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

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. [ABSTRACT FROM AUTHOR]