THE NEGATIVE BINOMIAL DISTRIBUTION IN
QUARK JETS WITH FIXED FLAVOUR
By Alberto Giovannini (Turin U. & INFN, Turin), Sergio Lupia (Munich,
Max Planck Inst.), Roberto Ugoccioni (Lund U.) LU TP 96-23, DFTT 50/96,
MPI-PhT/96-80, Sep 1996. 15 pp.
Submitted to Pysics Letters B.
We show that both the multiplicity distribution and the ratio of factorial cumulants over factorial moments for 2-jet events in $e^+e^-$ annihilation at the $Z^0$ peak can be well reproduced by the weighted superposition of two negative binomial distributions, associated to the contribution of $b\bar b$ and light flavoured events respectively. The negative binomial distribution is then suggested to describe the multiplicity distribution of 2-jet events with fixed flavour.
The shoulder structure of charged particles multiplicity distributions (MD's) in full phase space in e+e- annihilation at the Z0 peak and the quasi oscillatory behavior of the ratio of factorial cumulants to factorial moments as a function of the order are quantitatively reproduced within a simple parametrization of the MD in terms of a weighted superposition of two negative binomial distributions, associated to two- and multi-jet production, i.e., to hard gluons radiation.
The dependence of the average number of partons per clan on virtuality and rapidity variables is analytically predicted in the framework of the Generalized Simplified Parton Shower model, based on the idea that clans are genuine elementary subprocesses. The obtained results are found to be qualitatively consistent with experimental trends. This study extends previous results on the behavior of the average number of clans in virtuality and rapidity and shows how important physical quantities can be calculated analytically in a model based on essentials of QCD allowing local violations of the energy-momentum conservation law, still requiring its global validity.
It is shown that the ratio of factorial cumulant moments to factorial moments for a multiplicity distribution truncated in the tail reveals oscillations in sign similar to those observed in experimental data. It is suggested that this effect be taken into account in the analysis of data in order to obtain correct physical information on the multiplicity distributions.
Clan structure analysis in rapidity intervals is generalized from the negative binomial multiplicity distribution to the wide class of Compound Poisson distributions. The link of the 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.
By considering clans as genuine elementary subprocesses, i.e., intermediate parton sources in the Simplified Parton Shower Model, a generalized version of this model is defined. It predicts analytically clan properties at parton level in agreement with the general trends observed experimentally at hadronic level and in Monte Carlo simulations both at partonic and hadronic level. In particular the model shows a linear rising in rapidity of the average number of clans at fixed energy of the initial parton and its subsequent bending for rapidity intervals at the border of phase space, and approximate energy independence of the average number of clans in fixed rapidity intervals. The energy independence becomes stricter by properly normalizing the average number of clans.
Theoretical basis of void scaling function properties of hierarchical structure in rapidity and pT intervals are explored. Their phenomenological consequences are analyzed at single jet level by using Monte Carlo methods in e+e- annihilation. It is found that void scaling function study provides an interesting alternative approach for characterizing single jets of different origin.
By using Jetset 7.2 as e+e- event generator at different c.m. energies, we studied single jet multiplicity distributions in different rapidity and pT intervals. Good NB behaviour is found and related clan structure analysis is performed. Observed differences in the behaviour of the 2- and 3-jet samples can be understood in terms of the relative contribution of single quark and gluon jet to the 3-jet sample, which are obtained by selecting event by event in this sample the highest and the lowest energy jet respectively.
Continuing our research program on the origin of the Negative Binomial (NB) regularity in high energy collisions we discuss virtuality evolution and rapidity structure of a simplified parton shower (SPS) assuming essentials of QCD in a correct kinematical framework. We find that multiplicity distributions (MD) of final partons in full phase space and in symmetric rapidity windows are well fitted by NBMD's. Using generalized Local Parton Hadron Duality as hadronization prescription, NB regularity shows up also for final hadron MD's. Clan structure analysis manifests both at partonic and hadronic level very interesting properties which are consistent with previous findings in experimental data on MD's and in standard Montecarlo simulations. It is striking that in the SPS model the average number of clans scales with virtuality in a fixed rapidity window.
MULTIPLICITY DISTRIBUTIONS AND FLUCTUATIONS
IN THE DISCRETE QCD MODEL
by R. Ugoccioni (Lund Univ.)
LU TP 96-25, to appear in the Proceedings of the 7th International
Workshop ``Correlations and Fluctuations'', Nijmegen, The Netherlands,
June 30-July 6, 1996.
Results are presented from calculations carried out in the discrete QCD model, concerning multiplicity distributions in rapidity intervals and fluctuations.
We show, via a simple parametrization of the multiplicity distribution of charged particles in $e^+e^-$ annihilation at the $Z^0$ peak in terms of the weighted superposition of two negative binomial distributions, that both the shoulder structure in the intermediate multiplicity range and the oscillation in sign of the ratio of factorial cumulants over factorial moments of increasing order are related to hard gluon radiation.
We present a cascading model for a single jet, inspired to QCD and to the phenomenological analysis of multiplicity distributions. The model, describing as it does a two dimensional evolution in virtuality and rapidity, allows analytical predictions for clan analysis parameters to be made.
The concept of clans as intermediate parton sources independently produced has been implemented in a simplified model of single parton shower by allowing local violations of the energy-momentum conservation law, still requiring its global conservation. Analytical calculations of the average number of clans in symmetric rapidity intervals show qualitative agreement with experimental results in e+e- annihilation. In this framework, a new scaling law for rescaled observables is discovered.
Clan concept is generalized to the whole class of Compound Poisson Distributions which are the natural framework for studying multiplicity distributions of a dynamical two step production process. It is shown that the probability to detect no particles in a given rapidity interval has an intriguing connection with the clan concept as well as with n-order normalized factorial cumulants. The need for analyzing data on multiplicity distributions in rapidity intervals both at Hera and Tevatron is pointed out.
By discussing recent results which are related to multiplicity distributions, it is shown how the study of the latter both in full phase space and in restricted regions of it can be very effective in order to deepen our understanding of challenging problems central to high energy physics, such as parton cascades, hadronization processes, rapidity gaps physics and final particles correlations.
After a brief review of the theoretical basis of void scaling function properties of hierarchical structure, we analyze the phenomenological consequences at single jet level in Monte Carlo e+e- annihilation events. We find an interesting alternative approach for characterizing quark and gluon jets.
Void analysis at single jet level is discussed and shown to provide a possible tool for distinguishing quark-jets from gluon-jets and for determining the multiplicity distributions.THE GENERALIZED SIMPLIFIED PARTON SHOWER MODEL.
We explore the consequences of considering clans real physical objects in the framework of a generalized version of the Simplified Parton Shower model for a single jet. We predict that the average number of clans at fixed energy grows linearly in rapidity and slowly decreases with energy in a fixed rapidity interval.
An integrated description of multiplicity distributions and correlation functions in terms of void probability analysis is suggested. The connection of void parameters with hierarchical models for correlation functions and with clan structure parameters is discussed.
We report on a study of multiplicity distributions in a single jet isolated from a sample generated by the Monte Carlo program Jetset 7.2 at different c.m. energies. The different behavior of single jets from the 2- and the 3-jet samples can be understood when, event by event, the lowest and the highest energy jets are selected.
The new findings on NB behavior at LEP by the DELPHI Collaboration on separated 2-, 3- and 4-jet contributions and our previous work on q-qbar and gg systems with the JETSET 6.3 Lund Parton Shower strongly suggest to study single parton shower properties, where a higher degree of simplicity is expected. We propose a Simplified Parton Shower (SPS) model based on essentials of QCD in a correct kinematical framework. NB regularity is found to occur in the SPS for single parton shower both in full phase space and in symmetric rapidity windows.
We discuss the important phases in the evolution of our understanding of multiplicity distributions in high energy collisions with particular emphasis to intermittent behavior and shoulder structure problem.
The hadronic properties of e+e- annihilation in the QCD Parton Shower Model can be understood in terms of the partonic properties by means of Local Parton Hadron Duality (LPHD). The partonic properties have been shown to result from the superposition of the two single - parton showers produced by the ``ancestor'' quark and antiquark. Continuing our previous work we have investigated the rapidity and multiplicity distributions of single b(b-bar) showers in the b-bbar system at c.m. energies 91 GeV, 200 GeV and 2000 GeV. The evolution of ancestor b(b-bar) has been studied from the initial to the final state at parton level and then to B-hadron at hadron level. The emerging structures are remarkably simple at high c.m. energy. At c.m. energy 91 GeV (LEP energy), due to the relatively low virtuality of the process the heavy flavor ancestor gives origin to unexpected features, which are of particular interest.
Continuing our previous work on the QCD Parton Shower Model for electron positron annihilation we have investigated the rapidity and multiplicity structures of the q-qbar and gg systems at c.m. energies 200 GeV and 2000 GeV in terms of the jets produced by the two initially created partons. The corresponding distributions of final partons are found to be remarkably simple and control the observed distribution by Local Parton Hadron Duality (LPHD). In this paper we discuss the single g-jet structure resulting by separating the gg system.
Void analysis at single jet level is discussed and shown to provide a possible tool for distinguishing quark- from gluon-jets and for determining their multiplicity distribution.
MULTIPARTICLE DYNAMICS. PROCEEDINGS, 24TH INTERNATIONAL SYMPOSIUM, VIETRI SUL MARE, SALERNO, ITALY, SEPTEMBER 12-19, 1994. By A. Giovannini, (ed.), S. Lupia, (ed.), R. Ugoccioni, (ed.) (Turin U. & INFN, Turin). 1995. Singapore, Singapore: World Scientific (1995) 614 p.
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Preprints from the Lund HEP group