Fluctuations and fermi-dirac correlations in e+e--annihilation

Abstract

In this talk I first present a short review of fluctuations in e + e − -annihilations. I then describe some new results on FD correlations. Experimental analyses of pp and ΛΛ correlations indicate a very small production radius. This result relies very strongly on comparisons with MC simulations. A study of the approximations and uncertainties in these simulations implies that it is premature to draw such a conclusion from the data.Keywords: Fluctuations; Correlations I. FLUCTUATIONSIn bremsstrahlung in QED, the emission of a photon does not change the current for subsequent emissions. This implies that the photon multiplicity is described by a Poisson distribution. As the gluons carry colour charge, the emission of a gluon in QCD changes the current relevant for the subsequent emissions. An initial hard gluon will radiate many softer gluons. This leads to a cascade with an exponential growth in multiplicity and to large fluctuations. The multiplicity distribution is approximately satisfying KNO scaling, with a width which is proportional to the average multiplicity. This short review is divided in three parts:A. Multiplicity distribution for partons B. Effects of hadronization C. Fractal structures A. Multiplicity distribution for partonsIn e + e − -ann. the emission of a gluon from a qq pair is a coherent emission from e.g. a red and an anti-red colour charge. The result corresponds to a colour dipole with the following distribution (in the leading log approximation):The phase space is a triangular region in (y, ln k 2 ⊥ )-space given by |y| < ln(s/k 2 ⊥ )/2, and within this triangle the density is just given byᾱ. The emission of a second (softer) gluon is given by two dipoles, one between the quark and the first gluon, and one between this gluon and the antiquark. The phase space is enlarged compared to that for the first gluon, and corresponds to the folded surface shown in Fig. 1a. For subsequent gluons the phase space is further increased, and corresponds to the fractal surface in Fig. 1b.We let P(N, L = ln s) denote the distribution in the number, N, of dipoles (which neglecting qq pair creation is equal to the number of gluons plus one). The Laplace transform P (γ, L) is given byThe phase space for gluon emission in e + e − -ann. is a triangular region in the (y, κ = ln k 2 ⊥ )-plane. The height of the triangle is given by L = ln s. When one gluon is emitted at (y 1 , κ 1 ) a second gluon with smaller k ⊥ can be emitted in the larger phase space shown in Fig. (a). Further softer emissions give the fractal phase space in Fig. (b).It is then relatively easy to derive the relation (see e.g. ref.[1])Expanding eq. (2) in a Taylor series in γ givesInserted in eq. (3) this also implies the following equations for the average and the variance of the multiplicity distribution:The solutions to these equations are Bessel functions, and for high energies we getWe see that the width of the distribution is proportional to N , in agreement with KNO scaling, and thus for large N the distribution is much wider than a Poi...