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2 Nuclear Physics A532 (1991) 489c-496c N U C L E A R. North-Holland, Amsterdam P H Y S IC S A INTERMITTEN CY AT INTERMEDIATE ENERGIES W. Kittel and R. Peschanski6* Experimental High-Energy Physics, University of Nijmegen, Toemooiveld 1, 6525ED Nijmegen, The Netherlands 6Theory Division, CERN, CH Geneve 23, Switzerland Abstract Apart from some misunderstandings of this summer, recent developments have led to considerable progress in this controversial but challenging field. Intermittency is now established as an effect in at least two dimensions, where it has turned out to be stronger and cleaner than in one. The effect is strongest in Ih and e+e~ collisions. In hh collisions, where the energy range covered is the largest, the effect is stronger at intermediate then at high y/a. Next to parton showering, intermittency is very sensitive to the soft phase, which will have to be studied with large statistics at intermediate energies, where it is not overshadowed by hard effects. In terms of the space-time development of a scattering process, the intermittency phenomenon and the determination of its parameters are shown to give a genuine information on the nature of hadronization at large distances compared to the hard interaction region. As such, it is expected to be sentitive to the nuclear environment of a deep-inelastic electron collision. In recent years events have been observed containing high particle density spikes in rapidity space. Fig.la shows the JACEE event [1 ] at a pseudo-rapidity resolution of 77= 0. 1 unit, with local fluctuations up to dn/srj us 300 and with a signal to background ratio of about 1 :1. The NA2 2 event [2] of Fig.lb contains a rapidity spike of dn/dy 100 at a rapidity resolution >y=0.1. This corresponds to 60 times the average density in this experiment. Also UA5 [3] has reported spikes of dn/dq up to 30 (10 times average) as early as JACEE, but found these to be in agreement with a short range cluster Monte Carlo. No doubt, local density fluctuations exist. The question is whether they are of statistical or dynamical origin, whether the underlying probability density is continuous or intermittent. To study these, Bialas and Peschanski [4] have suggested to study the dependence of scaled factorial multiplicity moments * On leave of absence from Saclay, Service de Physique Théorique, France /91/$ Elsevier Scicncc Publishers B.V. All rights reserved.

3 490c W. Kittcl, R, Peschanski I ntennittency at intermediate energies 300 t JACEE dn dy NA Pseudo-rapidity n J 3 Fig.la. The JACEE event [1 ] Fig.lb. The NA22 event [2] J r, \ 1 1 )... ( n m t + 1 ))... ~ M ( n m) 1 * ' as a function of the resolution Sy. Here, nm is the (charged) particle multiplicity in bin m (m = l... M) of size Sy Ay/M, with M the number of bins into which an original region Ay is divided. The averages under the sum are over the events in the sample. High order moments resolve the large nm tail of the multiplicity distribution and are, therefore, particularly sensitive to density fluctuations at the various scales Sy used in the analysis. The authors show that the (F,) are independent of Sy if the y distribution is smooth (probability density continuous), but follow the power law (JFi) a Sy~fl, fi > 0 (2) if the distribution is fractal (probability density is intermittent ). The powers fi (slopes in a log-log plot) are related to the anomalous dimensions d{=fi/(i 1 ) giving the deviation dj d d{ of the fractal dimension (df) from the integer one (d), due to fluctuations. The first step has, therefore, been the search for a more or less linear increase of ln (f i) as a function of hi Sy. Within a surprisingly short time, this search has been peiformed in e+e~ [6-9], ftp [10], va [1 1 ], hh [12,13], ha [14,15] and AA [14,16,17,18,19] collisions. Intermittency is indeed seen in all types of collision. Recent reviews are given in refs.2 0 and 2 1. Anomalous dimensions d,- fitted over 0. 1 < 7/(77) <1-0 are compared to each other in Fig.2 [20]. They typically lie between 0.01 and 0.1, so that the fractal dimensions are close to one. They arc larger and grow faster with increasing order i in fip and e^e- (Fig.2 a) than in hh (Fig.2b) collisions and arc small and almost independent of i in heavy ion collisions (Fig.2c). Within hh collisions, the growth is considerably faster for NA22 (at y/s=22 GeV) than for UAl (at 630 GeV). F\irthermore, the d{ are larger and intermittency is cleaner when studied in two dimensions.

4 W, Kittel, R. Peschanski / Intermittency at intermediate energies 491c a)»mp 010 i t NA22 UA1 n b ) 005,0 A * * i i Fig.2 Anomalous dimension </» as a function of the order t, for a) ftp and e+e~ collisions b) NA22 and UAl, c) KLM [20]. I What do presently used models say about intermittency? As shown in ref.21, presently used models for particle production in hadron-hadron collisions do not reproduce the intermittency observed in the data. In Fig.3a the EMC data [10] (4 < W < 20 GeV) are compared to what is expected from an extrapolation of conventional short range and long range correlations [22]. At low Sy, the data are consistently above these expectations. In Fig.3b, the slopes ƒ,- of the same data are seen to exceed considerably the expectations from the Webber and Lund models. Similarly, Fig.3c shows too low I n ^ ) from Lund for une and vd-i data [11] ({W ) =6.5 GeV). So, also presently used lepton-hadron models cannot reproduce the intermittency observed in these types of collision. In e+e" collisions, the situation is far less unanimous. While a first (indirect) analysis [6] of HRS data and an analysis of the TASSO data [7] has shown a similar deviation from model expectations as observed in Ih and hh data, recently CELLO [8] and in particular DELPHI [9] show reasonable agreement of their data with the parton shower version of the Lund Monte Carlo. If we forget about the discrepancy between CELLO and TASSO for the moment, it could be thinleable that the parton shower (which is a cascade process and therefore expected to give intermittency) is too short in the models at PE P/PE T R A energies and only fully developed at LEP. A comparison of the log-log plots on parton and hadron level in Figs.4a,b [23], however, shows that in the standard JETSET 6.3 version the increase of In(JFi) at large In Sy is not due to the parton shower, but to hadronization! Only if the parton shower is allowed to continue down to very low Q\ values (Fig.7c,d for Q'q OA GeV2, implying local hadron parton duality), intermittency is becoming visible also at the parton level. On the other hand, intermittency seems to be fully developed on the parton level already at 91 GeV in the Webber model, and even smeared out by hadronization there [24].

5 492c W. Kittel, R. Peschanski 11ntermittcncy at intermediate energies L ė a ) The sensitivity to the cut-off for the perfrurbative QCD decade and the role of both hard and soft phases has also been 2 1 discussed in terms of the Ariadne dipole radiation model [25]. Intermittency can be increased in the soft phase by an increase of the 7T/ p ratio also required from i - t - i l- L !/ÌY 40 direct measurements by NA22 [26] and EMC [27]. The direct pions resolve the underlying parton structure better than the more massive resonances. From a tunneling production mechanism, these pions are expected to have smaller pr than other particles, a property which has been neglected in the MC programs until now. In any case, it will be necessary to approach the problem on the exact origin of intermittency in shower MC s from two sides: * o s o i,,necctfoto A D* CC dolo 06 * order I 5 T 1. With the limitations mentioned above, parton showering has become a good candidate to explain intermittency in the hard phase and should be tried for other types of collision. Before that, 0.4 c ) however, its exact origin will have to be 02 come clear in the Monte Carlo versions and the model will have to be checked 0 on more sensitive distributions to be dis -à WA59 cussed below. Further studies are, therefore, needed at high energies where parton ln Äy showering is fully developed and expected to dominate over the soft phase., r i, 2. On the other hand, intermittency Fig.3 a) EMC results [10] compared to tums out to be parlkulail sensitive to ^pcctetions from ref.22 h) slopes f, for the ^ tieatment ot the ioft hase. EMC data as well as Webber and Lund models, c) va data [11] in comparison to Lund model expectations. This phase will have to be studied in de- tail with, high statistics at intermediate energies, where it is not dominated by par- ton showering in the hard phase. What is needed there is <t fully three dimensional analysis, with in particular the study of the transverse momentum dependence of the effect, and an analysis in terms of identical and non-identical particles to isolate the role of Bose-Einstein correlation. At the level of the physical interpretation of the intermittency phenomenon, it is convenient to introduce the space-time description of a scattering process, such as represented in Figs*5. In Fig.Sa, one considers the conventional picture of quark

6 W. Kittel, R. Peschanski / Intermittency at intermediate energies 493c A Ll " V a ) JETSET 6.3 coherent branching n/s«91 GeV b) T r T T C) T T O G«V* T T T d > parton level particle level parton level rtfwüffim portici«level 1 I * A O _ I r f o D 1 A i * 2 -I a a «_. A O a I ~2-1 o o. I -In 6y Fig.4 la(i<) as functions of Infy for JETSET 6.3 parton shower at ^ = 9 1 GeV [23a] at the a) parton, b) hadron level, both with cut-off Qfc 1 GeV2, c) d) with cut-off Qq=0.4 GeV2. hadronization; once created with the speed of light, a quark (or an antiquark) comes out of the interaction region by emitting hadrons. This region is typically of order 1 fermi in the proper-time of the system. As a consequence, this scale is expected to show up in the fluctuation pattern. By contrast, the intermittency phenomenon in principle requires the absence of typical scales of fluctuations, at least in some range from 1 fermi to 10 fermis or more. As a qualitative illustration of such a physical process Fig.5b shows a cascading mechanism which has been proposed [4] to conciliate the scale-invaxiance of intermittent fluctuation patterns with the correlation length of 1 fermi which is reasonably supposed to remain a feature of the fundamental interaction. Fluctuations are expected when the interacting system, submitted to the relativistic expansion, becomes larger than the correlation length. However, instead of emitting hadrons directly, the two (or more) sub-systems may in turn expand and breaks later into pieces, generating a new, superimposed, fluctuation and so on, so forth. Indeed, such an hypothesis has been shown [28] to be realized in the semi-classical approximation of qq pair creation by a Schwinger tunnelling mechanism. In this case, the tunnelling rate (within one-fermi distance) is not enough to transform the whole energy of the color field into qq pairs, leaving room to subsequent cascading steps of pair formation in a fractal type process. Note, however, that such a longer p 'ocess has not to be present in all the events, or in the whole of one event. It is a fluctuating mechanism, which can (and in fact should) keep the average unchanged, and well represented by Fig.5a. If this physical interpretation is correct, it can have far-reaching consequences on the study of hadronization mechanisms at long space-time distances. Indeed, the existance of intermittency patterns suggests that the density fluctuations of hadrons

7 494c W. Kittel, R. Peschanski / Intermittency at intermediate energies Random cascade in an expanding ultra-retatwislic system a) in-out picture of hadron production 8y# t/t Correlation length Resolution 6y probes (proper) time T- /6y Fig.5. Space-time interpretation of intermittent fluctuations. In Fig.5a-b> one shows the space-time frame in which a collision takes place, at least projected on the (t, z) plane, namely (time, longitudinal distance). The causal conus ( i± s > 0,i > 0) is displayed, together with causal hyperbolae, r = (t2 z2)1' 2 = csie, where r is the proper time. a) In-out picture of hadron production The Figure shows the conventional picture of hadron production after a high-energy collision. At the origin (O) an interacting string of partonic or hadromc matter (hatched region) is found. After a proper time t (of order 1 fermi) of relativistic expansion, the "string breaks into pieces measured (in average) by the conventional correlation length giving rise to hadrons. b) Random cascading in space-time In the (1+1) space-time frame, Fig.5b shows the (random) generation of intermittent fluctuations following the logical structure of the a-models [4], namely a random cascading model of hadronization. At each step of the cascade, one iterates the process shown in Fig.5a. A system (or string) is stretched by the relativistic expansion to a length larger than the correlation length, and breaks into A pieces with fluctuating density p. The fluctuations appear at successive proper-time values r3 leading to structures at different values of rapidity size Sy /t s. In the Figure, one has chosen I 2, t s = 24, and random factors W+ and W^. axe sensitive to a succession of different scales. Thus, the intermittency parameters, such as the anomalous dimensions, are expected to teach something on the deep nature of the hadronization mechanism. It has already been proposed [29] that the intermittent fluctuation and correlation patterns may distinguish the presence of a phase transition such as the quark-gluon plasma formation in heavy-ion collisions from a genuine cascading mechanism. In the same spirit, we want to suggest that intermittency features give an interesting and complementary information on the space-

8 W. Kittel, R. Peschanski / Intermittency at intermediate energies 495c time development of hadronization in a nuclear environment. The example of deep- inelastic scattering of electrons on nuclei at an energy of 20 GeV or more deserves a particular study; it is quite possible that the study of hadronic density fluctuations in the final state represents a valuable tool of physical investigation for quarks in nuclei. As a final comment, the intermediate range of energies compared with the one usually considered for intermittency studies which is involved in the present case calls for some remarks. It is true that high multiplicities are welcome for a better determination of the anomalous dimensions of the fluctuations. However, the factorial moment and correlator method has been proven to give interesting information at rather low values of the average multiplicity [30]. However, it seems required to develop better tools for handling statistical deviations, such as the empty-bin effect which appears when the maximal number of particles per bin is of the order of the factorial moment under study. Work is under way along this line, either by simulation [31] or by evaluation of extrapolation procedures for high moments [32]. It is our opinion that further progress will be made in the intermittency analysis at intermediate energies, giving its price to the fluctuation study of hadronization. We would like to thank warmly B. Frois and the organizers of the Dourdan Conference, for their support and interest in the subject. R eferences 1. T.H. Burnett et al. (JACEE), Phys. Rev. Lett. 5 (1983) M. Adamus et al. (NA22), Phys. Lett B185 (1987) P. Carlson (UA5), 4th Topical Workshop on pp Collider Physics, Bern, March 1983 and G.J. Alner et al. (UA5), Phys. Rep. 154 (1987) A. Bialas and R. Peschanski, Nucl. Phys. B273 (1986) 703 and B308 (1988) P. Lipa and B. Buschbeck, Phys. Lett. B223 (1989) 465; R. Hwa, Phys. Rev. D41 (1990) B. Buschbeck, P. Lipa and R. Peschanski, Phys. Lett. B215 (1988) 788; S. Abachi et al. (HRS), Study of Intermittency in e+e- Annihilations at 29 GeV, preprint ANK-HEP-CP W. Braunschweig et al. (TASSO), Phys. Lett. B231 (1989) H.-J. Behrend et al. (CELLO), Intermittency in Multihadronic e+e Intermit- tency in Multihadronic e^e Annihilations at 35 GeV, paper submitted to the V Int. Conf. on High Energy Physics, Singapore P. Abreu et al. (DELPHI), Phys. Lett. B247 (1990) 137; A. De Angelis, Study of Intermittency in Hadronic Z Decays, CERN-PPE/ (to be publ. in Mod. Phys. Lett. A). 10. I. Derado, G. Jancso, N. Schmitz and P. Stopa (EMC), Z. Phys. C47 (1990) L. Verluyten et al. (WA59 and E180), Study of Factorial Moments in Neutrino Neon and Deuteron Charged Current Interactions, paper submitted to the V Int. Conf. on High Energy Physics, Singapore I.V. Ajinenko et al. (NA22), Phys. Lett. B222 (1989) 306 and B235 (1990) 373; W. Kittel (NA22), Low pt Intermittency in ir+p and K +p Collisions at 250

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