Large Jets at Hadron Colliders at Order $\alpha_s^3$: A Look Inside

Davison E. Soper; Stephen D. Ellis; Zoltan Kunszt

REVIEW

Large Jets at Hadron Colliders at Order $\alpha_s^3$: A Look Inside

Stephen D. Ellis, Zoltan Kunszt, Davison E. Soper

High Energy Physics - Phenomenology · Submitted 26 Aug 1992 · 10 references

Pith's verdict: The first order-αs³ look inside the jet, and the paper is candid that its shape agreement comes from a tuned merging cutoff rather than a prediction.

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arXiv IDhep-ph/9208249 (version 1)

Pith ID7Z5H7SQV, Pith's permanent record of this paper and its review

CitationDavison E. Soper, Stephen D. Ellis, Zoltan Kunszt. "Large Jets at Hadron Colliders at Order $\alpha_s^3$: A Look Inside." arXiv:hep-ph/9208249 (1992). https://pith.science/paper/7Z5H7SQV

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Editor's note

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Worth your time, with one caveat up front: the shape agreement is tuned, and the paper says so itself. That candor is why I'd still take it seriously rather than file it as curve-fitting. What's new: EKS point their published NLO jet calculation (Ref. [1]) at the inside of the jet instead of the rate. At order αs³ a jet can hold two partons, so the inclusive cross section picks up R dependence and you can compute the fractional transverse energy profile F(r,R,ET). Neither appears in the earlier EKS or Aversa et al. work, which was about rates. The diagnostic payoff is real: the R slope is strongly scale-dependent even where the rate is stable, R ≈ 0.7 comes out as the sweet spot, and the raw calculation puts too much energy at the cone edge. That last observation is what opens the merging question, and it is why this paper gets cited. Where it's soft: Table 1. The untuned run (μ = ET/2, R_sep = 2R) matches A and C but misses B badly, 0.13 against 0.28, and the authors say so in the text. The row that agrees — μ = ET/4, R_sep = 1.3R — uses two knobs against three fitted numbers, and it improves B while nudging A down. Fig. 2 shows the same pattern: the untuned curves sit systematically below the data, and the dot-dash curve that fixes it is the tuned one. So the shape claim is consistency, not prediction. The stress-test is right about that, though I'd soften the framing: the paper never presents the tuned row as a prediction, and the untuned agreement on A and C is a genuine independent success. Two mechanical points. The theory curves carry no error bands; the 10% scale, 20% PDF and 6 GeV/ET power estimates live in prose only. And the merging justification is deferred to Ref. [8], listed as in preparation — that is the one citation I'd want to see before treating R_sep as physics rather than a fitting parameter. The F comparison uses preliminary CDF data (Ref. [7]), which the authors flag. Who it's for: anyone working on jet substructure, and anyone who wants to watch a founding calculation handle a discrepancy it cannot yet resolve. Recommend peer review, yes. The flaws are acknowledged, the physics is sound, and this is the Letter that opened the program.

Referee report

Significance

If the central claim holds, this is an influential result: it is one of the first demonstrations that fixed-order QCD at order α_s³ describes not only the jet rate but the distribution of energy inside a jet, and it identifies the merging prescription in cone algorithms as a physically consequential ingredient rather than a technical detail. The observation that the perturbative series is best behaved near R ≈ 0.7, and that the R-dependence of the cross section is a genuine order-α_s³ prediction with no Born analogue, has lasting value for jet-physics methodology. The paper is also commendably candid about what is and is not under control: it states the residual scale, PDF and power-suppressed uncertainties (∼10%, ∼20%, ∼6 GeV/E_T), and it labels the R_sep discussion as preliminary. The calculation rests on matrix elements published in Ref. [1], so the technical machinery is not at issue here; what is at issue is the strength of the data comparison.

Major objections (4)

[Table 1 and the paragraph beginning 'We have seen that the R dependence and the B parameter suggest'] The last row of Table 1 (μ = E_T/4, R_sep = 1.3R, giving A = 0.49, B = 0.22, C = 0.19) is presented as the resolution of the B-parameter discrepancy, and the accompanying sentence states that 'the extra constraint of R_sep has ensured that there is approximately the observed fraction of E_T near the edge of the cone.' Two quantities (μ and R_sep) have been varied against three fitted data numbers, and the tuned row improves B while moving A away from the data (0.49 versus 0.52 untuned, against 0.54 measured) and leaving C essentially unchanged. As it stands the tuned row demonstrates that a two-parameter family can pass through the data; it does not by itself demonstrate that the merging physics is right. I would ask the authors to reword this passage as a consistency check rather than a verification, and to state plainly that μ and R_sep were chosen with reference to these data.
[Fig. 2 and the R-dependence discussion on p. 2] The text argues that for R > 0.9 the order-α_s³ cross section 'is dominated by the order α_s³ real emission process' and that 'the usefulness of fixed order perturbation theory is compromised,' yet Fig. 2 compares the profile F(r,R,E_T) to CDF data at R = 1.0, i.e. inside the regime the authors themselves flag as unreliable. The systematic offset of the theory curves below the data in Fig. 2 is then given a physical reading (too much E_T near the cone edge). Either the R > 0.9 caveat should be softened, or the profile comparison should be shown at the R ≈ 0.7 value the paper identifies as best behaved, or the authors should explain why the R = 1.0 profile is nevertheless a meaningful test. This is load-bearing because Fig. 2 is one of the two pillars of the internal-structure claim.
[The R_sep prescription, p. 4, and Ref. [8]] The merging cutoff R_sep is the novel physical ingredient of the paper, and its detailed justification is deferred to Ref. [8], listed as 'in preparation.' The Letter gives only the qualitative statement that a real experiment would resolve two separated sprays as two jets, and the specific value 1.3R is not derived. Since the headline agreement in Table 1 and the dot-dash curves of Figs. 1 and 2 depend on this value, I would ask for at least a sketch of the physics of R_sep in this Letter and a statement of how the results vary over a plausible range (say 1.0R to 2R), so that the reader can judge whether 1.3R is singled out or merely representative.
[Figs. 1 and 2, and the uncertainty estimates on p. 1] The text quotes a residual scale uncertainty of ∼10%, a PDF uncertainty of ∼20%, and a nonperturbative correction of order ∼6 GeV/E_T, but neither figure carries an uncertainty band, and the differences being interpreted (the B parameter, the F profile offset) are of the same order as these quoted uncertainties. The authors also note that the total cross section falls by ≤10% as R_sep is reduced from 2R to 1.3R, which is within the quoted scale uncertainty, so the R_sep effect on the rate is not resolvable while its effect on the shape parameters is. Adding indicative bands, or stating explicitly in the captions which comparisons survive the quoted uncertainties, would make the strength of the agreement much easier to assess.

Minor objections (7)

[Table 1 caption] The caption reads '3 parameter fits to data and calculated curves in Figs. 1,3 and 4,' but the manuscript contains only Figs. 1 and 2. The figure numbers should be corrected.
[Table 1, first row] The data row is attributed to Ref. [7], which is the CDF jet-shapes paper, whereas the R-dependence of the inclusive cross section shown in Fig. 1 is compared to Ref. [3]. Please check that the A, B, C values for the data are attributed to the correct CDF paper.
[Eq. (2)] The fitting form σ = A + B ln R + C R² is introduced without motivation, and the paper does not state the range of R over which the fit is performed or the number of data points entering it. A sentence on both would let the reader judge how much weight the B and C comparisons can bear.
[p. 3, 'optimal size' for the jet cone] The conclusion that R ≈ 0.7 is the 'optimal size' follows from the value at which the residual μ dependence is smallest. That is a stability criterion for the truncated series, not an independent prediction, and the phrase 'in some sense the perturbation theory is telling us' could be sharpened to say so.
[Fig. 1 caption and p. 2] The sentence 'This figure is essentially Fig. 3 of Ref. [3] but with the correct theoretical result' reads as a criticism of the CDF theory curve; a neutral phrasing such as 'with the present order-α_s³ result' would be preferable in print.
[p. 2, overlapping-jet rule] The rule that one-parton jets overlapping a two-parton jet are discarded is stated but its quantitative impact on the normalization is not estimated. A brief remark on the size of this effect would be useful, particularly since it enters the same normalization as the R_sep study.
[Author affiliations] The second affiliation is printed as 'Eidgenossosche Technische Hochshule' and the city as 'Z¨ urich'; these should read 'Eidgenössische Technische Hochschule' and 'Zürich.'

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The paper

The paper claims

A third-order QCD calculation claims to describe not just how often jets are produced at a hadron collider, but how their energy is distributed inside the jet cone, and it agrees with CDF measurements at 100 GeV once a hand-set merging rule

The reading

The paper asks whether fixed-order perturbative QCD can say something about the inside of a jet, not merely about how many jets are made. At order $alpha_s^2$ a jet is a single parton and has no internal structure at all; at order $alpha_s^3$ a jet can contain two partons, so the calculated cross section begins to depend on the cone radius $R$ and on how energy is spread within the cone. The authors compute the inclusive single-jet cross section in $pp$ collisions at $sqrt{s}=1800$ GeV and $E_T=100$ GeV, together with a fractional transverse-energy profile $F(r,R,E_T)$ that records what share of a jet's energy lies inside an inner cone of radius $r

What carries the argument

The load-bearing object is the fractional transverse-energy profile $F(r,R,E_T)$: for a sample of jets of transverse energy $E_T$ defined by a cone of radius $R$, it is the average fraction of the jet's energy falling inside a concentric inner cone of radius $r<R$. The authors compute the complement $1-F$ rather than $F$ itself, because the energy in the annulus between $r$ and $R$ is free of the collinear singularity at $r=0$; the numerator is a purely order $alpha_s^3$ integral of the three-parton cross section over the annulus, normalized to $E_T$ times the Born cross section. The second piece of machinery is the merging rule. The default algorithm merges any two partons that can be covered by one cone, up to a separation of $2R$, which counts two partons sitting $2R$ apart with a valley between them as a single jet centered on the valley. The authors add a cutoff $R_{sep} leq 2R$ so that partons separated by more than $R_{sep}$ are not merged, and they set $R_{sep}=1.3R$ as a simple stand-in for what an experimental jet algorithm would do.

Extended reading notes

Core claim

The central claim is that an order $alpha_s^3$ perturbative calculation of single-jet production, equipped with a jet definition that includes a merging cutoff, reproduces the CDF measurements of both the jet rate versus cone radius and the distribution of transverse energy inside the jet, at $sqrt{s}=1800$ GeV and $E_T=100$ GeV. The authors show that the order $alpha_s^3$ cross section is far less sensitive to the renormalization/factorization scale $mu$ than the Born result, with residual scale dependence estimated at about 10%, and that its dependence on the cone radius $R$ is qualitatively right: stable near $R approx 0.7$, dominated by real emission at large $R$, and dominated by the virtual correction at small $R$. Fitting both data and theory to $sigma = A + B ln R + C R^2$, they find the $R$-independent piece $A$ agrees well, the piece $C$ associated with radiation far from the jet axis agrees at $mu = E_T/2$, but the piece $B$ associated with radiation near the jet direction comes out too small. Lowering the scale to $E_T/4$ raises $B$ toward the data while overshooting $C$; adding the merging cutoff $R_{sep}=1.3R$ at $mu = E_T/4$ then restores $C$ and leaves both parameters in reasonable agreement with the measured values.

Load-bearing premise

The agreement on jet shapes depends on a merging cutoff $R_{sep}=1.3R$ that is chosen by hand to imitate the experimental jet algorithm, with its detailed justification deferred to a paper still in preparation.

Editorial extensions

If this is right

Fixed-order QCD at order $alpha_s^3$ becomes a usable tool for jet physics at hadron colliders, since the residual scale dependence drops to roughly 10% and the predicted radius dependence of the cross section has the right shape.
The cone radius $R approx 0.7$ is singled out as the size at which the order $alpha_s^3$ result is most stable, which is the size actually used in the CDF inclusive jet analysis.
The comparison of the fitted parameters $B$ and $C$ against data gives a diagnostic for how much radiation is emitted near the jet axis versus far from it, and hence a handle on the effective strong coupling.
Because the merging rule changes the predicted energy profile at the cone edge by a sizeable amount while changing the total cross section by at most about 10%, details of jet definition matter more for jet substructure than for jet rates.
Differences in internal jet structure are suggested as a possible way to separate quark-initiated from gluon-initiated jets.

Reading between the lines

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

If the merging cutoff is really standing in for finite angular resolution and fragmentation spread, then its value should scale with $E_T$ and with the detector's angular resolution rather than stay fixed at $1.3R$; a measurement of the profile at several $E_T$ would test that.
The pattern of a correct $A$, a too-small $B$ and a correct $C$ at $mu = E_T/2$ reads like a mismatch in the angular distribution of radiation rather than an overall normalization error, which points at higher-order corrections that populate the collinear region rather than at the parton distributions.
Since the profile is computed at lowest nontrivial order, its scale dependence is monotonic, so the fact that data prefer a lower scale than the cross section does is itself a signal that the two observables need different effective scales at this order.
A natural extension is to use the same profile machinery on photon-tagged or $Z$-tagged jet samples, where the parton flavour of the jet is fixed at leading order, to see whether the merging prescription is flavour-independent.

Original abstract

Results from the study of hadronic jets in hadron-hadron collisions at order $\alpha_s^3$ in perturbation theory are presented. The focus is on various features of the internal structure of jets. The numerical results of the calculation are compared with data where possible and exhibit reasonable agreement.

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, 2,847 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.38 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.49 · mentions code or data availability 0.03.

The abstract, sentence by sentence

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

Results from the study of hadronic jets in hadron-hadron collisions at order $\alpha_s^3$ in perturbation theory are presented. The focus is on various features of the internal structure of jets. The numerical results of the calculation are compared with data where possible and exhibit reasonable agreement.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Results from the study of hadronic jets in hadron-hadron collisions at order $\alpha_s^3$ in perturbation theory are presented.	contribution 0.99	0.89	0.12	asserted 0.98
s1	The focus is on various features of the internal structure of jets.	contribution 0.98	0.57	0.07	asserted 0.93
s2	The numerical results of the calculation are compared with data where possible and exhibit reasonable agreement.	result 0.94	0.89	0.36	asserted 0.84

What kind of paper the classifier reports this is: theoretical result 0.68 · evidence from simulation or computation 0.96.

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.** Inclusive jet cross section versus R for √ s = 1800 GeV, ET = 100 GeV and 0.1 < |η| < 0.7 with µ = ET /4, ET /2, ET compared to data from CDF[3]; the dot-dash curve is explained in the text. Also indicated for the three µ choices are the values of the R-independent Born cross section

Figures reproduced from arXiv:hep-ph/9208249 by the authors.

References

Works cited (10)

[1]

S.D. Ellis, Z. Kunszt and D.E. Soper, 64 2121 90 ; 40 2188 89 ; 62 726 89 ; preprint OTIS 495, UW/PT-92-08, ETH-PT/92-22 (unpublished)

View on Pith
[8]

CDF Collaboration, F. Abe, et al. , ``A Measurement of Jet Shapes in p p Collisions at s =1.8 TeV,'' preprint FERMILAB-PUB/92/167-E (1992)

View on Pith 1992
[7]

P.N. Harriman, et al. , 42 798 90

View on Pith
[2]

F. Aversa, M. Greco, P. Chiappetta and J. Ph. Guillet, 65 401 90 ; C46 253 90 ; B327 105 89 ; 211B 465 88 ; 210B 225 88

View on Pith
[3]

CDF Collaboration, F. Abe, et al. , 68 1104 92

View on Pith
[4]

See, for example, J. C. Collins and D. E. Soper,

View on Pith
[5]

See, for example, ``Towards a Standardization of Jet Definitions'', J.E. Huth et al. , Proceedings of the 1990 DPF Summer Study on High Energy Physics , Snowmass, 1990

View on Pith 1990
[6]

For a compilation of jet algorithms see B. Flaugher and K. Meier, Proceedings of the 1990 DPF Summer Study on High Energy Physics , Snowmass, 1990

View on Pith 1990
[9]

S.D. Ellis, Z. Kunszt and D.E. Soper, in preparation

View on Pith
[10]

J. Pumplin,``Variables for Distinguishing Between Quark Jets and Gluon Jets'', Proceedings of the 1990 DPF Summer Study on High Energy Physics , Snowmass, 1990; ``How to tell Quark Jets from Gluon Jets'', preprint MSUTH91/3(1991)

View on Pith 1991

Forward citations

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How the literature uses it

32 arXiv papers cite this in 37 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 · 22 sentences without a confident label

Builds on it 1 paper

This corresponds to the jet shape [ref] but defined with respect to the WTA axis instead of the standard jet axis.

Phenomenology with a recoil-free jet axis: TMD fragmentation and the jet shape · Introduction · 0.83

Compares with it 1 paper

This is in contrast to, e.g., the integrated jet shape $Ψ(R)$ [ref] , which requires a detailed calculation even for its leading order behavior [ref] .

Angular Scaling in Jets · Calculating the average · 0.98

Cites it as background 13 papers

The jet shape [ref] is one of the classic jet substructure observables and it probes the transverse energy profile inside a jet.

Towards the understanding of jet shapes and cross sections in heavy ion collisions using soft-collinear effective theory · Introduction · 0.99
Jet shapes [ref] are defined as the normalised transverse momentum flow as a function of the distance to the jet axis.

Determination of the $b$-quark mass $m_b$ from the angular screening effects in the ATLAS $b$-jet shape data · Introduction · 0.97
With such additional handles, we might have a chance to distinguish boosted $$ signal from the QCD background even at a smaller integrated luminosities.[ There are other approaches dealing with a similar situation in a different perspective in recent literature [ref] .] We discuss jet substructure later in the text.

Top Jets at the LHC · Introduction · 0.88

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

S.D. Ellis, Z. Kunszt and D.E. Soper, 64 2121 90 ; 40 2188 89 ; 62 726 89 ; preprint OTIS 495, UW/PT-92-08, ETH-PT/92-22 (unpublished)

View on Pith

[2] [8]

CDF Collaboration, F. Abe, et al. , ``A Measurement of Jet Shapes in p p Collisions at s =1.8 TeV,'' preprint FERMILAB-PUB/92/167-E (1992)

View on Pith 1992

[3] [7]

P.N. Harriman, et al. , 42 798 90

View on Pith

[4] [2]

F. Aversa, M. Greco, P. Chiappetta and J. Ph. Guillet, 65 401 90 ; C46 253 90 ; B327 105 89 ; 211B 465 88 ; 210B 225 88

View on Pith

[5] [3]

CDF Collaboration, F. Abe, et al. , 68 1104 92

View on Pith

[6] [4]

See, for example, J. C. Collins and D. E. Soper,

View on Pith

[7] [5]

See, for example, ``Towards a Standardization of Jet Definitions'', J.E. Huth et al. , Proceedings of the 1990 DPF Summer Study on High Energy Physics , Snowmass, 1990

View on Pith 1990

[8] [6]

For a compilation of jet algorithms see B. Flaugher and K. Meier, Proceedings of the 1990 DPF Summer Study on High Energy Physics , Snowmass, 1990

View on Pith 1990

[9] [9]

S.D. Ellis, Z. Kunszt and D.E. Soper, in preparation

View on Pith

[10] [10]

J. Pumplin,``Variables for Distinguishing Between Quark Jets and Gluon Jets'', Proceedings of the 1990 DPF Summer Study on High Energy Physics , Snowmass, 1990; ``How to tell Quark Jets from Gluon Jets'', preprint MSUTH91/3(1991)

View on Pith 1991