The distribution of measured decay lengths for the 257 selected events is plotted in Fig. 4(a). The average decay length is extracted from a maximum likelihood fit using an exponential decay distribution function convoluted with a Gaussian resolution function:
where the are the measured decay lengths, is the parent decay length, and is a scale factor on the calculated decay length errors . The are determined on an event by event basis from a projection of the three-dimensional vertex error matrix along the event thrust axis, combined with a small contribution from the uncertainty in the transverse beam position. The mean value of the decay length error for these events is calculated to be m. From the fit, the average decay length is found to be mm, with a scale factor . The solid curve in Fig. 4(a) represents the fit function with these values, normalized to the data points. As a measure of the goodness of the fit, the per degree of freedom for the normalization is 0.8.
Figure 4(b) shows the decay length distribution for 2846 Monte Carlo tau-pair events generated with a mean lifetime of 305 fs and passing the same event selection criteria as the data. The fit yields an average decay length mm, corresponding to a mean lifetime of fs, in good agreement with the generated value. The average decay length error for these events is m, and the fitted scale factor is . Since the maximum likelihood fit to the Monte Carlo events returns the generated lifetime, no bias is attributed to the method and no associated systematic error is assigned.
A Monte Carlo study using a large number of event samples showed that the fitted value of the scale factor from the data is consistent with a statistical fluctuation from unity, as the distribution of is skewed toward larger values. As can be seen in Fig. 4, is sensitive primarily to events with relatively large decay length, while is determined mainly from decay lengths near zero, so that the lifetime result is fairly insensitive to the value of the scale factor. This was checked in the data by repeating the maximum likelihood fit with a fixed scale factor of unity. From the corresponding change in the fitted average decay length, a conservative systematic error of 0.8%in the lifetime is assigned.
The effect of the track quality requirements and the cut on the vertex fit probability was studied in the data. From the changes in the result when these quantities were varied, we derive a systematic error of 1.0%. The sensitivity of the measurement to the event selection requirements was studied in the Monte Carlo, and the systematic uncertainty was found to be negligible. As seen in Table i, the only significant background found to survive all the analysis cuts comes from multihadron final states at a level of about 0.3%of the 1-3 sample. It was determined that the average lifetime of such events is consistent with zero, and that including this background reduces the fitted decay length by 0.3%. Due to the uncertainty in the modeling of low-multiplicity hadronic events, a conservative systematic error equal to the full amount of this correction is assigned.
To study the effect of a possible misalignment between the VXD and CDC, the data were divided into four samples by azimuthal quadrants of the detector where decay vertices were found. Lifetimes determined separately for the four samples were the same within statistical errors, and the associated systematic uncertainty is estimated to be negligible. The effect of non-Gaussian tails in the distribution of beam positions has been estimated to be negligible in this measurement. The two-dimensional projection of the decay length using the thrust axis was checked by using instead the net momentum of the three-prong to define the tau direction, and results were found to be consistent. The uncertainty in the calculation of initial- and final-state radiation in the Monte Carlo was estimated to contribute a systematic error of 0.3%in the average boost of the taus. Finally, the effect of uncertainty in the SLC beam energy and energy spread was studied by varying these quantities in the Monte Carlo for tau events and observing the change in the average tau momentum. A systematic uncertainty in the lifetime of 0.3%is estimated from this source.
The systematic effects discussed above are summarized in Table ii. From the Monte Carlo, we find for the average boost of the tau in these events: . Applying this factor, and including background correction and systematic errors, the decay length method yields a tau lifetime This measurement was checked by an independent analysis in which the decay length was computed in three dimensions and considerable attention was paid to reducing the errors on the longitudinal position of the interaction point. A consistent result was obtained.