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The Resummation of Soft Gluon in Hadronic Collisions

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S. Catani, M.L. Mangano, P. Nason, L. Trentadue

High Energy Physics - Phenomenology · Submitted 18 Apr 1996 · 34 references

Pith's verdict: The Minimal Prescription is a real fix for a real problem, and the paper is honest about what it does and does not prove; the one soft spot is an over-strong interpretive sentence, not the math.

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arXiv IDhep-ph/9604351 (version 2)
Pith IDU24ED4TL, Pith's permanent record of this paper and its review
CitationL. Trentadue, M.L. Mangano, P. Nason, S. Catani. "The Resummation of Soft Gluon in Hadronic Collisions." arXiv:hep-ph/9604351 (1996). https://pith.science/paper/U24ED4TL
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The paper

The paper claims

Soft-gluon resummation is fixed by moving the Mellin contour

The reading

Near the kinematic threshold of a hadron collision, soft gluon emission multiplies the cross section by large logarithms of the Mellin moment $N$, and these logarithms must be summed rather than expanded. The paper asks what happens when the summed expression is converted back to momentum-fraction ($x$) space while the formally subleading terms are dropped, and shows that exactly those dropped terms were cancelling a factorial growth: the truncated $x$-space formula produces coefficients growing like $k!$ and forces cutoffs of several GeV on radiation that should be soft. The authors instead propose a Minimal Prescription, keeping the resummation in $N$ space and running the inverse Mellin integral along a contour placed to the right of the parton-density singularities but to the left of the Landau pole, $2 < C_{MP} < N_L = e^{1/(2 b_0 alpha_S)}$. With that choice the expansion coefficients grow far slower than factorially, and the residual ambiguity of the asymptotic series is exponentially small, of order $e^{-H Q(1-tau)/Lambda}$, so that genuinely non-perturbative physics is confined to the region $1-tau$ of order $Lambda/Q$ where it belongs. Applied to Drell-Yan, heavy-flavour and dijet production, the prescription gives modest corrections except quite close to threshold.

What carries the argument

The load-bearing object is the Minimal Prescription integral, $sigma_{res}(tau) = (1/2pi i) int_{C_{MP}-iinfty}^{C_{MP}+iinfty} F_N^2(Q^2) Delta_N(Q^2) tau^{-N} dN$ with $2 < C_{MP} < N_L = e^{1/(2 b_0 alpha_S)}$, where $Delta_N$ is the next-to-leading-logarithmic exponentiated coefficient built from the functions $g_1$ and $g_2$ of $lambda = b_0 alpha_S log N$. Placing the contour in this window keeps the parton luminosities on their usual Regge behaviour while staying left of the Landau singularities at $N_L$ and $N_L^2$. The supporting machinery is a Cauchy-inequality bound on the remainder of the Taylor expansion of $h_N(lambda)$ in $lambda$, applied separately to the near and far regions of the deformed $N$ contour; the far region is exponentially suppressed, and the near region yields coefficients growing no faster than $(2 b_0 log n / log(1/tau))^n$.

Extended reading notes

Core claim

The central claim is that the large threshold enhancements reported in $x$-space soft-gluon resummations are largely an artefact of an incomplete transformation, not a physical prediction. Exponentiation of threshold logarithms is natural in $N$ space, where longitudinal-momentum conservation factorizes; converting the result to $x$ space at leading or next-to-leading logarithmic accuracy discards terms that, though formally subleading, are precisely the ones that cancel the factorial growth of the retained series. The paper demonstrates this in a fixed-coupling double-logarithmic example, where the exact $N$-space integral has an infinite radius of convergence while the truncated $x$-space form diverges, and then in the running-coupling case, where the truncated $x$-space integral generates $k! C^k (b_0 alpha_S)^k$ behaviour corresponding to spurious power ambiguities as large as $(Lambda/Q)^{0.16}$ for gluon fusion. The remedy is the Minimal Prescription: evaluate the inverse Mellin transform of the next-to-leading-logarithmic exponentiated coefficient along a contour $2 < C_{MP} < N_L$, with all singularities except the Landau poles to its left. The authors prove in Appendix A that this representation is asymptotic to its formal $alpha_S$ expansion, that its coefficients do not grow factorially, and that truncating at the minimal term leaves an error of order $e^{-H Q(1-tau)/Lambda}$.

Load-bearing premise

Appendix A proves the absence of factorial growth only for the leading-logarithmic function $f(lambda)$, and merely states that the proof can be easily generalised to subleading accuracy, while the $Delta_N$ actually used contains $g_2$ with additional branch points at $lambda = 1/2$ and $lambda = 1$.

Editorial extensions

If this is right

  • Threshold resummation can be implemented without any cutoff on soft radiation: the resummed partonic cross section stays smooth and integrable down to $1-rho$ of order $Lambda/m_Q$, the scale where non-perturbative physics actually sets in.
  • For top-quark production at the Tevatron the resummation correction beyond next-to-leading order is about 1%, and negligible at the LHC; for bottom production at HERA-B it is about 12%, well inside the scale uncertainty of the fixed-order calculation.
  • The large-$K$ behaviour of the dijet invariant-mass distribution at the Tevatron is dominated by plain next-to-leading order, with soft-gluon resummation adding roughly 10% at dijet masses near 1 TeV, which makes it unlikely that resummation explains the reported jet excess.
  • Because the prescription respects low moments (no correction at $N=1$), momentum sum rules are not violated, so the resummed coefficient can be matched to fixed-order results by simple subtraction of the first few orders.
  • The same contour prescription is argued to be applicable to the resummed anomalous dimensions used in parton evolution, and is simplest in the $overline{MS}$ scheme where those anomalous dimensions carry no Landau-pole singularity.

Reading between the lines

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

  • If the no-factorial-growth property holds at the accuracy actually used, then published threshold enhancements obtained with $x$-space cutoffs of several GeV should be revisited: the spread among cutoff choices would be an artefact rather than a legitimate uncertainty band.
  • A natural test is to extend the Appendix A bound to the full $Delta_N$ containing $g_2$, whose branch points at $lambda = 1/2$ and $lambda = 1$ sit inside the region where the Cauchy bound is applied; a numerical study of the large-order coefficients at next-to-leading-logarithmic accuracy would settle whether the generalisation asserted in the text is safe.
  • The argument suggests a general diagnostic for any resummation scheme: check whether the truncated transform respects the sum rules of the observable, since a violation there is a symptom that the dropped subleading terms were carrying the cancellation.
  • The same reasoning should apply to transverse-momentum resummation, where the transform back from impact-parameter space plays the role of the inverse Mellin transform; the paper explicitly limits itself to the jet-pair mass variable and leaves jet $p_T$ aside.
Original abstract

We compute the effects of soft gluon resummation for the production of high mass systems in hadronic collisions. We carefully analyse the growth of the perturbative expansion coefficients of the resummation formula. We propose an expression consistent with the known leading and next-to-leading resummation results, in which the coefficients grow much less than factorially. We apply our formula to Drell--Yan pair production, heavy flavour production, and the production of high invariant mass jet pairs in hadronic collisions. We find that, with our formula, resummation effects become important only fairly close to the threshold region. In the case of heavy flavour production we find that resummation effects are small in the experimental configurations of practical interest.

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, 4,827 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.81 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.79 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We compute the effects of soft gluon resummation for the production of high mass systems in hadronic collisions. We carefully analyse the growth of the perturbative expansion coefficients of the resummation formula. We propose an expression consistent with the known leading and next-to-leading resummation results, in which the coefficients grow much less than factorially. We apply our formula to Drell--Yan pair production, heavy flavour production, and the production of high invariant mass jet pairs in hadronic collisions. We find that, with our formula, resummation effects become important only fairly close to the threshold region. In the case of heavy flavour production we find that resummation effects are small in the experimental configurations of practical interest.

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 compute the effects of soft gluon resummation for the production of high mass systems in hadronic collisions. contribution 1.00 0.95 0.06 asserted 0.99
s1 We carefully analyse the growth of the perturbative expansion coefficients of the resummation formula. contribution 0.63 0.95 0.04 asserted 0.90
s2 We propose an expression consistent with the known leading and next-to-leading resummation results, in which the coefficients grow much less than factorially. contribution 0.83 0.93 0.28 hedged possibility 0.54
s3 We apply our formula to Drell--Yan pair production, heavy flavour production, and the production of high invariant mass jet pairs in hadronic collisions. contribution 0.77 0.92 0.04 asserted 0.96
s4 We find that, with our formula, resummation effects become important only fairly close to the threshold region. result 0.99 0.97 0.45 asserted 1.00
s5 In the case of heavy flavour production we find that resummation effects are small in the experimental configurations of practical interest. result 0.99 0.97 0.21 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.90 · evidence from simulation or computation 0.54.

What was examined, and what was not

sources read
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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. Integration contour for N for the determination of asymptotic properties. large, we get two contributions to our integral, from the circle and from the disconti￾nuity along the negative axis. The integral on the circle is bounded by the expression C log2k N0 /k! for some value of C. The discontinuity integral is instead given by (replacing N → −N) 36 k! Z ∞ N0 exp[N log τ − 2 log(N(N − 1)(N − 2))] Disc[log2k (−N)] dN ≤ 36 k! Z ∞ N0 exp[N log τ + 2k log(π + log N)] dN . (3.18) By saddle point integration of the right hand side we immediately see that the above expression cannot grow…
Figure 2. Drell–Yan pair production cross section in pp¯ collisions, normal￾ized to the Born result. subprocesses (given in ref. [2]). The dotted curve is the NL-resummed cross section, without the inclusion of the exact O(αS) result. The full curve is obtained by adding
Figure 3. Partonic cross section for the production of a 175 GeV heavy quark pair. qq¯ initial state (left) and gg initial state (right). The dotted lines are the Born result, the dashed lines the NLO result and the solid line the resummed result. Λ MS 5 =152 MeV
Figure 4. Contribution of gluon resummation at order α 4 S and higher, relative to the NLO result, for the individual channels and for the total, for bottom production as a function of the CM energy in pp collisions
Figure 5. Contribution of gluon resummation at order α 4 S and higher, rela￾tive to the NLO result, for the individual subprocesses and for the total, as a function of the top mass in pp¯ collisions at 1.8 TeV
Figure 6. Contribution of gluon resummation at order α 4 S and higher, relative to the NLO result, for the individual channels and for the total, as a function of the CM energy in pp collisions
Figure 7. Contribution of gluon resummation at order α 4 S and higher, relative to the NLO result, for the individual channels and for the total, for charm production as a function of the CM energy in pp collisions. factorization scales from mb/2 to 2mb. Thus the upper band is 80% higher than the central value, to be compared with a 10% increase from the resummation effects. This result is much less dramatic than the results of ref. [20]. In fig. 7 we also show the effect of resummation in charm production. Due to the large uncertainties that plague charm production [26], this plot should onl
Figure 8. Comparison of the partonic cross section at O(α 3 S ) for the MP resummation formula (solid line), for the resummation formula with the term proportional to γE included (dotted line), and for the exact O(α 3 S ) formula of ref. [21] with µ = mt/2, mt and 2mt (dashed lines)
Figure 9. Contribution of gluon resummation at order α 3 S and higher, rel￾ative to the LO result, for the invariant mass distribution of jet pairs at the Tevatron
Figure 10. Contribution of gluon resummation at order α 4 S and higher, relative to the truncated O(α 3 S ) result, for the invariant mass distribution of jet pairs at the Tevatron

Figures reproduced from arXiv:hep-ph/9604351 by the authors.

References
How later papers used it

150 later papers cite it

Evidence: substantial. 144 of the citing papers share no author with it, and 84 of those carry a confident label from Pith's citation classifier.

Pith read 29 of the 246 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 23 of 84 papers with this label · 1 by the same authors

By the same authors

How the literature uses it

137 arXiv papers cite this in 219 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 · 91 sentences without a confident label

Contests it 1 paper
Builds on it 1 paper
Uses its method or tool 79 papers
Compares with it 1 paper
Cites it as support 4 papers
Cites it as background 21 papers

Discussion

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