  Long-range correlation plays an important role in analyses of pionic
Bose-Einstein correlations (BECs). In many cases, such correlations are
phenomenologically introduced. In this investigation, we propose an analytic
form. By making use of the form, we analyze the OPAL BEC and the L3 BEC at
$Z^0$-pole and the CMS BEC at 0.9 and 7 TeV using our formulas and the
$\tau$-model. The parameters estimated by both approaches are found to be
consistent. Utilizing the Fourier transform in four-dimensional Euclidean
space, a number of pion-pair density distributions are also studied.
