The three techniques used in this analysis are illustrated in Fig. 3. Figure 3(a) shows the principle of the decay length method. Using tau-pair events with a 1-3 topology, selected as described above, a vertex fit in three dimensions is performed on the three-prong side to determine the decay point of the tau. Because of the relatively large beam size in the longitudinal direction, a two-dimensional decay length is calculated for each event using the precisely measured transverse beam position and the projection of the three-prong vertex in the transverse plane. The result is then translated into a three-dimensional decay length using the event thrust axis.
Figure 3(b) illustrates the impact parameter method. In this technique, events in the 1-1 data sample are used. For each event, either one or two measurements are extracted according to whether one or both of the two charged tracks in the event satisfy the selection requirements described in section III.C. The impact parameter of a daughter track is related to the transverse decay length of the parent tau as follows (see Fig. 3(b)): where is the angle between the track and the direction of the parent tau, and the polar angle of the tau direction.
The impact parameter difference technique  is illustrated in Fig. 3(c). Using events with a 1-1 topology, selected as described in section III.C, this method exploits the correlation which exists between the difference of the impact parameters of the two tracks in the event and their acoplanarity. This is expressed by the following equation: where and are the impact parameters of the two tracks, is the acoplanarity of the tracks, and is the polar angle of the tau direction.
Directly or indirectly, all three methods involve the measurement of an average decay length . The tau lifetime is related to by: , where is the average boost of the tau and is determined from Monte Carlo. In all three methods, the tau direction is taken to be the event thrust axis, which is determined on an event-by-event basis from charged tracks and isolated energy clusters in the calorimeter. Monte Carlo studies show this to be a very good approximation.
In the decay length (DL) method, the result is derived from an unbinned maximum likelihood fit to the decay length distribution using an analytical function given below. In both the impact parameter (IP) and impact parameter difference (IPD) techniques, we make use of a binned maximum likelihood fit using Monte Carlo to parameterize the experimental quantities that are used in the two measurements: impact parameter for the IP method, and impact parameter difference and acoplanarity for the IPD method.