Analysis of the L3 BEC at Z${}^0$-pole -- Comparison of the conventional formula against the $\tau$-model

Abstract

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