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Resummed event shapes at hadron-hadron colliders

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Andrea Banfi, Gavin P. Salam, Giulia Zanderighi

High Energy Physics - Phenomenology · Submitted 24 Jul 2004 · 73 references

Pith's verdict: A careful, genuinely new set of global dijet event-shape definitions with NLL resummations; the weak point is validation, not construction, and the authors say so themselves.

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arXiv IDhep-ph/0407287 (version 3)
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CitationAndrea Banfi, Gavin P. Salam, Giulia Zanderighi. "Resummed event shapes at hadron-hadron colliders." arXiv:hep-ph/0407287 (2004). https://pith.science/paper/VQ3Z6IOA
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The paper

The paper claims

The authors define a family of hadron-collider dijet event shapes engineered to be sensitive to radiation in every direction, so that they can be resummed to next-to-leading logarithmic accuracy with an automated tool while still being meas

The reading

Event shapes — single numbers summarising how far an event's energy flow departs from a two-jet Born configuration — are precision tools in electron-positron and deep-inelastic scattering, but at hadron colliders they have barely been used, in part because the one shape measured at the Tevatron is not 'global': it ignores radiation close to the beam, which blocks the standard all-orders resummation machinery. This paper defines a set of dijet event shapes and jet-resolution thresholds that are global by construction, computes their next-to-leading logarithmic resummed distributions with the automated resummation program CAESAR, and checks whether the resulting predictions live in the region where a detector's finite rapidity coverage can be ignored. The authors organise the definitions into three families: shapes built directly from all final-state momenta, shapes measured centrally but given an exponentially suppressed forward term, and shapes measured centrally with a missing-transverse-momentum recoil term supplying indirect sensitivity to the rest of the event. If the construction holds, hadron-collider dijet events become a quantitative laboratory for the strong coupling, for the colour-evolution matrices that first appear with four hard partons, and for underlying-event and hadronisation models, with observables whose sensitivity to beam fragmentation can be dialled up or down by design.

What carries the argument

The carrying idea is continuous globalness, expressed through the single-emission parametrisation $V = d_ell (k_t/Q)^{a_ell} e^{-b_ell eta} g_ell(phi)$ with a common exponent $a$ across all four hard legs. That common $a$ is exactly what allows the observable's dependence on multiple emissions to collapse into a single function $F(R')$ of $R' = -partial_{L} L g_1(alpha_s L)$, which CAESAR computes numerically (or which is known analytically, $F(R') = e^{-gamma_E R'}/Gamma(1+R')$, for additive observables such as the transverse thrust and the sum of jet masses). The second piece of machinery is the bookkeeping that makes the finite detector reach a controlled approximation: because a soft emission at rapidity $eta$ contributes a factor $e^{-b eta}$, cutting the measurable region at $eta_max$ costs at most of order $e^{-(a + b_{min}) eta_max}$, so the resummed prediction survives wherever $ln v gtrsim -(a+b)(eta_max + <ln X>)$. The third is the choice of the rescaled logarithm $ln 1/(X v)$, with $ln X = -(1/n) sum_ell ln(bar d_ell)$ and $ln bar d_ell = ln d_ell + int (d phi/2 pi) ln g_ell(phi)$, which cancels the arbitrary normalisation of the observable so as not to introduce spurious subleading logarithms.

Extended reading notes

Core claim

The paper's claim is that one can define event shapes for hadron-hadron dijet production that are simultaneously measurable at the Tevatron and the LHC and continuously global, meaning that a soft emission anywhere in the event — including at large rapidity along the incoming beams — contributes to the observable with the same power of transverse momentum, so that the same exponent $a$ applies to every hard leg in $V = d_ell (k_t^{({ell})}/Q)^{a} e^{-b_ell eta^{({ell})}} g_ell(phi)$. Continuous globalness is what current automated resummation can handle, and the authors use CAESAR to produce explicit next-to-leading logarithmic results for each observable, tabulating the leg-by-leg parameters $a_ell$, $b_ell$, $g_ell(phi)$, $d_ell$ and the multiple-emission function $F(R')$ that the resummation needs. For the directly global transverse thrust, thrust minor and three-jet resolution threshold $y_{23}$, the incoming legs have $b=0$ and the observable is uniformly sensitive to transverse momentum out to the beam; for the second family the authors add a forward term $E_{bar C} = (1/Q_{perp,C}) sum_{i notin C} q_{perp i} e^{-|eta_i - eta_C|}$, which raises $b$ to equal $a$ on the incoming legs, doubling the range of $ln v$ over which a finite detector reach is harmless and removing the linear growth of the underlying-event shift with $eta_max$. The third family replaces the forward term with the vector recoil $R_{perp,C}$ of the central region, which by momentum conservation equals the transverse momentum carried outside it; this keeps the observable global using only central particles, at the price that the multiple-emission function $F(R')$ develops a divergence at a finite $R'_c$ where cancellations between emissions become the cheaper way to suppress the observable. The paper's practical claim is that for most of these observables the bulk of the cross section falls inside the region $ln v gtrsim -(a + b)(eta_max + <ln X>)$ with $eta_max = 3.5$, so that comparisons of the fully global theoretical definition with partially global data remain meaningful.

Load-bearing premise

The load-bearing premise is that the automated resummation is correct for these newly defined observables, and the only independent check reported is a comparison of the expansion against a fixed-order program at leading order, which the authors say was not accurate enough at next-to-leading order to be meaningful — so the four-parton colour structure that distinguishes these observables from the electron-positron case is not independently verified.

Editorial extensions

If this is right

  • Hadron-collider dijet event-shape distributions become available at next-to-leading logarithmic accuracy for a whole family of observables rather than one at a time, because the definitions are chosen to fit the automated resummation rather than the resummation being done by hand per observable.
  • The channel-by-channel distributions separate according to the colour factors multiplying the leading logarithms, so a measured distribution can in principle discriminate gluon-rich from quark-rich underlying scattering channels at fixed jet transverse energy.
  • Because four hard partons are present at Born level, these observables expose the colour-evolution (soft anomalous dimension) matrices that are trivial or absent in three-parton final states, giving a phenomenological handle on quantum colour flow in QCD amplitudes.
  • The two forward-term variants give a designed knob on sensitivity to beam fragmentation: the exponentially suppressed version keeps the underlying-event shift of order $langle k_perp^{u.e.} rangle/(E_{perp,1}+E_{perp,2})$ with no $eta_max$ enhancement, while the directly global transverse thrust shifts by an amount proportional to $eta_max$ and is therefore suited to studying the underlying event
  • The indirectly global observables, defined purely from central particles plus a missing-transverse-momentum recoil term, sidestep the detector rapidity limit entirely, at the cost of losing reliability in a region of very small observable values where the multiple-emission function diverges.

Reading between the lines

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

  • The paper's design logic — pick the observable so that the resummation tool applies, rather than forcing the tool to fit a familiar observable — is a template that later automated-resummation work could apply to other multi-jet environments, for instance three-jet or vector-boson-plus-multijet final states where colour evolution is even richer.
  • The recoil-based family suggests a general recipe for making any central detector-level observable global: add a quantity that momentum conservation ties to the unmeasured region. The same trick should work for observables the authors did not list, such as energy-energy correlations restricted to a central calorimeter region.
  • Since the divergence in $F(R')$ for the recoil observables sits at a calculable $R'_c$ tied to the two-dimensional nature of transverse-momentum cancellation, the practical reach of these observables in $ln v$ is predictable in advance from the colour channel, which makes it possible to state before measuring which channels will be usable.
  • The comparison to exact fixed-order calculations is the natural next test: agreement of the expansion of the resummation with the logarithmically enhanced terms of a full next-to-leading order four-jet calculation would confirm the colour structure that the leading-order check cannot reach.
Original abstract

This article introduces definitions for a number of new event shapes and jet-rates in hadron-hadron dijet production. They are designed so as to be measurable in practice at the Tevatron and the LHC, and to be global so that they can be resummed with currently available techniques. We explain how to vary their sensitivity to beam fragmentation, limiting its impact for purely perturbative studies, or deliberately enhancing it so as to focus on non-perturbative effects. Explicit next-to-leading logarithmic resummed results are presented, as obtained with CAESAR.

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, 3,589 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.07 · mentions code or data availability 0.04.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

This article introduces definitions for a number of new event shapes and jet-rates in hadron-hadron dijet production. They are designed so as to be measurable in practice at the Tevatron and the LHC, and to be global so that they can be resummed with currently available techniques. We explain how to vary their sensitivity to beam fragmentation, limiting its impact for purely perturbative studies, or deliberately enhancing it so as to focus on non-perturbative effects. Explicit next-to-leading logarithmic resummed results are presented, as obtained with CAESAR.

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 This article introduces definitions for a number of new event shapes and jet-rates in hadron-hadron dijet production. contribution 1.00 0.90 0.13 asserted 1.00
s1 They are designed so as to be measurable in practice at the Tevatron and the LHC, and to be global so that they can be resummed with currently available techniques. contribution 0.86 0.59 0.08 asserted 0.98
s2 We explain how to vary their sensitivity to beam fragmentation, limiting its impact for purely perturbative studies, or deliberately enhancing it so as to focus on non-perturbative effects. method 0.64 0.77 0.07 asserted 0.93
s3 Explicit next-to-leading logarithmic resummed results are presented, as obtained with CAESAR. result 0.78 0.87 0.18 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.84 · evidence from proof or derivation 0.83.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
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. Dependence gℓ(φ) of τ⊥,g on the azimuthal angle φ of a single emission collinear to each leg ℓ. eq. (3.1) has already been used as brief example of the capabilities of caesar in [18], but here we discuss it more ex￾tensively. In the 2 + 2 jet limit it tends to T⊥,g = 1. Accordingly the observable that one resums is τ⊥,g ≡ 1−T⊥,g. Its de￾pendence on soft and collinear emissions is given by eq. (2.6) with the parameters shown in table 1. The fact that bℓ = 0 for the incoming legs ℓ = 1, 2 means that along the beam direction the observable is sensitive to the emitted transverse mo￾me
Figure 2. The differential cross section dΣH/d ln τ⊥,g separated into the different hard-scattering channels
Figure 8. The distribution of ρH,E in the different hard-scattering channels. shown, channel-by-channel in fig. 8. Com￾pared, say, to an observable like Tm,g, the distributions extend to relatively low val￾ues, a consequence of the smaller LL terms (since b = a). However, as mentioned at the beginning of this section, the fact that b1,2 = a = 1 means that the limit of appli￾cability of the resummation, in the presence of an experimental rapidity cut, is given by ln ρH,E & −2ηmax. Thus all channels for the distribution are comfortably contained within this region. 4.3 Jet Broadenings With th
Figure 9. The distribution of BW,E for separate hard-scattering channels
Figure 11. F(R′ ) for τ⊥,R
Figure 13. The distribution of τ⊥,R for separate hard-scattering channels

Showing 6 of the paper's 13 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/0407287 by the authors.

References
How later papers used it

60 later papers cite it

Evidence: substantial. 52 of the citing papers share no author with it, and 5 of those carry a confident label from Pith's citation classifier.

Pith read 13 of the 86 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 2 of 4 papers with this label
How the literature uses it

59 arXiv papers cite this in 84 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 · 45 sentences without a confident label

Uses its method or tool 4 papers
Compares with it 1 paper
Cites it as support 1 paper
  • As pointed out in [ref] large spurious subleading logarithms may otherwise be introduced, that are associated with the overall normalization of observables, rather than radiative corrections.

    A critical appraisal of NLO+PS matching methods · Comment on the NLL accuracy of the POWHEG formula · 0.96

Cites it as background 31 papers
  • An extensive list of hadronic event-shape definitions can be found in [ref] .

    Event shapes at hadron colliders · Event-shape variables at hadron colliders · 0.98

  • Resummation with Next-to-leading-logarithmic (NLL) accuracy matched to NLO QCD predictions has been achieved for global event shapes [ref] .

    NNLO QCD corrections to event shapes at the LHC · Introduction · 0.89

  • The transverse thrust event shape $T_⊥$ [ref] , is designed for hadron collisions, but has also been calculated for $e^+e^- →2$ jets [ref] ,

    Calculating Soft Radiation at One Loop · Transverse thrust in $e^+e^-$ · 0.89

Discussion

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