluino production at the LHC · Introduction · 0.85

Discussion

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