Analysis of the L3 BEC at Z$^0$-pole -- Comparison of the conventional formula against the $τ$-model

Abstract

The L3 Collaboration reported data on 2-jet and 3-jet Bose--Einstein correlations (BECs) with the results obtained through the $τ$-model in 2011. In this study, we analyze these correlations using the conventional formula with the Gaussian long-range correlation (${\rm CF_I\times LRC_{(Gauss)}}$). The estimated ranges of interactions for 2-jet and 3-jet, $R_{\rm 2-jet}$ and $R_{\rm 3-jet}$, are almost the same magnitude as those by $τ$-model: $R_{\rm 2-jet}=0.83\pm 0.05$ (stat) fm and $R_{\rm 3-jet}=1.09\pm 0.04$ (stat) fm. The anticorrelation in BEC (less than 1.0) observed by the L3 Collaboration is related to the partially negative density profile ($ρ_{\rm τ, BE}(ξ)$ in $ξ$ space) in the $τ$-model and the ${\rm LRC_{(Gauss)}}=1/(1+αe^{-βQ^2})$ in ${\rm CF_I\times LRC_{(Gauss)}}$, respectively. Remarkably, the probability $P_τ(ξ)$ calculated from the Levy canonical form exhibits partially negative behavior.