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Jets at Hadron Colliders at Order α3

arXiv:hep-ph/9208249v1 26 Aug 1992UW/PT-92-01DOE/ER/40614-16August 1992Jets at Hadron Colliders at Order α3s: A Look InsideStephen D. EllisDepartment of PhysicsUniversity of Washington, Seattle, WA 98195, USAZoltan KunsztEidgenossosche Technische HochshuleCH-8093 Z¨urich, SwitzerlandDavison E. SoperInstitute of Theoretical ScienceUniversity of Oregon, Eugene, OR 97403, USAPACS numbers 12.38, 13.87AbstractResults from the study of hadronic jets in hadron-hadron collisions at orderα3s in perturbation theory are presented. The focus is on various features of theinternal structure of jets.

The numerical results of the calculation are comparedwith data where possible and exhibit reasonable agreement.PREPARED FOR THE U.S. DEPARTMENT OF ENERGYThis report was prepared as an account of work sponsored by the United States Government.Neither the United States nor the United States Department of Energy, nor any of their em-ployees, nor any of their contractors, subcontractors, or their employees, makes any warranty,express or implied, or assumes any legal liability or responsibility for the product or processdisclosed, or represents that its use would not infringe privately-owned rights. By acceptance ofthis article, the publisher and/or recipient acknowledges the U.S. Government’s right to retaina nonexclusive, royalty-free license in and to any copyright covering this paper.

Recent advances, both theoretical[1, 2] and experimental[3], in the study of jet produc-tion in hadron collisions have made possible detailed comparisons of theory with experiment.In the Standard Model our general understanding of the high energy collisions of hadronssuggests that jets arise when short distance, large momentum transfer interactions gener-ate partons (quarks and gluons) that are widely separated in momentum space just afterthe hard collision. In a fashion that is not yet quantitatively understood in detail theseconfigurations are thought to evolve into hadronic final states exhibiting collimated spraysof hadrons, which are called jets.

These jets are then the observable signals of the shortdistance parton configurations.This general qualitative picture is characteristic of both perturbative QCD and thedata. When one proceeds to a quantitative confrontation of theory and experiment, a precisedefinition of a jet must be supplied, and measured jet cross sections depend on the definitionused.

For instance, when one defines a jet as consisting of all the particles whose momenta lieinside a cone of radius R, then measured jet cross sections depend on R. On the theoreticalside, in an order α3s calculation a jet can consist of two partons instead of just one. At thislevel, then, a jet can have internal structure and jet cross sections calculated at order α3swill depend on the jet definition applied.

Two questions arise. First, how well does thedependence on the jet definition exhibited by the theoretical jet cross section match thatof the experimental jet definition?

Second, how well does the internal structure calculatedat order α3s compare to the internal structure of experimentally observed jets? We addressthese questions in this Letter.We consider the inclusive single jet cross section in pp collisions.

This cross section isa function of the physical variables s, the total energy, ET, the transverse energy of the jet(ET = E sin θ), and η, the pseudorapidity of the jet (η = ln cot(θ/2)) and, as suggestedabove, the definition of the jet. The theoretical inclusive jet cross section also depends onthe unphysical renormalization/factorization scale µ and on the specific choice of partondistribution functions, fa/A(x, µ).

In a Born level (α2s) calculation the dependence on thescale µ arises from both the parton distribution functions and the parton scattering crosssection through the dependence of the latter on the strong coupling constant αs(µ). At orderα3s explicit factors of ln(µ) appear that serve to cancel a part of this µ dependence.

If higherorder contributions were calculated, they would tend to eliminate, successively, more of theµ dependence. At any fixed order in perturbation theory the residual µ dependence actsas an estimator of the theoretical uncertainty associated with the truncated perturbationseries.We find that the situation for the jet cross section at order α3s is a major improvementover the order α2s Born result both because the µ dependence is reduced and because thecross section exhibits a reasonable dependence on the jet definition.

While the Born crosssection exhibits monotonic dependence on µ, the higher order result is relatively insensitiveto the value of µ in a broad region near µ ≃ET/2.We estimate[1] that the residualtheoretical uncertainty, as indicated by the residual µ dependence, is ∼10%. The uncertaintyin the cross section due to the current uncertainty in the parton distribution functions isestimated[1] to be somewhat larger, ∼20%.

In the perturbative calculations described here,1

the effects of the long distance fragmentation processes and of the soft interactions of the“spectator” partons are ignored. We estimate[1] that these uncalculated power suppressedeffects constitute a correction of order ∼6 GeV/ET to the cross section.

Thus for jet ET’sof order 100 GeV, as discussed here, the nonperturbative uncertainty is of the same order orsmaller than that due to perturbative effects.At hadron colliders jets are typically defined[4] in terms of the particles n whose mo-menta −→pn lie within a cone centered on the jet axis (ηJ, φJ) in pseudorapidity η and azimuthalangle φ, [(ηn −ηJ)2 + (φn −φJ)2]1/2 < R. The jet angles (ηJ, φJ) are the averages of theparticles’ angles,ηJ=Xn∈conepT,nηn/ET,J ,φJ=Xn∈conepT,nφn/ET,J(1)with ET,J = Pn∈cone pT,n. This process is iterated so that the cone center matches the jetcenter (ηJ, φJ) computed in Eq.

(1). It is important to note that this jet algorithm is not yetfully defined since jets can overlap.

In particular, it is possible for the constraints above tobe satisfied by configurations where some particles are common to more than one cone. Inthe α3s calculation it is possible for a one parton jet to lie within the cone of a two parton jet.In the theoretical calculations described here[1], we use the rule that only the two partonjets are included and the overlapping one parton jets are discarded.

The precise definitionsused by the various experiments differ to a greater or lesser extent from this form[5].While the Born cross section with only a single parton per jet is R independent, theorder α3s cross section can have 2 partons inside a jet and is R dependent. The theoreticalexpectation for the R dependence is shown in Fig.

1 along with results from CDF[3]. Theinclusive single jet cross section is evaluated at ET = 100 GeV using the HMRS(B)[6] partondistributions.

Since dependence on R is not present in the Born cross section, this dependenceis a lowest order result at order α3s, that is, dσ/dR = 0 + O(α3s). One therefore expects that,although the cross section itself is relatively µ independent, its slope dσ/dR will be quitestrongly µ dependent.

Just this behavior is indicated in Fig. 1 by the curves for the crosssection versus R for 3 different µ values.

(This figure is essentially Fig. 3 of Ref.

[3] but withthe correct theoretical result. The fourth, dot-dash curve and the parameter Rsep will beexplained below.

The values of the corresponding R-independent but strongly µ-dependentBorn cross section are also indicated.) A correlated feature is that the µ dependence of thejet cross section changes as we vary R. While the order α3s jet cross section is relativelyindependent of µ for R ≈0.7, for large R, R > 0.9, it is dominated by the order α3s realemission process and becomes a monotonically decreasing function of µ much like the Bornresult.

At small R, R < 0.5, the µ dependence of the order α3s cross section is dominated bythe negative contribution from the virtual correction associated with the collinear singularityand becomes a monotonically increasing function of µ. In either regime we expect higherorder corrections to be important so that the usefulness of fixed order perturbation theoryis compromised.

Thus we conclude that, at this order in perturbation theory, the results2

are most stable for R ≈0.7. It is precisely this size that was used in the inclusive single jetcross section analysis published by CDF[3].

In some sense the perturbation theory is tellingus that R = 0.7 is the “optimal size” for a jet cone, at least from the standpoint of makingcomparison with the order α3s result.The comparison with the data in Fig. 1 suggests that, while the agreement betweentheory and data for R = 0.7 is quite good, the strong dependence on R exhibited by thedata favors a small µ value, i.e., larger αs and more radiation.

To make this comparisonmore quantitative we can characterize both the data and the theory curves in terms of 3parameters,σ = A + B ln R + CR2 . (2)The parameterizations of the data and the theory for µ = ET /2 and µ = ET/4 are indicatedin the first three rows of Table 1 (the parameter Rsep will be defined below).

We see thatthe theoretical value for the R-independent A parameter is not too sensitive to µ. This isexpected since it is a true one loop quantity, containing both the order α2s contribution andcontributions from real and virtual graphs at order α3s.

The B and C terms, however, aresubject to larger theoretical uncertainty since these terms express the R dependence of thecross section and this appears first at order α3s. This is indicated by the sensitivity to µfound in the Table.

We can naively associate the B term with correlated (approximatelycollinear) final state parton emission that is important near the jet direction and the C termwith essentially uncorrelated initial state parton emission that is important far from the jetdirection.The theoretical value of A agrees quite well with the experimental value of A. Theagreement between the data and the µ = ET/2 theory for C is also quite good, but theagreement for B is worse than one would expect. This suggests that for the µ = ET/2theory, the amount of initial state emission far from the jet direction is about right but thatthere is not enough correlated radiation near the jet center.

If we change µ to ET /4, thenthe effective αs is larger and there is more radiation in all parts of phase space. Now B islarger, although still smaller than indicated by the data, while C is larger than indicated bythe data.To examine this issue further and to analyze the internal structure of jets in detail, it isuseful to consider the fractional ET profile, F(r, R, ET) (we suppress the dependence on thejet direction (ηJ, φJ)).

Given a sample of jets of transverse energy ET defined with a coneradius R, F(r, R, ET) is the average fraction of the jets’ transverse energy that lies insidean inner cone of radius r < R (concentric with the jet defining cone). Said another way,the quantity 1 −F(r, R, ET) describes the fraction of ET that lies in the annulus betweenr and R. It is this latter quantity that is most easily calculated in perturbation theory asit avoids the collinear singularities at r = 0.

Computing the ET weighted integral of thepp →3 partons +X cross section over the annulus and normalizing to ET,J times the Borncross section yields the order αs contribution to 1 −F (the numerator is purely order α3swhile the denominator is purely order α2s). The result for F is plotted in Fig.

2 versusthe inner radius r with R = 1.0 for ET = 100 GeV and compared to preliminary CDFdata[7].Again curves for three choices of µ are exhibited. (The fourth, dot-dash curve3

and the parameter Rsep will be explained below.) As with the R dependence of the crosssection discussed above, F is being calculated to lowest nontrivial order and thus exhibitsmonotonic µ dependence.

While there is crude agreement between theory and experiment,the theory curves are systematically below the data for all interesting values of µ. Thissituation suggests that the theoretical jets have too large a fraction of their ET near the edgeof the jet (r ≃R).We have seen that the R dependence and the B parameter suggest the importance ofhigher orders to increase the level of associated radiation, at least near the center of thecone. At the same time our detailed considerations of the parameter C and of F suggestthat the data favor a reduction of the ET fraction near the edge of the cone.

Althoughthese conclusions seem contradictory, there may be a consistent explanation based on adetailed but important physical point concerning how the jets are defined. We will presenta preliminary discussion of this point here and present a more detailed study in a separatenote.

[8] The issue is that of merging, how close in angle should two partons be in order tobe associated as a single jet. In a real experiment such a situation is presumably realizedas two sprays of hadrons, each with finite angular extent due to both fragmentation effectsand real experimental angular resolution effects.

If the angular separation is large enough,there is a valley in the ET distribution between the two sprays and experimental jet findingalgorithms will tend to recognize this situation as two distinct jets. Recall that we expectfor jets of ET > 100 GeV that the angular extent of fragmentation effects will be smallcompared to the defined jet cone sizes.

However, the theoretical jet algorithm we are usingwill merge two partons into a single jet whenever it is mathematically possible. This includesthe limiting configuration when two equal transverse energy partons (each with ET /2) arejust 2R apart.The calculation counts this as a single jet of transverse energy ET withits cone centered between the two partons, i.e., centered on the valley.

The treatment ofthis configuration in a real experiment will depend in detail on the implementation of thejet algorithm. To simulate the experimental algorithm in a simple way we add an extraconstraint in our theoretical jet algorithm.

When 2 partons, a and b, are separated by morethan Rsep(≤2R), Rab = [(ηa −ηb)2 + (φa −φb)2]1/2 ≥Rsep, we no longer merge them intoa single jet. The theoretical jet algorithm used above corresponds to Rsep = 2R as notedin Table 1.

As an example, the results of calculating both the R dependence and the ETfraction F with Rsep = 1.3R and µ = ET/4 are illustrated by the dot-dash curves in Figs.1 and 2. Clearly the extra constraint of Rsep has ensured that there is approximately theobserved fraction of ET near the edge of the cone while the reduced µ value has increased theamount of associated radiation near the center of the cone and produced a larger variationwith R. This conclusion is also verified by the last line in Table 1 where we observe that,compared to the first line of theoretical results, B has increased while C has remained thesame.

The values of both B and C that arise with these parameter choices are in reasonableagreement with the data. The jet cross section itself is relatively insensitive to the parameterRsep, decreasing by ≤10% as Rsep is reduced from 2R to 1.3R with fixed µ for ET = 100GeV.In summary, the agreement between data and QCD perturbation theory at order α3s for4

the question of the dependence on the jet definition is a vast improvement over the situationthat obtained at the Born level. There is good agreement between theory and experiment,at least for ET ≥50 GeV and R near 0.7.

On the question of the detailed structure withinjets the qualitative agreement is good but there are important quantitative issues that seemto be dependent on the details of the implementation of the jet definition, especially thequestion of jet merging. Further study, both theoretical and experimental, is required toobtain a full understanding of this problem[8].

This is particularly interesting since thereis some indication that such detailed internal jet structure can be invoked to differentiatequark jets from gluon jets[8, 9].We thank the members of the CDF Collaboration’s QCD group, and in particular J.Huth and N. Wainer, for discussions concerning the CDF jet measurements.This workwas supported in part by U.S. Department of Energy grants DE-FG06-91ER-40614 andDE-FG06-85ER-40224.References[1] S.D. Ellis, Z. Kunszt and D.E.

Soper,Phys. Rev.

Lett. 64, 2121 (1990);Phys.

Rev.D40, 2188 (1989); Phys. Rev.

Lett. 62, 726 (1989); preprint OTIS 495, UW/PT-92-08,ETH-PT/92-22 (unpublished).

[2] F. Aversa, M. Greco, P. Chiappetta and J. Ph. Guillet, Phys.

Rev. Lett.

65, 401 (1990);Zeit. Phys.

C46, 253 (1990);Nucl. Phys.

B327, 105 (1989);Phys. Lett.

211B, 465(1988); Phys. Lett.

210B, 225 (1988). [3] CDF Collaboration, F. Abe, et al., Phys.

Rev. Lett.

68, 1104 (1992). [4] See, for example, “Towards a Standardization of Jet Definitions”, J.E.

Huth et al., Pro-ceedings of the 1990 DPF Summer Study on High Energy Physics, Snowmass, 1990. [5] For a compilation of jet algorithms see B. Flaugher and K. Meier, Proceedings of the 1990DPF Summer Study on High Energy Physics, Snowmass, 1990.

[6] P.N. Harriman, et al., Phys.

Rev. D42, 798 (1990).

[7] CDF Collaboration, F. Abe, et al., “A Measurement of Jet Shapes in pp Collisions at√s = 1.8 TeV,” preprint FERMILAB-PUB/92/167-E (1992). [8] S.D.

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

[9] J. Pumplin,“Variables for Distinguishing Between Quark Jets and Gluon Jets”, Proceed-ings of the 1990 DPF Summer Study on High Energy Physics, Snowmass, 1990; “How totell Quark Jets from Gluon Jets”, preprint MSUTH91/3(1991).5

Table 1: 3 parameter fits to data and calculated curves in Figs. 1,3 and 4ABCCDF data[7]0.540.280.22µ = ET/2, Rsep = 2R0.520.130.19µ = ET/4, Rsep = 2R0.470.190.30µ = ET/4, Rsep = 1.3R0.490.220.19Figure CaptionsFig.

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 curveis explained in the text. Also indicated for the three µ choices are the values of theR-independent Born cross section.Fig.

2: F(r, R, ET) versus r for R = 1.0, √s = 1800 GeV, ET = 100 GeV and 0.1 < |η| <0.7 with µ = ET/4, ET/2, ET compared to data from CDF[7]; the dot-dash curve isexplained in the text.6

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