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Parton-hadron duality in event generators

arXiv:hep-ph/9208258v2 4 Sep 1992PRA-HEP-92/14August 11, 2021Parton-hadron duality in event generatorsJ.Ch´yla and J.RameˇsInstitute of Physics, Czechoslovak Academy of SciencesPrague, Czechoslovakia1AbstractThe validity of local parton-hadron duality within the framework of HER-WIG and JETSET event generators is investigated. We concentrate on e+e−annihilations in LEP 2 energy range as these interactions provide theoreticallythe cleanest condition for the discussion of this concept.1Postal address: Na Slovance 2, 180 40 Prague 8, Czechoslovakia1

1IntroductionThe concept of parton hadron duality (PHD) and in particular its local version attemptsto answer the question of the relation between the properties of experimentally observedhadrons and the assumed underlying parton dynamics. A few years ago the St.Petersburgschool has gone beyond the original global version of the duality idea and has argued infavour of much closer relation between the single particle inclusive spectra of partons andhadrons [1].

They have developed powerful theoretical tools to calculate within pertur-bative QCD partonic spectra in great detail, taking into account various subtle effects(for comprehesive review of this topic see, for instance, [2]). In converting their resultsinto the statements concerning hadrons they, however, crucially rely on two important as-sumptions.

First, the independent fragmenation model is used to hadronize the partonicconfigurations originating from perturbative cascading. Secondly, partonic cascades areallowed to evolve down to rather small (timelike) virtualities of the order of pion mass.As neither of these assumptions is incorporated in the currently widely used generatorsbased on either the string or cluster fragmentation, we have undertaken a detailed in-vestigation of the PHD within two distinct event generators which successfully describevast amount of experimental data from various collisions.

For related work see also [3, 4].Both HERWIG (we use its 5.3 version) and JETSET (version 7.2) use in their respectivehadronization stages algorithms which do not allow a sensible, i.e. reasonably unambigu-ous interpretation of produced hadrons as being fragments of a particular single parton.In our study we have addressed two closely related questions:• to what extent do the hadronic distributions reflect those of the perturbativelyproduced partons• how much do the partonic spectra in HERWIG and JETSET differ from each otherIn the local PHD picture the first question has a very simple answer, at least if thefragmentation function advocated by the authors of [1] is employed: hadronic spectra areproportional to those of the partons with the proportionality factor of the order of unity.This is not the case in either HERWIG or JETSET generator.

These generators are bothbased on the partonic cascade as described by perturbative QCD so that the startingpoint is in principle the same as in [1]. The partonic cascades in HERWIG and JETSETare, however, neither identical nor fully equivalent to the analytic calculations of [1] sothat differences do appear already on the parton level.

Moreover, the influence of thehadronization stage turns out to be very important in its effects on the hadronic spectra.What we basicaly observe is that quite different configurations of partons yield much thesame hadronic spectra. Indeed while HERWIG and JETSET are quite different as far aspartonic spectra are concerned they yield remarkably similar results for hadrons.

Thisindicates that the interplay between the perturbative and hadronization stages in hardscattering processes is very important, nontrivial and model dependent. Because of thisambiguity in the relation between the partonic and hadronic characteristics the conceptof local PHD looses much of its intuitive appeal and predictive power.2

In order to investigate these questions in relatively ”clean” conditions, we have con-centrated on the e+e−annihilations into hadrons at 200 GeV center of mass energy, i.e.on LEP 2 energy range. There the perturbative cascades, though model dependent, arealready rather well developed and so the whole perturbation theory machinery seems tobe well justified and under control.Any comparison between several sets of results depends on the quantities selected forthat purpose.

We have chosen the following ones:• multiplicity distribution• single particle inclusive distributions in the standard variables z and p2t wherez = ln 1xp! ;xp = 2p√s(1)where pt denotes the transversal component of particle momenta with respect to thethrust axis.• factorial moments of particle multiplicities (intermittency measure)The first two types of them are those discussed in [1] while the factorial moments ofparticle multiplicities in small regions of phase space [5, 6] are commonly considered as aclear manifestation of underlying partonic cascading [7].2A few remarks on generatorsWe briefly recall some of the important features of HERWIG and JETSET generators andin particular those of their parameters which will play an essential role in the followingdiscussion of local PHD.

In HERWIG the event simulation proceeds in four stages:1. hard scattering, i.e. in our case e+e−→qq2.

evolution of parton showers on all partonic legs, here qq3. formation of colorless clusters out of partonic final state qq pairs4.

decay of these clusters into hadronsIn JETSET the first two steps are in principle similar although the details of showerevolution differ from HERWIG. The main difference between these generators concerns,however, the hadronization stage.

Instead of the formation of colorless clusters JETSETspans relativistic string on the products of partonic cascade which then breaks into ob-servable hadrons.There are many parameters which determine the details of each of these steps, butthe following ones are essential for fixing the relative importance and interplay betweenthe parton shower and hadronization stages in HERWIG:3

• QCDLAM: the usual QCD Λ-parameter• V QCUT, V GCUT: parameters setting, when added to parton masses, the minimalparton virtuality in timelike cascades. The default options are such that for bothquarks and gluons the minimal timelike virtuality is about 0.9 GeV.• CLMAX: the decisive parameter for the hadronization stage of HERWIG.

It forcesthe colorless clusters, produced in the perturbative cascade, with mass squaredabove CLMAX2 + (mq + m¯q)2 to split, before hadronization, into lower mass ones.The default value is 3.5 GeV. Although in typical events the action of CLMAXparameter is limitted it plays quite an essential role in defining the relative role ofpartonic cascades and hadronization stages.

As will be shown below it is particularlyimportant in low parton number events.In JETSET analogous role is played by the parameters:• PARJ(81): analogue of QCDLAM• PARJ(82): invariant mass cutoffon parton virtualities, similar in effect to V GCUT,V QCUT in HERWIG.• there is no direct analogue of CLMAX parameter.In JETSET the user can choose betweeen the leading log parton showers and exact fixedorder matrix element approaches. In HERWIG parton showers cannot be straightfor-wardly switched offand they always accompany the hard scattering subprocess, be ite+e−→qq or e+e−→qqg.

Nevertheless to study the relation between partons andhadrons in theoretically cleanest conditions we have concentrated in both generators onthe e+e−→qq subprocess and selected the parton shower option in JETSET. In ourquest to understand the relation between the parton and hadron spectra we have• compared the hadron as well as parton characteristics, specified above, for the cur-rently ”best” sets of parameters of HERWIG and JETSET• compared parton distributions with the corresponding hadronic ones, in HERWIGas well as in JETSET• looked separately on events characterized by different number of partons.

”Tagging”on this parton number allows us to see clearly under what circumstances are thehadronic characteristics direct manifestations of the underlying partonic ones.• varied the values of V QCUT, V GCUT, CLMAX and looked at the changes in therelation between partonic and hadronic spectra4

3Discussion of the resultsThe basic results of extensive simulations with HERWIG and JETSET at LEP2 energyare presented in Fig.1-7. We stress that neither generator has been tuned especially forour purposes and we have taken the currently ”best” sets of their respective parameters.3.1Multiplicity distributionsFig.1 displays the comparison between the partonic as well as hadronic multiplicity dis-tributions in HERWIG and JETSET.

Already this simplest quantity signals the basicmessage: while for the hadrons these models predict results which are rather close to eachother (typically within 10-15 %) they differ vastly on the level of partons! Despite thelarge difference in the average number of perturbatively produced partons (9 in HERWIGvs.

15.5 in JETSET) the hadronic multiplicity distributions are much closer in shape aswell in average values: 42.5 in HERWIG vs. 48 for JETSET. This large difference betweenthe average parton and hadron multiplicities is in sharp contrast with the results of [1],where the number of partons is much closer to the number of hadrons.3.2Single particle inclusive spectraThe same message can be read offFig.2, where the single particle spectra in z and p2t, fromboth HERWIG and JETSET, are compared on partonic as well as hadronic levels.

Thepronounced difference in z distributions of partons in the region close to zero comes fromthe region of phase space where one of the partons carries nearly all available momentum.The most important difference between HERWIG and JETSET partonic distributionsis, however, observed for large z and small p2t, i.e. for soft partons which dominate thetotal multiplicity.

We see that HERWIG z-distribution is significantly lower than thatof JETSET down to z = 2 and practically vanishes for z > 5. Despite these dramaticdifferences on the level of partons, the corresponding hadronic distributions are quitesimilar even in the large z region.

For p2t distributions of partons the dramatic differencein the region close to p2t = 0 is a direct manifestation of the cut on minimal virtualityof partons as set by the parameters V QCUT, V GCUT. The position of the maximumin HERWIG spectrum is in fact proportional to them.

In JETSET there does not seemto be an analogous effect. This difference between HERWIG and JETSET is, however,again not reflected in the corresponding hadronic p2t distributions, which look practicallyindistinguishable.

The slightly higher hadronic multiplicity in JETSET is then reflectedin somewhat higher values of hadronic z spectra in the region around z = 4.In Fig.3 we compare the partonic and hadronic spectra obtained with both generatorsby plotting (as solid lines) the ratiar(w) = 1NeventsdNdwpartons! ,1NeventsdNdwhadrons!

;w = z, p2t(2)of appropriately normalized partonic over hadronic distributions, which should be approx-imately constant if local PHD holds. Clearly rather large deviations from constancy in5

most of the phase space and for both z and p2t are observed, the pattern of these violationsbeing, except for z close to the upper limit, similar in HERWIG as in JETSET. In bothgenerators a large part of this effect can be traced back to the fact that their partonicshowers are stopped at much higher virtualities than in [1].

This is demonstrated by dot-ted lines in Fig.3 which correspond, for HERWIG as well as JETSET, to lower virtualitycut-offQ0 = 0.2 GeV. For technical reasons this low virtuality cut-offrequires, in bothgenerators, simultaneous lowering of the QCD Λ-parameter.

In our case we have taken itto be 0.04 GeV.The most dramatic effect occurs for the ratio r(z) which for Q0 = 1.0 GeV was rapidlydecreasing function of z in the whole phase space, while now this ratio is nearly constantin the large interval z ∈(1, 6). Similar effect is observed for p2t spectra, in particularon low p2t region.

There is thus no doubt that local PHD is much better reproduced inboth HERWIG and JETSET for small virtuality cut-offQ0 = 0.2 GeV. However, neitherHERWIG nor JETSET can accommodate such low values of Q0 and still describe theavailable experimental data as accurately as with the Q0 in the region of 1 GeV.

Inthe regions of xp close to 0 and 1 the deviations from constancy are very large even forQ0 = 0.2 GeV but this is not surprising as these are also the regions where the analyticcalculation of [1] contain subtle effects not included in the generators. An interestingdifference between HERWIG and JETSET z distributions is observed in large z region,i.e.

for xp →0, which is populated by soft partons, and where the ratia behave quiteoppositely. All this suggests that the validity of local PHD relies heavily on evolving thepartonic showers to rather small scales comparable to the pion mass.In order to understand the differences between HERWIG and JETSET in more detailwe have furthermore subdivided all events in three classes according to the number ofperturbatively produced partons.• small number of partons: 2-6• moderate number of partons: 7-11• large number of partons: more than 11In Fig.4 the comparison between partonic z and p2t spectra from HERWIG and JETSET isdone for each of the classes separately.

We see that the differences increase with decreasingnumber of underlying partons. For all three classes of events the corresponding hadronicspectra (not displayed) are, however, again much closer as in the case of the full samples.We now come to the interplay between the parton shower and cluster decay stagesin HERWIG event generator.

In Fig.5 we plot separately for each of the three classesof events the z and p2t distributions for partons and hadrons. In the case of hadronicdistributions we moreover show two sets of curves corresponding to two different values ofthe CLMAX parameter.

Beside the default value CLMAX = 3.5 GeV we plot also theresults corresponding to CLMAX = 50 GeV. This large value effectively means that wedo not force large mass clusters to split into smaller ones before hadronization and thuscome close to the original formulation of HERWIG.

We conclude that6

1. Small value of CLMAX is more effective in events with small number of partons.For them perturbative branching itself leads on average to small number of heavyclusters.

Once small CLMAX is taken, all these heavy clusters are first split intosmaller ones and only then allowed to hadronize. After this step the cluster massdistributions are essentially the same irrespective of the original parton number.The only trace of the originally different parton numbers then remains in the smallercluster multiplicity.2.

The differences between the shapes as well as magnitudes of parton and hadronspectra are large in the whole phase space and depend sensitively on the number ofpartons.3. The differences between the two hadronic spectra, corresponding to CLMAX = 3.5GeV and CLMAX = 50 GeV show that for events with small parton numbers theinitial part of the hadronization stage i.e.

the cluster splitting is crucially important.This shows why and how the hadronization stage may significantly influence therelation between the parton and hadron spectra.Taken together the message contained in the preceding observations is simple andclear: While on the level of hadrons different models give very similar results, theseresults can originate from very different partonic distributions. Moreover, the effects andimportance of the hadronization stage are nontrivial and do depend on particular partonicconfiguration.

All this demonstrates that the concept of local PHD does not hold withineither HERWIG or JETSET event generators and is thus more ambiguous than as claimedin [1].3.3Intermittency analysisThe intermittency phenomenon, as quantified by the factorial moments in small phasespace regions [5, 6] is commonly regarded as a convincing evidence for the underlyingpartonic cascading process with the selfsimilarity property [7]. As such it should alsoprovide the evidence for the local PHD.

To check this assumption we have carried outseveral simple tests using both HERWIG and JETSET event generators. The question weask ourselves is similar as before: to what extent is the observed intermittency behaviouron the level of hadrons a direct consequence of the underlying partonic cascade?To find the answer we have calculated the conventional factorial moments of the i−thrank in two dimensions (rapidity versus azimuthal angle)Fi(y, φ) =1NeventsXeventsPnbinsk=1 ⟨nk(nk −1) · · ·(nk −i + 1)⟩(⟨n⟩/nbins)i(3)where ⟨n⟩is the average number of particles in the full ∆y −∆φ region accepted (we havetaken full azimuthal coverage and y ∈(−2, 2)) and nbins denotes the number of bins inthis two-dimensional space, given as 4b with ndiv = 2b defining the number of divisionsin each of the two directions.

We have constructed these moments for partons as well ashadrons as functions of b or ndiv. Then we have compared for i = 2, 3, 4, 57

• the results from HERWIG and JETSET (Fig.6a)• the results for partons with those of hadrons (Fig.6b)• the results corresponding to events with different number of partons (Fig.7)On the basis of the plots displayed in Figs.6,7 we draw the following conclusions, whichpoint in the same direction as those of the preceding paragraphs:1. Despite the large differences on the partonic level HERWIG and JETSET give verysimilar results for the hadronic factorial moments (3).

This conclusion holds wellfor the full sample of events as well as for all three classes of events defined above(corresponding plots similar to Fig.6a are omitted), i.e. is independent of the numberof underlying partons.2.

For HERWIG we find a remarkable agreement between the partonic and hadronicfactorial moments in the region where the former show the appropriate rising be-haviour. It is clear that this region can not be large as the number of partons ismuch smaller than that of the hadrons.

In order to see the effect on the partonlevel clearly we have therefore chosen finer steps in the division of ∆y-∆φ intervalfor partonic moments. We have, however, seen that much the same behaviour ofhadronic factorial moments results even for the case where there are so little par-tons that their own factorial moments vanish.

So again we see that the propertyof hadrons usually considered as a direct consequence of the presence of partoniccascade can equally well result from the effects of the hadronization stage. In otherwords there is some kind of duality but of different sort that originally proposed in[1], namely the duality between the partonic and hadronization stages within thewhole event generation.3.

Except for F2 the factorial moments in HERWIG as well as JETSET are practicallyindependent of the number of partons in the event. In Fig.7 we present results fromJETSET, but the same picture is obtained for HERWIG as well.

For the purposeof this comparison we have somewhat changed the definition of the three classes ofevents (2-10,11-18,19 and more) which is more appropriate for JETSET due to itshigher average parton multiplicity. This at first sight surprising observation againshows that there is no simple relation between the partonic and hadronic properties.The fact that events with small number of partons give smaller F2 than those withmany partons may be a simple reflection of lower overall hadronic multiplicity inthese events.

In HERWIG the approximate indepedence of the factorial momentshas a simple explanation in the interplay between the partonic cascade and thehadronization stage. What happens is that in the case of small parton number theaction of the CLMAX parameter creates out of the small number of heavy clus-ters, produced by purely perturbative cascade, much larger number of lighter ones.Instead of partonic cascade we have ”cluster” cascade, which, however, producesessentially the same behaviour of hadronic factorial moments.

In JETSET we canoffer no such simple explanation but the effect holds as well.8

4Summary and conclusionsIn this paper we have discussed the relation between partons and hadrons in two different,widely used event generators, HERWIG and JETSET. We concentrated on e+e−annihi-lations at LEP 2 energies as these conditions provide theoretically the cleanest place forinvestigation of such a relation.

Our simulations show that this relation is complicated andgenerator dependent. The idea of local PHD as suggested in [1] is not realized in eitherof these models.

To large extent, this is due to the fact that the analytical calculationsof [1] rely in a crucial way on the use of independent fragmenation model coupled withthe assumption that partonic showers are allowed to evolve down to virtualities of theorder of the pion mass, while both HERWIG and JETSET stop their respective cascadesat much higher virtualities.On the other hand we have found evidence for the strong interplay between the effectsof parton showers and those of the hadronization stage. Clear example of such an interplayis provided by factorial moments in narrow phase space region where the intermittentbehaviour originates equally from perturbative parton branching as well as from colorlesscluster splitting.

This indicates that we can not check details of partonic evolution withoutthe knowledge of hadronization mechanism and vice versa.References[1] Ya.I.Azimov, Yu.L.Dokshitzer, V.A.Khoze, S.I.Troyan: Z. Phys. C - Particles andFields 27 (1985) 65[2] Yu.L.Dokshitzer, V.A.Khoze, A.H.Mueller, S.I.Troyan: Basics of Perturbative QCD,Editions Frontieres, ed.

J.Tran Thanh Van (1991)[3] B.Andersson, P.Dahlqvist, G.Gustafson: Z. Phys. C - Particles and Fields 44 (1989)455[4] B.Andersson, P.Dahlqvist, G.Gustafson: Z. Phys.

C - Particles and Fields 44 (1989)461[5] A.Bialas, R.Peschanski: Nucl. Phys.

B273 (1986) 703[6] A.Bialas, R.Peschanski: Nucl. Phys.

B308 (1988) 857[7] W.Ochs, J.Wosiek: Phys. Lett.

B214 (1988) 6179

Figure captions.Fig.1: Multiplicity distributions of partons (a) and hadrons (b) from HERWIG andJETSET.Fig.2: Single particle distributions in z and p2t of hadrons (a) and partons (b) fromHERWIG and JETSET.Fig.3: Ratia r(z), r(p2t) from JETSET (a) and HERWIG (b) for two values of the lowervirtuality cut-offQ0.Fig.4: Single particle distributions of hadrons from HERWIG and JETSET, as functionsof z and p2t, for the three classes of events defined in the text.Fig.5: Partonic and hadronic distributions in z and p2t for the three classes of events definedin the text as obtained from HERWIG. For the hadrons the spectra corresponding to twodifferent values of CLMAX parameter are displayed.Fig.6: Comparison of factorial moments Fi(y, φ) between HERWIG and JETSET (a), andbetween partons and hadrons within HERWIG (b).Fig.7: Dependence of factorial moments Fi(y, φ) on the number of partons as obtainedfrom JETSET.10

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