As described in section III.C, the event selection yielded a sample of
642 events for this analysis. The impact parameter difference and acoplanarity
distributions for these events are plotted in Fig. 6(a)
and 6(b),
respectively[17]. The data (represented by the dots)
and the Monte Carlo (by the histograms) are seen to be in good agreement.
In Fig. 7(a), the scatter plot of impact parameter difference
versus acoplanarity for the data is shown. A clear correlation is evident, and
the tau lifetime can be extracted from the slope of the distribution.
This is even better illustrated in Fig. 7(b) where each data point
represents the mean impact parameter difference over a given bin
in
.
A straightforward way to extract the tau lifetime is to fit the distribution in Fig. 7(b) to a straight line over a range in acoplanarity where the correlation is linear, together with a trimming procedure in order to remove from the fit tails that are not well modeled. In our case, we have chosen an alternative method based on a binned maximum likelihood method. This is motivated by the fact that in any acoplanarity bin, the impact parameter difference distribution is asymmetric and non-Gaussian (it is exponentially distributed) and any truncation would result in a bias in the lifetime.
The maximum likelihood technique used here is similar to that described in
section IV.B for the single-impact parameter method; the only difference is that
the data are described in two-dimensional bins in impact parameter difference
and acoplanarity. Just as in the impact parameter method, a single Monte Carlo
sample was generated, and several samples with different lifetimes were
simulated using a weighting technique with a weight defined by:
where
is mean lifetime in the original Monte Carlo sample,
the
alternative lifetime, and
and
the decay times of the two taus in
the event.
Before performing the fit, ten events lying in the tails of the impact
parameter and acoplanarity distributions were removed (this corresponds to a
1.4%trim). This was done by defining a fit region represented by a
parallelogram surrounding the core of the scatter plot in Fig. 7(a).
In addition, in order to reproduce the distribution of impact parameter
difference vs. acoplanarity as provided by Monte Carlo, the binning used in
the maximum likelihood fit was chosen to be very fine in the middle of the
scatter plot and progressively coarser towards the tails.
In order to account for background, Monte Carlo events from the various
sources of contamination were merged with the tau Monte Carlo sample used in
the fit. The result of the fit for the tau lifetime using this technique is:
The error is statistical only. A comparison between the scatter plot in
Fig. 7(a) and its counterpart from Monte Carlo yields a
per degree of freedom of 1.2.
The largest source of systematic errors comes from the fitting procedure. We assign a systematic error of 1.6%due to the sensitivity of the measurement to the chosen bin size. This is due mainly to the tails of the impact parameter difference and acoplanarity distributions where the finite statistics in the Monte Carlo do not allow a very fine binning. In the Monte Carlo sample, we require a minimum of 5 events in each square bin in impact parameter difference and acoplanarity in order to compute the likelihood probability for that bin. When this is not satisfied, entries from neighboring bins are progressively combined until the minimum number of events is reached. We have varied the number of entries required from 1 to 10 and observe no change in the measured lifetime. We have also studied the effect of the size of the fit region which we have varied over a wide range. This lead to assigning a systematic error of 1.0%. Furthermore, we estimate a systematic error of 0.7%due to Monte Carlo statistics, determined from the spread in the measured lifetime as a function of the size of the analyzing Monte Carlo sample used in the maximum likelihood fit.
Various analysis cuts were varied to study the effect of the event
selection and the track quality criteria. We observe a 0.9%change in the
measured lifetime when we vary the cuts that had the most effect at reducing
the remaining background in the 1-1 sample, namely the cuts on the two-prong
mass, the maximum track momentum, and . By varying the
minimum track momentum and the minimum number of VXD and CDC hits on a track,
we assign a systematic error of 0.5%due to track
quality cuts. The sensitivity to the detector
alignment was studied by dividing the data into bins in both the azimuth and
polar angle; no significant effect was observed.
Similarly to the IP method, we assign a systematic uncertainty of 0.2%due to possible variations in the impact parameter distribution associated with each one-prong decay channel of the tau. Furthermore, we estimate a contribution of 0.3%from both initial- and final-state radiation and the uncertainty in the beam energy and the beam-energy spread.
Other systematic effects were investigated. The analysis was performed on data samples from different run periods; the results were all consistent. An important systematic check of the impact parameter difference technique was done by running on event samples with known zero lifetimes, namely Bhabha scattering and muon-pair final states. This is illustrated in Fig. 8 which shows the impact parameter difference versus acoplanarity for (a) muon-pair and (b) wide-angle Bhabha events, both taken from the data. The two distributions are flat as expected, and the majority of the events are clustered at the origin[18]. As a result, this method is relatively insensitive to background.
Replacing the data sample by several Monte Carlo data sets with the same number of events resulted in a spread in the measured lifetimes consistent with the statistical errors, while their mean value was very close to the input value in the Monte Carlo. This is a check that the method does not bear any systematic bias. This was further confirmed using several generator-level Monte Carlo samples with different lifetimes; the value of the tau lifetime with which the sample was generated was reproduced in each case.
The systematic errors are summarized in Table iv. Adding all
contributions in quadrature leads to an overall systematic error of 2.3%.
Thus, the result from the impact parameter difference method is:
where the first error is statistical and the second is systematic.