BEYOND GAUSSIAN APPROXIMATION (EXPERIMENTAL) W. KITTEL Radboud University Nijmegen

Some History A.Wróblewski (ISMD’77): BEC does not depend on energy BEC does not depend on type of particle (except AB) However: BEC does depend on statistics of experiment! (R increases with increasing N ev ) G.Thomas (PRD77), P.Grassberger (NP77): ρ-decay will make it steeper J.Masarik, A.Nogová, J.Pišút, N.Pišútova (ZP97): more resonances even power-like

Some More History B.Andersson + W.Hofmann (PLB’86): string makes it steeper than Gaussian (approximately exponential) A.Białas (NPA’92, APPB’92): power law if size of source fluctuates from event to event and/or the source itself is a self-similar (fractal-type) object (see also previous talk)

UA1 (1994)NA22 (1993) Power law, indeed!

Higher Orders NA22 (ZPC’93): multi-Gaussian fits according to M.Biyajima et al. (’90, ’92) as a function of Q 2 : “reasonable” fits As a function of –lnQ 2 : steeper than Gaussian

UA1 (ZPC’93, Eggers, Lipa, Buschbeck PRL’97) compared to Andreev, Plümer, Weiner (1993) Gaussian clearly excluded => power law

Edgeworth and Laguerre Expansion Csörgő + Hegyi (PLB 2000): Edgeworth: R 2 (Q) = γ ( 1 + λ * exp(-Q 2 r 2 ) [ 1 + κ 3 H 3 (2 ½ Qr)/3! + · · · ] with κ 3 = third-order cumulant moment H 3 = third-order Hermite polynomial Laguerre: Replace Gaussian by exponential and Hermite by Laguerre

fits by Csörgő and Hegyi: dashed = Gauss full = Edgeworth

fits again by Csörgő and Hegyi: dashed = exponential full = Laguerre but: low-Q points still systematically above and power law equally good (with fewer parameters)

Higher Dimensionality L3 (PLB 1999): Bertsch-Pratt parametrization (Q L, Q side, Q out ) Gaussian : CL = 3% Edgeworth : = 30%

Lévy-stable Distributions Csörgő, Hegyi, Zajc (EPJC 2004): (see also Brax and Peschanski 91) Lévy-stable distributions describe functions with non-finite variance which behave as f(r) = |r| -1-  for |r|  ∞ ( 0  µ  2) Particularly useful feature: “characteristic function” (i.e. the Fourier transform) of a symmetric stable distribution is F(Q) = exp(iQδ - |γQ| μ )  R 2 (Q) = 1 + exp(-|rQ| μ ) with r = 2 1/μ γ (see following talks )

Conclude Correlation functions at small Q in general steeper than Gaussian Edgeworth (and Laguerre) better, but what is the physics? Power law not excluded Lévy-stable functions allow to interpolate. Are they a solution?

Questions Elongation ( r side /r L < 1) Q inv versus directional dependence r out  r side Boost invariance m T dependence (also in e + e - ) factor 0.5 from m π to 1GeV. Space-momentum correlation non-Gaussian behavior Edgeworth, power law, Lévy-stability Connection to intermittency 3-particle correlations Phase versus higher-order suppression Strength parameter λ

Source image reconstruction Overlapping systems (WW, 3-jet, nuclei) HBT versus string Dependence on type of collision (no, except for heavy nuclei) Energy (virtuality) dependence (no, except for r L ) Multiplicity Dependence r increases λ decreases effect on multiplicity and single-particle distribution