Transverse and longitudinal Bose-Einstein correlations in hadronic Z$^0$ decays

Bose-Einstein correlations in pairs of identical charged pions produced in a sample of 4.3 million Z$^0$ hadronic decays are studied as a function of the three components of the momentum difference, transverse (“out" and “side") and longitudinal with respect to the thrust direction of the event. A significant difference between the transverse, r$_{t_{side}}$, and longitudinal, r$_l$, dimensions is observed, indicating that the emitting source of identical pions, as observed in the Longitudinally CoMoving System, has an elongated shape. This is observed with a variety of selection techniques. Specifically, the values of the parameters obtained by fitting the extended Goldhaber parametrisation to the correlation function ${\mathrm C'}~=~{\mathrm C^{DATA}}/{\mathrm C^{MC}}$ for two-jet events, selected with the Durham algorithm and resolution parameter y$_{cut}$ = 0.04, are r$_{t_{side}}$ = (0.809 $\pm$ 0.009 (stat) $^{+0.019}_{-0.032}$ (syst)) fm, r$_l$ = (0.989 $\pm$ 0.011 (stat) $^{+0.030}_{-0.015}$ (syst)) fm and r$_l$/r$_{t_{side}}$ = 1.222 $\pm$ 0.027 (stat) $^{+0.075}_{-0.012}$ (syst). The results are discussed in the context of a recent model of Bose-Einstein correlations based on string fragmentation. The results of a unidimensional analysis are also presented