Shape analysis of HBT correlations T. Csörgő, S. Hegyi and W. A. Zajc (KFKI RMKI Budapest & Columbia, New York) Bose-Einstein Correlations for Lévy Stable Sources Model independent analysis: Edgeworth and Laguerre expansions Adding plane wave approximation: Central limit theorem, Gaussians Generalized CLT, Lévy stable laws Two and three particle HBTcorrelations Fits to data

Really Model Independent Method HBT or B-E: often believed to be dull Gaussians In fact: may have rich and interesting stuctures Experimental conditions for model independent study: C(Q) const for Q infinity C(Q) deviates from this around Q= 0 Expand C(Q) in the abstract Hilbert space of orthogonal functions, identify the measure in H with the zeroth order shape of C(Q). Method works not only for Bose-Einstein correlations but for other observables too. T. Cs, S. Hegyi, Phys. Lett. B489 (2000) 489, hep-ph/

Principle of Expansions Measure and complete orthonormal set of functions: Identify the measure with the approximate 0th order shape of the correlations, using t=QR

Laguerre Expansion Expansion around an approximately exponential shape

Edgeworth Expansion Expansion around an approximately Gaussian shape (generalization to multidimensional systems is possible) Hermite polynomials

2d Edgeworth expansions Example of NA35 S+Ag 2d correlation data, and a Gaussian fit to it (lhs) which misses the peak around q=0. The rhs shows a 2d Edgeworth expansion fit to E802 Si+Au data at AGS (upper panel) compared to a 2d Gaussian fit for the same data set (lower panel). Note the difference in the vertical scales.

3d Edgeworth expansions Easy for factorizable sources, have a peak around Q=0, see Fig. for illustration

Plane wave approximation

Extra Assumption: ANALYTICITY

Limit distributions, Lévy laws The characteristic function for limit distributions is known also in the case, when the elementary process has infinite mean or infinite variance. The simplest case, for symmetric distributions is: This is not analytic function. The only case, when it is analytic, corresponds to the  = 2 case. The general form of the correlation function is where 0 <  <= 2 is the Lévy index of stability. 4 parameters: center x 0, scale R, index of stability  asymmetry parameter 

Examples in 1d Cauchy or Lorentzian distribution,  = 1 Asymmetric Levy distribution, has a finite, one sided support,  = 1/2,  = 1

Asymmetry parameter and three-particle correlations

3d generalization The case of symmetric Levy distributions is solved by with the following multidymensional Bose-Einstein correlations and the corresponding space-time distribution is given by the integral, related to the R -1, the inverse of the radius matrix

Next to do check the existence of the Lévy exponent in collisions at RHIC, by fitting the measured correlation functions relate a to the properties of QCD