As described in section III.C, the event selection yielded a total of 1556 tracks for this lifetime determination. The distribution of impact parameters measured for these tracks is shown in Fig. 5. The impact parameter is assigned a positive (negative) value if the extrapolated track crosses the event thrust axis before (after) its point of closest approach to the interaction point. Negative impact parameters result from finite tracking errors and uncertainties in the beam position determination.
To extract a lifetime from the impact parameter distribution, a binned maximum likelihood fit to the data is performed. The fit function is represented by the impact parameter distribution from the Monte Carlo, corrected for background and normalized to the number of events in the data. The likelihood probability is expressed as follows: where N is the number of bins in the distribution used in the fit, is the number of entries in the data bin, and is the normalized content of the bin in the Monte Carlo. The bin width chosen is 10 m, except that in the tails of the Monte Carlo distribution the bins are widened as required to include at least 10 events. The data are then binned the same as the Monte Carlo.
A single Monte Carlo sample was generated with an input lifetime of fs. A weighting technique is then used to generate the impact parameter distributions corresponding to alternative lifetimes. A track originating from a tau decay with proper time is taken to have been produced from a sample with different lifetime by weighting its assigned probability with the following ratio: Weighted impact parameter distributions are formed with seven different lifetimes in the range . These distributions are corrected for background by adding the normalized impact parameter distribution for background tracks passing all cuts. A likelihood probability is calculated for each distribution, and a fourth-order-polynomial fit is performed to the seven values as a function of lifetime. The best fit lifetime is taken to be the value corresponding to the maximum of this curve, with a statistical error assigned by taking the values where the likelihood has decreased by 0.5 from the maximum. The result is The impact parameter distribution corresponding to the best fit lifetime in the Monte Carlo, including the background correction, is shown as the solid histogram in Fig. 5. The agreement between data and Monte Carlo is very good, and the comparison between the two distributions gives a of 0.9 per degree of freedom.
This analysis was checked for systematic bias using several different Monte Carlo samples of 1000 tracks each in place of the data. In every case the generated lifetime was reproduced within the statistical errors. A similar check was made with several groups of Monte Carlo events using the true generated tau direction in place of the event thrust axis, and results were consistent within statistics. The statistical error obtained from the likelihood fit was found to be consistent with the fluctuations in the results from the different Monte Carlo samples.
The sensitivity of the fitting technique to bin size and Monte Carlo statistics was studied extensively. The analysis was repeated with nominal bin sizes a factor of two larger and smaller than the chosen m, and the minimum number of events required in a Monte Carlo bin was also varied. The corresponding variations in the result lead to a systematic uncertainty of 0.3%. A study using several Monte Carlo samples of different sizes lead to an estimated systematic error of 0.9%due to finite Monte Carlo statistics. The range of impact parameters covered by the fit was varied, equivalent to a symmetric trimming of the sample from zero to 5%, both for the data and Monte Carlo. The results fluctuate within statistical errors, and no associated systematic error is assigned.
All track quality cuts were varied in order to study their effect on the measurement. From the observed changes in the measured lifetime, we assign a conservative systematic error of 1.0%. The uncertainty in the background correction was checked by varying the background fraction in the Monte Carlo, and a corresponding systematic error of 0.1%is assigned.
The possibility of detector misalignment was checked as described above for the decay length method, and no statistically significant effect was observed. The effect of non-Gaussian tails in the beam position determination was studied in the Monte Carlo by introducing an additional smearing in the beam position in a fraction of events. The change in the measured lifetime was small and a conservative systematic error of 0.4%is assigned. We have also investigated the effect of uncertainties in the tau one-prong branching ratios used in the Monte Carlo. We estimate a systematic error of 0.2%from this source. Finally, as mentioned above for the decay length method, we assign a systematic error of 0.3%due to initial- and final-state radiation, and another 0.3%due to the uncertainty in the beam energy and beam-energy spread.
A summary of the systematic errors is given in Table iii. Including systematic errors, the direct impact parameter method yields a tau lifetime