m / GeV
5.3 Invariant Mass Spectra
m/GeV
Number of Events
OPAL K−π−π+π0 Background
m/GeV
Number of Events
Signal−Bkg
0 2 4 6 8 10 12 14 16 18 20
0.5 1 1.5 2 2.5 −1
0 1 2 3 4 5 6 7 8
(a) (b)
9 10
0.5 1 1.5 2 2.5
Fig. 5.7: Invariant mass spectra of the K−π+π−π0ντ final states. Plot (a) shows the measured invariant mass spectrum. The dots are the data, the open histogram is the Monte Carlo signal and the shaded area is the background. Plot (b) shows the background subtracted spectrum.
The same procedure as for the K−π+π−ντ channel is used. In addition, one identifiedπ0 meson with an energy of more than 2 GeV is required. The invariant mass spectrum can be seen in Figure 5.7. From this selection, 14 events are seen in the data with a contribution of 10 events from background. The selection efficiency is of the order of 1%. The main background contribution comes from τ− →K−π+π−ντ decays, where one fake neutral pion was identified.
Since the number of signal events in this final state is not significantly different from zero, this channel is not considered any further in this analysis. For the spectral function, the Monte Carlo prediction has been used instead.
m/GeV m/GeV
Efficiency
Number of Events
OPAL K−π0 Background
m/GeV
Number of Events
Signal−Bkg
0 0.02 0
0.04
20
0.06
40
0.08
60
0.1
80
0.12
100 120 140 160 180
0.5 1 1.5 2 2.5
0.5 1 1.5 2 2.5
(a) (b) (c)
0 20 40 60 80 100 120
0.5 1 1.5 2 2.5
m/GeV
Number of Events
OPAL K0Sπ− Background
m/GeV
Number of Events
Signal−Bkg
0 20 40 60 80 100 120 140
0.5 1 1.5 2 2.5
(a) (b)
0 20 40 60 80 100
0.5 1 1.5 2 2.5
Efficiency
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
0.5 1 1.5 2 2.5
m/GeV (c)
Fig. 5.8: Invariant mass spectra of the two meson final states. In the first (second) row, the plots for the K−π0ντ
(K0π−ντ) final states are shown. Plot (a) shows the measured invariant mass spectrum. The dots are the data, the open histogram is the Monte Carlo signal and the shaded area is the background. Plot (b) shows the background subtracted spectrum and plot (c) the selection efficiency as function of the invariant mass.
m/GeV
Number of Events
OPAL K−π+π− Background
m/GeV
Number of Events
Signal−Bkg
0 20 40 60 80 100 120 140
0.5 1 1.5 2 2.5
m/GeV
(a) (b) (c)
−10 0 10 20 30 40 50 60 70
0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5
Efficiency
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
m/GeV
Number of Events
OPAL K0π−π0 Background
m/GeV
Number of Events
Signal−Bkg
0 5 10 15 20 25 30 35 40
0.5 1 1.5 2 2.5 −4
−2 0 2 4
(a) (b) (c)
6 8 10 12 14 16 18 20
0.5 1 1.5 2 2.5
m/GeV
Efficiency
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
0.5 1 1.5 2 2.5
Fig. 5.9: Invariant mass spectra of the three meson final states. In the first (second) row, the plots for the K−π+π−ντ
(K0π−π0ντ) final states are shown. Plot (a) shows the measured invariant mass spectrum. The dots are the data, the open histogram is the Monte Carlo signal and the shaded area is the background. Plot (b) shows the background subtracted spectrum and plot (c) the selection efficiency as function of the invariant mass.
Physical observables like the invariant mass spectra or angular distributions are usually distorted by the measurement procedure for several reasons. The probability of observing an event in the detector is usually less than one due to acceptance effects. The response of the measuring device might not be linear and thus distorting the measured observable, or the limited resolution of the detector leads to a smearing of the observed quantity. The measured observable therefore substantially depends on the properties of the detector. In order to be able to compare the results obtained from different experiments, to combine invariant mass spectra with different mass resolution or to theoretically evaluate the obtained results, a procedure to correct for these detector effects is necessary. This procedure is usually called unfolding, deconvolution or unsmearing. Apart from high energy physics, correcting measured observables from biasing effects is applied in various fields like medical imaging, radio astronomy or crystallography, to give some examples.
Any correction procedure has to fulfill certain quality criteria. It has to produce numerically stable results.
An enhanced sensitivity to statistical fluctuations, either in the measured observable or in the Monte Carlo sample used in the correction procedure, is otherwise likely to produce artifacts in the ‘corrected’ spectrum.
They could erroneously be interpreted as information from physics processes, or hide the wanted information.
The result of the correction should be as independent as possible from the dynamics of the physics process used in the Monte Carlo simulation. In particular in those cases, where the dynamics in the data events are basically unknown, like in the case of invariant mass spectra in strange hadronicτ decays. The stability of the result when using Monte Carlo samples containing different models of the process under investigation, is vital for a useful physical interpretation. In addition, a good knowledge of the detector effects biasing the measurement is important. This means that the calibration and the resolution of the detector have to be well described by the Monte Carlo simulation.
Several correction procedures have been used in the analysis of high energy particle physics data. There are simple migration corrections, where each bin is assumed to be independent, unfolding procedures based on Singular Value Decomposition [58], methods using spline interpolations to parametrize the detector response [59] or methods based on Bayes Theorem [60]. Any correction procedure however in principle has a bias towards the model used in the Monte Carlo simulation. This bias has to be minimized, e.g. by
‘regularization’ of the result or by using iterative algorithms, where the model of the corresponding process in the Monte Carlo simulation is refined according to the results obtained after each iteration step. The remaining systematic uncertainty associated with the procedure chosen has to be reliably estimated.
In this analysis, a Matrix Unfolding method was used, where the inverse detector matrix was determined directly from Monte Carlo simulation, which avoids the instabilities of numerical matrix inversion. This correction procedure is used in an iterative algorithm, which leads to stable results within a few iteration steps. The result obtained for the corrected spectrum is independent of the dynamics of the physics process assumed in the Monte Carlo simulation within the statistical uncertainties of this analysis.
This chapter is organized as follows. It starts with a mathematical formulation of the unfolding problem followed by a brief discussion of other unfolding procedures. The Matrix Unfolding is introduced in Chapter 6.2 and the iteration procedure, applied in order to reduce the bias toward the model used in the Monte Carlo, is explained. Tests using events including full simulation of theOpaldetector and assuming various resonance structures for the invariant mass spectra is presented in 6.2.3. Finally the results obtained from the unfolding of the spectra fromτ data events are presented in Chapter 6.3.