m / GeV
6.3 Unfolding Data Events
6.2.3 Test with τ Monte Carlo
To be able to apply the unfolding method explained above to theOpaldata, special Monte Carlo samples were generated. For the signal channelsτ−→K−π0ντ,τ− →K0π−ντandτ−→K−π+π−ντ 200.000 events were generated for each decay mode. For theτ− →K0π−π0ντ, due to the much lower selection efficiency, 600.000 events are produced. These events were processed through the fullOpaldetector simulation.
The events in these samples are generated between the lower kinematic limit (which depends on the signal channel under investigation) and the upper limit of m2τ = 3.154 GeV2 with a flat mass distribution. This allows for a precise determination of the inverse detector response matrix even in mass regions where the standard τ Monte Carlo does not provide a sufficient number of events. The uncertainty on the inverse detector response matrix due to the Monte Carlo statistic in these sample is of the order of a (3−4)% for the diagonal elements of the matrix and below 10% for the two first off diagonal elements. These events are also used for the determination of the selection efficiency.
In order to test the method, a flat mass distribution and a mass distribution according to phasespace was used to set up the inverse detector matrix. The latter is obtained from the flat distribution by reweighting the Monte Carlo events. Then, a subsample test was performed. For each signal channel, 200 subsamples of the size expected in the data for the corresponding channel were selected. The events were then unfolded using the iterative procedure as explained above.
The result of the unfolding procedure can be seen in Figures 6.3 and 6.4. In the first row in each figure, the results obtained after each iteration step are displayed using a phasespace distribution as initial guess to set up the inverse detector matrix. In the second row the same results are shown, now using a flat mass distribution as initial guess. In each row in the first plot, the mass distribution used as initial guess can be seen on generator level (dotted line) and detector level (dashed line). The dots show the distribution to be unfolded averaged over all subsamples. The next plots in the row, labeled ‘0th Iteration Step’, ‘1st Iteration Step’ and so on, compare the results of the corresponding iteration step (dots) to the original distribution on detector level (dashed line) and generator level (dotted line).
Using a phasespace distribution as initial guess, for all final states considered here, an agreement between the unfolded spectrum and the corresponding distribution on generator level of better than the expected statistical uncertainty was obtained after the second iteration step. Applying on additional iteration step changes the result only on the percent level, which can be seen by comparing the last and the second-to-last plot in the first row for each corresponding channel. Using a flat mass distribution as initial guess for the unfolding matrix, one additional iteration step is necessary to obtain a result which reproduces the corresponding tree distribution on the same level as for the phase space distribution. Also in this case, the obtained result is stable against one additional iteration step.
In all plots discussed here, the error bars do not correspond to the statistical uncertainty of the spectrum.
They represent the spread obtained over the 200 subsamples.
6.TheMassCorrectionProcedure
K π
m/GeV
Entries
Initial Guess (I.G.) phasespace I.G. (tree) I.G. (det)
m/GeV
Entries
0th Iteration Step MC (det)
m/GeV
Entries
1st Iteration Step
m/GeV
Entries
2nd Iteration Step
m/GeV
Entries
3rd Iteration Step
m/GeV
Entries
m/GeV
Entries
0th Iteration Step
m/GeV
Entries
1st Iteration Step
m/GeV
Entries
2nd Iteration Step
m/GeV
Entries
3rd Iteration Step
100 20 3040 50 6070 80 90
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53
100 20 3040 5060 7080 90
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53
K
0π
-m/GeV
Entries
Initial Guess (I.G.) phasespace Detector I.G. (tree) I.G. (det)
m/GeV
Entries
0th Iteration Step MC (tree) MC (det)
m/GeV
Entries
1st Iteration Step
m/GeV
Entries
2nd Iteration Step
m/GeV
Entries
3rd Iteration Step
m/GeV
Entries
Initial Guess (I.G.) flat
m/GeV
Entries
0th Iteration Step
m/GeV
Entries
1st Iteration Step
m/GeV
Entries
2nd Iteration Step
m/GeV
Entries
5th Iteration Step
100 2030 4050 60 7080 90
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53
100 2030 40 5060 70 8090
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53 0
20 40 60 80 100 120 140
0.28 1.405 2.53
6.3:Subsampletestforthetwomesonfinalstates.Foreachsignalchannel,inthefirstrow,theresultoftheunfoldingprocedureusingaphasespacedistributionasinitialguessisdisplayed.Inthesecondrow,aflatmassdistributionwasused.Forfurtherdetails,seeexplanationinthetext.
UnfoldingDataEvents91
K π π
m/GeV
Entries
Initial Guess (I.G.) phasespace I.G. (det)
m/GeV
Entries
0th Iteration Step
m/GeV
Entries
1st Iteration Step
m/GeV
Entries
2nd Iteration Step
m/GeV
Entries
3rd Iteration Step
m/GeV
Entries
Initial Guess (I.G.) flat
m/GeV
Entries
0th Iteration Step
m/GeV
Entries
1st Iteration Step
m/GeV
Entries
2nd Iteration Step
m/GeV
Entries
3rd Iteration Step
0 5 10 15 20 25 30 35 40
0.28 1.405 2.53 1005 15 20 25 30 35 40 45
0.28 1.405 2.53 1005 15 20 25 30 35 40 45
0.28 1.405 2.53 1005 15 20 25 30 35 40 45
0.28 1.405 2.53 1005 15 20 25 30 35 40 45
0.28 1.405 2.53
0 5 10 15 20 25 30 35 40
0.28 1.405 2.53 05
10 15 20 25 30 35 40 45
0.28 1.405 2.53 05
10 15 20 25 30 35 40 45
0.28 1.405 2.53 05
10 15 20 25 30 35 40 45
0.28 1.405 2.53 05
10 15 20 25 30 35 40 45
0.28 1.405 2.53
K
0π
-π
0m/GeV
Entries
Initial Guess (I.G.) phasespace Detector I.G. (tree) I.G. (det)
m/GeV
Entries
0th Iteration Step MC (tree) MC (det)
m/GeV
Entries
1st Iteration Step
m/GeV
Entries
2nd Iteration Step
m/GeV
Entries
3rd Iteration Step
m/GeV
Entries
Initial Guess (I.G.) flat
m/GeV
Entries
0th Iteration Step
m/GeV
Entries
1st Iteration Step
m/GeV
Entries
2nd Iteration Step
m/GeV
Entries
3rd Iteration Step
0 1 2 3 4 5 6 7 8 9
0.28 1.405 2.53 0
2 4 6 8 10 12 14
0.28 1.405 2.53 0
2 4 6 8 10 12 14
0.28 1.405 2.53 0
2 4 6 8 10 12 14
0.28 1.405 2.53 0
2 4 6 8 10 12 14
0.28 1.405 2.53
0 1 2 3 4 5 6 7 8 9
0.28 1.405 2.53 0
2 4 6 8 10 12 14
0.28 1.405 2.53 0
2 4 6 8 10 12 14
0.28 1.405 2.53 0
2 4 6 8 10 12 14
0.28 1.405 2.53 0
2 4 6 8 10 12 14
0.28 1.405 2.53
6.4:Subsampletestforthethreemesonfinalstates.Foreachsignalchannel,inthefirstrow,theresultoftheunfoldingprocedureusingaphasespacedistributionasinitialguessisdisplayed.Inthesecondrow,aflatmassdistributionwasused.Forfurtherdetails,seeexplanationinthetext.
m/GeV
m/GeV
m/GeV
m/GeVm/GeV
m/GeV
Number of Events
m/GeV
Number of EventsNumber of Events
0.2
0.2 0.3
0.3 0.4
0.4 0.5
0.5 0.6
0.6 0.7
0.7 0.8
0.8
0 10 20 30 40 50 60 70
0.9
0.9 1
1
0 0
20 20
40 40
60 60
80 80
100 100
120 120
140 140
160 160
0.5 0.5 0.5
1 1 1
1.5 1.5 1.5
2 2 2
2.5 2.5 2.5
m/GeV
m/GeVm/GeV
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
m/GeV
K ππ
K
0π π
0K
0π K π
00.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1 1.5
1 1.5
1.4 1.6
0.8 1 1.2 0.6 0.8 1 1.2 1.4 1.6 0.6 0.8 1 1.2 1.4 1.6
0.8 1 1.2 1.4 1.6
1 1.5
1 1.5
2 3 4 5 6 7 8 9
0.5 1 1.5 2 2.5
Number of Events
0 1
m/GeV
Fig. 6.5: Unfolded spectra and correlation matrices for the final states K−π0ντ, K0π−ντ, K−π+π−ντand K0π−π0ντ. On the left side the unfolded spectra are displayed. On the right sight, the correlation matrices are given.
In this chapter, the results obtained from the measured invariant mass spectra are presented. At first, new branching fractions for the decay channels τ− → K−π0ντ and τ− → K−π+π−ντ are determined in a simultaneous fit. Since the measured values change the present world averages, improved averages are calculated for these final states. In Section 7.2, the strangeness spectral function is presented and the systematic uncertainties associated with it are discussed. Spectral moments are then determined from the spectral function. In addition, the weighted difference of strange and non-strange spectral moments and their ratio are calculated. Finally, in Section 7.5, the mass of the strange quark is determined from this weighted difference and the obtained result is compared to previous analyses.