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BEYOND GAUSSIAN APPROXIMATION (EXPERIMENTAL)

BEYOND GAUSSIAN APPROXIMATION (EXPERIMENTAL). W. KITTEL Radboud University Nijmegen. Some History A.Wr óblewski (ISMD’77): BEC does not depend on energy BEC does not depend on type of particle (except AB) However:

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BEYOND GAUSSIAN APPROXIMATION (EXPERIMENTAL)

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  1. BEYONDGAUSSIAN APPROXIMATION (EXPERIMENTAL) W. KITTEL Radboud University Nijmegen

  2. 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 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

  3. Some More History • B.Andersson + W.Hofmann(PLB’86): • string makes it steeper than Gaussian • (approximately exponential) • 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)

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

  5. Higher Orders NA22 (ZPC’93): multi-Gaussian fits according to M.Biyajima et al. (’90, ’92) as a function of Q2 : “reasonable” fits As a function of –lnQ2 : steeper than Gaussian

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

  7. 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

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

  9. 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)

  10. Higher Dimensionality L3 (PLB 1999): Bertsch-Pratt parametrization (QL, Qside, Qout) Gaussian : CL = 3% Edgeworth : = 30%

  11. 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)  R2(Q) = 1 + exp(-|rQ|μ) with r = 21/μγ (see following talks )

  12. 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?

  13. 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

  14. 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

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