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The rapidity gap probability is investigated in varying symmetric bin sizes in proton-nucleus interactions at 800 GeV. It shows a scaling behavior and confirms the linked pair approximation for the n-particle cumulant correlation functions to high order. A comparison of the scaled rapidity gap probability is made with the negative binomial distribution. We also interpret the rapidity gap probability in the clan production model of hadronization and reanalyze the clan parameters in its light.

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... Therefore, in this case, the events could be an admixture of CNO events and peripheral AgBr events. It has been specifically pointed out that [14][15][16] events with 2 ≤ N h ≤ 5 are pure CNO events and events with 6 ≤ N h ≤ 8 are CNO events with an admixture of events generated from peripheral collisions between the projectile and AgBr target. In this analysis, we have not considered those events that have 2 ≤ N h ≤ 8. ...

A study of centrality dependence of target fragmentation and multiparticle production has been carried out in 16O – AgBr, 22Ne – AgBr and 28Si - AgBr interactions at 4.1-4.5 AGeV/c for the total disintegrated (TD) events. Average multiplicity of black particles increases linearly with decreasing centrality while average multiplicity of grey and shower particles decreases with decreasing centrality for the total disintegrated events (TD events). The decrease of average multiplicity of grey particles is linear on the other hand in case of shower particles the increase is non-linear. With the increase of the mass of the projectile beam average multiplicity of black particles decreases and average multiplicity of grey particles increases for the TD events. This observation has been explained on the basis of fireball model. However, in case of shower particles average multiplicity increases with increase of projectile mass.

... Therefore, in this case, the events could be an admixture of CNO events and peripheral AgBr events. It has been specifically pointed out that272829 events with 2 N h 5 are pure CNO events and events with 6 N h 8 are CNO events with an admixture of events generated from peripheral collisions between the projectile and AgBr target. In this analysis, we have not considered those events, which have 2 N h 8. ...

We have presented a systematic study of two-particle rapidity correlations in terms of investigating the dynamical fluctuation observable in the forward–backward pseudo-rapidity windows by analyzing the experimental data of interactions at 4.5 AGeV/c, interactions at 4.1 AGeV/c, and interactions at 4.5 AGeV/c. The experimental results have been compared with the results obtained from the analysis of event sample simulated (MC-RAND) by generating random numbers and also with the analysis of events generated by the UrQMD and AMPT model. Our study confirms the presence of strong short-range correlations among the produced particles in the forward and the backward pseudo-rapidity region. The analysis of the simple Monte Carlo-simulated (MC-RAND) events signifies that the observed correlations are not due to mere statistics only; explanation of such correlations can be attributed to the presence of dynamical fluctuations during the production of charged pions. Comparisons of the experimental results with the results obtained from analyzing the UrQMD data sample indicate that the UrQMD model cannot reproduce the experimental findings. The AMPT model also cannot explain the experimental results satisfactorily. Comparisons of our experimental results with the results obtained from the analysis of higher energy emulsion data and with the results of the RHIC data have also been presented.

... A similar trend of average clan multiplicity N with pseudo-rapidity interval η Table 2. Represents the values of average clan multiplicity N and the average number of particles per clan n c for different pseudo-rapidity intervals η in the case of 22 Ne projectiles on interactions with AgBr and CNO targets at an incident momentum of 4.1 GeV/c. has also been observed in hadron–nucleus interactions [28, 29]. If we have a careful look at tables 2–5, it can be seen that for a given value of η, the value of average clan multiplicity increases with the increase in target mass. ...

A detailed study of clan model parameters and their target dependence has been carried out in the light of void probability scaling for heavy (AgBr) and light (CNO) groups of targets present in the nuclear emulsion using 22Ne (at an incident momentum of 4.1 GeV/c), 28Si (at an incident momentum of 4.5 GeV/c), 16O (at an incident momentum of 4.5 GeV/c) and 32S (at an incident momentum of 4.5 GeV/c) projectiles. The variation of scaled rap-gap probability with the single moment combination has been studied for all of the interactions. The experimental points are found to lie approximately on the NBD curve for all of the interactions, indicating a scaling behavior. According to the two-source model of particle production, in the case of 32S–AgBr interactions at 4.5 GeV/c, the pion production mechanism has been found to be almost chaotic. On the contrary, for all of the other interactions at 4.1–4.5 GeV/c, pion production is predominantly chaotic. Average clan multiplicities () for all of the interactions are found to increase with the increase in the pseudo-rapidity interval Δη. The values of for the AgBr target are larger than those for the CNO target. The average number of particles per clan () increases initially with the increase in pseudo-rapidity interval. An approximate saturation of the values of is observed for 22Ne and 16O projectiles around Δη = 6. With the increase in projectile size, the saturation of seems to be wiped out. It can also be observed from the tables that for a particular target the average number of particles per clan () increases with the increase in the size of the projectile nucleus. A comparison with the results of the SPS data has also been presented.

We study scaling behaviour of statistics of voids in the context of the halo
model of nonlinear large-scale structure. The halo model allows us to
understand why the observed galaxy void probability obeys hierarchical scaling,
even though the premise from which the scaling is derived is not satisfied. We
argue that the commonly observed negative binomial scaling is not fundamental,
but merely the result of the specific values of bias and number density for
typical galaxies. The model implies quantitative relations between void
statistics measured for two populations of galaxies, such as SDSS red and blue
galaxies, and their number density and bias.

Multiplicity distributions of charged particles produced in non single-diffractive collisions between protons and antiprotons at centre of mass energies of 200 and 900 GeV are presented. The data were recorded in the UA5 streamer chambers at the CERN Collider, which was operated in a pulsed mode between the two energies. A new method to correct for acceptance limitations and inefficiencies based on the principle of maximum entropy has been used. Multiplicity distributions in full phase space and in intervals of pseudorapidity are presented in tabular form. The violation of KNO scaling in full phase space found by the UA5 group at an energy of 546 GeV is confirmed also at 200 and 900 GeV. The shape of the 900 GeV distribution in full phase space is narrower in the peak region than at 200 GeV but exhibits a pronounced high multiplicity tail. The negative binomial distribution fits data at 200 GeV in all pseudorapidity intervals and in small intervals at 900 GeV. In large intervals at 900 GeV, however, the negative binomial distribution. Fits to the partially coherent laser distribution are also presented as well as comparisons with predictions of the Dual Parton, the Fritiof and the Pythia models.

The multiplicity distributions of charged particles in restricted rapidity intervals inZ
0 hadronic decays measured by the DELPHI detector are presented. The data reveal a shoulder structure, best visible for intervals of intermediate size, i.e. for rapidity limits around 1.5. The whole set of distributions including the shoulder structure is reproduced by the Lund Parton Shower model. The structure is found to be due to important contributions from 3-and 4-jet events with a hard gluon jet. A different model, based on the concept of independently produced groups of particles, clans, fluctuating both in number per event and particle content per clan, has also been used to analyse the present data. The results show that for each interval of rapidity the average number of clans per event is approximately the same as at lower energies.

The relation of combinants to various statistics characterizing the fluctuation pattern of multihadron final states is discussed. Scaling laws are derived for count probabilities and combinants in the presence of homogeneous and clustered monofractal density fluctuations. It is argued that both types of scaling rules are well suited to signal quark-gluon plasma formation in a second-order QCD phase transition.

We derive and display relations which can be used to express many quantitative measures of clustering in terms of the hierarchy of correlation functions. The convergence rate and asymptotic behaviour of the integral series which usually result is explored as far as possible using the observed low-order galaxy correlation functions. On scales less than the expected nearest neighbour distance most clustering measures are influenced only by the lowest order correlation functions. On all larger scales their behaviour, in general, depends significantly on correlations of high order and cannot be approximated using the low-order functions. Bhavsar's observed relation between density enhancement and the fraction of galaxies included in clusters is modelled and is shown to be only weakly dependent on high-order correlations over most of its range. The probability that a randomly placed region of given volume be empty is discussed as a particularly simple and appealing example of a statistic which is strongly influenced by correlations of all orders, and it is shown that this probability may obey a scaling law which will allow a test of the small-scale form of high-order correlations.

The multiplicity distributions of charged secondaries in proton-proton interactions at
$\sqrt s = 31$
, 44 and 62 GeV have been measured with high statistics using the split field magnet (SFM) detector at the CERN ISR. The multiplicity distributions for narrow bins in rapidity depend on the rapidity variable. The validity of KNO scaling is discussed. The negative binomial function fits well the charged and negative multiplicity data for full phase space and for central rapidity windows. The energy and the rapidity dependence of the negative binomial parameters,
$\bar n$
andk, are presented. We also analyse their implications for a cascade model in terms of the average number of «clans» and of the average number of particles per «clan».

The low-PT region of secondary pions, the sea and valence quark distributions in the projectile and the target, and rapidity correlations among secondary pions are investigated in interactions of 50-GeV/c π- with nuclear emulsion placed in a strong magnetic field. The existence of a sharp peak in the region of PT2≤0.12 (GeV/c)2 for positive and negative pions in the forward and backward hemispheres in the center-of-mass system is explained on the basis of resonance formation. The production of secondary pions is explained on the basis of the quark-antiquark-recombination model. We have used the valence and sea quark distributions in the pion for explaining the π+/π- ratio. We find an enhancement in the number of sea quarks with an increase in nuclear-target size. Single-particle and two-particle inclusive rapidity distributions have been studied. The rapidity-difference distributions and the normalized correlation function R have also been investigated. It is found that no positive correlations among secondary pions are evident at this primary energy.

Multiplicity distributions, the dependence on n of ψn=σn/σ, are discussed. Within the framework of the Amati-Fubini-Stanghellini model, a cluster expansion for the moments of ψn is derived. This same expansion is then derived as a consequence of asymptotic dominance of inclusive reactions by an isolated, factorizable Regge pole. Such an expansion furnishes a systematic way of describing the shape of ψn. It is argued that a Poisson distribution for multiple particle production can not be expected to occur, even for very high energies.

Data on 200 and 400 GeV proton interactions with nuclear emulsion have
been analyzed. It is found that the multiplicity distributions of the
shower particles in the restricted rapidity intervals are well described
by the negative binomial distribution (NBD). The dependences of the NBD
parameters on rapidity interval, energy and target size have been
studied. The results have also been discussed in terms of Giovannini and
Van Hove’s clan model of multiparticle production.

The intermittency effect is investigated using the assumption that the observed events can be decomposed into randomly distributed self-similar clusters. It is demonstrated that the factorial moments of a single cluster can be extracted from data using the probability that a certain phase-space box is empty of secondaries. The validity of the linked-pair approximation is shown to correspond to independent superposition of monofractal clusters. It is suggested that certain scaling properties of the void probability could signal a thermal phase-transition from the quark-gluon plasma.

Multiplicity distributions of charged particles produced in proton-proton, proton-argon and proton-xenon inelastic collisions at 200 GeV/c in various rapidity intervals are presented. Nuclear target data are analysed separately for the forward and the backward hemispheres. Results of a similar analysis for only negative particles are shown. The data are well parametrized in terms of negative binomial distributions. However, such a parametrization fails in describing thep-nucleus multiplicity distributions of charged particles for large rapidity intervals comprising both hemispheres. Parameters characteristic for the clan model of Giovannini and Van Hove are also presented.

This paper concerns the results recently obtained by the UA5 Collaboration on charged particle multiplicity distributions at the CERN
p[`(p)]p\bar p
collider
(Ös = 540 GeV)(\sqrt s = 540 GeV)
and their comparison with lower energy data [1, 2]. Our aim is to interpret in general terms the experimental finding that the distributions are of negative binomial (NB) form both for total multiplicities and in finite pseudorapidity intervals. This can be understood in terms of partial stimulated emission of bosons, or of a simple form of cascade process, or (more artificially) with both mechanisms. The cascade interpretation appears to be the most attractive. It leads to a new concept of clustering, the empirical data imply unexpected properties for the clusters, and a novel type of high energy asymptotics seems to emerge. We briefly discuss the relation with QCD-based models for hadron production and the case ofe
+
e
– annihilation.

We report on an analysis of the multiplicity distributions of charged particles produced ine
+
e
– annihilation into hadrons at c.m. energies between 14 and 46.8 GeV. The charged multiplicity distributions of the whole event and single hemisphere deviate significantly from the Poisson distribution but follow approximate KNO scaling. We have also studied the multiplicity distributions in various rapidity intervals and found that they can be well described by the negative binomial distribution only for small central intervals. We have also analysed forward-backward multiplicity correlations for different energies and selections of particle charge and shown that they can be understood in terms of the fragmentation properties of the different quark flavours and by the production and decay of resonances. These correlations are well reproduced by the Lund string model.

Theoretical basis of void scaling function properties of hierarchical structure in rapidity andp
T intervals are explored. Their phenomenological consequences are analyzed at single jet level by using Monte Carlo methods ine
+
e
– annihilation. It is found that void scaling function study provides an interesting alternative approach for characterizing single jets of different origin.

In this paper we examine the similarity between clan structure and self-similar cascading in high energy proton-nucleus interactions at 70 GeV/c in emulsion. The investigation shows that the negative bionomial fitting parameterk can be used for the evaluation of factorial moments in intermittency studies and that these exhibit similar behaviour with those derived from nonstatistical procedure. The results of our study signify compatibility between clan cascading and self-similar cascading in soft hadroization processes.

The rapidity gap probability, i.e., the probability of detecting no particles in a given rapidity interval is investigated in proton-antiproton collisions at CERN collider energies. A scaling behaviour is found in the central rapidity domain, similar to the scaling of the void probability in the Perseus-Pisces supercluster region of galaxies. This observation confirms that the recently proposed linked-pair approximation for the N-particle cumulant correlation functions holds valid to high order with linking coefficients slightly smaller than the negative binomial values. In the outer region of rapidity deviations are found from the scaling law which possibly is caused by the violation of translation invariance of the correlation functions. The correspondence to the clan production picture of hadronization is outlined: it is shown that the clan-model parameters can be obtained directly from the “hole probability”.

The presence or absence of an excess of filamentary structure in the galaxy distribution has important implications for the origin of such structure. In an attempt to answer this question, many statistics have been devised to quantify the visual impression of filaments seen in galaxy maps, but few of these are at all well understood. Two of these, a quadrupole elongation statistic and a nearest-neighbor alignment statistic, are analyzed in the common language of the irreducible N-point correlation functions of the galaxy distribution, and it is argued that there has as yet been no success in isolating filaments in particular from clustering in general.

Knowledge of N-point correlation functions for all N allows one to invert and obtain the probability distribution of mass fluctuations in a fixed volume. The hierarchical sequence of higher order is applied to correlations with dimensionless amplitudes suggested by the BBGKY equations. The resulting distribution is significantly non-Gaussian, even for quite small mean square fluctuations. The qualitative and to some degree quantitative results are to a large degree independent of the exact sequence of amplitudes. An ensemble of such models compared with N-body simulations fails in detail to account for the low-density frequency distribution.

The probability that a volume of the universe of given size and shape spaced at random will be void of galaxies is used here to study various models of the origin of cosmological structures. Numerical simulations are conducted on hot-particle and cold-particle-modulated inflationary models with and without biasing, on isothermal or initially Poisson models, and on models where structure is seeded by loops of cosmic string. For the Pisces-Perseus redshift compilation of Giovanelli and Haynes (1985), it is found that hierarchical scaling is obeyed for subsamples constructed with different limiting magnitudes and subsamples taken at random. This result confirms that the hierarchical ansatz holds valid to high order and supports the idea that structure in the observed universe evolves by a regular process from an almost Gaussian primordial state. Neutrino models without biasing show the effect of a strong feature in the initial power spectrum. Cosmic string models do not agree well with the galaxy data.

We show that the increase of bin-averaged factorial moments with decreasing size of the rapidity bin δy (the so-called ‘‘intermittency’’ phenomenon) can be understood on the basis of conventional short-range correlations and a simple linked-pair Ansatz for higher-order correlations.

Clan structure analysis in rapidity intervals is generalized from negative binomial multiplicity distribution to the wide class of compound Poisson distributions. The link of generalized clan structure analysis with correlation functions is also established. These theoretical results are then applied to minimum bias events and evidentiate new interesting features, which can be inspiring and useful in order to discuss data on rapidity gap probability at TEVATRON and HERA. Comment: (14 pages in Plain TeX plus 5 Postscript Figures, all compressed via uufiles) DFTT 28/94

- P. Carruthers