1 Department of Physics and Astronomy, Iowa State University, Ames IA 50011-3160, USA2 Physics Department, Univ. Instelling Antwerpen, Universiteitsplein 1, B-2610 Wilrijk, Belgiumand IIHE, ULB-VUB, Pleinlaan 2, B-1050 Brussels, Belgiumand Faculte des Sciences, Univ. de l’Etat Mons, Av. Maistriau 19, B-7000 Mons, Belgium
3 Physics Laboratory, University of Athens, Solonos Str. 104, GR-10680 Athens, Greece4 Department of Physics, University of Bergen, Allegaten 55, N-5007 Bergen, Norway5 Dipartimento di Fisica, Universita di Bologna and INFN, Via Irnerio 46, I-40126 Bologna, Italy6 Centro Brasileiro de Pesquisas Fisicas, rua Xavier Sigaud 150, RJ-22290 Rio de Janeiro, Braziland Depto. de Fisica, Pont. Univ. Catolica, C.P. 38071 RJ-22453 Rio de Janeiro, Braziland Inst. de Fisica, Univ. Estadual do Rio de Janeiro, rua Sao Francisco Xavier 524, Rio de Janeiro, Brazil
7 Comenius University, Faculty of Mathematics and Physics, Mlynska Dolina, SK-84215 Bratislava, Slovakia8 College de France, Lab. de Physique Corpusculaire, IN2P3-CNRS, F-75231 Paris Cedex 05, France9 CERN, CH-1211 Geneva 23, Switzerland
10 Centre de Recherche Nucleaire, IN2P3 - CNRS/ULP - BP20, F-67037 Strasbourg Cedex, France11 Institute of Nuclear Physics, N.C.S.R. Demokritos, P.O. Box 60228, GR-15310 Athens, Greece12 FZU, Inst. of Physics of the C.A.S. High Energy Physics Division, Na Slovance 2, 180 40, Praha 8, Czech Republic13 Dipartimento di Fisica, Universita di Genova and INFN, Via Dodecaneso 33, I-16146 Genova, Italy14 Institut des Sciences Nucleaires, IN2P3-CNRS, Universite de Grenoble 1, F-38026 Grenoble Cedex, France15 Research Institute for High Energy Physics, SEFT, P.O. Box 9, FIN-00014 Helsinki, Finland16 Joint Institute for Nuclear Research, Dubna, Head Post Office, P.O. Box 79, 101 000 Moscow, Russian Federation17 Institut fur Experimentelle Kernphysik, Universitat Karlsruhe, Postfach 6980, D-76128 Karlsruhe, Germany18 Institute of Nuclear Physics and University of Mining and Metalurgy, Ul. Kawiory 26a, PL-30055 Krakow, Poland19 Universite de Paris-Sud, Lab. de l’Accelerateur Lineaire, IN2P3-CNRS, Bat. 200, F-91405 Orsay Cedex, France20 School of Physics and Chemistry, University of Lancaster, Lancaster LA1 4YB, UK21 LIP, IST, FCUL - Av. Elias Garcia, 14-1o, P-1000 Lisboa Codex, Portugal22 Department of Physics, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, UK23 LPNHE, IN2P3-CNRS, Universites Paris VI et VII, Tour 33 (RdC), 4 place Jussieu, F-75252 Paris Cedex 05, France24 Department of Physics, University of Lund, Solvegatan 14, S-22363 Lund, Sweden25 Universite Claude Bernard de Lyon, IPNL, IN2P3-CNRS, F-69622 Villeurbanne Cedex, France26 Universidad Complutense, Avda. Complutense s/n, E-28040 Madrid, Spain27 Univ. d’Aix - Marseille II - CPP, IN2P3-CNRS, F-13288 Marseille Cedex 09, France28 Dipartimento di Fisica, Universita di Milano and INFN, Via Celoria 16, I-20133 Milan, Italy29 Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark30 NC, Nuclear Centre of MFF, Charles University, Areal MFF, V Holesovickach 2, 180 00, Praha 8, Czech Republic31 NIKHEF, Postbus 41882, NL-1009 DB Amsterdam, The Netherlands32 National Technical University, Physics Department, Zografou Campus, GR-15773 Athens, Greece33 Physics Department, University of Oslo, Blindern, N-1000 Oslo 3, Norway34 Dpto. Fisica, Univ. Oviedo, C/P. Perez Casas, S/N-33006 Oviedo, Spain35 Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK36 Dipartimento di Fisica, Universita di Padova and INFN, Via Marzolo 8, I-35131 Padua, Italy37 Rutherford Appleton Laboratory, Chilton, Didcot OX11 OQX, UK38 Dipartimento di Fisica, Universita di Roma II and INFN, Tor Vergata, I-00173 Rome, Italy39 CEA, DAPNIA/Service de Physique des Particules, CE-Saclay, F-91191 Gif-sur-Yvette Cedex, France40 Istituto Superiore di Sanita, Ist. Naz. di Fisica Nucl. (INFN), Viale Regina Elena 299, I-00161 Rome, Italy41 Instituto de Fisica de Cantabria (CSIC-UC), Avda. los Castros, S/N-39006 Santander, Spain, (CICYT-AEN93-0832)42 Inst. for High Energy Physics, Serpukov P.O. Box 35, Protvino, (Moscow Region), Russian Federation43 J. Stefan Institute and Department of Physics, University of Ljubljana, Jamova 39, SI-61000 Ljubljana, Slovenia44 Fysikum, Stockholm University, Box 6730, S-113 85 Stockholm, Sweden45 Dipartimento di Fisica Sperimentale, Universita di Torino and INFN, Via P. Giuria 1, I-10125 Turin, Italy
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46 Dipartimento di Fisica, Universita di Trieste and INFN, Via A. Valerio 2, I-34127 Trieste, Italyand Istituto di Fisica, Universita di Udine, I-33100 Udine, Italy
47 Univ. Federal do Rio de Janeiro, C.P. 68528 Cidade Univ., Ilha do Fundao BR-21945-970 Rio de Janeiro, Brazil48 Department of Radiation Sciences, University of Uppsala, P.O. Box 535, S-751 21 Uppsala, Sweden49 IFIC, Valencia-CSIC, and D.F.A.M.N., U. de Valencia, Avda. Dr. Moliner 50, E-46100 Burjassot (Valencia), Spain50 Institut fur Hochenergiephysik,Osterr. Akad. d. Wissensch., Nikolsdorfergasse 18, A-1050 Vienna, Austria51 Inst. Nuclear Studies and University of Warsaw, Ul. Hoza 69, PL-00681 Warsaw, Poland52 Fachbereich Physik, University of Wuppertal, Postfach 100 127, D-42097 Wuppertal, Germany53 On leave of absence from IHEP Serpukhov
Received: 27 August 1996
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Abstract. Event shape and charged particle inclusive dis-tributions are measured using 750000 decays of theZ tohadrons from the DELPHI detector at LEP. These pre-cise data allow a decisive confrontation with models of thehadronization process. Improved tunings of the JETSET,ARIADNE and HERWIG parton shower models and theJETSET matrix element model are obtained by fitting themodels to these DELPHI data as well as to identified parti-cle distributions from all LEP experiments. The descriptionof the data distributions by the models is critically reviewedwith special importance attributed to identified particles.
1 Introduction
Precision measurements at LEP using the hadronic finalstate, such as determinations of the strong coupling constantαs from event shapes, the forward backward asymmetriesfor quarks, theZ mass and width, or at higher energies theW± mass, require a satisfactory model for the properties ofthe corresponding final states. Perturbative QCD cannot pro-vide full theoretical insight into the transition from primaryquarks to observable hadrons, the so-called fragmentation orhadronization process, since only the part of this transitioninvolving large momentum transfer, mainly the radiation ofhard gluons or the evolution of a parton shower, is calcula-ble perturbatively. The final formation of hadrons is hiddenby the increase of the strong coupling constantαs at smallmomentum transfer and the ensuing failure of perturbationtheory.
Guidance towards a better understanding of the hadro-nization process must therefore come from detailed exper-imental investigations of the hadronic final state, includingattempts to describe this process by phenomenological mod-els inspired by QCD. LEP I provides a unique and unrivaledopportunity to pursue these studies. The clean well-definedinitial state ine+e− annihilation provides an excellent testingfield, since the event rate at theZ is very high, the energyis large, and the capabilities of the experimental apparatusare much improved with respect to previous experiments.
This paper attempts to determine, for the most frequentlyused hadronization models, parameters which give an opti-mal description of a) the observed hadronic event shapes andcharged particle inclusive distributions as measured with theDELPHI experiment at LEP, as well as b) the available in-formation on identified particles from all LEP experiments.The latter allows precise determination of more model pa-rameters than the event shapes alone, and also a check of
the internal consistency of the models. The performance ofthese models is compared and critically reviewed.
This paper is organized as follows. Section 2 gives a briefoverview of the relevant detector components and describesthe experimental procedure applied to determine event shapeand inclusive distributions and the related systematic errors.Section 3 discusses the models employed and the relevantparameters. Section 4 describes the optimization strategy ap-plied to obtain the best parameters for the fragmentationmodels and justifies the choice of the distributions used inthe fits: the fits are then discussed in detail and the result-ing optimized parameters and their errors are presented. InSect. 5, the model predictions are compared with the ob-served event shape distributions, charge particle spectra, andidentified particle data. Finally, Sect. 6 summarizes.
The appendices contain the definitions of the variablesused throughout this paper (Appendix A), followed by tablesof the model parameter settings in the Delphi Monte Carloused to correct the event shape distributions (Appendix B),of the inclusive charged particle and event shape distribu-tions fitted (Appendix C), of the sensitivities of the fittedparameters to the different distributions used (Appendix D),and of the results of the fits (Appendix E). Finally, AppendixF presents a set of figures comparing the data distributionswith the model predictions after the fits.
2 Detector and data analysis
This analysis uses the 1991, 1992 and 1993 data taken withthe DELPHI detector at LEP. The determination of the eventshape distributions uses charged particles measured in thesolenoidal 1.2 T magnetic field of DELPHI and showerscaused by neutral particles in the electromagnetic or hadroniccalorimeters. The following detectors [1] are relevant to theanalysis:
− the Vertex Detector, VD, measuring charged particle co-ordinates in the plane perpendicular to the beam withup to three layers of silicon micro-strip detectors at radiibetween 6.3 cm and 11 cm and covering polar angles,θ,to thee+-beam between 37 and 143;
− the Inner Detector, ID, a cylindrical jet chamber withθcoverage from 17 to 163;
− the Time Projection Chamber, TPC, the principal track-ing detector of DELPHI, which has 6 sector plates, eachwith 16 pad rows and 192 sense wires, in the forwardand backward hemispheres, inner and outer radii of 30cm and 122 cm, and a polar angle coverage from 20 to160;
14
− the Outer Detector, OD, a five layer drift chamber at 192cm radius covering polar angles between 43 and 137;
− two sets of forward planar drift chambers, FCA and FCB,with 6 and 12 layers respectively and overall polar anglecoverages of 11 to 35 and 145 to 169;
− the High density Projection Chamber, HPC, a lead-gaselectromagnetic calorimeter with a very good spatial res-olution located inside the DELPHI coil between 208 cmand 260 cm radius and covering polar angles between43 and 137;
− the Forward Electro-Magnetic Calorimeter, FEMC, com-prising two lead-glass arrays, one in each endcap, eachconsisting of 4500 lead glass blocks with a projectivegeometry, and covering polar angles from 10 to 36.5and from 143.5 to 170 to the beam;
− the HAdron Calorimeter, HAC, an iron-gas hadroniccalorimeter outside the coil, consisting of at least 19 lay-ers of streamer tubes and 5 cm thick iron plates also usedas flux return, whose overall angular coverage is from11.2 to 168.8.
The performance of the detector is described in [2].Because the event statistics available for this analysis
are very large, the final experimental error is dominated bythe systematics. Therefore the selections ensure that the ma-jor components of the event were measured in DELPHI withoptimal efficiency and resolution, as well as minimizing sec-ondary interactions in the detector material. However, carehas been taken not to bias the measured distributions signif-icantly by these cuts.
Charged particles were accepted in the analysis if theysatisfied the following criteria:
− momentump ≥ 200 MeV/c,− ∆p/p ≤ 1,− 20 ≤ θ ≤ 160,− measured track length≥ 50 cm,− impact parameter with respect to the nominal interaction
point within 2 cm perpendicular to or 5 cm along thebeam.
Furthermore, charged particles with large momenta (p ≥ 25GeV/c) within the geometrical acceptance of the OD or FCBhad to be measured in these detectors as well as in the ID orVD. This requirement ensured a good momentum resolutionfor high momentum particles.
Energy clusters reconstructed in the calorimeters and notassociated with charged particles were accepted as beingdue to neutral particles (photons or neutral hadrons) if theirreconstructed energy exceeded 1 GeV for clusters in theHAC, or 0.5 GeV for clusters in the HPC or FEMC.
Events were selected if:
− there were at least 5 charged particles selected,− the total energy of the charged particles exceeded 15
GeV, and also exceeded 3 GeV in each half of the de-tector defined by the plane perpendicular to the beamdirection,
− the polar angle of the thrust axis was either between 50and 85, or between 95 and 130,
− the momentum imbalance of the event along the beamdirection satisfied|∑ pz|/
√s ≤ 0.15.
About 750000 events were selected. The contamination ofbeam gas events,γγ-events, and leptonic events other thanτ+τ− is expected to be less than 0.1% and has been ne-glected. The influence ofτ+τ− events, which have a pro-nounced 2-jet topology and contain high momentum parti-cles, has been determined by a simulation study using eventsgenerated by the KORALZ model [3] and treated by the fullsimulation of the DELPHI detector, DELSIM [2], and thestandard data reconstruction chain. Theτ+τ− contributionshave been subtracted from the measured data according totheir relative abundance ((0.16±0.03)% of hadronic events).
Differential inclusive single particle and event shape dis-tributions normalized to the number of hadronic Z decaysare presented in Appendix C.1 and C.2 as a function of thephysical observables defined in Appendix A. Two differenttypes of result are given: (i) distributions measured fromcharged particles only and corrected to refer to the part ofthe final state consisting of charged hadrons only, and (ii)distributions measured from charged plus neutral particlesand corrected to refer to the full hadronic final state.
The distributions presented have been corrected for kine-matic cuts, limited acceptance and resolution of the detector,and effects due to reinteractions of particles inside the detec-tor material. The correction was calculated using simulatedevents, generated by JETSET 7.3 PS [4] using the parametersettings given in Appendix B, and treated by the full simu-lation and analysis chain. For each bin of each distribution,a correction factorCi was calculated as the ratio betweenthe generated and observed distributions. Particles with aproper mean lifetime bigger than 1 ns were considered asstable particles in the generated distributions. The correctionfor initial state photon radiation was determined separately,using events generated by JETSET 7.3 PS, with and with-out initial state radiation as predicted by DYMU3 [5]. Theoverall correction factors are displayed in the upper insetsof the figures presented in Appendix F.
The above simple unfolding by correction factors in gen-eral leads to biases of the final results when the detectorsmearing is bigger than the bin width used, and when themodel used to determine the correction factors does not de-scribe the data well [6]. In our case, the model has beentuned to DELPHI data as described in [7] and in this paper,and is in good agreement for all distributions considered. Tokeep residual biases small with respect to other systematicerrors, the bin width of all distributions presented is at leastas big as the detector smearing.
To account for possibly imperfect representation of theDELPHI detector or of secondary processes in the simulationprogram DELSIM, the cuts given above were varied over awide range, i.e. including smaller polar angles for the eventaxis, demanding events with more than 7 charged particles,etc. Further cuts were imposed to exclude the boundaries ofthe TPC sectors for high momentum particles, where the de-tector effects are known to be less well modelled in the sim-ulation. Systematic uncertainties were deduced as the root-mean-square deviations with respect to the central value.As the systematic error is expected to grow in proportionto the deviation of the overall correction factor from unity,an additional systematic uncertainty, assumed to be 10% ofthis deviation, was added quadratically to the above value.A further systematic error was added in quadrature for a
15
Table 1. Setting of JETSET PS switches used
Variable Value Variable Value Variable ValueMSTJ(11) 3 MSTJ(12) 3 MSTJ(46) 3
few bins where the results of the individual data sets corre-sponding to the different years of data-taking were found tobe statistically incompatible. This error was calculated suchas to reduce theχ2 per degree of freedom for the mergingto unity when it was considered in addition to the statisticalerrors. As a final step, the systematic uncertainties for eachindividual variable were smoothed.
The data points and their statistical and systematic errorsare given in Tables 11 – 44 in Appendix C. The uncertain-ties shown in the graphs in Appendix F comparing data andmodels are the final experimental uncertainties used in thefits, obtained by adding the statistical and systematic uncer-tainties in quadrature. In the fits, they were treated in thesame way as purely statistical uncertainties. They are there-fore reflected in the statistical errors on the fitted parametersquoted in Appendix E. The systematic errors quoted therewere evaluated differently, as described below in Sect. 4.2,by varying the choice of distributions fitted.
3 Fragmentation models
A comprehensive overview of fragmentation models can befound in [8]. This paper considers the most frequently usedfragmentation models, namely JETSET 7.3 PS, JETSET 7.4PS and JETSET 7.4 ME [4], ARIADNE 4.06 [9], and HER-WIG 5.8 C [10]. Recently ARIADNE 4.06 has been updatedto version 4.08. It has been checked that, with the parame-ters given below, this new version and version 4.06 predictalmost identical shape and inclusive distributions.
HERWIG and JETSET are complete models describingthe parton shower evolution or the QCD matrix element cal-culation, the hadronization of partons into hadrons, and thesubsequent decays of short lived particles. ARIADNE mod-els only the parton shower, the subsequent hadronization anddecays are treated by JETSET.
3.1 JETSET 7.3 / 7.4 parton shower model
JETSET with the parton shower option was used with theswitches set as in Table 1. Other switches were left at theirdefault values.
The JETSET parton shower algorithm is a coherent lead-ing log approximation (LLA) with angular ordering. Theshower evolves in the centre of mass frame of the partons,obeying energy and momentum conservation at each stepof the shower. In order to represent the 3-jet cross sectioncorrectly at the same time as the 4-jet and multi-jet crosssections, the lowest order 3-jet cross-section is reproducedby rejecting some of the first branchings of the initialqqsystem that are predicted by the LLA formalism. Angularordering of the branchings is explicitly imposed and gluonhelicity effects can be included. The value ofαs is running,with a scale given by the squared transverse momentum ofthe branching. The shower evolution is stopped at a mass
scaleQ0, then fragmentation takes over:Q0 andΛQCD (i.e.αs) are the parameters of the parton shower part of JETSET.
The fragmentation is performed using the Lund stringscheme. This can be formulated as an iterative procedure. Astring stretches between the oppositely coloured quark andantiquark via the gluon colour charges. Two gluons nearby inphase space act like a single gluon with equal total momen-tum, so the string model is infrared safe. The longitudinalmomentum fractionz of a hadron is determined using theLund symmetric fragmentation function:
f (z) =(1− z)z
a
· exp
(−b ·m2
t
z
)wherem2
t = m2 + p2t is the transverse mass squared of the
hadron, anda and b are parameters of the fragmentationfunction. In principleb is universal, anda can depend onthe quark flavour. But for heavy quarks, we use instead thePeterson fragmentation function [11]:
f (z) =1
z(
1− 1z − εq
1−z)2
with parameterεq being εb or εc, since this gives a betterdescription of heavy quark fragmentation. The transversemomenta of the hadrons are determined from thept valuesof their constituent quarks, which in turn are chosen froma tunneling process. This leads to a Gaussianp2
t behaviour.The relevant parameter is the standard deviation of this dis-tribution, σq.
The tunneling also determines the quark flavour gener-ated in the string breakup, leading to a dependence pro-portional to exp (−m2
q), and thus to negligible heavy quarkproduction in the fragmentation. Due to the higher mass,even strangeness production is strongly suppressed:P (uu) :P (dd) : P (ss) = 1 : 1 :γs, whereγs ≈ 0.3. Mesons are pro-duced according to their quark content in the six multipletswith smallest mass, i.e. in the states:1S0, 3S1, 1P1, 3P0, 3P1and 3P2. Contrary to the standard formulation implementedin JETSET, we have defined individual production proba-bilities P (2s+1Lj) for these multiplets, and also for light,strange, charm and bottom flavours. Except for the light3P0multiplet, the probabilities for theP -multiplets are takento be proportional to (1− P (1S0)− P (3S1)) · (2j + 1). In a(2 dimensional) string picture, production of particles withnon-zero angular momentum (i.e.P -states) is suppressed andexpected to be small (about 10% [12]).
Using additional mass relations, the tunneling mecha-nism is also applied to baryon production (replacing a quarkby a di-quark). Parameters related to baryon production arethe relative di-quark production rateP (qq)/P (q), an extrastrange di-quark suppression [P (us)/P (ud)]/γs, and an ex-tra suppressionP (qq1)/P (qq0) of spin 1 di-quarks relativeto spin 0 ones leading tos = 3/2 and s = 1/2 baryons.Furthermore, it turned out to be necessary to include an ex-tra suppression of leading baryons, as implemented in JET-SET. However, this extra suppression was not used in heavyquark fragmentation, nor in the simulation of heavy particledecays by fragmentation, because the baryon spectra becametoo soft and there was too strong a suppression of all heavybaryons.
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Table 2. Setting of JETSET 7.4 ME switches used
Variable Value Variable Value Variable ValueMSTJ(11) 3 MSTJ(12) 3 MSTJ(46) 3MSTJ(101) 2 MSTJ(111) 1
The parameter related to baryon-meson-baryon produc-tion, the so-called ‘popcorn’ parameter, was left at its defaultvalue, since experimental determinations of this parameterare not fully conclusive (see [13] and references therein).Furthermore we did not include simulation of Bose Einsteininterference. Although, with properly chosen parameters, theBose Einstein simulation procedure results in a good repre-sentation of the correlation functions for identical particles,as well as in a strongly improved description of light mesonresonance lineshapes [14, 15], the energy momentum rescal-ing performed in the current procedure (subroutine LUBOEI)is somewhat unphysical, strongly influences angular distri-butions between particles and multi-jet rates, and leads towidely different model parameters, some of which also tendto become unphysical. For example, compare the resultingparameters given in Tables 48 and 49 with the parametersof the DELPHI simulation (Table 10), which includes BoseEinstein interference.
For the tuning of JETSET 7.3 and ARIADNE 4.06, thedescription of heavy particle decays, and in particular oftheir branching fractions, has been modified on the basis ofrecent data. Throughout this paper these modified decays arereferred to as ‘DELPHI decays’. The modifications are sim-ilar to those implemented as default in JETSET 7.4. Similarmodifications, based on earlier data, were already imple-mented in the JETSET 7.3 PS Monte Carlo versions usedhere for modelling detector effects (Appendix B).
3.2 JETSET 7.4 matrix element model
In the historically older matrix element version of JETSET,the parton shower simulation is replaced by the exact sec-ond order matrix element calculation which provides up to4 partons. Two calculations, GKS [16] and ERT [17], areavailable in JETSET. This paper considers only the defaultERT option because it is expected to be more exact [18]. AtPETRA/PEP, the predicted 4-jet rate turned out to be toosmall [18] for a given 3-jet rate. This has been connectedwith higher order terms missing in the second order calcula-tion. These terms can be partially accounted for by choosinga suitable scaleµ (Q2 = µs, µ ≤ 1) according to the “op-timal perturbation theory” description [19]. The ME scaleparameterµ replaces the PS cutoff parameterQ0 in the fit,sinceQ0 is not relevant in the ME model. The JETSET 7.4ME switches used are given in Table 2.
3.3 ARIADNE 4.06 parton shower model
ARIADNE is a particularly elegant formulation of a partonshower based on colour dipoles [9, 20]. The emission of agluon from a colour dipole (i.e. the initial quark-antiquarkpair) creates two new dipoles, one in between the quarkand the gluon and one between the gluon and the antiquark,
Table 3. Setting of ARIADNE 4.06 switches used
Variable Value Variable Value Variable ValueMSTA(1) 1 MSTA(3) 0 MSTA(5) 1MSTJ(11) 3 MSTJ(12) 3 MSTJ(41) 0MSTJ(46) 3 MSTJ(105) 0
and each in turn can independently radiate further gluons. Itautomatically includes ordering in angle (or transverse mo-mentum), as well as azimuthal dependences, in a proper way.The dipole chain resembles the Lund string. Parameters arethe QCD scale parameterΛQCD and the cut-off scalepQCDt .The latter corresponds toQ0 in the JETSET parton showermodel. The evolution variable is the transverse momentumsquared. As in JETSET, the first order 3-jet cross sectionis reproduced in ARIADNE. The ARIADNE switches wereset as given in Table 3.
3.4 HERWIG 5.8 C parton shower model
The evolution of the parton shower in HERWIG is based onthe Coherent Parton Branching formalism, an extension ofthe LLA. It accounts for the leading and sub-leading loga-rithmic terms arising from soft and collinear gluon emission.HERWIG pays special attention to the simulation of QCDinterference phenomena [18]. The most important parame-ters of the parton shower algorithm areΛQCD (QCDLAM),the quark masses (RMASS(1-6)), and the effective gluonmass (RMASS(13)), which provides the shower cutoff. Theparton shower in HERWIG 5.8 C is matched with the firstorder 3-jet cross section. At the end of the parton showerevolution, gluons are split non-perturbatively intoqq-pairs.
The hadronization in HERWIG proceeds via the so-called ‘cluster algorithm’, based on the preconfinement char-acteristic of QCD. The colour charge of a parton is com-pensated to leading order by an anti-colour object which isnearby in phase space. Low mass colour-neutral clusters areformed by combining colour and anti-colour objects. Highermass clusters are further split into two lighter ones. The split-ting is controlled by the parameters CLMAS and CLPOW.In the decay of a cluster containing a quark from the pertur-bative phase, the direction of this quark is remembered. AGaussian smearing is applied, controlled by the parameterCLSMR.
Hadrons are then formed in 2-body cluster decays, ac-cording to phase space and spin factors. The particle andhadron transverse momenta are thus produced dynamically,as a consequence of the cluster mass spectrum. Particleproduction in cluster decays can be modified by changingthe a priori weights for the individual hadron types. Theseare VECWT, TENWT 2 and DECWT for vector or ten-sor mesons and decuplet baryons respectively, and PWT forquarks.
Light particle decays are simulated in HERWIG usingdecay tables. Particles including heavy quarks decay via thedecay of the heavy quark and subsequent fragmentation.
Relevant parameter settings used for the tuning of HER-WIG are given in Table 4.
2 The 1+− and 0++ meson multiplets are not included in HERWIG 5.8 C
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Table 4. Setting of HERWIG parameters used
Variable Value Variable Value Variable ValueIPROC 100 SUDORD 1 CLDIR 1
4 Fit of models to experimental data
Classical optimization strategies, like hill-climbing methods,generally fail to converge if they are applied directly to theoptimization of a Monte Carlo model. This is because thephysical observables predicted by the model is defined onlyon a statistical basis (and thus are known only within the sta-tistical errors). Moreover, this straight-forward strategy re-quires very much computer time, and therefore cannot easilybe repeated with changed input data in order to check theinfluence of systematic errors of the data, etc. Therefore, asin previous work [21, 22, 23], the dependence of the physi-cal observables on the model parameters was approximatedanalytically. For each bin of each distribution, the quadraticapproximation
f (p0 + δp, x) = a(0)0 (x) +
n∑i=1
a(1)i (x)δpi
+n∑i=1
n∑j=i
a(2)ij (x)δpiδpj
≈ MC(p0 + δp, x) (1)
was used. In this equation,n is the number of parameters tobe fitted, andMC(p0 + δp, x) denotes the distribution of aphysical observablex predicted for a given set of parametervaluesp0+δp, wherep0 is a central parameter setting andδpiis the deviation of parameteri from this setting. The thirdterm in the expansion includes correlation terms between themodel parameters. These terms are not present in a linearapproximation, such as that used in [23].
The m = 1 +n + n(n + 1)/2 coefficientsa(0,1,2) of theexpansion were determined by fitting Eq. (1) to` referencesimulation distributions ( ≥ m), generated with differentparameter settings. This fit is equivalent to solving a systemof linear equations:
P · a = MC (2)
whereMC is the vector of model predictions correspondingto the parametersp0 + δp(k), a is the vector of coefficientsa(k)i(j)(x), andP is a matrix in which columnk contains the
parameter variations of model setk :
Pk,1..m = (1, δp(k)1 , . . . , δp(k)
n , δp(k)21 , . . . , δp(k)2
n ,
δp(k)1 · δp(k)
2 , . . . , δp(k)n−1 · δp(k)
n )
andk runs from 1 to`. The optimal solution, for an over-constrained linear system (` > m), was obtained using astandard singular value decomposition method [24, 25].
The parameters of thereference models (generated withequal statistics) were randomly chosen in parameter spacearound the central pointp0. Assuming thea priori chosenparameter intervals to be renormalized to±1, it turns outto be unimportant, except for minor differences in the sta-tistical precision, whether this volume is a hypercube or ahypersphere or whether the points are placed throughout its
volume or on its surface. Our choice should ensure that theprecision of the fitted linear function is roughly constantwithin the hypercube.
All simulated sets were generated with equal statistics,large enough that the overall statistical error was small com-pared with the experimental uncertainty of the data. There-fore the statistical errors of the simulated data sets have beenneglected.
The optimum values of the parameterspi, their errorsσi,and their correlation coefficients%ij were then determinedfrom a standardχ2 fit of the analytic approximation (1) tothe corresponding data using MINUIT [26]. The fit was donesimultaneously for all distributions and all bins considered.Note that the overall analytic approximation containsnbins ·m coefficientsa(0,1,2)
i(j) (x).In order to minimise the large number of coefficients
to be fitted, if the sensitivity of a distribution to a givenmodel parameterpi was found to be negligible comparedwith the sensitivity to other parameters (see also next sectionand Tables 45–47 in Appendix D), the dependence of thequadratic expansion (1) on this parameter was suppressedby omitting it from the linear system (2). This led to a betterconvergence of the minimization and more robust results.
The method was tested by generating` + 1 = 51 simu-lated data sets, with 6 important fragmentation parameters,and then simultaneously fitting these 6 parameters to eachdata set in turn, after using the other 50 data sets to de-termine the coefficientsa(k)
i(j)(x). In this case, the statistical
error on the coefficientsa(0,1,2)i(j) (x) of the quadratic expansion
(1) was negligible compared with the statistical error of thetest set. The pull distributions (pfiti − ptruei )/σi were foundto be approximately standard normal distributions, showingthat the fitting method is self-consistent and unbiased andproduces correct errors [14].
4.1 Choice of distributions
Most parameters of a fragmentation model have a well-defined physical meaning. However, some parameters aredirectly coupled, likea, b, andσq in the Lund fragmenta-tion function, while the effect of some parameters on phys-ically observed quantities is obscured by other processes,like decays. Therefore the best choice of distributions fortuning the model parameters is not always evident. Conse-quently, from the many possible distributions, some havebeen chosen to determine the central fitted values, and al-ternative choices have been used to estimate the systematicuncertainties [7, 22, 23, 27].
In practice, to keep the influence of statistical errors assmall as possible, it is clear that the models should be fittedto the distributions that show the strongest dependence onthe parameter under consideration and least dependence onothers. For each distributionMC(x), its sensitivity to a givenmodel parameter, i.e. the quantity:
Si(x) =δMC(x)MC(x)
∣∣∣pi
/δpipi
≈ ∂ lnMC(x)∂ ln |pi|
∣∣∣pi
was therefore calculated, whereδMC(x) is the change of thedistributionMC(x) when the model parameterpi is changed
18
by δpi from its central value. Using the fractionδpi/pi givesall parameters the same normalization.
Figures 1 and 2 show the sensitivities of the main JET-SET PS parameters to some single charged particle and eventshape distributions respectively. For a more comprehensiveoverview, Tables 45 and 46 show the averaged absolute val-ues of the sensitivities to the JETSET 7.4 and HERWIG 5.8model parameters respectively. However, note that the av-eraging tends to dilute the sensitivities (compare Fig. 1 withTable 45). The sensitivities for ARIADNE are similar tothose for JETSET [14]. For the definitions of the variables,see Appendix A.
The sensitivity to all model parameters tends to be largerfor single particle inclusive distributions than for event shapedistributions (compare for example Figs. 1 and 2). It can beseen that there is almost no sensitivity to the parton showercut off Q0 in the event shape distributions. For this parame-ter, thexp spectrum at largexp is the most important. How-ever, correlations among the individual parameters in theinclusive spectra are very strong, as can be seen by the sim-ilar (or opposite) behaviour of the sensitivities for differentparameters. The opposite, almost symmetric, behaviour ofthe sensitivities to Lund fragmentation function parametersa andb illustrates the strong correlation between these twoparameters, and why it is possible to find good descriptionsfor many different choices ofa and b. As expected,σq isbest determined by thepoutt spectrum or related quantities.Another important quantity is the charged particle multiplic-ity, which is known to high precision. Obviously it dependson very many model parameters and also on details of theparticle decays.
The differential 3-jet rate as measured byDDurham2 or
DJade2 is best suited to determineΛQCD because the sensi-
tivities to other parameters are negligible for moderately lowvalues ofycut, see Fig. 2. Thus the determination ofΛQCDcan be almost completely decoupled from the determinationof other parameters. This also underlines the reliability ofαs determinations from differential jet rates.
Other event shape distributions measuring the overallshape, like the thrustT , sphericityS, majorM , or the highhemisphere mass variableM2
high/E2vis, also depend mainly
onΛQCD (see Table 45), except in the 2-jet region (left handsides of the plots in Fig. 2), where fragmentation effects arealso relevant.
Distributions measuring the aplanarity of the events, likethe aplanarityA, minor m, or the lower hemisphere massvariableM2
low/E2vis, tend to show increased sensitivities to
ΛQCD, but also high sensitivities to other parameters. Thisillustrates why quantities measuring differences of eventshape variables, like the oblatenessO, hemisphere massdifferenceM2
diff/E2vis, and the asymmetry of the energy
energy correlationAEEC, contrary to widespread belief,depend not only onΛQCD but also on many other fragmen-tation parameters.
The above discussion indicates that, for a determinationof the general parton-shower and fragmentation parameters,the models should be fitted to:
− the inclusive charged particle distributions as a functionof xp, pint andpoutt with respect to the thrust or sphericityaxis,
− the differential 3-jet rateD3, or alternativelyR3,− a combination of event shape distributions likeT , M ,
m, or S, A, P , or M2high/E
2vis andM2
low/E2vis.
Any combination of event shape distributions or event axesfor thept spectra is in principle equivalent. Varying the com-bination can be used to estimate the stability of the fit andthe systematic errors of the parameters determined, or can beviewed as a check of the “predictive power” of the models.However, the strong correlations among the model parame-ters need to be considered.
The action of the model parameters linked directly toidentified particle production (like the strange quark suppres-sion,γs, or the relative probability to form aqq pair to makea baryon,P (qq)/P (q), or others) usually follows the physi-cal interpretation more directly. However, identified particlespectra also have high sensitivity to fragmentation parame-ters (see Tables 45 and 46). Therefore the scheme describedabove to determine the optimal dependences is used also inthis case.
4.2 Strategy of the fit
4.2.1 JETSET & ARIADNE.Fits of the general fragmenta-tion parameters (ΛQCD; Q0 or µ or pQCDt ; a, b and σq)were first performed to the charged particle inclusive dis-tributions and global event shape distributions. In order todetermine the coefficientsa(0,1,2)(x) in (1), 50 simulated setswith 100 000 events each were used for each fit. Parametersrelated to identified particles were set to values similar tothose used for the DELPHI simulation (see Table 10), sincethese were already known to describe identified particle ratesand spectra well.
The average scaled energies< xE > of charm andbeauty particles in the models depend only onεc, εb andΛQCD. However, the stable hadron spectra depend onlyweakly on the heavy quark fragmentation parameters. Thefollowing procedure was adopted to reduce further the de-pendences of the stable particle distributions on the heavyquark parameters and at the same time to ensure correct< xE > values forD∗ and B mesons. The< xE >values forD∗ andB mesons were determined within themodel for several choices ofΛQCD and εc(b). Then thedependence ofεc(b) on < xE >D∗(B) and ΛQCD wasparametrized. This allowed the method to choose, for eachmodel set and its givenΛQCD value, corresponding val-ues ofεc(b) such that the model set reproduced within theirerrors the average heavy meson scaled momenta as mea-sured by the LEP experiments:< xE >D∗= 0.504± 0.009,< xE >B= 0.701± 0.008 [28].
The models were then fitted to several combinations ofinclusive charged particle and event shape distributions. Theresults forΛQCD andσq were relatively stable, those forQ0
or pQCDt less so, and there were many different solutionsfor a andb, because of the strong correlation between thesetwo parameters. Therefore only one central value ofb wasused in later fits and onlya was treated as a variable. Theparametera was preferred because it is less directly coupledto σq in the Lund fragmentation function.
Parameters relevant only to the production of specificparticles were then adjusted to the related data. In overview:
19
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 2 4 6 8 10 12 14 16pt
in
sens
itivi
typt
in
ΛQCD Q0
a b
σq
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 0.5 1 1.5 2 2.5 3 3.5pt
out
sens
itivi
ty
ptout
ΛQCD Q0
a b
σq-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1xp
sens
itivi
ty
xp
ΛQCD Q0
a b
σq
a b c
Fig. 1. Sensitivity of the JETSET PS parametersΛQCD , Q0, a, b, andσq to the single particle variablespint , poutt , andxp. The curves are to guide theeye
− the extraη and η′ suppressions were adjusted to datafrom [29, 30];
− the probabilities for producing the differentB mesonmultiplets were adjusted to agree with recent measure-ments [31, 32] and the correspondingD meson proba-bilities were interpolated between theB and light mesonvalues;
− [P (us)/P (ud)]/γs, the strange baryon suppression, wasadjusted to the ratio ofΛ0 to proton production [13, 33,34, 35, 36];
− the leading baryon suppression was adjusted to the highmomentum tail of the proton andΛ0 spectra [13, 33, 34,35, 36];
− P (qq1)/P (qq0), the spin 1 di-quark suppression, was ad-justed to the ratio ofΣ(1385) toΛ0 or proton production[13, 33, 34, 36, 37].
Then a simultaneous fit of 10 important parameters (seeTables 48 – 52 in Appendix E) was prepared, by gener-ating 100 simulated data sets of 100 000 events each. Theanalytical approximation obtained from these simulated datasets was then fitted to various choices of inclusive chargedparticle, event shape, and identified particle distributions.
The inclusive charged particle and event shape distribu-tions measured by DELPHI in this analysis were used forthese fits in various combinations, as shown in Table 5. Themean charged particle multiplicity at theZ used in the fits,< Nch >= 20.92± 0.24, was an average of all availableresults [38].
The identified particle distributions used included resultsfrom all LEP experiments. The combinations chosen areshown in Table 6. They were selected to take into accountthe discrepancies observed for the proton data from differ-ent experiments and the imperfect representation of theKspectra in the models.
The proton data were used in the fit only to determinethe di-quark suppression parameterP (qq)/P (q) and were ex-cluded from theχ2 calculation for the fits of the remainingparameters. Baryons at intermediate and large momenta arelikely, according to the models, to be primary produced par-ticles. Therefore these data strongly influence the primary
Table 5. Overview of the combinations of inclusive charged particle andevent shape distributions used for the fits of JETSET, ARIADNE and HER-WIG. Distributions used are marked by•. The central results presentedcorrespond to the choiceS6. The other combinations were used to provideestimates of the systematic errors
Table 6. Overview of the combinations of identified particle data used forthe JETSET and ARIADNE fits. Data used are marked by•. The centralresult presented corresponds to the choiceP0. The other combinations wereused to provide estimates of the systematic errors
fragmentation function and the related parameters. Sincesome experimental discrepancies are present in the protonspectra, and because it had proved necessary to modify thefragmentation function by an extra suppression at large mo-menta, this strong impact on the fit results was considered
20
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35ycut
sens
itivi
tyD2
Durham
ΛQCD Q0
a b
σq
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51-T
sens
itivi
ty
1-Thrust
ΛQCD Q0
a b
σq
a b
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 0.1 0.2 0.3 0.4 0.5 0.6M
sens
itivi
ty
Major
ΛQCD Q0
a b
σq
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4m
sens
itivi
ty
Minor
ΛQCD Q0
a b
σqFig. 2. Sensitivity of the JETSET PS pa-rametersΛQCD , Q0, a, b, andσq to theevent shape variablesDDurham
2 , 1−T ,M , andm. The curves are to guide theeyec d
to be unphysical and therefore was excluded. TheΛ0 datawere also excluded from the final fits for similar reasons, andbecause they are not described well enough by the models.
Separate fits were performed for all possible combina-tions of the 12 choices of inclusive and event shape distri-butions in Table 5 with the 5 possible choices of identifiedparticle data in Table 6. A full MINOS [26] error estimatewas performed for all parameters.
To check the stability of the fits, the optimization wasstarted with 6 random start values of the fragmentation pa-rameters. The fits were stable and converged to the same
solution in 95% of cases. The differences occurred mainlyfor two strongly correlated parameters. Where the fits gavemore than 1 solution, the one with the betterχ2 was kept.
The central results presented correspond to the combina-tion S6P0. This combination was chosen because it containsevent shape distributions linear and quadratic in the particlemomenta, the charged particle multiplicity, and all relevantidentified particle information, so it should result in a com-plete overall description. The other combinations were usedto provide estimates of the systematic errors. The results
21
are given, with their statistical and systematic errors, in Ap-pendix E, Tables 48–52.
The results for some of the parameters are strongly cor-related. Typical correlation coefficients for JETSET 7.4 (de-fault decays) and ARIADNE 4.06 (DELPHI decays) aregiven in Appendix E, Tables 54 and 55. Besides these statis-tical correlations, further correlations exist due to the differ-ent possible choices of input data. The upward (downward)systematic error quoted is the root-mean-square spread ofthe results obtained from all combinations of input data thatgave parameter values bigger (smaller) than the central fit.
The following observations were made by comparing theresults of the different input data choices for the JETSET PSfits.
As should be expected, the results forΛQCD, Q0 andσqwere almost independent of the choice of identified particledata. If justD2 was included in the fit, the resulting valueof ΛQCD did not depend on the algorithm used (JADE orDURHAM). However, if the higher jet rates (D3, D4) wereincluded, theΛQCD value was somewhat bigger (≈ 8%) us-ing the JADE algorithm. The values ofΛQCD andQ0 werepositively correlated, implying that the number of final par-tons is more stable within the models than might be expectedfrom the error ofQ0 alone. The parametersΛQCD, a andσqwere anticorrelated, due to a compensation of the transversemomenta generated in the parton shower and fragmentationphases of the model.
The results forγs were higher if theK± data were in-cluded as well as theK0 data.P (qq)/P (q) was larger forthe ALEPH proton spectrum than for the OPAL one. Produc-tion parameters for strange vector and pseudoscalar mesonswere anticorrelated. If the charged particle multiplicity wasnot included, and therefore the predicted multiplicity wassmaller than the measured result, the primary productionprobabilities for light pseudoscalar and vector mesons wereP (1S0)ud ≈ 0.40 andP (3S1)ud ≈ 0.26 respectively. If themultiplicity was fixed toNch ≈ 20.9, these values were0.28 and 0.29. This implies substantial production probabil-ities for light p-wave mesons of 0.36 to 0.43.
Most of these observations also apply to ARIADNE.However, in this case the different choices ofD2, D3, etc.led to stable results forΛQCD. There was a tendency toobtain slightly bigger values forΛQCD from the JADE al-gorithm (0.245) than from the Durham one (0.237). This isagain consistent with the bigger values forpQCDt obtainedfrom JADE (≈ 0.9 GeV/c) than from DURHAM (≈ 0.6GeV/c).
For JETSET ME,ΛQCD was found to be anticorrelatedwith the scaleµ. The fragmentation parameter values differfrom those from JETSET PS. Especiallyσq ≈ 0.48 GeV/c ismuch bigger. This partially compensates the missing higherorders in the ME model. The values ofγs andP (qq)/P (q)are smaller than for the PS case and depend less on thechoice of input data.
4.2.2 HERWIG.HERWIG uses fewer parameters than JET-SET, especially in the hadronization sector of the model.Therefore a simultaneous fit of all model parameters whichare found to be important is easily performed. These param-eters (see Table 46) are:
− the QCD scale parameter QCDLAM and the gluon massRMASS(13) as major parameters of the parton showerphase,
− the cluster fragmentation parameters CLMAX, CLPOW,and CLSMR (CLDIR=1),
− thea priori weights PWT(3) for strange quarks, PWT(7)for di-quarks, and DECWT for decuplet baryons.
Besides the particle spectra, the cluster parameters alsostrongly influence the stable charged particle and event shapedistributions. Fits were therefore performed to combinationsof inclusive charged particle and event shape distributions(see Table 5) and identified particle data (see Table 7). Theaverage scaled momenta of heavy mesons are important forthe determination of CLMAX and CLPOW.
The identified particle data sets were again chosen (seeTable 7) such that systematic differences in the data would bereflected in the systematic errors of the fitted parameters. Thecentral fit result for HERWIG (see Table 53) corresponds tothe combinationS6P5. The other combinations were used toprovide estimates of the systematic errors.
For HERWIG, comparing the results of the different in-put data choices showed that the values obtained for QCD-LAM were in general bigger (by≈ 0.01) when fittingDURHAM rather than JADE jet rates. Contrary to the JET-SET case, QCDLAM was smaller (by≈ 0.005) when thedifferential 2-, 3-, and 4-jet rates were all fitted, rather thanthe 2-jet rate only. The gluon mass RMASS(13) showedsome dependence on the identified particle information se-lected, and on the charged particle multiplicity. The clusterparameters CLMAX and CLPOW depended on the identifiedparticle spectra only when the multiplicity was not includedin the fit. CLPOW was higher for JADE than for DURHAMjet rates. CLSMR depended only on the inclusive chargedparticle and event shape information. Including the baryondecuplet data tended to spoil the description of the octetsector.
5 Comparison of models to data
The fits of the individual fragmentation models are comparedwith corrected DELPHI data (this analysis) in Figs. 3–38 col-lected in Appendix F. Figs. 3–32 and 37–38 show data mea-sured from charged particles only, while Figs. 33–36 com-pare the models with a few selected distributions measuredfor charged and neutral particles. All distributions are cor-rected to the corresponding final states. The lower insets ofthe plots depict the relative deviation of the models from thedata. Also shown, as shaded areas in these insets, are the to-tal experimental errors obtained by adding quadratically thesystematic and statistical error in each bin.
The different decay treatments lead to negligible differ-ences for the stable charged particle and event shape distri-butions. Comparisons are made with the following models:
− JETSET 7.3 with DELPHI decays labeled JT 7.3 PS− JETSET 7.4 default decays labeled JT 7.4 PS− ARIADNE 4.06 with DELPHI decays labeled AR 4.06− HERWIG 5.8 C default decays labeled H 5.8 C− JETSET 7.4 ME default decays labeled JT 7.4 ME
22
Table 7. Overview of the combinations of identified particle data used for the HERWIG fits. Data used are markedby •. The central fit result corresponds to the choiceP5. The other combinations were used to provide estimates ofthe systematic errors
The observations below are made from the model to datacomparisons.
5.1 Inclusive charged particle spectra
All models describe the general trends of the data well. Fewdiscrepancies show up from direct model to data compar-isons. More quantitatively, the comparisons of model anddata (lower insets) show the following.
Thexp spectrum (Fig. 4) forxp < 0.4 is almost perfectlydescribed by the ARIADNE and JETSET PS models. Atlargexp these models slightly underestimate the data. Thistrend is reduced if the multiplicity is left free in the fit. TheHERWIG and JETSET ME predictions alternate betweenbeing too high and too low with respect to the data. Thisbehaviour is also reflected in the rapidity distributions (seeFigs. 5, 6).
For all models, thepint distribution (Fig. 7) agrees withthe data within errors. Only the largepint tail is slightlyunderestimated.
Also for all models, the predictedpoutt distribution (Fig. 8)for poutt > 0.8 GeV falls off more rapidly than the data andis about 30% below the data at largepoutt . The largepoutt tailis due mainly to gluon radiation. For the ME model, a dis-crepancy might be expected because of missing higher orderterms in the second order matrix element calculation. Thefailure of the parton shower models can possibly be tracedback to missing large angle terms in the basic LLA formal-ism used by the models. If these reasons for the failure ofthe models are correct, a matching of the second order cal-culation and the LLA formalism should lead to an improveddescription ofpoutt and related distributions.
5.2 Event shape distributions
The general event shape distributions, 1− T , S, C andBsum (Figs. 11, 21, 31 and 29), are well described within
Table 8. χ2/bin for the model/data comparisons of event shape and inclu-sive charged particle distributions. Only thepoutt distributions are badlydescribed by all models
χ2/bin
ARIADNE JETSET JETSET HERWIG JETSET4.06 7.4 PS 7.3 PS 5.8 C 7.4 ME
the small experimental errors (typically 2–3%) by all theparton shower models. The HERWIG predictions tend to lieslightly above the data for large values of these observables,the JETSET ME predictions tend to oscillate around the data.
The description of distributions sensitive to transversemomenta in the event plane, likeM , Bmax or M2
h/E2vis
(Figs. 12, 27 or 25), is only slightly less good (typically bet-ter than 5%).
Turning to distributions sensitive to transverse momentaout of the event plane, namelym, O, A, M2
l /E2vis, and
Bmin (Figs. 13, 14, 22, 26, 28), the following pattern is ob-served: ARIADNE generally describes the data well, whilefor higher values of the observables, HERWIG tends to over-estimate the distributions and JETSET PS to underestimatethem. The latter is to be expected from the underestimationof the poutt distribution by all PS models.
A similar pattern is also observed for the jet rates. Thedifferential 2-jet rateD2 (Figs. 15, 16) is well described byall models. ARIADNE describes also the higher jet rateswell (Figs. 17–20), HERWIG overestimates and JETSET PSunderestimates them. This behaviour is similar for the JADEand DURHAM jet algorithms.
The event shape distributions are less well described bythe JETSET ME model than by the PS models. The extreme2-jet region, the multi-jet rates, and observables sensitiveto radiation out of the event plane are also not describedquite so well. Nevertheless, in general, the description bythe JETSET ME model is reasonable.
To permit a comprehensive comparison of the differentmodels, Table 8 shows the meanχ2/bin obtained from thecomparison of the individual data and model distributions.The χ2/bin values shown should be used only for relativecomparison, not to infer absolute confidence levels. In thelimit of large statistics, even a slightly imperfect model de-scription leads to a very largeχ2/bin. Furthermore, in prin-ciple, correlations between the different distributions shouldbe taken into account.
5.3 Identified particle rates
Table 9 compares the particle rates predicted by the mod-els with the current measured LEP averages. If the meancharged multiplicity< Nch > is included in the fits, it iswell described by the models. Neglecting this constraint, themultiplicity predicted by ARIADNE and JETSET PS is toolow (20.2–20.4) and that predicted by JETSET ME is toohigh (22.7). The HERWIG prediction is correct without con-straint.
The meson rates, with the exception of theK± and theη rates, are described fairly well. TheK± rate is sensitiveto heavy quark decays (see below). The octet baryons alsoagree reasonably. Only HERWIG overestimates theΞ− rateby about a factor 2. But it gives the best prediction forthe Ω− rate. There are some discrepancies in the decupletbaryon sector.
5.4 Meson momentum spectra
Figures 39, 40 compare the models to the identified kaonspectra [13, 33, 34, 35, 36, 37] as a function ofξp = log 1
xp
from different LEP experiments. For a more quantitativecomparison which also allows the agreement among the dif-ferent experiments to be judged, the relative deviation be-tween the individual data sets and each model is shown bythe lines in the figures on the right. In the fragmentationregion at largeξp (i.e. smallxp), where theK± data aremore precise and therefore dominate the fit, theK0 rates areoverestimated by≈ 10–20% by all models. At more cen-tral momenta, all kaons are well described. But in the range1.0 < ξp < 2.5, theK± rates are underestimated. This in-consistency between data and models causes the relativelylarge systematic error onγs.
Kaons from heavy particle decays tend to contributemainly in this momentum range, so wrong descriptions ofthe decay ofb-hadrons are a likely cause of this discrepancy.A recent DELPHI measurement [46] of the inclusive particleproduction inb-events found the difference between theK±andK0 multiplicities to be 0.58± 0.51. The correspondingvalue in the models is≈ 0.30 for JETSET and ARIADNEand 0.04 for HERWIG. Improvement will be possible assoon as precise measurements of the momentum spectra ofkaons for identifiedbb and light quark events become avail-able.
Finally, at very smallξp (i.e. largexp) there is an indica-tion that the models underestimate the K production slightly.
The fragmentation functions of vector mesons [35, 39,41, 42, 47], which are likely to be primary particles, arecompared to the data in Figs. 41, 42, 43. Within the largeerrors of these resonance measurements, all models describethe spectra very well (but note here that the parameterizationof resonance production probabilities used here differs fromthe JETSET default, see Sect. 3). There is only a tendencyto predict a somewhat harder fragmentation than measured.This is most evident for theΦ(1020) spectrum.
Figure 44 shows good agreement of the model pre-dictions with the measurements of thef0(980) scalar andf2(1270) tensor meson [39]. With large production proba-bilities for p-wave resonances, all models describe the datawell. Note that high production of p-wave states (> 10%) isnot expected in the string picture [12].
5.5 Baryon momentum spectra
The proton spectra are compared in Fig. 45 [33, 34, 36].There is a severe discrepancy between the OPAL and ALEPHresults at smallξp. Since both data sets are used for the cen-tral fits, the fits interpolate between them.
For both data sets, the rate predicted by HERWIG is toohigh at smallξp and too low at highξp. At small ξp, i.e.high momentum, the behaviour of JETSET PS and ARI-ADNE is similar to that of HERWIG if the extra baryonsuppression is not used. The need for an extra suppressionof high momentum baryon production may indicate a differ-ent production mechanism for baryons than for mesons. Itcould also be partly due to orbitally excited states missingin the models [8]. However, these states have not yet beenobserved ine+e− annihilation.
The experimental situation for theΛ0 spectrum [13, 35,48] also shows some discrepancies (see Fig. 46). JETSET PSand ARIADNE describe or slightly overestimate the smallξp
24
Table 9. Single particle production rates for the different generators compared to LEP I data
JETSET JETSET ARIADNE JETSET HERWIG LEP7.3 PS 7.4 PS 4.06 7.4 ME 5.8 C [38, 43, 44, 45]
region. The HERWIG prediction is again too high. At largeξp, where the measurements agree, all models underestimatethe Λ0 production by≈ 15%. Better descriptions may beobtainable by restricting to the proton andΛ0 spectra ofspecified experiments or by fitting all parameters relevant tobaryon production simultaneously [50].
Figure 47 shows in a comprehensive overview a compar-ison of the octet and decuplet baryon production [13, 35, 48,49] with the model predictions. The discrepancies discussedabove are hidden here, due to the large scales. JETSET andARIADNE describe the gross features of the octet and de-cuplet baryon production well. HERWIG predicts too harda baryon fragmentation, and also predicts the relative pro-duction rates of the different multiplet states less well.
6 Summary
Precise fully corrected inclusive charged particle and eventshape distributions have been determined from 750 000e+e−→ Z → hadrons events measured by the DELPHI experi-ment.
A systematic quantitative study has been undertaken todetermine the optimal choice of distributions to tune frag-mentation models. Semi-inclusive charged particle and iden-tified particle distributions constrain the hadronization partof the models, whereas 3-jet rate distributions and most eventshape distributions mainly control the parton shower param-eters (especiallyΛQCD) in the models.
Optimum parameter values have been determined for theARIADNE 4.06, HERWIG 5.8 C and JETSET 7.3 and 7.4parton shower models and for the JETSET 7.4 matrix el-ement model. The models were fitted to the event shapeand inclusive charged particle distributions measured in thisanalysis and to the mean charged particle multiplicity andidentified particle data measured by all the LEP experiments.The fit algorithm employed allowed a simultaneous fit of upto 10 model parameters. Statistical and systematic errors aswell as correlations of the model parameters have been de-termined.
All models describe the inclusive charged particle andevent shape distributions reasonably well. The data measuredfrom charged particles and charged plus neutral particlesyield consistent results when compared with the correspond-ing model predictions. All models underestimate the tail ofthepoutt distribution by more than 25%. With this exception,the best overall description is provided by ARIADNE 4.06.HERWIG 5.8 C tends to overestimate and JETSET 7.3/7.4to underestimate the production of 4 or more jet events.The tails of event shape distributions sensitive to particleproduction out of the event plane are overestimated (un-derestimated) by HERWIG (JETSET). The matrix elementmodel JETSET 7.4 ME with optimized scale also providesreasonable predictions. However, it shows the expected dis-crepancies in the extreme 2-jet and multi-jet regions due tothe missing higher order terms.
Identified meson spectra are fairly well described by allmodels. It has been found that strong production of p-wave
25
resonances (25–40%) has to be considered. This is not ex-pected in a string fragmentation picture. The gross featuresof baryon production are described by JETSET and ARI-ADNE. HERWIG shows stronger discrepancies, with thepredicted fragmentation functions being too hard. In JET-SET and ARIADNE, a similar tendency has been correctedby applying an extra leading baryon suppression.
Acknowledgements.We are greatly indebted to our technical collaboratorsand to the funding agencies for their support in building and operating theDELPHI detector, and to the CERN-SL Division for the superb performanceof the LEP collider. We would like to thank G. Heindl, I. Knowles, L.Lonnblad, M. Seymour and T. Sjostrand for useful discussions.
A Definition of variables
This paper uses the following definitions of event shape andinclusive particle variables:
A.1 Inclusive single particle variables
Scaled momentum,xp, ξpThe scaled momentum,xp, is the absolute momentum,|p|, ofa particle scaled to the beam momentum, whileξp = log 1
xp.
Transverse momenta,pt, pint , poutt
With respect to the thrust axis, the component of the trans-verse momentumpt of a particle in the event plane ispint =p·nMajor and that out of the event plane ispoutt = p·nMinor.For the values with respect to the sphericity axis, the axesdefined by the eigenvectors of the quadratic momentum ten-sor are used instead.
RapidityyThe rapidity is given by:
y =12· log
E + p‖E − p‖
where p‖ is the particle momentum parallel to the eventthrust axisnThrust for yT , or the sphericity axisnSphericityfor yS .
A.2 Event shape variables
ThrustT , majorM , minorm, oblatenessOThe Thrust [51],T , and the Thrust axis,nThrust, are definedby:
T = maxnThrust
Nparticle∑i=1
|pi · nThrust|Nparticle∑
i=1|pi|
nThrust is a unit-vector along the Thrust axis. Major andMinor are defined similarly, replacingnThrust by nMajor,perpendicular tonThrust, andnMinor = nMajor × nThrustrespectively. The Oblateness isO = M −m.
SphericityS, AplanarityA, PlanarityPOrdering the eigenvaluesλ of the quadratic momentum ten-sor:
Mαβ =Nparticle∑
i=1pαi p
βi (α, β = 1, 2, 3)
λ1 ≥ λ2 ≥ λ3 λ1 + λ2 + λ3 = 1
The Sphericity isS = 32(λ2 +λ3), the Aplanarity isA = 3
2λ3,and the Planarity isP = 2
3(S − 2A) [52]. The Sphericityaxis is parallel to the eigenvector corresponding toλ1. Asthe momenta enter quadratically, the Sphericity axis is influ-enced more strongly by large momentum particles than theThrust axis.
C andD parametersThe C and D Parameters are defined through the eigenvaluesλ of the linear momentum tensor [53]:
Θi,j = 1Nparticle∑
k=1
|pk|·Nparticle∑
k=1
pikpjk
|pk|
C = 3 · (λ1λ2 + λ2λ3 + λ3λ1) D = 27 · λ1λ2λ3
Hemisphere massesM2high/E
2vis,M
2low/E
2vis,M
2diff/E
2vis
(also called jet masses)Particles are ordered in the two hemispheres of an eventseparated by the plane normal tonThrust. Then
M2high
E2vis
=1
E2vis
·max
∑
pk·nThrust>0
pk
2
,
∑pk·nThrust<0
pk
2
For M2low
E2vis
the maximum in the above formula is replaced by
the minimum, andM2diff
E2vis
=M2high
E2vis
− M2low
E2vis
. This definition
differs from the original one by Clavelli [54], but is easierto calculate and therefore preferred by many experiments.
Hemisphere broadeningBmax, Bmin, Bsum, Bdiff (alsocalled jet broadening)In each hemisphere, defined in the same way as for thehemisphere masses, the momenta transverse to the thrustaxis are summed and divided by the corresponding sum ofthe absolute momenta [55], i.e.
B± =
∑±pi·nThrust>0
| pi × nThrust |
2∑i
|pi|
Bmax = max (B+, B−) ; Bmin = min (B+, B−) ;
Bsum = B+ +B− ; Bdiff = |B+ −B−| .Differential jet ratesDi(y)Jets are reconstructed using cluster finding algorithms. Forthe JADE [56] or Durham [57] algorithm respectively, thescaled invariant mass or transverse momentumyij :
yJadeij = 2EiEj/E2vis · (1− cosθij)
yDurhamij = 2 min(E2i , E
2j )/E2
vis · (1− cosθij)
is evaluated for each pair of particlesi and j in an event,whereEi, Ej are the energies andθij the angle between
26
the momentum vectors of the two particles. The particlepair with the lowest value ofyij is replaced by a pseudo-particle with 4-momentum (pi+pj), reducing the multiplicityby one. The procedure is repeated until the scaled invariantmasses of all pairs of (pseudo-)particles exceed a given res-olution y. The remaining (pseudo-)particles are called jets.The differential 2-jet rateD2 is derived from the 2-jet rateR2 = N2−jets/N [58]:
D2(y) =R2(y +∆y)−R2(y)
∆y
Higher differential jet rates follow from the recursion:
Dn(y) =Rn(y +∆y)−Rn(y)
∆y+Dn−1(y)
Energy-energy-correlationEEC and asymmetryAEECThe Energy-Energy-CorrelationEEC is the histogram ofangles,χij , between all particles weighted by their scaledenergies [59]:
EEC(cosχ) =1N
1∆ cosχ
·N∑
events
∑i<j
Ei
Evis
Ej
EvisΘ(∆ cosχ− | cosχ− cosχij |)
where cosχ and∆ cosχ are the lower edge and width of abin respectively, andΘ is the step function. For cosχ ≥ 0,the asymmetryAEEC of theEEC is:
AEEC(cosχ) = EEC(− cosχ)− EEC(cosχ)
B Parameter settings of the DELPHI Monte Carlo(Table 10)
The DELPHI tuning of JETSET has been obtained by tun-ing the model to charged particle data from the 1991 and1992 data taking. Care has been taken to describe the ob-servables relevant to standard precision analyses which arethe charged multiplicity, the momentum spectrum, 2-jet rateand Thrust and Sphericity distribution. The simulation alsoincludes Bose Einstein interference (by LUBOEI) to obtaina correct description of two particle correlations and lightresonance line shapes. The BE parameters are taken fromthe DELPHI measurement [60]. Particle spectra have beenadjusted and partially fitted to available data [45]. Heavyparticle decays have been adjusted.
C Tables of inclusive charged particle and event shapedistributions (Tables 11–44)
The normalized differential distributions were determinedfrom charged particles only and from charged plus neu-tral particles. The tables show first the measurement fromcharged particles corrected to the charged hadronic finalstate and then the measurement from charged plus neu-tral particles corrected to the full hadronic final state. Forthe inclusive charged particle distributions, only the relevantevent axes have been evaluated for the corresponding finalstates. The statistical and systematic errors are also given.Computer-readable files of the data presented in these tablesare available on the HEPDATA database [61].
Table 10.DELPHI parameter settings of JETSET 7.3 PS for the December1993 tuning used for modelling detector effects for 1993 data and for theSeptember 1994 tuning used for modelling detector effects for 1994 data.For 1994 the meson spin parameters, PARJ(11) to PARJ(17), have been re-placed by the more detailed parameters,P (2s+1Lj )q , whereq is the heavierquark type
Table 45. Average sensitivities (×100) for JETSET 7.4 PS with standard decays. Sensitivities for ARI-ADNE (with pQCDt in place ofQ0) and JETSET 7.3 PS are similar. The identified particle sensitivitieshave been calculated for theξp distributions. The quoted values have statistical errors
model parameteru/d quarks s quarks
property a b σq ΛQCD Q0 γs P (qq)P (q) P (1S0) P (3S1) P (1S0) P (3S1)
Table 46.Average sensitivities (×100) for HERWIG 5.8 C with standard decays. The identified particle sensitivitieshave been calculated for theξp distributions. The quoted values have statistical errors
P (qq)/P (q) PARJ(1) 0.1 0.08 - 0.11 0.087±0.002 +−
0.0080.002
[P (us)/P (ud)]/γs PARJ(3) 0.4 0.657 adj. to data
P (qq1)/P (qq0) PARJ(4) 0.05 0.07 adj. to data
extra baryon supp. PARJ(19) 0. 0.5 adj. to data, only for uds
extraη supp. PARJ(25) 1.0 0.65 0.65± 0.06
extraη′ supp. PARJ(26) 1.0 0.23 0.23± 0.05
Table 53. Parameter settings and fit results for HERWIG 5.8 C
Parameter Default Range gen. Fit Result
Value stat. sys.
QCDLAM 0.18 0.155 - 0.205 0.163±0.001+−
0.0040.005
RMASS(13) 0.75 0.64 - 0.96 0.65±0.01 +−
0.020.01
CLMAX 3.35 2.35 - 4.35 3.48±0.04 +−
0.200.13
CLPOW 2.0 1.0 - 2.0 1.49±0.04 +−
0.160.08
CLSMR 0. 0.0 - 1.0 0.36±0.04 +−
0.060.01
DECWT 1.0 0.0 - 1.0 0.77±0.08 +−
0.280.24
PWT(3) 1.0 0.0 - 1.0 0.83±0.02 +−
0.030.09
PWT(7) 1.0 0.0 - 1.0 0.74±0.09 +−
0.370.25
45
Table 54. Table of Correlation Coefficients for JETSET 7.4 PS Fit
a σq ΛQCD Q0 γs P (1S0)ud P (3S1)ud P (1S0)s P (3S1)sa 1.00 0.33 -0.60 0.01 -0.10 0.27 -0.11 0.06 0.03
σq 1.00 -0.69 -0.08 0.43 -0.41 0.11 -0.26 -0.29
ΛQCD 1.00 0.18 -0.26 0.19 -0.10 0.15 0.20
Q0 1.00 0.20 0.06 -0.38 -0.35 -0.13
γs 1.00 -0.04 0.19 -0.73 -0.65
P (1S0)ud 1.00 0.37 -0.29 0.12
P (3S1)ud 1.00 -0.10 -0.26
P (1S0)s 1.00 0.41
P (3S1)s 1.00
Table 55. Table of Correlation Coefficients for ARIADNE 4.06 PS Fit
a σq ΛQCD pQCDt γs P (1S0)ud P (3S1)ud P (1S0)s P (3S1)sa 1.00 0.11 -0.18 0.90 -0.29 0.71 -0.04 0.29 0.07
σq 1.00 -0.60 -0.07 0.12 -0.47 -0.06 0.02 0.05
ΛQCD 1.00 0.16 0.02 0.18 0.03 -0.03 -0.06
pQCDt 1.00 -0.21 0.65 -0.02 0.26 0.05
γs 1.00 -0.27 -0.02 -0.41 -0.47
P (1S0)ud 1.00 -0.11 -0.03 0.13
P (3S1)ud 1.00 0.08 -0.03
P (1S0)s 1.00 0.02
P (3S1)s 1.00
F Figures of data and model comparisons
The figures in this section compare corrected DELPHI dataand identified particle spectra to the following fragmentationmodels:
− JETSET 7.3 with DELPHI decays labeled JT 7.3 PS (fullcurves)
− JETSET 7.4 with default decays labeled JT 7.4 PS(dashed curves)
− ARIADNE 4.06 with DELPHI decays labeled AR 4.06(dotted curves)
− HERWIG 5.8 C with default decays labeled H 5.8 C(dot-dashed curves)
− JETSET 7.4 ME with default decays labeled JT 7.4 ME(widely spaced dots)
The references for the identified particle data can be foundin Table 7. The total correction factor for each distributionis shown in the upper inset. The lower insets of the plotsshow the relative deviation of the models from the data. Alsoshown as a shaded area in these insets is the total experimen-tal error obtained by adding quadratically the systematic andstatistical error in each bin. The error should be interpretedlike a statistical one standard deviation uncertainty.For the ξp distributions of identifiedK0, K±, p and Λ0
(Figs. 39, 40, 45, 46), we also show separate plots compar-ing the models with each data set separately (ALEPH dashedcurve, DELPHI full curve, OPAL dotted curve).
46
F.1 Inclusive single charged particle distributions
ξp=log(1/xp)co
r. fa
c.
0.60.8
11.21.4
10-2
10-1
1
10
ξp=log(1/xp)1/
N d
N/d
ξp
DELPHI
charged particles
JT 7.3 PS
JT 7.4 PS
AR 4.06
H 5.8C
JT 7.4 ME
ξp
(MC
-Dat
a)/D
ata
-0.1
0
0.1
0 1 2 3 4 5ξp
(MC
-Dat
a)/D
ata
-0.1
0
0.1
0 1 2 3 4 5
Fig. 3. ξp = log(1/xp)
xp
cor.
fac.
0.60.8
11.21.4
10-2
10-1
1
10
10 2
xp
1/N
dN
/dx p
DELPHI
charged particlesJT 7.3 PSJT 7.4 PSAR 4.06H 5.8CJT 7.4 ME
xp(M
C-D
ata)
/Dat
a
-0.1
0
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1xp(M
C-D
ata)
/Dat
a
-0.1
0
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Fig. 4. Scaled Momentum,xp
yT
cor.
fac.
0.5
1
1.5
0
2
4
6
yT
1/N
dN
/dy T
DELPHI
charged particlesJT 7.3 PSJT 7.4 PSAR 4.06H 5.8CJT 7.4 ME
yT
(MC
-Dat
a)/D
ata
-0.1
0
0.1
0 1 2 3 4 5 6yT
(MC
-Dat
a)/D
ata
-0.1
0
0.1
0 1 2 3 4 5 6
Fig. 5. Rapidity,yT
yS
cor.
fac.
0.5
1
1.5
0
2
4
6
yS
1/N
dN
/dy S
DELPHI
charged particlesJT 7.3 PSJT 7.4 PSAR 4.06H 5.8CJT 7.4 ME
yS
(MC
-Dat
a)/D
ata
-0.1
0
0.1
0 1 2 3 4 5 6yS
(MC
-Dat
a)/D
ata
-0.1
0
0.1
0 1 2 3 4 5 6
Fig. 6. Rapidity,yS
47
ptin
Thr.co
r. fa
c.
0.60.8
11.21.4
10-4
10-3
10-2
10-1
1
10
ptin
Thr.1/
N d
N/d
p tin
DELPHI
charged particlesJT 7.3 PSJT 7.4 PSAR 4.06H 5.8CJT 7.4 ME
ptin [GeV/c]
(MC
-Dat
a)/D
ata
-0.2
-0.1
0
0.1
0.2
0 2 4 6 8 10 12 14pt
in [GeV/c]
(MC
-Dat
a)/D
ata
-0.2
-0.1
0
0.1
0.2
0 2 4 6 8 10 12 14
Fig. 7. Transverse momentum,pint , with respect to the Thrust axis
ptout
Thr.
cor.
fac.
0.60.8
11.21.4
10-2
10-1
1
10
ptout
Thr.
1/N
dN
/dp tou
t
DELPHI
charged particlesJT 7.3 PSJT 7.4 PSAR 4.06H 5.8CJT 7.4 ME
ptout[GeV/c]
(MC
-Dat
a)/D
ata
-0.3
-0.2
-0.1
0
0 0.5 1 1.5 2 2.5 3 3.5pt
out[GeV/c]
(MC
-Dat
a)/D
ata
-0.3
-0.2
-0.1
0
0 0.5 1 1.5 2 2.5 3 3.5
Fig. 8. Transverse momentum,poutt , with respect to the Thrust axis
< ptout> vs. xp
cor.
fac.
0.60.8
11.21.4
0.1
0.2
0.3
0.4
< ptout> vs. xp
<p tou
t > [G
eV/c
]
DELPHI
charged particles
JT 7.3 PS
JT 7.4 PS
AR 4.06
H 5.8C
JT 7.4 ME
xp
(MC
-Dat
a)/D
ata
-0.1
0
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1xp
(MC
-Dat
a)/D
ata
-0.1
0
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Fig. 9. < poutt > vs. xp
< pt> vs. xp
cor.
fac.
0.60.8
11.21.4
0.5
1
< pt> vs. xp
<p t>
[GeV
/c]
DELPHI
charged particles
JT 7.3 PS
JT 7.4 PS
AR 4.06
H 5.8C
JT 7.4 ME
xp
(MC
-Dat
a)/D
ata
-0.1
0
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1xp
(MC
-Dat
a)/D
ata
-0.1
0
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Fig. 10.< pt > vs. xp
48
F.2 Event shape distributions
1-Thrustco
r. fa
c.
0.60.8
11.21.4
10-3
10-2
10-1
1
10
1-Thrust1/
N d
N/d
(1-T
)
DELPHI
charged particlesJT 7.3 PSJT 7.4 PSAR 4.06H 5.8CJT 7.4 ME
Fig. 38. Asymmetry of the energy energy correlation, AEEC
55
F.3 Identified particle distributions
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 1 2 3 4 5
ξp
1/N
dn/
dξp
ALEPH
DELPHI
OPAL
JT 7.3 PS
JT 7.4 PS
AR 4.06
H 5.8 C
JT 7.4 ME
K0
(MC
-DA
TA
)/M
C JT 7.3 PS
-0.1
0
0.1
-0.1
0
0.1JT 7.4 PS
-0.1
0
0.1AR 4.06
-0.1
0
0.1H 5.8 C
-0.1
0
0.1
0 1 2 3 4 5
ξp
JT 7.4 ME
Fig. 39. ξp distribution of K0. Thecurves on the right refer to the differentexperiments given on the left plota b
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 1 2 3 4 5
ξp
1/N
dn/
dξp
ALEPH
DELPHI
OPAL
JT 7.3 PS
JT 7.4 PS
AR 4.06
H 5.8 C
JT 7.4 ME
K±
(MC
-DA
TA
)/M
C JT 7.3 PS
-0.1
0
0.1
-0.1
0
0.1JT 7.4 PS
-0.1
0
0.1AR 4.06
-0.1
0
0.1H 5.8 C
-0.1
0
0.1
0 1 2 3 4 5
ξp
JT 7.4 ME
Fig. 40. ξp distribution of K±. Thecurves on the right refer to the differentexperiments given on the left plota b
56
10-1
1
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8xp
1/N
dn/
dxp
K *±K*0
ALEPH
DELPHI
OPAL
JT 7.3 PS
JT 7.4 PS
AR 4.06
H 5.8 C
JT 7.4 ME
K*0 & K *±
Fig. 41.xE distribution ofK∗0 andK∗±
10-2
10-1
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8xE
1/N
dn/
dxE
ALEPH
DELPHI
OPAL
JT 7.3 PS
JT 7.4 PS
AR 4.06
H 5.8 C
JT 7.4 ME
Φ(1020)
Fig. 42.xE distribution ofφ
10-2
10-1
1
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8xp
1/N
dn/
dxp
ALEPH
DELPHI
L3
JT 7.3 PS
JT 7.4 PS
AR 4.06
H 5.8 C
JT 7.4 ME
ρ0(770)
ω(783)*0.1
Fig. 43.xp distribution ofρ andω
10-2
10-1
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8xp
1/N
dn/
dxp
f0 f2
DELPHI
JT 7.3 PS
JT 7.4 PS
AR 4.06
H 5.8 C
JT 7.4 ME
f2(1270)
f0(980) * 0.1
Fig. 44.xp distribution off0(980) andf2(1270)
57
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
ξp
1/N
dn/
dξp
ALEPH
DELPHI
OPAL
JT 7.3 PS
JT 7.4 PS
AR 4.06
H 5.8 C
JT 7.4 ME
p
(MC
-DA
TA
)/M
C JT 7.3 PS
-0.2-0.1
00.10.2
-0.2-0.1
00.10.2 JT 7.4 PS
-0.2-0.1
00.10.2 AR 4.06
-0.2-0.1
00.10.2 H 5.8 C
-0.2-0.1
00.10.2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
ξp
JT 7.4 ME
Fig. 45. ξp distribution of protons. Thecurves on the right refer to the differentexperiments given on the left plota b
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
ξp
1/N
dn/
dξp
ALEPH
DELPHI
OPAL
JT 7.3 PS
JT 7.4 PS
AR 4.06
H 5.8 C
JT 7.4 ME
Λ0
(MC
-DA
TA
)/M
C JT 7.3 PS
-0.2-0.1
00.10.2
-0.2-0.1
00.10.2 JT 7.4 PS
-0.2-0.1
00.10.2 AR 4.06
-0.2-0.1
00.10.2 H 5.8 C
-0.2-0.1
00.10.2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
ξp
JT 7.4 ME
Fig. 46. ξp distributionΛ0. The curveson the right refer to the different exper-iments given on the left plota b
58
Octet-Baryons
10-3
10-2
10-1
1
10
10-2
10-1
1xp
1/N
dn/
dxp
p
Λ
Ξ±
ALEPH
DELPHI
OPAL
JT 7.3 PS
JT 7.4 PS
AR 4.06
H 5.8 C
JT 7.4 ME
Decuplet-Baryons
10-2
10-1
1
10-1
1xp
1/N
dn/
dxp
∆++(1232)
Σ±(1385)
Ξ0(1530)
Fig. 47.xp distributions of octet and de-
cuplet baryonsa b
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