HEN362 (1993)
IIHE93.01
FIAN/TD09/93
update July 1995
SCALING LAWS FOR DENSITY CORRELATIONS AND
FLUCTUATIONS IN MULTIPARTICLE DYNAMICS
E.A. De Wolf^{1}^{1}1 Dept. of Physics, Universitaire Instelling Antwerp, B2610 Wilrijk and InterUniversity Institute for High Energies, Universities of Brussels, B1050 Brussels, Belgium, I.M. Dremin^{2}^{2}2 P.N. Lebedev Institute of Physics, Acad. of Sciences of the Russia, 117 924, Moscow, Russia, W. Kittel^{3}^{3}3 University of Nijmegen/NIKHEF, NL6525 ED Nijmegen, The Netherlands
Abstract
Experimental data are presented on particle correlations and fluctuations in various highenergy multiparticle collisions, with special emphasis on evidence for scalinglaw evolution in small phasespace domains. The notions of intermittency and fractality as related to the above findings are described. Phenomenological and theoretical work on the subject is reviewed.
Contents
 1 Introduction
 2 Formalism
 3 Experimental survey on correlations

4 Multiplicity fluctuations and intermittency
 4.1 Prelude
 4.2 Normalized factorial moments
 4.3 Higher dimensions
 4.4 Dependences of the effect
 4.5 Factorial cumulants
 4.6 Factorial correlators
 4.7 Multifractal analysis
 4.8 Density and correlation stripintegrals
 4.9 Correlations in invariant mass
 4.10 Genuine higherorder correlations
 4.11 Summary and conclusions
 5 Theoretical description
 6 Conclusions
 7 Figure Captions
Of the achieved triumph pangs and tricks Are just tightly stretched bowstrings.
B. Pasternak
Chapter 1 Introduction
Recent years have witnessed a remarkably intense experimental and theoretical activity in search of scaleinvariance and fractality in multihadron production processes, for short also called “intermittency”. These investigations cover all types of reactions ranging from ee annihilation to nucleusnucleus collisions, up to the highest attainable energies. The creation of soft hadrons in these processes, a major fraction of the total cross section, relates to the strongcoupling longdistance regime of Quantum Chromodynamics (QCD), at present one of the least explored sectors in the whole of highenergy particle physics.
A primary motivation is the expectation that scaleinvariance or selfsimilarity, analogous to that often encountered in complex nonlinear systems, might open new avenues ultimately leading towards deeper insight into longdistance properties of QCD and the unsolved problem of colour confinement.
History shows that studies of fluctuations have often triggered significant advances in physics. In the present context, it was the observation of “unusually large” particle density fluctuations, reminiscent of intermittency spikes in spatiotemporal turbulence, which prompted the pioneering suggestion to investigate the pattern of multiplicity fluctuations in multihadron events for ever decreasing domains of phasespace. Scaleinvariance or fractality would manifest itself in powerlaw behaviour for scaled factorial moments of the multiplicity distribution in such domains. It is important to stress here that, in practice, one deals with the problem of evolution of particle number distributions for ever smaller bins and intermittent behaviour implies that, for small phase space bins, the distributions become wider in a specific way. The same problem can be stated as an increasing role of correlations within a small phase space volume.
Through a multitude of increasingly sophisticated experimental studies of factorial moments, much new information has been gathered in a surprisingly short time. This work indeed confirms approximate power behaviour down to the experimentally possible resolution, especially when carried out in two and threedimensional phase space.
The proposal to look for intermittency also has triggered a thorough revival of interest in the old subject of particle correlations, by experimentalists and theorists alike. The need for greater sensitivity in measurements of correlation functions has directly inspired important work on refined analysis techniques. A promising and long overdue systematic approach to correlation phenomena of various sorts, including BoseEinstein interferometry, is finally emerging.
The large body of experimental observations now available is calling for satisfactory explanation and, indeed, theoretical ideas of all sorts abound.
The level of theoretical understanding is quite different for the various types of collision processes. In ee annihilation, parton cascade models based on leadinglog QCD have met considerable success and good overall description of multiplicity fluctuations is claimed. Closer inspection, nevertheless, reveals potentially serious deviations from the data, thus requiring further study.
For other processes, in particular hadron initiated collisions, models are faced with large and partly unexpected obstacles. This may be a reflection of insufficient knowledge of the reaction dynamics, although present evidence points to hadronization as the main culprit.
Within the framework of perturbative QCD, results of considerable interest on the emergence of power behaviour and multifractality have been obtained. However, these are asymptotic in nature and most likely quite unrelated to presentday experiment. Being related to the mechanism of confinement, not surprisingly, the role of hadronization remains unclear.
Random selfsimilar multiplicative branching models have inspired much of the original work on intermittency. Among many scaleinvariant physical systems, the cascade process is a particularly natural candidate for the description of strong fluctuations selfsimilar over a wide range of scales. It finds support in the cascade nature, not only of perturbative QCD, but also of the subsequent hadronization. However, further work is needed to help understand the details of the process.
Alternatively, “classic” and extensively studied possibilities are scaleinvariant systems at the critical point of a highorder phase transition. This subject has attracted particular attention in view of potential application to quarkgluon plasma formation in heavyion collisions.
This paper contains a review of the present status of work on intermittency and correlations as performed over the last years. In Chapter 2 we introduce the necessary formalism and collect useful results and relations widely scattered in the literature. Chapter 3 describes experimental data on correlations in various experiments and discusses predictions of popular models. Chapter 4 is devoted to data and models on the subject of particle fluctuations and the search for power laws. Chapter 5 gives an overview of the many theoretical ideas related to the problem of multiplicity scaling and fractality. Conclusions are summarized in Chapter 6.
Chapter 2 Formalism
2.1 Definitions and notation
In this section, we compile and summarize definitions and various relations among the physical quantities used in the sequel of this paper. No originality is claimed in the presentation of this material. It merely serves the purpose of fixing the notation and assembling a number of results scattered throughout the literature.
2.1.1 Exclusive and inclusive densities
We start by considering a collision between particles a and b yielding exactly particles in a subvolume of the total phase space . Let the single symbol represent the kinematical variables needed to specify the position of each particle in this space (for example, can be the c.m. rapidity^{1}^{1}1 The rapidity is defined as , with the energy and the longitudinal component of momentum vector p along a given direction (beamparticles, jetaxis, etc.); pseudorapidity is defined as variable of each particle and an interval of length ). The distribution of points in can be characterized by continuous probability densities ; . For simplicity, we assume all finalstate particles to be of the same type. In this case, the exclusive distributions can be taken fully symmetric in ; they describe the distribution in when the multiplicity is exactly .
The corresponding inclusive distributions are given for by:
(2.1)  
The inverse formula is
(2.2)  
is the probability density for points to be at , irrespective of the presence and location of any further points. The probability of multiplicity zero is given by
(2.3) 
This suggests to define in (2.1). It is often convenient to summarize the above results with the help of the generating functional^{2}^{2}2 The technique of generating functions has been known since Euler’s time and was used for functionals by N.N. Bogoliubov in statistical mechanics already in 1946 [1]; see also [2]
(2.4) 
where is an arbitrary function of in . The substitution
(2.5) 
gives through (2.1) the alternative expansion
(2.6) 
and the relation
(2.7) 
From (2.4) and (2.7) one recovers by functional differentiation:
(2.8) 
and
(2.9) 
To the set of inclusive numberdensities corresponds a sequence of inclusive differential cross sections:
(2.10) 
(2.11) 
Integration over an interval in yields
(2.12) 
where the angular brackets imply the average over the event ensemble.
2.1.2 Cumulant correlation functions
The inclusive particle densities in general contain “trivial” contributions from lowerorder densities. Under certain conditions, it is, therefore, advantageous to consider a new sequence of functions as those statistical quantities which vanish whenever one of their arguments becomes statistically independent of the others. It is well known that the quantities with such properties are the correlation functions–also called (factorial) cumulant functions–or, in integrated form, Thiele’s semiinvariants [3]. A formal proof of this property was given by Kubo [4] (see also Chang et al. [5]). The cumulant correlation functions are defined as in the cluster expansion familiar from statistical mechanics via the sequence [6, 7, 8]:
(2.13)  
(2.14)  
(2.15)  
and, in general, by
(2.16)  
Here, is either zero or a positive integer and the sets of integers satisfy the condition
(2.17) 
The arguments in the functions are to be filled by the possible momenta in any order. The sum over permutations is a sum over all distinct ways of filling these arguments. For any given factor product there are precisely [7]
(2.18) 
terms. The complete set of relations is contained in the functional identity:
(2.19) 
where
(2.20) 
It follows that
(2.21) 
The relations (2.16) may be inverted with the result:
(2.22)  
In the above relations we have abbreviated to ; the summations indicate that all possible permutations have to be taken (the number under the summation sign indicates the number of terms). Expressions for higher orders can be derived from the related formulae given in [9].
It is often convenient to divide the functions and by the product of oneparticle densities. This leads to the definition of the normalized inclusive densities and correlations:
(2.23)  
(2.24) 
From expression (2.19) it can be deduced that, at finite energy, an infinite number of will be nonvanishing: The densities vanish for , where is the maximal number of particles in allowed e.g. by energymomentum conservation. As a consequence, the functional is a “polynomial” in . This in turn requires the exponent in (2.19) to be an “infinite series” in . In other words, the higherorder correlation functions must cancel the lowerorder ones that contribute to a vanishing density function. Phenomenologically, this implies that it is meaningful to use correlation functions only if the number of correlated particles in the considered phasespace domain is considerably smaller than the average multiplicity in that region [2]. These conditions are not always fulfilled in presentday experiments for very small phasespace cells, with the exception of perhaps collisions.
2.1.3 Correlations for particles of different species
The generating functional technique of Sect. 2.1.1 can be extended to the general situation where several different species of particles are distinguished. This will not be pursued here and we refer to the literature for details [2, 10, 11, 12]. Considering two particle species a and b, the twoparticle rapidity correlation function is of the form:
(2.25) 
with
(2.26) 
Here, and are the c.m. rapidities, the inelastic cross section and a, b represent particle properties, e.g. charge.
The normalization conditions are:
(2.27) 
(2.28) 
where for the case when a and b are particles of different species and for identical particles, and and are the corresponding particle multiplicities.
Most experiments use
(2.29) 
so that the integral over the correlation function (equal to the ratio of the negative binomial parameters [13]) vanishes for the case of a Poissonian multiplicity distribution. Other experiments use
(2.30) 
to obtain a vanishing integral also for a nonPoissonian multiplicity distribution.
To be able to compare the various experiments, we use both definitions and denote the correlation function when following definition (2.29) and when following definition (2.30). We, furthermore, use a reduced form of definition (2.30),
(2.31) 
The corresponding normalized correlation functions
(2.32) 
follow the relations
(2.33) 
and is defined as . These are more appropriate than when comparisons have to be performed at different average multiplicity and are less sensitive to acceptance problems.
The correlation functions defined by expressions (2.25)(2.33), contain a pseudocorrelation due to the summation of events with different charge multiplicity and different semiinclusive singleparticle densities .
The relation between inclusive and semiinclusive correlation functions has been carefully analyzed in [14]. Let be the topological cross section and
(2.34) 
The semiinclusive rapidity single and twoparticle densities for particles a and b are defined as
(2.35) 
The inclusive correlation function can then be written as
(2.36) 
where
(2.37) 
(2.38) 
with and . In (2.37) is the average of the semiinclusive correlation functions (often misleadingly denoted as “shortrange”) and is more sensitive to dynamical correlations. The term (misleadingly called “longrange”) arises from mixing different topological singleparticle densities.
A normalized form of can be defined as
(2.39) 
and and their normalized forms and are defined accordingly, with the averages and replaced by and , respectively.
Analogous expressions may be derived for threeparticle correlations. They are discussed in Sect. 3.4.
2.1.4 Factorial and cumulant moments
When the parametric function is replaced by a constant , the generating functionals reduce to the generating function for the multiplicity distribution. Indeed, the probability for producing particles is given by
(2.40) 
and we have
(2.41)  
(2.42)  
(2.43) 
The are the unnormalized factorial (or binomial) moments
(2.44)  
This relation can (formally) be inverted. If for then an approximation for is given by:
(2.45) 
and is included between any two successive values obtained by terminating the sum at and , respectively.
In (2.44) denotes the multiplicity in and the average is taken over the ensemble of events. All the integrals are taken over the same volume such that . Using the correlationfunction cluster decomposition, one further has
(2.46) 
The are the unnormalized factorial cumulants, also known as Mueller moments [8]
(2.47) 
the integrations being performed as in (2.44). The quantities and are easily found if is known:
(2.48)  
(2.49)  
and 

(2.50) 
Using Cauchy’s theorem, this can also be written as
(2.51) 
where the integral is on a circle enclosing . Equation (2.51) is sometimes useful in deriving asymptotic expressions for in terms of factorial moments or cumulants [15, 8].
As a simple example, we consider the Poisson distribution
for which
(2.52) 
showing that for . In that case one has:
(2.53) 
The expressions of density functions in terms of cumulant correlation functions, and the reverse relations, are duplicated for their integrated counterparts. They follow directly from the equations:
(2.54)  
or 

(2.55) 
by expanding either the exponential in (2.54) or the logarithm in (2.55) and equating the coefficients of the same power of . One finds [9]:
The latter formula can also be written as:
(2.58) 
(with , ) and is wellsuited for computer calculation. An equivalent relation was derived in [8]. The (ordinary) moments:
(2.59) 
may be derived from the moment generating function
(2.60) 
since
(2.61) 
We note the useful relations
(2.62)  
(2.63) 
Moments and factorial moments are related to each other by series expansions. From the identities [16]:
(2.64)  
(2.65) 
where and are Stirling numbers of the first and second kind, respectively, follows directly:
(2.66)  
(2.67) 
Cumulants can be defined in terms of the moments in the standard way [17, 9]. They obey relations identical to (2.56). The cumulants are integrals of the type (2.47) of differential quantities known as density moments. These are discussed in [18, 19]. Relations expressing central moments in terms of factorial moments via noncentral Stirling numbers are derived in [20].
2.1.5 Cellaveraged factorial moments and cumulants; generalized moments
In practical work, with limited statistics, it is almost always necessary to perform averages over more than a single phasespace cell. Let be such a cell (e.g. a single rapidity interval of size ) and divide the phasespace volume into nonoverlapping cells of size , independent of . Let be the number of particles in cell . Different cellaveraged moments may be considered, depending on the type of averaging.
Normalized factorial moments [21, 22], which have become known as vertical moments, are defined as^{3}^{3}3 Here and in the following we consider rapidity space for definiteness
(2.69) 
The full rapidity interval is divided into equal bins: ; each is within the range and .
One may also define normalized horizontal moments by
(2.70) 
with .
Horizontal and vertical moments are equal if . Vertical moments are normalized locally and thus sensitive only to fluctuations within each cell but not to the overall shape of the singleparticle density. Horizontal moments are sensitive to the shape of the singleparticle density in and further depend on the correlations between cells. To eliminate the effect of a nonflat rapidity distribution, it was suggested to either introduce correction factors [23] or use “cumulative” variables which transform an arbitrary distribution into a uniform one [24, 25].
Likewise, cellaveraged normalized factorial cumulant moments may be defined as
(2.71) 
They are related [26] to the factorial moments by^{4}^{4}4 The higherorder relations can be found in [26]
(2.72) 
In and higherorder moments, “bar averages” appear. They are defined as .
Besides factorial and cumulant moments, other measures of multiplicity fluctuations have been proposed. In particular, moments [27]—known in statistics as frequency moments [9]—were extensively used to investigate whether multiparticle processes possess (multi)fractal properties [28, 29]. moments are defined as
(2.73) 
Also here, is the number of particles in bin , the absolute frequency; is the total multiplicity in an initial interval and is the number of bins at “resolution” . Bins with zero content (“empty bins”) are excluded in the sum, so that can cover the whole spectrum of real numbers. For negative, is sensitive to “holes” in the rapidity distribution of a single event. Note that in (2.73) is not a probability but a relative frequency or “empirical measure” in modern terminology. For small , moments are very sensitive to statistical fluctuations (“noise”), especially for large M. This seriously limits their potential. In attempts to reduce this noisesensitivity, modified definitions have been proposed in [30].
2.1.6 Multivariate distributions
The univariate factorial moments characterize multiplicity fluctuations in a single phasespace cell and thus reflect only local properties. More information is contained in the correlations between fluctuations (within the same event) in two or more cells. This has led to consider multivariate factorial moments. For nonoverlapping cells, the 2fold factorial moments, also called correlators, are defined as:
(2.74) 
where () is the number of particles in cell (cell ). A normalized version of the twofold correlator is discussed in [21] and defined as:
(2.75) 
For reasons of statistics, these quantities are usually averaged over many pairs of cells, keeping the “distance” () between the cells constant^{5}^{5}5 In onedimensional rapidity space, is defined as the distance between the centers of two rapidity intervals; in multidimensional phase space a proper metric must first be defined. This averaging procedure requires the same precautions regarding stationarity of single particle densities as for their singlecell equivalents.
Multifold factorial moments are a familiar tool in radio and radar physics and in quantum optics [31]. There, they relate to simultaneous measurement of photoelectron counts detected in, say , timeintervals, or in space points, leading to a joint probability distribution . The importance of multifold moments derives from the fact that, e.g. in the simplest case of two cells, is directly related to the autocorrelation function of the radiation field and obeys, for small cells, the Siegertrelation [31], whatever the statistical properties of the field. The higherorder moments are sensitive to higherorder correlations and to the phase of the field.
Factorial moments and factorial correlators are intimately related quantities. In terms of inclusive densities one has:
(2.76) 
where is the inclusive density of order . The integrations are performed over two arbitrary (possibly overlapping) phasespace cells and , separated by a “distance” .