Cluster models against new and old experimental data on multiparticle production

Nuclear Physics B107 (1976) 2 6 2 - 2 8 4 © North-Holland Publishing Company

CLUSTER MODELS AGAINST NEW AND OLD EXPERIMENTAL DATA ON MULTIPARTICLE PRODUCTION A. ARNEODO and G. PLAUT

Physique ThEortque, Universtt~de Ntce* Received 12 January 1976 (Revised 3 March 1976)

We analyze m the cluster model framework not only recent data on zones, rapidity gap d:str:but:ons and charge transfer correlations m multiparticle production, but also data on "older" quantities, m particular on inclusive and semi-inclusivelongitudinal correlations We show that a global analysis of experimental results strongly constrains the duster characteristics. High mtracluster multlplicaties are defimtely excluded whale about 50 to 60% of clusters have to carry an electric charge; these conclusions show that wellknown mesomc resonance production may indeed account for a large part of clustering.

1. Introduction Various types of correlation measurements in high-energy reclusive experiments have been at the origin of numerous theoretical works on multiple productxon dynarmcs [1 ]. The possibility that particles are produced in clusters was considered early, in order to account for the short-range behaviour of longitudinal inclusive correlations; nevertheless it has never been possible to specify the precise characteristics of clusters [ 2 - 5 ] , nor even to prove their exxstence [6]. However, the cluster models allowed a very race understanding of semi-inclusive longitudinal correlations. For large enough multlphclties, these correlahons are free from contarrunatlon coming from diffractive production and allow one to extract m a rather clean way a short-range component corresponding to both parhcles coming from the same cluster [ 7 - 9 ] . As it was recently observed at the ISR [ 1 0 - 1 2 ] and at FNAL [13], it is noteworthy that the shape of th:s short-range part is almost independent of b o t h energy and multiphclty, which allows a good determination of the cluster width and gives some indications on the cluster decay multlphcity distribution. Recently, Krzywxcki and Weingarten [14,15] put forward the local charge corn-

* Equlpe de recherche assoclde au CNRS. Postal address Physique Th~or:que, IMSP, Parc Valrose 06034 Nice Cedex, France. 262

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pensatlon (LCC) hypothesis an order to gave account - v t a umtarlty - of the energy dependence of charge exchange cross sectaons. They Introduced the charge transfer correlation function D ( Y l , Y2) = ( Z ( Y l ) Z ( Y 2 ) ) - (Z(Y 1))(Z(Y2)),

(1)

where Z ( y ) is the charge transfer n

Z(y) = AQ(y) = ~

1=1

Qt O(y - Yt) - QbeamO(Y - Ybeam) -- QtargetO(Y - Ytarget) "

(2) Here Qt and Yt are the charge and the rapidity of the lth partacle and O(y) is the usual step function given by 0 for y < 0 and 1 elsewhere. They also introduced the notxons of gap and zone. According to the LCC hypothesas, D ( y 1, Y2) should be a rapidly decreasing function of ly 1 - Y2[, the zone characterastacs should not depend upon energy, while the average number of zones should be increasing with energy hke Ins. Thas new formalism has been at the origin of several experamental analyses of pp collisions, the purpose of whach was to test the LCC hypothesis the French-Sovaet 69 GeV/c collaboration [16] measured the zone characteristics and their statistical distrabution; a samilar work was done at FNAL by Bromberg et al. [17] who also computed D ( y t, Y2); the gap dlstr~butions of the Michigan-Rochester collaboration were also gaven at FNAL [18]. All these recent results have been also analyzed an a few theoretical papers, with the aam of extracting new mformat~on on the cluster concept and the cluster properties, let us summarize the most outstanding conclusions of these papers 0) Plrilg, Thomas and Quagg [18] (hereafter referred to as PTQ) concentrated on the various gap distrlbutaons between final charged particles, their interest was motivated by the fact that the large gap behavlour of these dlstrlbutaons characterizes the exchanges between the clusters. They assumed that clusters are produced independently and they neglected the problems of kinematical constraants and of leading particles (or clusters); in order to rrummlze the so introduced bias, the end gaps were dascarded from their analysis. They did not attempt to fit the multlplacity dastributaon, enforcing only the average multaplicaty Under these statements, they found that the rapad~ty density of clusters Ocl--~ 1, so that clusters contain on average ~c -~ 2 charged particles. Moreover they argued that the non-observation of large gaps carrying charge transfer I/XQI~> 2 suggests a production mechamsm with limited charge exchange (hereafter referred to as LCEX) between clusters of charge 0, +1, whale the observation of large IAQI = 1 gaps excludes models with only neutral clusters. (ii) Thas work was craticized by Ludlam and Slansky [19] who, on the contrary, pointed out the ~mportance of kinematical constraants, and the necessaty of fitting the charge multapllcaty d~stnbution in any analysis of the charge structure of the final states of pp collisions. But Ludlam and Slansky did not make a specafic hy-

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pothesis on the cluster charges, which rejects the idea o f short-range order In cluster production (this idea is a basic assumption o f models with limited charge exchange like PTQ's model). Moreover their clusters are generated in such a way that their average characteristics will show a priori no dependence on the total charged multiphclty, whereas one has indications [ 7 - 1 2 ] (both in ICEM and from experiments) that e.g. (nc(n c - l))n/(nc) n increases with n * . Ludlam and Slansky no more precise their assumptions about leading particles (possible neutron production, x spectrum, etc ). According to their conclusions, the gap distributions observed at F N A L do not distinguish among models with widely varying cluster sizes: clusters may as well contain an average of 2 or 4 charged particles. In order to go further they defined a serm-lnclusive fluctuation parameter an in terms o f left-right fluctuation density [20], and the comparison o f their theoretical values with experimental data o n K n led them to prefer large cluster sizes ~c = 3 - 4 , In disagreement with the conclusions of PTQ [181. 011) Blal'as et al. [21 ], and more recently Baler and Bopp [22], also came to concluslons which differ from those of PTQ. Starting from the data on inclusive oneparticle density, they were able to compute the quantity D2(y) = D ( y , y' = y ) in a neutral cluster model. They showed that one can get a result in good agreement with experimental data provided one takes into account the leading charge flow, which led them to conclude in favour o f neutral cluster models. According to Baier and Bopp, the leading charge flow in a neutral cluster model could also account for the abundance of large IAQI = 1 gaps. So, the theoretical situation does not look very clear since the conclusions of the above analyses differ from one paper to another. However, one should notice that all o f these works rely on rather small subsets of experimental results, a comprehensive study o f data is still lacking. In particular, none o f these works explicitly checks the (a bit older) longitudinal inclusive and serm-inclusive rapidity correlations. In this work, we will show the Importance of these quantities when setthng the cluster properties and the necessity - if one intends to better understand the multiparticle production mechanisms - to ensure that a model introduced to fit new data, also reproduces older but well established experimental results. In our opinion, the sahent question is whether, in the framework of cluster models, it Is possible to give a coherent description of the whole of experimental data on multiparticle production **: e.g. various kinds of correlations (inclusive and semi-inclusive longitudinal correlations, charge transfer correlations D ( y 1, Y2) for any Yl, Y2), various gap distributions (between charged particles, negative ones; with given charge transfer), zone characteristics, etc. If such a description proves to be

* However we found, after this work was achieved, that this dependence was much weaker than commonly beheved [26 ]; but this has to be attributed to the intracluster multiphcity distribution P(nc, ~c), whereas the way in which Ludlam and Slansky generate their events remains in contradiction with the sptrit of ICEM. ** Here we mean a study of all the experimental data m the rapidity space once integrated over transverse momenta

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impossible, it will be the time to throw away once for all the cluster models; otherwise, it will be attractive to constrain to a maximum the cluster properties by taking care of all kinds of data. In tins work, we try such a comprehensive study In the framework of a model as realistic as possible, this model relies on LCEX and LCC hypotheses and allows a global study of available experimental data on pp collisions. We explicitly take into account energy-momentum conservation at the cluster production stage, leading particle effects, and charged multiphclty distributions. The introduction of such constraints, in particular those concerning the leading particles, cannot proceed without some arbitrariness; so it is useful to state precisely our point of view: we do not intend to describe correctly e.g. the leading particle production mechamsm, however since the existence of leading particles strongly influences (vta e.g. kinematical constraints and leading charge flow) multiparticle production in the central region, we think it IS essential to take leading particles Into account - even through an hypothesis wtuch is necessarily a bit arbitrary. We may reasonably hope that the details of the so introduced hypotheses will not influence too much our final results in the central region, anyway the philosophy of this work consists in taking constraants into account, even if this requires some amount of arbitrariness. We come to the conclusion that it is possible to explain all the experimental results an a LCEX cluster model, with clusters of charges 0, -+1, decaying on average into nc = 2 charged particles. Moreover the data require the production of a nonnegligible percentage of charged clusters (we shall estimate boundaries on this percentage). Indeed though it is possible to fit specific experimental results with a model including leading particles plus neutral clusters, it IS quite impossible with such models to reproduce all the experimental data (even if the flow of leading charges is taken into account). Both these conclusions - namely ~c -~ 2 and the percentage of around 5 0 - 6 0 % of charged clusters - suggest that clusters have properties very similar to fhose of known mesonic resonances and may indeed be such resonances. Although our conclusmns agree with those of PTQ, we shall show that at FNAL energies the gap distributions alone do not exclude heavier clusters with nc "~ 3 or 4, and that the arguments of PTQ are asymptotlc arguments [23]; among present experimental results, only semi-inclusive longitudinal correlations are clearly inconsistent with so high multiplicities within clusters. Our paper is organized as follows. In sect. 2, we give a detailed description of our model. In sect. 3 we describe the reformation one gets from gap distrIbutmns. In sect. 4 we turn to the zone characteristics and the charge transfer distributions. In sect. 5 we complete the comparison with experiment by &scusslng the inclusive and semi-inclusive longitudinal correlations, and we give our conclusions on cluster nature and production mechanism. We end with a few remarks and suggestions m sect. 6.

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2. Description of our model In this section we will make exphclt our motivations and give a detailed description of the model and techniques we use to describe pp collisions. We started from the basic idea of PTQ of a limited charge exchange between the clusters [18]. Indeed the non-occurrence of large gaps with a charge transfer IAQt ~> 2, the fact that the zones are narrow and the rapid decrease o f D ( Y l , Y2) with lYl - Y2[ induce one to choose such a model (a LCEX model) between clusters carrying charges 0, -+1. Let us point out that there exists a possible alternative, namely a model with charges randomly distributed among the clusters (not necessarily 1 3, 1 1 for charges - 1 , 0, +1, respectively); In the followlngwe with probabilities 3, wall call such models RCC models. However we will see that wxth RCC models the cluster characteristics have to depend on energy, which runs against the idea of an energy-independent production mechanism in the central region. So we choose to stay in the framework of LCEX models, with lsotroplcally decaying clusters leading to the decay rapidity distribution nc

N(Ycl; Y l, Y2 .... Yn c) "" t=[-I1 S(Ycl - Yi)

(3)

for a cluster of rapidity Ycl decaying into n c charged particles of raplditles Yr" For S(Ycl - y ) we take the usual Gausslan shape S(Ycl - y ) = (2rr82) -1/2 exp [ - ( Y c l - Y)2/(282)1 ,

(4)

with 8 "~ 0.7 in agreement with recent semi-inclusive correlation measurements [10-13]. The results are not very senmive to the details of the lntracluster charged particle multiplicity dlstnbution P(nc, nc), and depend especially on the average number nc of charged particles within clusters and on the width of the distribution p(n c, nc)" To specify, we wlU use Polsson-type chstrlbutions hke PTQ [ 1 8 ] , and narrower than Poisson distributions as suggested by semi-inclusive longitudinal correlations [10]; the details of these distributions are given in appendix A. In order to compute the consequences of our model, we generate sets of events with the help of a Monte Carlo technique. We start from very simple (although rough) laws such as. flat x spectrum for the leading particles, Polsson law for the number of clusters at a given massing mass, with umform density in rapidity. We correct these by reqmnng exact energy-momentum conservation between clusters and leading particles, and enforcing the experimental charged particle multiplicity distrlbutlon. To be more precise, at a given energy x/s, we first randomly choose the leading particle Feynman variables x 1 and x 2 with a flat spectrum between x = 0.2 and the kmematlcal limit. We then make the assumption that clusters are uniformly dlstrlb-

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uted on a plateau, the width of which is defined from the square of the missing mass M 2 = s(1 - Xl) (1 - x2) by the relation yeff(Xl, x2) = In M 2 - ln(OclMcl±) 2 ,

(5)

where Pcl and MclI are the rapidity density and the transverse mass of clusters, respectively. Ttus plateau is centred at the rapidity

_i

/1 - x l ~

~ - g ln(1_ - ~ 2 ) .

(6)

The probabihty to produce N clusters is given by a Poisson distribution of mean Pcl Yeff, PYeff (N) -

(PclYeff) N N! exp(-Pcl Yeff) -

(7)

Let us point out that ttus distribution is only a starting hypothesis; in particular we will reject zero-cluster events (those we regard as elastic events) and we will deal In a special way with one-cluster events (see appendix B); moreover the constraint to reproduce the experimental charged particle multiplicity distribution will certainly somewhat alterate the distribution (7). For N cluster events (N > 1) we randomly choose N rapldlties and we translate and dilate them in order that the two leading particles and the N clusters with rapidlties Yz(i = 1 ..... N) exhibit exact energy-momentum conservation. The transverse mass of clusters is taken to be Meli = gnc/~ 3- 1 where ~± = 0.45 GeV/c is the average transverse mass of plons. This simple formula approximates within a few percent the exact formula of Hayot et al. [24] if clusters have not too large transverse momenta (indications, coming from azimuthal correlations, that (/(-2)..~ 0.6 (GeV/c) 2 are given in ref. [25]). The cluster density Pcl is inversely proportional to nc: e.g. Pcl -~ 0.9 for h c = 2 and Pcl~- 0.45 f o r f c = 4. Once the N clusters are distributed and arranged in ascending order in rapidity, we assign charges to them, limiting the charges exchanged between them to 0, +1. For this purpose we use a method which ensures that on average one has Fin ~- 0.6 neutron and hp ~- 1.4 proton per event (we will see later the importance of talong into account neutron production [26]). Although we use conditional vertex probabilities like PTQ, we proceed in a way which is somewhat different from theirs: we first randomly choose the charges carried by the end gaps which relate the leading particles to the cluster chain, In order to get the required values for nn and f p . Then we are left with the task of randomly choosing the charges of the ( N - l) central gaps between clusters; to ensure the symmetry of pp collisions we build this random chain by using, alternatively from left to right and from right to left, the vertex conditlonal probability V(q i, qt+l)" V(qz, qi÷l) is the probabIhty that a link with charge qt turns to a link of charge qi+l by emitting a cluster of charge qi - q~+l" This vertex probability depends on two parameters a 0 and a I (0 ~< a 0, a I ~< 1) defined as the probabtlitles that a link of charge 0 (resp. +1) ermts a neutral cluster and proceeds as a link of charge 0 (resp. -+1):

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I1 V(q,, qt+l) =

al (1 - a0) 0

1 - a1 a0 1 -a 1

0 1 ~(1 - a0) ,

(8)

a1

where the rows and columns are labeled as charge 1,0, - 1 *. For given values o f a 0 and a 1, it is possible to estimate from (8) the asymptotic average percentage of charged produced clusters: Rca ,

(number of charged produced clusters) -, (number of produced clusters) 2

1

But this asymptotic estimate does not agree with "experiment" at FNAL energies; hence we will characterize our model by the values of the parameters a 0 and a 1, and by the average "measured" R c at a gwen energy. There remains to describe the decay of each cluster by choosing randomly its charged decay multiphcity n c according to the probability distribution p(n c, nc) (see appendix A), and by randomly assigning n c rapidiUes to the n c plons according to the law (3) ** Finally we constrain the charged multsplicity distribution to reproduce the experimental charged multiplicity distribution. This constraint is of special importance for one-cluster events, the number of which turns out to be Increased or decreased with respect to the original law according as ~c is larger or smaller. The detailed handhng of these events is given in appendix B. Their rapidity configuration looks somewhat like that of diffractive events, and for nc -~ 2 the modifications, through the constraint of reproducing the experimental charged mulhpliclty distribution, lead finally to a rather realistic description of diffraction, in particular we get a reasonable peak near x = 1 in the proton x-spectrum whereas this peak gets clearly too sharp with heavier clusters. Let us point out that we do not claim to get m such a way a realistic description of diffraction (m particular we neglect possible leading clusters); however it is natural to refuse too irrealistic leading particle spectra. Anyhow we will try to rule out the bias coming from diffraction and low missing masses * Let us remark that once the end gaps between the leading particles and the cluster chain are fixed, both ends of the cluster chain are constrained The constraints at the beginning of the chain are automatlcaUy taken into account by using the conditional vertex probability. On the other hand, the characteristics of the end of the chain are determined by the constraints coming from the choice of the end leading particle and not by the conditional vertex probabthty. ** When assxgmng rapidlties to the pions issued from the cluster according to the Gaussian law (4), we only prescribe average energy-momentum conservation at the cluster decay stage. Let us recall that neutral plon production is taken into account through the 3 factor in the formula giving Mcl±.

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by using, whenever it is possible, experimental senu-inclusive data #. By the way, it would be very interesting to have more systematic semi-inclusive analyses at one's disposal. In order to avoid misunderstandings, let us finally specify the philosophy of our work: it essentially consx~ts m searching for a correct description of multlproduction in the central region. Such a description has to take into account various types of constraints; m particular the leading particles are not considered with the object of getting a correct descriptmn of diffraction, or of their x-spectrum, or of neutron production for themselves, but only in order to get a correct description o f their influence upon phenomena in the central region (through leading charge flow or kinematical constraints for instance). Since we assume such a point o f view, we will exclude from our study end gaps and end zones; as regards correlations, we will stay in the central region (see appendix B). 3. Rapidity gap distributions

3.1. Gap distributions between negative or charged particles In the case of strictly independent cluster production, the probability of occurrence o f large rapidity gaps r between final pions (i.e. gaps with r ~ > 5 , where 5 is the cluster width in rapidity) has the same r-dependence as the gap distribution bet tween clusters:

G(r) ~_ exp(-Pclr),

(10)

where Pcl is the cluster rapidity density. Starting from this fact PTQ [18] concluded from experimental F N A L gap distributions between charged pions m pp collisions that the cluster density Pd ~ 1. This conclusion seems to be borne out by the fact that the exponential law (10) also fits the gap distributions between negative particles down to r = 0; this suggests that two negative particles are almost always issued from two different clusters and that clusters do not give rise to more than two charged particles on the average (nc ~- 2 corresponds to a rapidity density Pd "~ 1). Our computations confirm the large gap behaviour (10), but only at ISR or higher energies for reasonable values o f ~ c *. On the other hand, at FNAL energies, the dependence on Pcl of the large gap distributions (between produced pions) is completely reversed (see fig. 1) **. This important finite energy effect essentially originates from the fact that events with more than one cluster become fewer *** if ~c increases, I.e. the cluster density Pd decreases; hence a more pronounced decrease # A systematic study, m the framework of our model, of available serm-inclusive experimental data will be given in ref. [26] * For still larger values of~ c (e.g ~c :> 6) the large gap behavior (10) is obtained only at still tugher energies. ** Although it was not pointed out exphcltly, this finite energy effect is also present m the resuits of Ludlam and Slansky [19 ]. *** The law (10) for large gaps originates from such events, do not forget we discard end gaps like PTQ [18].

A Arneodo, G. Plaut / Multzparticle productzon

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of the distribution G(r) at large r when Pcl is smaller, contrary to (10). In fact, we can fit F N A L gap distributions between charged (and between negative) plons for a rather wide range of values o f h c ~< 4; the fit does not depend on the shape of the intracluster charged multiplicity distribution p(n c, nc)" These results on the gap distributions bear out Ludlam and Slansky's conclusions and show the danger of using at FNAL (or lower) energies arguments which are only valid asymptotically.

3.2. Fixed charge transfer gap distributions We now turn to the distributions of gaps carrying a given charge transfer AQ = 0, -+i, -+2, where AQ may be computed using formula (2). The comparison with experimental results does not bring any new information on the nature of clusters (as compared to information drawn from gap distributions between charged or negative particles), but shows that one must have some percentage of charged clusters as long as one confines oneself to realistic rapidity widths for clusters (8 ~ 0 . 6 - 0 . 7 [ 1 0 - 1 3 ], see the discussion m sect. 5). Thereby we exclude models with leading particles plus neutral clusters *, as can be seen in fig. 2 in the particular case with multiplicity nc = 2 at 405 GeV/c. These conclusions stay valid over the whole energy range of FNAL wlth any other possible choice Of~c(~ c ~< 4) and p(n c, nc)"

4. Zones and charge transfer correlations In this section, we will focus our attention on 405 GeV/c pp data. The discussion about the energy dependence of the results will be given in the next section.

4.1. Central zone characteristics The experimental characteristics of the central zones have been measured with a good accuracy, at 405 GeV/c [17] O~z) = 0.76 -+ 0.01 ,

(n z) = 2.77 -+ 0 . 0 3 ,

(11)

where (~z) is the average width of a zone and (n z) the average number of charged partlcles wittun a zone. Tlus leads us to work in the (Xz), (n z) plane in order to test our model. If we fix the cluster width (8 ~ 0.7) and the multiplicity distribution within clusters (p(n c, nc) of Poisson type, see appendix A), our model depends on two parameters a 0 and a 1. To each value of the couple of parameters (a 0, al), there corresponds a value of the couple ((~z), (nz)) and therefore a point in the (3,z), (n z) * Since we take into account neutron production, we mean here a model where all clusters are neutral except for those compensating possible neutron production, wlueh accounts for the leading charge flow. Such models are characterized by the fact that ao "~ 1 and depend on only one parameter a l-

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4.2, Chargetransfercorrelations If, for a given no, we choose the parameters of our model so as to have a good description of the zone characteristics, we also get a good description of the charge transfer correlation function D(Yl, Y2): see fig. 5 for nc "- 3 and fig. 9 later for nc = 2. We also show in these figures the charge transfer correlations corresponding to models with two leading particles plus neutral clusters (see footnote, subsect. 3.2); for such models, with realistic rapidity cluster widths (5 ~ 0.7), it is clear that on the one hand D 2 ( y ) -- D ( y , y ) is too small and that on the other hand D ( y 1, Y2) has too steep a slope in the variable Ay = [Yl - Y2 [; b°th these features are independent o f ~ c and of the shape of p(ne, no)" Such results seem to contradict the conclusions of Bia~as et al. [21 ] and Baler and Bopp [22], according to which experimental results on D 2 ( y ) = D(y, y) agree with a leading particle plus neutral cluster model if

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2

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y~-v, Fig. 5. Charge transfer correlations D(y 1, Y2) for Y l = -1.2 at 400 GeWc. Data from ref [ 17 ]. Model calculations with nc = 3. (a) p(n c, nc) narrower than Poisson and 6 = 0.7. Continuous line: a model with charged and neutral clusters (ao = 0.25, al = 0.80; R c "" 0.54). Shaded area" neutral duster models. (b) P(ne, ~c) of Polsson type and 6 ---0.85. Continuous hne: a model with charged and neutral dusters (ao = 0.6, al = 1, R c ~ 0.35). Shaded area: neutral cluster models. one takes into account the leading charge flow. However, if we choose a cluster width c5 -~ 0.85 like Bia~'as et al. [21 ], we get correct values forD2(y) (see fig. 5), while the slope o f D O ' l , Y 2 ) remains too steep. These features illustrate that one should not limit oneself to the study of D(Yl,Y2) for identical values of the arguments Yt =Y2, and show the danger in drawing conclusions from a small subset o f the data. Of course it is perhaps possible to fit the data on central zones and D(Yl, Y2) in the framework o f a leading particle plus neutral cluster model with a clever choice o f ~c and 6 ; but such fits will always require a rapidity width of clusters 6 > > 0.7 which clearly corresponds to unrealistic semi-inclusive longitudinal correlations (see discussion in sect. 5). Let us summarize the conclusions of these last two sections: the gap distributions, the zone characteristics and the charge transfer correlations do not allow one to draw definite conclusions on the charged multiplicity within clusters. However the

A. Arneodo, G. Plaut /Multtparttcle production

275

data on any of these quantities exclude the models with leading particles plus neutral clusters, at least if clusters have a realistic rapidity width. Let us recall that we should have got quite different quantitative results if we had neglected neutron productlon (e.g. with regard to zones).

5. Discussion For our study to be complete, it is necessary to check that we get a satisfactory description of inclusive and semi-inclusive longitudinal correlations. Longitudinal semi-inclusive correlations are investigated in the framework of our model in ref. [26] but we want here to report the most outstanding conclusions of this study, in view of their importance for the determination of cluster properties: The longitudinal semi-inclusive correlations allow one to exclude unambiguously rapidity widths larger than ~i ~--0.6-0.7 [10-13]. These correlations are not compatible with charge multiplicity within clusters nc ~> 3 [10-121. We saw in sects. 3 and 4 that it was possible to fit gap distributions, charge transfer correlations and zone characteristics with nc ~- 3,4 by appropriately choosing the other cluster characteristics (see figs. 1,4, 5); however if we use such cluster characteristics to compute longitudinal semi-mcluswe correlations, we always obtain quite unsatisfactory results. This feature is illustrated in fig. 6 where we plot the semiinclusive charged-charged correlation [27]:

1 don Cn(Yl'Y2) =n(n - 1)o n dYldY 2

1 don 1 don no n dy 1 no n dy 2 .

(12)

With the choice ~c = 2, 8 = 0.7 one can see that our results are in good agreement with data, whereas for nc = 3 our computations lead to much too high values for Cn(O, O) (and Rn(O, 0)) and for 8 = 0.85 they lead to a much too wide short-range part in Cn(Yl, Y2), which confirms the results from phenomenological analyses of refs. [10-13]. This demonstrates the importance of the semi-inclusive longitudinal correlations which, contrary to gap distributions, zone characteristics and charge transfer correlations, allow one to draw definite conclusions about the cluster width ~ 0.6-0.7 and the size of the clusters nc ~ 2.0. With such cluster characteristics, it is possible to fit almost all semi-lncluswe and reclusive data on pp collisions at 102 and 405 GeV/c alike (these are the extreme FNAL energies), provided that one has an average percentage of charged clusters witlun the bounds 0.45 ~
(13)

In order to dlustrate these features we computed with the above mentioned cluster

276

A. Arneodo, G. Plaut /Multiparttcle productton

oo~C8 [o, y~]

[]

~c=2,6=07

I=~IAQI

I

I•

l

Iltl

I

IQiIA-I

-001 [] -s

[]

o o~T.~ -~¢= 3,6=0.7 /+FX _'~o,o,l-~CY

00~

F-I

~

s ,=

~c=3,6=_0.85 Y2

-O.Ol Fig. 6. Longitudinal semi-incluswe charged-charged correlations Cn(Yl, Y2) for y: = 0 for 200 GeV/c 8-prong events. Data from ref. [27 ]. Model calculations: (a) n c = 2, 8 = 0.7, p(n c, nc) narrower than Poisson. Co) nc = 3, 8 = 0.7, p(n c, nc) narrower than Poisson. (c) nc = 3, 6 = 0.85, p(n c, nc) of Poisson type. properties the various quanttties of physical interest at 102 and 405 GeV/c: the gap distribution between negative and charged particles are shown in fig. 7, the fixed charge transfer gap distributions in fig. 8, and the charge transfer correlations in fig. 9. This description is completed by the exhibition in fig. 10 of the zone characteristtcs in the plane (Xz), (n z) (see sect. 4). To specify, we used here a distribution P(nc, ne) narrower than Poisson, the details of whtch are given in appendix A. Although there are indications that p(n c, nc) is mdeed narrower than Poisson, we found that the choice o f p ( n c, nc) is not unique and we do not claim to have found the charged multiplicity distribution within clusters. The bounds (13) on the average percentage R c of charged clusters could probably be improved if one knew the semi-mcluswe charge transfer correlations * and gap distributions. Anyhow it is noteworthy that the leading particle plus neutral cluster model (see footnote, subsect. 3.2) is definitely excluded by all the inclusive and semi-inclusive experimental data (see e.g. the neutral cluster model predictions in figs. 2, 9, 10). Let us point out that our conclusions about the average multiplicity within * Moreover these correlations would permit one to remove the bias introduced in D(yl, Y~) by our description of diffraction; this b:as may be seen m fig. 9 at 102 GeV/c for Yi or Y2 > 1.2.

A. Arneodo, G. Plaut /Multtparticle productton

277

lOnt- - - ' I ' l ~k nc=2 o , 4 0 0 Ge-" ~s,~ '~" 1 0 2 v/c

f

I(~L-

10

~-

charged secondaries

ne

1] 0

,

I 1

,

J 2

r

"-]

Fig. 7. Dzstributions of rapidzty gaps between produced particles in pp collisions at 102 and 400 GeV/c. Data from ref. [18]. Model calculations with: n c = 2, p(n c, nc) narrower than Poisson, 6 = 0.7 and a0 = 0.36, az = 0.68 (R c ~-- 0.55).

ld

-

'

I

'

I

lo' -

1 0 2 GeV/c o AQ=O • IAQI= 1

i

,%

'

400

'

r

~.c= 2 - "

o ~Q=O • I/IQI--- 1 l O~

2

• IAol=

r

GeV/c

,o

lu

10

10

1

1 0

1

r

2

0

1

r

2

Fzg. 8. Distributions o f rapidity gaps carrying a specified charge at 102 (a) and 405 GeV/c (b). Data from ref. [ 1 8 ] . Model calculations like in fig. 7.

278

A. Arneodo, G. Plaut / Multtparticle productton I

I

I

102 fieV/c

10

i

|

~c=2

05

10

\.

Y, -I 2

05

01

•

,

,

4 0 0 GeV/c

tI

_ffc=2

1.0 05

10 0.5

01

,

,

0

1

2

3

Y~-Y, Fig. 9. The charge transfer correlation function D(yl, Y2) evaluated at Yl = 0 and Yl = - 1 . 2 at 102 and 405 GeV/c. Data from ref. [17]. Same model calculations as in fig. 7. Shaded area' neutral cluster model predictions for y I = - 1 . 2 .

clusters ~ c = 2) run against the results obtained by Ludlam and Slansky [20] from the study of their left-right fluctuation parameter Kn(~c ~ 3-4). However we will show in ref. [26] that the cluster characteristics we have deternuned (in partict, lar nc ~- 2) are quite compatible with experimental data on rn when the neutron production )s taken into account. To close this discussion, let us remark that the independent cluster emission models, with no restriction on the exchanged charges (RCCM), also allow an accurate description of experimental data at FNAL energies with the same cluster charac teristlcs as those determined within the LCEX model framework, namely nc ~--2, 6 ~ 0.6-0.7 and R c ~--0.55. However if the LCC hypothesis proves to be exact and if consequently the central zone characteristics become independent of energy (whxch can be reasonably conjectured in vaew of FNAL experimental results [17]), such models will require R c to be a decreasing function of energy; in other words, the cluster characteristics will have to depend on energy just to account for energyindependent characteristics of multiproductlon in the central region!

A. Arneodo, G. Plaut / Multipartzcle productton 10

0.9

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0.8

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~! I

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i

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,

06

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2.6

2.7

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i

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~

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[]

25

23 10

2-_-_-2.

, I

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24 I

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279

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nc = 2

~ - - - - _ - S

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(A,>

0.7 0.6

,

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I

2.6

/

2.7

!

28

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2.9

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30

Ftg. 10. Representatzon in the (hz>, (n z) plane of the central zone characteristics in 102 (a) and 405 GeV/c (b) pp collisions, nc = 2, ti = 0.7, p(n c, nc ) narrower than Poisson. Rectangles E experimental data from ref. [17]. Shaded area: neutral cluster models. 6. Conclusions In view o f the i n c o n s i s t e n c y b e t w e e n several recent analyses w i t h i n the framework o f d u s t e r models o f n e w experimental results on gap distributions [18], zone characteristics [17], and charge transfer correlations [17] in pp colhsxons, we undert o o k in this paper a comprehensive study of experimental data from FNAL. We did n o t restrict ourselves to n e w results and we also checked older b u t well established data such as inclusive and senu-inclusive longitudinal correlations, which are o f prime i m p o r t a n c e in a s t u d y o f cluster properties. We worked in the framework o f an as realistic as possible cluster m o d e l relying on LCEX a n d LCC hypotheses; the model incorporated leading particle effects * * We wall dzscuss in ref. [26] the influence of leading charges on correlations. In partzcular, neutron production will have important effects upon Kn.

280

A. Arneodo, G. Plaut / Multtpartwle production

and energy-momentum conservation, and was constrained to reproduce the experimental charged multiplicity distribution. If the cluster model remains able to fit all the experimental data, it appears in this work that clusters are strongly constrained by these data. On the one hand, the longitudinal semi-inclusive correlations require that the clusters decay into two charged particles on the average and have a rapidity width 8 ~ 0.6-0.7. On the other hand, with such clusters, all the data on new quantities (gaps, zones, charge transfer correlations, etc.) cannot be reproduced if one does not admit a non-negligible percentage of charged clusters (about 50 or 60%). Therefore, we conclude that clusters are rather light ~ c ~- 2), narrow in rapidity, and that a large percentage of clusters are charged. It appears from such conclusions that clusters bear much resemblance to mesonic resonances; such a resemblance was already pointed out in refs. [11, 18, 28], but the hypothesis that clusters are resonances has never been checked against all the experimental data; moreover the conclusions of Ludlam and Slansky [20] seemed to rule out such an hypothesis. Our results show that the interpretation of clusters as resonances remains an attractive possibility. Our conclusions agree with previous estimates o f ~ c and ~ relying on a semiphenomenologlcal separation of a short-range part in semi-inclusive longitudinal correlations [10-13 ]. To be complete, we compare them with those of the theoretical papers quoted m sect. 1. lake F r Q [18], we worked in the LCEX hypothesis framework for cluster production; this hypothesis appears to be at least a very appropriate one in building models for multiproduction. But we found that the gap distributions alone do not allow one to draw as precise conclusions on nc as PTQ did (these distributions merely exclude very large values o f ~ c > 4); we foundthat only semi-inclusive longitudinal correlations permit to conclude that nc ~ 2. lake Ludlam and Slansky [20], we should emlahasize the importance of kinematical constraints: withouttaking energy-momentum conservation into account, most of our results would have been quite different. But the parameter Kn of Lud Ludlam and Slansky, which describes the senu-inclusive left-right rapidity fluctuations, appeared to be very sensitive to the details of several mechanisms (to energy momentum constraints, but also to leading particle effects, and to the intracluster multiphcity distribution), and therefore does not seem to be very appropriate to determine nc" Anyway, we show in ref. [26] that Kn may be quite well reproduced with light clusters. Perhaps our most outstanding conclusion is that gap distributions, zone characteristics, and charge transfer correlations definitely exclude neutral cluster models, whereas BiaIas et al. [21] and Baler and Bopp [22] showed that these models could work for the quantity D2(y) provided one takes the leading charge flow into account.

A. Arneodo, G. Plaut / Multiparttcle production

281

We are grateful to M. Le BeUac, P. Grassberger and J . L Meunier for useful discussions and to P. Grassberger for a careful reading of the manuscript.

Appendix A We considered in this paper two kinds of charged multiplicity distributions within clusters: (i) rather wide distributions of the Polsson type as PTQ [18] did; (ii) narrower distributions, as suggested by phenomenologlcal analyses of semiinclusive longitudinal correlations [10,12].

A. 1. Distribution o f the Poisson type The probability p(n c, nc) for a neutral cluster decaying on the average into nc charged particles to decay into n c charged particles is ~nc II0(2/3)[(~nc)!] 2 , P(nc,~e)

for n c even,

/ 0,

for n c o d d ,

whtle for a singly charged duster it is 0,

P(nc, no) =

for n c even

13,% , 1 -1-~-t 1 I1(2/3 ) (~(n c + 1)).Q(n c - 1))!

for n c odd

The parameters/~ and 13' are adjusted to make the average number of charged particles within clusters equal to tic"

A.2. Distributions narrower than the Poisson type Since the results are not very sensitive to the details o f p ( n c , no), we used very simple distributions, namely, (i) nc = 2 Neutral clusters: p(0, 2) = 0.1 ,

p(2, 2) = 0 . 8 ,

p(4,2) = 0.1 ,

p(n c, 2) = 0 elsewhere. Singly charged clusters: p(1,2) = 0.5,

p(3, 2) = 0.5 ,

p(nc, 2) = 0 elsewhere.

282

A. Arneodo, G. Plaut /Multtparttcle production (ii) nc --- 3 Neutral clusters: p(0, 3) = 0 . 0 5 ,

p(2, 3) = 0 . 4 5 ,

p(4, 3) = 0 . 4 5 ,

p(6, 3) = 0 . 0 5 ,

P(nc, 3) = 0 elsewhere.

Singly charged clusters: p(1, 3) = 0.1 ,

p(3,3) = 0.8,

p(5, 3) = 0.1 ,

p(n c, 3) = 0 elsewhere.

Although these particular distributions were chosen for their ability to fit experimental results, we do not claim that they are the best possible choice. Experimental results leave indeed a fair amount of freedom in the choice of the distributions P(n c, ne). * From another point of view, let us remark that these distributions are used at the beginning of our Monte-Carlo program and may be somewhat altered by the constraints which we assign to our cluster model, in particular by the constraint of reproducing the charged experimental multiphclty distribution.

Appendix B Dealing with one-cluster events Let us first notice that we have a priori to deal in a special way with the kinematics of one-cluster events: In our model, where the cluster mass is taken fixed, it is not possible to build a given missing mass with only one cluster. So for such events we keep only one of the randomly chosen variables x, and we define the other leading particle and the cluster kinematical variables so as to verify energy-momentum conservation. It appears finally that the one-cluster events involve always one fast leading particle (x close to 1) and have a rapidity configuration which looks like that of diffractive events. Now, when constraining the charge multlphcity distribution, we change the weight of one-cluster events: indeed, for large Hc (e.g. Hc ~ 4) one will get originally very few two or four-prong events; in order to fit the prong multiplicity distribution, one will have to increase the number of such events, which essentially consists in increasing the number of one-cluster events (most of two and four-prong events are one-cluster events ifH c ~ 4). Finally, as a consequence of our constraint on multi* In fact, after this work was achieved, we got from the systematic study of semi-reclusive correlations in the central region that p(n c, ~c) had indeed to be much narrower than Poisson type.

A. Arneodo, G. Plaut /Multtparttcle production

283

pliclty distributions, we generally increase the number of one-cluster events and in this way strengthen the leading parhcle x-spectrum near x = 1 ; we may hope to get in this way somehow a qualitative description of diffraction although we are fully aware that it is at best an ad hoc description. Let us note at this point that it would not be reasonable to accept events with a "diffractive" neutron 0.e. one neutron separated by a large rapidity gap from produced pions); those events we reject, keeping only events with a "diffractive" proton. This way of deahng with onecluster events finally yields a peak around x = 1 in the proton x-spectrum and a dip m the neutron spectrum at the same place. It is quite race that, for clusters with nc "~ 2, one gets in such a way a rather satisfactory description of chffraction, whereas the peak becomes clearly too sharp vath heavier clusters. However we would not hke that our conclusions depend too much on ttus ad hoc way of introducing chffraction, so that we constrain the charged particle multiplicity distribuhon only for a number of prongs >~4: on the one hand it would not be reasonable to try to interpret every two-prong event as an event with two leading particles plus one cluster (m particular we neglect possible leading clusters); on the other hand one can omit some two-prong events without problem since they do not contribute either to the gap distributions (if one discards the end gaps) or to central zones [ 17], or to the central region behavaour of various inclusive correlations (but by normalizations.)

References [1] P. Darnulat, talk at the 6th Int. Colloq. on multiparticle reactions, Oxford, 1975, and references therein. [2] P. Plrtla and S. Pokorski, Phys Letters 43B (1973) 502. [3] E.L. Berger and G.C. Fox, Phys. Letters 47B (1973) 162. [4] F. Hayot and A. Morel, Nucl. Phys. B68 (1974) 323. [5] M. Le BeUac, H.1. Miettmen and R.G. Roberts, Phys. Letters 48B (1974) 115. [6] A. Blab'as,Proc. 6th Int. Symp. on multlpamcle hadrodynamlcs, Pavia, 1973 (INFN, 1974) p. 511 [7] A. Morel and G. Plaut, Nucl. Phys. B78 (1974) 541. [8] E.L. Berger, Phys. Letters 49B (1974) 369, Nucl. Phys B85 (1975) 61. [9] F Hayot and M. Le BeUac, NucL Phys B86 (1975) 333. [10] L. Foa, Phys. Reports 22,(1975) 1. [11] K. Eggert et al., Nucl. Phys. B86 (1975) 201. [ 12 ] S.R. Amendoha et al., Measurement of two-particle semi-reclusiverapidity distributions at the CERN-ISR, Nuovo Omento to be pubhshed. [ 13] T. Kafka et al, Inclusive and semi-reclusivetwo-particle correlations, SUNY SB-BCG-1/75, unpubhshed. [14] A. Krzywlckl and D. Wemgarten, Phys. Letters 50B (1974) 265. [15 ] A. Krzywickl, Local compensation of quantum number and shadow scattering, Ptoc. 10th Rencontre de Moriond, 1975. [16] J. Derre et al., French-Soviet Collaboration, paper presented at 17th Int. Conf. on highenergy physics, London, 1974. [17] C. Bromherg et al., Phys. Rev. D12 (1975) 1224.

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A. Arneodo, G. Plaut / Multiparticle production

[18] P. Piril~i, G.H. Thomas and C. Quigg, Phys. Rev. D12 (1975) 92. [19] T. Ludlam and R. Slansky, Phys. Rev. D12 (1975) 65. [20] T. Ludlam et al., Phys. Letters 48B (1974) 449, T. Ludlam and R. Slansky, Phys. Rev. D12 (1975) 59. [21] A. Bl~as, K. Fiaikowskl, M. Jesabek and M. Zielinski, Acta Phys. Poh B6 (1975) 59. [22] R. Baier and F.W. Bopp, The flow of leading charges, preprint Bi-75/11, 1975. [23] A. Arneodo and G. Plaut, Charge structure in rapidity space and duster properties, Nice preprint NTH 75/5, 1975, unpublished. [24] F. Hayot, F.S. Henyey and M. Le Bellac, Nucl Phys. B80 (1974) 77. [25] A. Arneodo and G. Plaut, Nud. Phys. B97 (1975) 51. [26] A. Arneodo and G. Plaut, to be published. [27 ] B.Y. Oh et al., Phys. Letters 56B (1975) 400. [28] F. Hayot, Nuovo Cimento Letters 12 (1975) 676.