Presented here is an overview of studies of the manifestations of the Bose-Einstein correlations in the hadronic decays of Z/sup 0/ boson, performed by the LEP experiments. The scope of the overview is confined to studies of two and three particle correlation functions for charged pions, with emphasis on analyses of transverse mass m/sub T/ and directional dependence of the two particle correlation function parameters. (17 refs)

Smirnova, O G

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BOSE-EINSTEIN CORRELATIONS AT THE Z0PEAKOXANA SMIRNOVAon behalf of the DELPHI CollaborationDepartment of Physics, Elementary Particle Physics,Box 118, 221 00 Lund, Sweden(on leave from JINR, Dubna, Russia)E-mail: oxana@quark.lu.sePresented here is an overview of studies of the manifestations of the Bose-Einsteincorrelations in the hadronic decays of Z0boson, performed by the LEP experi-ments. The scope of the overview is conned to studies of two- and three-particlecorrelation functions for charged pions, with emphasis on analyses of transversemass mTand directional dependence of the two-particle correlation function pa-rameters.1 IntroductionStudies of Bose-Einstein correlations between particles produced in hadronicdecays of a virtual or Z0boson : e+e ! ; Z0! hadrons, have beenperformed by almost all e+e annihilation experiments1at dierent center-of-mass energies. Hadronic decays of the Z0boson are of particular interestbecause they allow profound studies of the hadronization process and of variousrelated fragmentation models, providing a clean sample of basically uniformevents. Data collected at LEP are unique from the point of view of amount ofregistered and analyzed events. The fact that the measurements were done byfour dierent detectors gives an unmatched possibility to estimate systematicuncertainties.This review will give an insight into the specics of the Bose-Einsteincorrelations (BEC in what follows) studies at the Z0peak. The summary of themeasurements of the two-particle correlation function in terms of the invariantfour-momentum dierence will be presented, as well as the latest achievementsin the multidimensional analysis of this correlation function. Results of thestudies of the genuine three-particle BEC will be overviewed as well.2 BEC in Z0hadronic decaysBEC is a quantum mechanical phenomenon, which reects the fact that thewave functions of bosons are symmetric. It manifest itself, in particular, asan enhanced probability for identical bosons to be emitted from a commonFigure 1: Scheme of a string decay in the Lund Model.boson source with small relative momenta. The width of this enhancementis directly connected to the size of the source2. Therefore, measurements ofmomenta of the produced in the Z0hadronic decays bosons and consecutiveanalysis of correlations between them is the way of estimating the size of thehadronization region. It is a particularly appealing task, as long as there isno analytical description for the hadronization process, and only semi-classicalmodels are available.In case of the Z0decay, the most appropriate description of the hadronsource appears to be the Lund Model3, incorporated into the JETSET4eventgenerator, used by all the LEP experiments. In the framework of this model,bosons are produced in the process of a string fragmentation (see Fig. 1). Theremight be several strings in a given event of the Z0decay, but in the simplestcase of the so-called 2-jet events, when Z0decays into a quark-antiquark pair,without a hard gluon radiation, one can talk about a single string, which isstretched between quark and antiquark, being a source of the observed bosons.Due to the kinematics of the process, strings are very much elongated, andcan be regarded as a basically one-dimensional objects, with a typical lengthof about 1 fm. Here comes the most obvious dierence between the bosonproduction in case of heavy ion collision and in e+e annihilation : while inthe former case one can approximate the boson source with a sphere, in thelatter, the longitudinal size of the source has to be bigger than the transverseone5.To test these and other model predictions, LEP experiments conductedvarious studies of the multiparticle correlations. However, due to the specicsof the experimental installations, there are many limitations imposed. Oneof the most important obstacle to a full-scale analysis is the low eciency ofneutral particles reconstruction in e+e experiments. Therefore, studies of theBEC at LEP are mostly performed for charged particles. Another dicultyis the particle identication : due to its poor eciency and purity, especiallyin the low momentum region, all registered particles are usually considered tobe pions. It is quite a valid approximation, taking into account that chargedpions constitute about 80% of the all measured charged particles at the Z0peak.Studies reviewed in the following sections are all performed using uniden-tied charged particles, assumed to be pions, unless stated otherwise.3 Denitions and early analysesThe most straightforward approach to study BEC is to measure a two-particlecorrelation function. Statistical physics denes this correlation function asC2(p1; p2) =P (p1; p2)P0(p1; p2); (1)where P (p1; p2) is the two-particle probability density, and P0(p1; p2) is thesimilar probability density in absence of correlations, with p1and p2denotingfour-momenta of particles. Strictly speaking, in the denominator should bethe product of single-particle probability densities : P (p1)P (p2). However,due to the phase space limitations, e+e annihilation experiments are unableto measure this product, which is thus being replaced by the more convenientobservable, P0.In terms of the four-momenta dierence Q which is dened asQ =p(~p1  ~p2)2  (E1  E2)2; (2)(~piand Eiare momenta and energies of particles), the correlation functionC2(Q) is measured asC2(Q) =d=dQdref=dQ: (3)Here d=dQ refers to the Q-distribution of like charge bosons, and dref=dQis the reference Q-distribution, i.e., one which is not aected by BEC. There aregenerally two kinds of the reference samples used by LEP experiments : eitherone constructed from the unlike charge pions, or the \mixed sample", whichconsists of pairs of like charge bosons, each being picked up from a dierentevent. Usually, the measured correlation function (3) is being normalized to theanalogous quantity, obtained from the Monte Carlo generated events withoutBEC.The most commonly used parameterization forC2(Q) is the Gaussian form,which stems from the model for a source to be a sphere of emitters distributedaccording to the Gaussian probability density :C2(Q) = N (1 + e R2Q2) : (4)Unlike ref. Mixed ref.Experiment  R; fm  R; fmALEPH 0:48 0:03 0:81 0:04 0:30 0:01 0:51 0:02DELPHI 0:31 0:02 0:83 0:03 0:24 0:02 0:47 0:03L3 0:30 0:01 0:94 0:04 0:20 0:02 0:58 0:05OPAL 0:87 0:03 0:93 0:02Table 1: Collated results for C2(Q) parameters. Results are shown for both unlike chargereference sample and the mixed one. Errors include both statistical and systematic ones.Parameter R is interpreted as the correlation width and is associated with thesize of a boson source. Parameter  basically describes the strength of thecorrelation. N is the overall normalization. Sometimes a linear term is beingadded as another multiplier in (4), to account for a background.All four LEP experiments measured C2(Q) using dierent reference sam-ples in (3). Table 1 shows the summary of their results6;7;8;9on R and .There is an evident systematic dierence between results obtained using dif-ferent reference samples. An enormous divergence of parameters measuredwith the unlike charge reference sample suggests that this reference sample isinappropriate for the BEC studies. Usage of the mixed reference sample givesmore uniform results, especially for the radius parameter R.Summarizing results on parameters of C2(Q) obtained by the LEP exper-iments at Z0peak, one comes to the conclusion that the observed size of theboson source is likely to be about 0:5 fm, which is twice smaller then thesupposed string size.4 Multidimensional analysisNext step towards more profound studies of BEC is to conduct the correlationfunction analysis in a multidimensional space. A convenient coordinate systemwas suggested for this purpose by Csorg}o and Pratt10: the LongitudinalCentre-of-Mass System, also referred sometimes to as the Longitudinal Co-moving System (LCMS). For each pair of particles it is the system where thesum of the two particle momenta is transverse to the jet direction, which isbasically the primary parton direction (see Fig. 2).In LCMS, Q is resolved into three orthogonal components : Qt;out, Qt;sideand Qlong. The longitudinal component Qlongis aligned with the jet direc-tion, hence in the two-jet conguration it is aligned with the color ow in thestring. Considering the Lund Model, and also aiming at the comparison withhydrodynamical models used for the heavy ion collisions, another interestingFigure 2: LCMS.variable is the transverse mass of the pair of particles :mT=qm21+ p2t;1+qm22+ p2t;22; (5)where m1, m2are masses of the particles and pt;1, pt;2{ their transverse mo-menta with respect to LCMS. Particular interest in studying the dependence ofthe two-particle correlation function parameters on mTarises from the resultsreported by the heavy ion collision experiments, which show the decrease ofthe correlation radii with increasing mT. This decrease is well described byhydrodynamical models, which can not be valid in the e+e annihilation case.The three-dimensional two-particle correlation function (4) will in the as-sumption of absence of cross-talk between directional components turn intothe following form :C2(Qt;out; Qt;side; Qlong) = N (1+e Q2t;outR2t;out Q2t;sideR2t;side Q2longR2long) (6)Even further simplication can be achieved by reducing the tree-dimensionalpicture to the two-dimensional one, considering the single transverse compo-nent Q?=qQ2t;out+Q2t;side. Using the notation Qkfor the longitudinal com-ponent, Eq.(6) can be rewritten asC2(Q?; Qk) = N (1 + e Q2?R2? Q2kR2k) (7)The two-dimensional analysis is more convenient because it gives smallerstatistical errors as compared to the three-dimensional one, and also because 0:249 0:004Rt;out; fm 0:550 0:010Rt;side; fm 0:420 0:010Rlong; fm 0:850 0:020Table 2: Three-dimensional two-particle correlation function parameters as measured by theDELPHI experiment. Only statistical errors are shown.the theoretical predictions5were done in this particular system. However, thethree-dimensional analysis allows to separate the boosted component Rt;out,which is also the only one coupled to the temporal dierence of the two bosonsemission.mT , GeVλmT , GeVR t,out , fmmT , GeVR t,side , fmmT , GeVR long , fm00.10.20.30.40.50.60.70.5 1 1.500.20.40.60.811.20.5 1 1.500.20.40.60.811.20.5 1 1.500.20.40.60.811.20.5 1 1.5Figure 3: Transverse mass dependence of the correlation function parameters. Closed points:DELPHI data. Open points : JETSET with BEC simulation and with the detector simula-tion applied. Errors are only statistical.The three-dimensional BEC analysis in hadronic decays of Z0is beingconducted within the DELPHI collaboration11. The preliminary results areshown in Table 2 and Fig. 3. No acceptance correction were made, sincethey rely totally on the JETSET simulation of the Z0hadronic decays, whichincludes only a very limited model for BEC12. The t was performed in theregion of jQt;outj; jQt;sidej; jQlongj < 0:5GeV .These results conrm the theoretical predictions that the transverse size ofthe boson source ought to be almost twice smaller than the longitudinal one.They also show existence of the transverse mass dependence of the correlationfunction parameters. Increase of  with increasing mTcorresponds to therelative increase of the promptly produced pions at high mT. Fast decreaseof Rt;outcomes in a big part from the boost to the LCMS. The clear decreaseof Rt;sideand Rlongwith growing mT, however, can not be explained so far.Surprising is also the similar behaviour of Rlongmanifested in the JETSETgenerated events.0.20.30.40.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.650.40.50.60.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65mTλmTR^ , fmmTR || , fm0.50.60.70.80.90.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65Figure 4: Transverse mass dependence of the correlation function parameters, L3 data.The complementary two-dimensional analysis of the two-particle correla-tion function is being done by L313and DELPHI groups. Fig. 4 shows thetransverse mass dependence of the two-dimensional correlation function pa-rameters as measured by L3. Table 3 compares preliminary results obtainedby two experiments. DELPHI results are presented without acceptance cor-rections, while L3 results include both acceptance and generator corrections.DELPHI t was performed in the region of Q?< 0:6GeV and jQkj < 0:8GeV .L3 DELPHI 0:28 0:09 0:27 0:04R?; fm 0:49 0:05 0:42 0:08Rk; fm 0:72 0:07 0:74 0:04Table 3: Two-dimensional two-particle correlation function parameters as measured by theL3 and DELPHI experiments (both preliminary). Errors include both statistical and esti-mated systematic uncertainties.The two-dimensional analysis done by both experiments shows the samebasic tendencies as in the three-dimensional case : the transverse radius ap-pears to be signicantly smaller then the longitudinal one, and there is a clearindication of the mT-dependence of the correlation radii. Also it is worthnoticing that results of both experiments agree very well within the errors.5 Three-particle correlationsThe BEC phenomenon is not limited to the two-particle correlations, but ap-plies to any number of bosons emitted with close space-time characteristics.Therefore, given a suciently high multiplicity of hadroproduction events, onemust be able to observe three-particle Bose-Einstein correlations as wella.Analogously to C2of Eq.(1), the correlation function for three particles isdened asC3(p1; p2; p3) =P (p1; p2; p3)P (p1)P (p2)P (p3): (8)However, knowing that there are two-particle correlations, one has to beaware that C3includes both \genuine" three-particle correlations, R3, andthose arising from two-particle correlations within a triplet, R03. The mainchallenge of the analysis is therefore to extract the component R3.The genuine 3-particle correlation function R3= C3 R03is given in termsof probability densities by the formR3=P (p1; p2; p3) + 3P (p1)P (p2)P (p3) Pi6=j 6=kP (pi; pj)P (pk)P (p1)P (p2)P (p3)(9)There are dierent methods used by dierent experiments to calculateterms containing products of probability densities. The DELPHI experimentused the mixing technique14, similar to that used for the C2analysis. TheaIt is rather surprising that no genuine three-particle correlationswere observed by the heavyion collision experiments so farFigure 5: Genuine three-particle correlation function as measured by DELPHI. Comparisonto JETSET with and without BEC simulation is also shown.OPAL collaboration evaluated those terms using the JETSET generated eventsand also utilizing mixed-charged combinations from the data, corrected to theCoulomb eects15.After being extracted, R3is parameterized in terms of the summary four-momenta dierence for all particles in the triplet, Q3:Q3=qQ212+Q223+Q231: (10)The parameterization is analogous to the Gaussian form of (4), with an ex-tra factor of 2, arising from the presence of the two possible pion exchangediagrams :R3(Q3) = N (1 + 23e r23Q23) : (11)Both DELPHI and OPAL observe a clear presence of the genuine three-particle correlations signal (see Fig. 5 and Fig. 6). Their results are shown inTable 4. While the amplitude of the eect is strikingly dierent, the width ofthe correlation appears to be very much the same within the errors.DELPHI OPAL30:28 0:09 0:50 0:04r3; fm 0:66 0:07 0:58 0:03Table 4: Parameters of the genuine three-particle correlation function as measured by DEL-PHI (mixed reference sample) and OPAL (Monte Carlo reference sample). Errors includeboth statistical and systematic uncertainties.Figure 6: Genuine three-particle correlation function as measured by OPAL.SummaryInvestigations into BEC at the Z0peak produced wide variety of results, be-ing of great use for studies of the multiparticle production and for tests ofcorresponding models in particular.Width of the two-particle correlation function C2(Q) at Z0peak is foundto be R  0:5 fm using the mixed reference sample and R  0:9 fm using theunlike charge one. Measured values of parameters appear to depend stronglyon the analysis method.Multidimensional analysis of the two-particle correlation function showsthat the longitudinal size of the boson source is almost twice as big as thetransversal one. Also, all components of R exhibit decrease with increasingtransverse mass of a boson pair : a similar eect was previously observed inthe heavy ion data analysis.There is a clear presence of the genuine three-particle correlations in hadronicdecays of the Z0boson. The width of the correlation function was found tobe r3 0:6 fm. This kind of correlations has been observed so far only in ppand e+e collisions.Not reviewed here, some other studies of two-particle BEC at the Z0peakwere done in past. Correlations between K0Swere studied by ALEPH, DEL-PHI and OPAL16, giving   1 and R = 0:65 0:90 fm. L3 reported studiesof BEC for neutral pions9, giving R  0:5 fm. OPAL collaboration showedthat the measured correlation width R slightly increases with multiplicity17,similarly to the results found for the heavy ion collisions.References1. S. Haywood: \Where Are We Going With Bose-Einstein { a Mini Re-view", RAL Report RAL-94-074.2. B. Lorstad, Int. J. Mod. Phys. A4 12, 2861 (1989).3. B. Andersson: \The Lund Model", Cambridge University Press 1998,ISBN 0 521 42094 6.4. T. Sjostrand, Comp. Phys. Comm. 28, 229 (1983);T. Sjostrand, Pythia 5.6 and Jetset 7.3, CERN-TH.6488/92 (1992).5. B. Andersson and M. Ringner, Phys. Lett. B 421, 283 (1998).6. OPAL Coll., P .D. Acton et al., Phys. Lett. B 267, 143 (1991).7. ALEPH Coll., D. Decamp et al., Z. Phys. C 54, 75 (1992).8. DELPHI Coll., P. Abreu et al., Phys. Lett. B 286, 201 (1992).9. L3 Coll., M. Acciarri et al.: \Measurement of Bose-Einstein CorrelationsFor Both Charged and Neutral Pions From Z Decays at LEP", NijmegenHEN-410, L3 Note 2272 (1998). Paper submitted to XXIX ICHEP ,Vancouver, 1998.10. T. Csorg}o and S. Pratt, in \Proceedings of the Workshop on RelativisticHeavy Ion Physics", KFKI-1991-28/A, p75.11. B. Lorstad, O. Smirnova: \Transverse Mass Dependence of Bose-EinsteinCorrelation Radii in e+e Annihilation at LEP Energies": Proceedingsof the 7th International Workshop on Multiparticle Production 'Correla-tions and Fluctuations', June 30 to July 6, 1996, Nijmegen, The Nether-lands.12. O. Smirnova, B.Lorstad, R.Muresan: \Tests of the JETSET Bose-Einstein Correlation Model in the e+e Annihilation Process": Pro-ceedings of the 8th International Workshop on Multiparticle Production'Correlations and Fluctuations', June 14-21, 1998, Matrahaza, Hungary.13. J.A. van Dalen et al.: \A Two-dimensional Study of Bose-EinsteinCorrelations in Z decays at LEP", Nijmegen HEN-408, L3 Note 2270(1998), also in Proceedings of the 8th International Workshop on Mul-tiparticle Production 'Correlations and Fluctuations', June 14-21, 1998,Matrahaza, Hungary.14. DELPHI Coll., P. Abreu et al., Phys. Lett. B 355, 415 (1995).15. OPAL Coll., K. Ackersta et al., Eur. Phys. J. C 5, 239 (1998).16. OPAL Coll., R.Akers et al., Z. Phys. C 67, 389 (1995),DELPHI Coll., P. Abreu et al., Phys. Lett. B 323, 242 (1994),ALEPH Coll., D. Buskulic et al., Z. Phys. C 64, 361 (1994).17. OPAL Coll., G. Alexander et al., Z. Phys. C 72, 389 (1996).

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