Volume 223, number 3,4 PHYSICS LETTERS B 15 June 1989 FROM STRONG TO WEAK INTERMITTENCY P. LIPA and B. BUSCHBECK Institut J~r Hochenergiephysikder Osterreichischen Akademie der Wissenschaften,A-1050 Vienna, Austria Received 31 March 1989 The basic ideas of the factorial moment analysis of intermittency in particle physics are shortly reviewed and the connection to a set of commonly used scaling indices for fractal objects, the generalized dimensions, is established. The effect of a random superposition of fractal distributions on the strength of the intermittency signal is studied by means of a simple multifractal one- parameter model. We observe a weakening of intermittency with increasing number of fractal sources within one event. This result might be useful for the interpretation of recent experimental data on this subject. 1. Introduction 2. The measurement of fluctuations Since the interest in intermittency properties of particle collisions rised steadily in the last year [ 1- 7 ] we feel the need to review the definition and to add some not so well known aspects of the factorial moment (FM) method proposed by Bia{as and Peschanski [ 8 ]. This method was originally designed for an analysis of rapidity fluctuations in high multi- plicity events of heavy ion and cosmic ray experi- ments, but soon turned out to be also suitable for the analysis of lower multiplicity collisions produced at present accelerator energies [ 9 ]. Up to now the measurement of multiparticle cor- relations were experimentally almost intractable; but now the FM analysis not only makes an experimental study of multiparticle correlations at short distances very attractive from the statistical point of view, but also relates eventual scaling behavior of such corre- lations to the physics of fractal objects. Thus this method provides an exciting new tool to study spe- cific questions in those parts of multiparticle physics which are concerned with short-range correlations: the nature of spikes [ 10 ], the existence of hadronic Cerenkov radiation [ 11 ], the formation of hot or cold QGP [4], the fractal structure of jets [ 12,3] and last but not least the properties of Bose-Einstein correla- tions [6 ]. Biatas and Peschanski propose in ref. [ 8 ] to divide the region of interest Ay (e.g. the rapidity plateau re- gion) in M bins of size 8y = Ay/M and to character- ize fluctuations by scaled moments of the probabili- tiespm(m= 1, ..., M) to find a particle in the mth bin: (C i)=: (Mpm) i , (1) 1 where i= 1, 2 .... is the order of the moment (the case i= 1 is trivial: ( C i) = 1 ) and the brackets ( ) denote the average over many events. The study of the de- pendence of ( C i) on the bin size ~y gives informa- tion on the structure, size and on possible scaling be- havior of rapidity spikes. There is one major difficulty in measuring the mo- ments as defined in ( 1 ): the probabilities pm can only be estimated by the ratio nm/N (nm is the number of particles in the ruth bin and N is the total number of particles in the interval Ay); this estimate is accurate enough only for large N and gets worse for smaller bin sizes 6y. Moreover, it is well known [ 13 ] that N par- ticles filled in a histogram with M bins according to the probabilities Pm (m = 1.... , M) follow a multi- nomial distribution; thus the observed fluctuations are composed of fluctuations of the probabilities Pm (which are called dynamical fluctuations) and of multinomial fluctuations of the ratios nm/N around their averages p,, (which are called statisticalfluctua- 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division ) 465