Multiplicity, Jet, and Transverse Mass dependence of Bose-Einstein Correlations in e + e − - Annihilation
MMultiplicity, Jet, and Transverse Mass dependenceof Bose-Einstein Correlations in e + e − - Annihilation ∗ Wesley J. Metzger , IMAPP, Radboud University, 6525 AJ Nijmegen, The NetherlandsOctober 15, 2018
Abstract
Bose-Einstein correlations of pairs of identical charged pions producedin hadronic Z decays are analyzed for both two- and three-jet events. Aparametrization suggested by the τ -model is used to investigate the depen-dence of the Bose-Einstein correlation function on track multiplicity, numberof jets, and transverse momentum. After a brief review of relevant previous results, new preliminary results are pre-sented on the dependence of the Bose-Einstein correlation function on track andjet multiplicity and transverse momentum, using a parametrization which has beenfound [1] to describe well Bose-Einstein correlations (BEC) in hadronic Z decay,namely that of the τ -model [2, 3], ∗ This talk was also given at
XLIV International Symposium on Multiparticle Dynamics ,Bologna, 8–12 September 2014. a r X i v : . [ h e p - e x ] N ov igure 1: λ and r at √ s = M Z found in the lep experiments [4–10]. The Bose-Einstein correlation function, R , is usually parametrized as R = γ (cid:104) λ exp (cid:16) − ( rQ ) (cid:17)(cid:105) (1 + (cid:15)Q ) , (1) and is measured by R ( Q ) = ρ ( Q ) /ρ ( Q ) , where ρ ( Q ) is the density of identi-cal boson pairs with invariant four-momentum difference Q = (cid:112) − ( p − p ) and ρ ( Q ) is the similar density in an artificially constructed reference sample, whichshould differ from the data only in that it does not contain BEC. Dependence on the reference sample
Two methods were frequently used at lep to construct ρ : unlike-sign pion pairs from the same event, and like-sign pairsfrom different events. The latter method is generally referred to as mixed events.However, it must be pointed out that the observed values of the parameters r and λ depend to a great extent on which reference sample is used. This is clearly seenin Fig. 1 where the values of λ and r found for charged-pion pairs from hadronic Z decays by the lep experiments aleph [4, 5], delphi [6], l [7] and opal [8–10]are displayed. Solid points are corrected for pion purity; open points are not. Thiscorrection increases the value of λ but has little effect on the value of r . All of theresults with r > . fm were obtained using an unlike-sign reference sample, whilethose with smaller r were obtained with a mixed reference sample. The choice ofreference sample clearly has a large effect on the observed values of λ and r . Incomparing results we must therefore be sure that the reference samples used arecomparable. r on the mass of the particle as determined at √ s = M Z from 2-particle BEC for charged pions [4–10], charged kaons [11,12] and neutralkaons [11,13,14] and from Fermi-Dirac correlations for protons [13] and lambdas[15, 16]. The curves illustrate a 1 / √ m dependence. Dependence on the particle mass
It has been suggested, on several grounds[17], that r should depend on the particle mass as r ∝ / √ m . Values of r foundat lep for various types of particle are shown in Fig. 2. Comparing only resultsusing the same type of reference sample (in this case mixed), we see no evidencefor a / √ m dependence. Rather, the data suggest one value of r for mesons anda smaller value for baryons. The value for baryons, about 0.1 fm, seems very small,since the size of a proton is an order of magnitude greater. If true it is telling ussomething unexpected about the mechanism of baryon production. Dependence on the transverse mass
However, r has been observed to de-pend on the transverse mass of the particle pair, [18, 19] as is shown in Fig. 3. Dependence on particle and jet multiplicity
The opal collaboration hasstudied the dependence of r and λ on the charged track multiplicity and on thenumber of jets [9]. They used an opposite-sign reference sample, which necessitatedthe exclusion of regions of Q where R was too strongly affected by resonances inthe reference sample. To describe the long-range correlations they introduced aquadratic term resulting in R ( Q ) = γ (cid:2) λ exp (cid:0) − ( rQ ) (cid:1)(cid:3) (cid:0) (cid:15)Q + δQ (cid:1) . (2)3ongitudinal side outFigure 3: The transverse mass dependence of the components of r in the LCMSfrom Refs. 18, 19. They observed a linear rise of r with charged track multiplicity as well as anincrease of r with the number of jets. The behavior of λ was the opposite. However,when only two-jet (or only three-jet) events were selected, r was approximatelyindependent of multiplicity. τ -model However, the “classic” parametrization of Eq. (1) is found to be inadequate, evenwhen it is generalized to allow for a L´evy distribution of the source: R = γ [1 + λ exp ( − ( rQ ) α )] (1 + (cid:15)Q ) , < α ≤ This was not realized for a long time because the correlation function was onlyplotted up to Q = 2 GeV or less. In Ref. 1 Q was plotted to 4 GeV , and it becameapparent that there is a region of anti-correlation ( R < ) extending from about Q = 0 . to . . This anti-correlation, as well as the BEC correlation are welldescribed by the τ -model.In the τ -model R is found to depend not only on Q , but also on quantities a and a . For two-jet events a = 1 /m t , where m t = (cid:112) m + p is the transverse massof a particle). Parameters of the model are the parameters of the L´evy distributionwhich describes the proper time of particle emission: α , the index of stability of theL´evy distribution; a width parameter ∆ τ ; and the proper time τ at which particleproduction begins.We shall use a simplified parametrization [1] where τ is assumed to be zero and a and a are combined with ∆ τ to form an effective radius R : R ( Q ) = γ (cid:104) λ cos (cid:16) ( R a Q ) α (cid:17) exp (cid:16) − ( RQ ) α (cid:17)(cid:105) (1 + (cid:15)Q ) , (4a) R α a = tan (cid:16) απ (cid:17) R α . (4b) Note that the difference between the parametrizations of Eqs. (3) and 4 is thepresence of the cos term, which accounts for the description of the anti-correlation.The parameter R describes the BEC peak, and R a describes the anti-correlation R for two-jet events. Thecurve corresponds to the fit of Eq. (4). Also plotted is ∆, the difference betweenthe fit and the data. The dashed line represents the long-range part of the fit, i.e. , γ (1 + (cid:15)Q ). The figure is taken from Ref. 1.5 egion. While one might have had the insight to add, ad hoc , a cos term to Eq. (3),it is the τ -model which predicts a relationship, Eq. (4b), between R and R a .A fit of Eq. (4) to l two-jet events is shown in Fig. 4, from which it is seenthat the τ -model describes both the BEC peak and the anti-correlation region quitewell. Also the three-jet data is well described [1], which is perhaps surprizing sincethe τ -model is inspired by a picture of fragmentation of a single string.It must also be pointed out that the τ -model has its shortcomings: The τ -modelpredicts that R depends on the two-particle momentum difference only through Q , not through components of Q . However, this is found not to be the case [1].Nevertheless, regardless of the validity of the τ -model, Eq. (4) provides a gooddescription of the data. Accordingly, we shall use it in the following.Since the results on the dependence of the BEC parameters on particle and jetmultiplicities and on transverse mass mentioned in Sect. 1.1 were obtained using theclassic Gaussian parametrization, Eq. (1), and since this parametrization has beenshown to be inadequate, in the rest of this paper we investigate these propertiesusing the τ -model parametrization, Eq. (4). The results are preliminary. Data
The data were collected by the l detector at an e + e − center-of-mass energy of √ s (cid:39) . . Approximately 36 million like-sign pairs of well-measured chargedtracks from about 0.8 million hadronic Z decays are used. This data sample isidentical to that of Ref. 1.The same event mixing technique is used to construct ρ as in Ref. 1.Using the JADE algorithm, events can be classified according to the numberof jets. The number of jets in a particular event depends on the jet resolutionparameter of the algorithm, y cut . We define y J23 as that value of y cut at which thenumber of jets in the event changes from two to three. Small y corresponds tonarrow two-jet events, large y to events with three or more well-separated jets. Preliminary
Results
The parameters of the Bose-Einstein correlation function have been found to dependon charged multiplicity, the number of jets, and the transverse mass. However thesequantities are related. Both the charged particle multiplicity and the transverse massincrease rapidly with the number of jets. This is seen in Fig. 5, where the averagetransverse mass and the average charged multiplicity are plotted vs. y J23 . In thefollowing we investigate the dependence of R and λ on these three quantities.An unfortunate property of the τ -model parameterization, Eq. (4), is that theestimates of α and R from fits tend to be highly correlated. Therefore, to stabilizethe fits, α is fixed to the value 0.44, which corresponds to the value obtained in afit to all events.While we show only the results using the JADE jet algorithm, we have alsoperformed the same analysis using the Durham algorithm. It is found to lead to thesame conclusions. DEntries 6007 9904592mkhists_j0_data94.hst < m t > ( G e V ) IDEntries 6005 804574 log y J23 < N d e t c h > Figure 5: The average transverse mass and the average charged multiplicity asfunction of y J23 . The short vertical line at log y J23 = − .
638 corresponds to thecut used to define 2- and 3-jet events. 7igure 6: R obtained in fits of Eq. (4) as function of detected charged multiplicityfor two-jet events ( y J23 < . y J23 > . R and λ on track and jet multiplicities The dependence of R and λ on the detected charge multiplicity, is shown in Figs. 6and 7, respectively, for two- and three-jet events as well as for all events.For all events R is seen to increase linearly with the multiplicity, as was observedfor R by opal . However, the same linear increase is also seen for two- and three-jetevents, with R for three-jet events and for all events being approximately equal and R for two-jet events shifted lower by about a 0.5 fm. This contrasts with the opal observation of little dependence of r on multiplicity for two- and three-jet events.For all events, as well as for two- and three-jet events, λ decreases with multiplic-ity, the rate of decrease becoming less for high multiplicity. It is higher for three-jetevents than for two-jet events, with the values for all events lying in between. Thiscontrasts with the opal observation that λ was higher for two-jet events, as wellas the opal observation that the decrease of λ with multiplicity is linear. R and λ on trasverse mass and jetmultiplicity The dependence of R and λ on track multiplicity is shown in Figs. 8 and 9, respec-tively, for various selections on the transverse momentum, p t , (or, equvilantly, m t )of the tracks. For two-jet events both R and λ are slightly higher when both tracks The charge multiplicity is approximately given by N ch ≈ . N detch . λ obtained in fits of Eq. (4) as function of detected charged multiplicityfor two-jet events ( y J23 < . y J23 > . have p t < . than when only one track is required to have so small a p t . Forthree-jet events the same may be true, but the statistical significance is less; thedifference decreases with multiplicity. When neither track has p t < . , thevalues of both R and λ are much lower for both two- and three-jet events. In allcases both R and λ increase with multiplicity, and the values for two-jet events areroughly equal to those for three-jet events. The dependence of R and λ for the τ -model parametrization is different from thatof r and λ found by opal for the usual Gaussian parametrization. However, it isunclear how much the differences depend on the use of different reference samplesand how much on the parametrization used.Multiplicity, number of jets, and transverse mass all affect the values of R and λ in the τ -model parametrization, Eq. (4). y J23 < .
023 JADE 3-jet, y J23 > . R obtained in fits of Eq. (4) as function of detected charged multiplicity(left) for two-jet events ( y J23 < .
23) and (right) for three-jet events ( y J23 > . p t : (cid:52) both tracks having p t < . (cid:5) atleast one track having p t < . (cid:79) one track with p t < . p t > . ◦ all tracks; (cid:3) both tracks having p t > . p t = 0 . m t = 0 .
52 GeV.JADE 2-jet, y J23 < .
023 JADE 3-jet, y J23 > . λ obtained in fits of Eq. (4) as function of detected charged multiplicity(left) for two-jet events ( y J23 < .
23) and (right) for three-jet events ( y J23 > . p t : (cid:52) both tracks having p t < . (cid:5) atleast one track having p t < . (cid:79) one track with p t < . p t > . ◦ all tracks; (cid:3) both tracks having p t > . p t = 0 . m t = 0 .
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