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IL NUOVO CIMENTO VOL. 110 A, N. 8 Agosto 1997 NOTE BREVI

Multifractal Laurent series expansions at high energies

A. BERSHADSKII

P.O. Box 39953, Ramat-Aviv 61398, Tel-Aviv, Israel (ricevuto l’1 Settembre 1997; approvato il 3 Novembre 1997)

Summary. — An effective Laurent series expansion, based on instability of

cascade-type multifractal spectra to complex pole appearance, is suggested. Good agreement between this approximation and experimental data on multiparticle production is established.

PACS 13.85.Hd – Inelastic scattering: many-particle final states.

1. – Since papers [1] intermittency of multiparticle production at high energies has been attracting the attention of theorists and experimentalists (see, for a recent review, [2] and references therein). Now there are numerous experimental evidences of the intermittent behavior obtained from e1_e2 _{annihilation, hadron,}

hadron-nucleus, and nucleus-nucleus interactions [2]. Interpretation of these experimental data is still an actual problem. The self-similar cascade mechanism is considered as one of the most relevant mechanisms of the intermittent multiparticle production (see, for instance, [3, 4]). This mechanism, however, is related to a very wide class of the generalized dimensions spectra, Dq, [5] and it is difficult to decide what concrete

multifractal spectrum is realized in each concrete case. Therefore in this short note we use some common property for all these spectra (some kind of instability to appearance of complex poles) to obtain a general analytic representation for these spectra in some representative interval of q.

2. – Let us recall some standard definitions. Let Dh be the pseudorapidity interval, subdivided into M bins, each of width dh 4Dh/M. Let N be the number of particles in one event in Dh interval and km be the number of particles in the m-th bin. The Gq

moments are defined as

Gq4

!

M pmq,

where pm4 km/N is the probability that particles are in the m-th bin for one event and q is any real number. The summation is carried over non-empty bins only. If the particle production process exhibits self-similar behavior, then the moment follows the power law GqP (dh)t(q). The generalized dimensions spectrum Dq4 t(q) /(q 2 1 ).

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3. – The first successful application of the self-similar cascade mechanism to an intermittent process was provided in paper [6] for turbulent energy dissipation field. It was shown in this paper that a simplest multifractal spectrum corresponding to such type of processes is

Dq4 const 1 ( 1 2 q)21log2[pq1 ( 1 2 p)q] ,

(1)

where p E1 is some constant. Then this spectrum was generalized in papers [5, 7] as Dq4 const 1 ( 1 2 q)21log

!

j njpjq,

(2)

where nj and pj are some positive constants, and

!

general class of Cantor set models of the self-similar cascade processes [5, 8]. Moreover, using the Lebesgue-Stielties integral this representation can be extended to [5, 7]

Dq4 const 1 ( 1 2 q)21log



(3)

with



4. – Let us consider a spectrum

Dq4 const 1 ( 1 2 q)21log2[ ( 1 1e)pq1 ( 1 2 ( 1 1 e) p)q]

(4)

such that lim

e K0D

(e)

q 4 Dq given by representation (1). While spectrum (1) and its

derivatives have no singularities in the complex plane (if q is considered as a complex variable), the first derivative of Dq(e)has a simple pole at q 4q0

q04

ln( 1 1e) ln( 1 1e21/p) . (5)

It is clear that lim

e K0q04 0, and the value of q0is complex for small enough e, due to p E1.

It can be shown that this kind of instability to appearance of the complex poles is also a property of the generalized spectra (2) and (3).

One can use this property to obtain a general approximation of these multifractal spectra in some interval of q. Indeed, one can expand dDq(e)/dq (with 1 c NeN) with

Laurent power series at q 4q0

dDq(e) dq 4 a₂₁(e) q 2q0 1

!

Then it follows from (5) and (6) that for NqND1 dD(e) q dq C a21(e) q 1n 40

!

!

in some interval 1 ENqNER. Using condition 1 cNeN one can apply this approximation also for Dq given by eq. (1) along the real axis. It should be noted, however, that the

MULTIFRACTAL LAURENT SERIES EXPANSIONS AT HIGH ENERGIES 889

case. The leading term of this approximation is

DqC D611 a21(6)ln NqN .

(9)

It is easy to check that already the leading term approximation (9) gives a good fit of the multifractal spectrum (1) in some representative interval of q. More important, however, approximation (9) is also applicable to the generalized spectra (2) and (3).

5. – To apply this approximation to real multifractal data, let us start from phase transitions from periodic attractors to chaos in the Feigenbaum scenario. It is interest-ing to note that in a recent paper [9] it is shown that there is a direct relationship between multifractality of Feigenbaum strange attractors and multifractality of the multiparticle production at high energies. In paper [10] the generalized dimensions Dq,

for critical strange sets which refer to the Feigenbaum-type attractors formed on criti-cal points of transitions to chaos in 1D iterative systems, were criti-calculated. Figure 1 (adapted from [10]) shows these generalized dimensions calculated for the map f(x) 4 1 2aNxNzfor z 43. The axes in this figure are chosen for comparison with (9). Upper set of data (dots) corresponds to negative values of q, whereas lower set of data corre-sponds to positive values of q. The straight lines are drawn for comparison with (9). One can see good agreement between the data and approximation (9) (it should be noted that Dqin this case is normalized on D0[10]).

In paper [4] the multifractal spectrum has been constructed using recent experimental data [11] on interaction of hadron with emulsion nuclei at an energy of

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Fig. 2.

800 GeV/c. The multifractal spectrum was constructed for various cuts on the number (Ng) of medium-energy (“grey”) particles. We show in fig. 2 (adapted from [4]) the data

corresponding to Ng4 0 (full circles) and to Ng4 1 , 2 (open circles). The upper sets of

data (both open and full circles) correspond to negative values of q, whereas the lower sets of data (both open and full circles) correspond to positive values of q. The straight lines are drawn for comparison with (9). One can see that parameters a(6)

1 are

practically independent of Ng.

* * *

The author is grateful to I. HOSOKAWA and to K. R. SREENIVASAN for information and encouragement.

R E F E R E N C E S

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