BEYOND GAUSSIAN APPROXIMATION (EXPERIMENTAL)

W. KITTELRadboud University Nijmegen

Some History

A.Wróblewski (ISMD’77):BEC does not depend on energyBEC does not depend on type of particle (except AB)However:BEC does depend on statistics of experiment! (R increases with increasing Nev)

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 Q2 : “reasonable” fitsAs a function of –lnQ2 : 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:

R2(Q) = γ ( 1 + λ*exp(-Q2r2) [ 1 + κ3H3(2½Qr)/3! + · · · ]

with κ3= third-order cumulant moment H3= 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 (QL, Qside, Qout)

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-finitevariance 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)

R2(Q) = 1 + exp(-|rQ|μ) with r = 21/μγ

(see following talks )

Conclude

Correlation functions at small Q in general steeper thanGaussian

Edgeworth (and Laguerre) better, but what is the physics?

Power law not excluded

Lévy-stable functions allow to interpolate. Are they asolution?

Questions

• Elongation ( rside/rL < 1) Qinv versus directional dependence

• rout rside

Boost invariance

• mT dependence (also in e+e-) factor 0.5 from mπ 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 rL)

• Multiplicity Dependence r increases λ decreases

• effect on multiplicity and single-particle distribution