Eur. Phys. J. A (2008) 37: 227-232
https://doi.org/10.1140/epja/i2008-10624-4

Regular Article - Theoretical Physics

Energy dependence of Lévy stability and multi-fractal spectrum in e⁺e⁻ collisions

¹ Physical Department, School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, China
² Institute of Particle Physics, Huazhong Normal University, Wuhan, 430079, China
³ Key Laboratory of Quark and Lepton Physics, (Huazhong Normal University), Ministry of Education, Wuhan, China

^* e-mail: chengang1@cug.edu.cn

Received: 10 May 2008
Revised: 23 June 2008
Published online: 30 July 2008

Abstract

The energy dependence of non-linear dynamical fluctuations in e⁺e⁻ collisions at high energy is studied using the Monte Carlo method. The Lévy indices μ are measured, and the generalized fractal dimensions D(q) and multi-fractal spectrum f(α) are presented at various energies. It is found that the Lévy index μ distributes within 1 < μ < 2 in consistence with the Lévy stability. It takes approximately the central limit value (μ = 2) at √s = 40 GeV and decreases with the increasing of the c.m. energy, departing farther and farther away from the central limit. The Rényi dimensions D(q) decrease with increasing q, lying in a narrow band for different energies. The multi-fractal spectrums are convex curves with maximum at q = 0. The maximum moves along the α-axis to the right with the increasing of energy. The fractal dimension D _F , i.e. D(q = 0), almost does not change with energy, while the information dimension D _I [D(q = 1)] and the correlation dimension ν [D(q = 2)] decrease with the increasing of the c.m. energy when √s < 80 GeV and approach saturation when √s > 80 GeV.