Anode Wire Plane
3 Hits
4.1.5 Adjustment of the Experimental Errors
After all the corrections described above, the systematic bias of the energy loss measurement in the τ environment could be reduced from O(10%) to O(1%). From the measured energy loss, its error and the expectation calculated using the Bethe-Bloch equation, χ2 probabilities are calculated that the measured energy deposition is in accordance with the expectation for a given particle type, as
P(χ2meas, ν) = Z ∞
χ2meas
Γ(χ2, ν) dχ2, (4.6)
where Γ(χ2, ν) is theχ2-p.d.f. and ν = 1 the number of degrees of freedom. Pion- and kaon-weights, Wπ
and WK are then calculated by taking one minus the value of this probability. These weights acquire a sign depending on whether the actual energy loss lies above or below the expectation for a certain particle hypothesis. This means that Wπ is expected to be close to −1 for kaons since their energy loss per unit length is smaller in the momentum range relevant in this analysis. For electron tracks,Wπ is expected to be close to +1 due to the higher energy loss in this case. Whenever these quantities are used in the selection, a cut on at least 20 dE/dxhits for this track is implicitly made.
To calculate the probability that the measured energy loss is in agreement with the expectation for a given particle type, an accurate description of the experimental uncertainty is necessary. In order to verify this and to determine correction factors if needed, distributions of residualsR are studied:
R = (dE/dx)meas−(dE/dx)exp
σdE/dxexp
, (4.7)
which is the measured energy loss for a given particle observed in the detector, minus the expectation as calculated from the Bethe-Bloch equation for a particular particle hypothesis, divided by the expected error of the measurement. For a sample of a single particle species and for the right particle hypothesis, this results in a unit Gaussian with mean zero and unit width. Any deviation in shape and position from this expectation is an indication for either the wrong particle hypothesis, a systematic shift in the energy loss measurement or an inaccurate description of the experimental uncertainties.
The track sample analyzed here contains three different particle species: pions, kaons and electrons. The distribution of residuals as defined in Equation 4.7 is therefore a superposition of three distributions. The number of muon tracks in the sample considered here is negligible. For those tracks, the probability for final state radiation and a subsequent conversion of the photon (to fake a 3-prong event) is very low. The rate is below 1% and these tracks are absorbed in the peak produced by the pions. Using pion hypothesis, one observes one Gaussian peak originating from pion tracks which is centered around zero. The peak from the kaons in the sample is situated left from the pion peak, since the energy loss for kaons is less than the one expected for pions in the momentum range considered here (see Figure 4.1). The contribution from electron tracks is expected right from the pion peak. The shape of the contributions from kaon and electron tracks are not Gaussian-like since the difference in energy loss for these two particle species is not constant as a function of the particle momentum relative to the expectation from pions. The individual contributions as described above are illustrated in Figure 4.13.
The residual under pion hypothesis is then analyzed in a χ2-fit as explained below. To assess possible correction factors, for every track in the sample the residual under pion hypothesisRπmeas is calculated
Rπmeas= dE
dx
meas
·Pscal− dE
dx π
exp
σmeasdE/dx·Perr· (dEdx)πexp (dEdx)meas
−Pmean. (4.8)
The parametersPscal andPerrare multiplicative scaling factors for the measured energy loss and the exper-imental error. Pmean is an additive correction factor, to allow for an overall shift of the whole distribution.
They are free parameters in the fit. After all corrections,Pscal andPmean are expected to be one. Since the bias of the dE/dxmeasurement has been substantially improved, the experimental uncertainty is expected to be smaller. Perr is therefore expected to be less than one, which is then used as scaling factor for the experimental uncertainty. For small enough momentum bins, the difference in energy loss is in good approx-imation constant and the expected distribution of measured residuals can be described by the sum of three Gaussians.
Dtheo= X
i∈{K,π,e}
Ciexp
−1 2
x2i σ2i
, (4.9)
C
ππ C
KK
C
ee
Residual 0
Fig. 4.13: Distribution of residuals as expected for the particles species considered here.
where xi is the expected residual as calculated from the Bethe-Bloch formula for each of the particle hy-potheses using the measured momentum of the track. The parameters Ci are normalization factors that represent the relative size of the contributions of each of the three particle species relevant here. Two of them, CK and Ce are free parameters of the fit. The fraction of pion tracks is not a free parameter since it can be calculated via Cπ = 1−CK−Ce. The errorσi is the expected width of the Gaussian, which is σπ= 1 for the central peak andσi = dEdxi
exp/ dEdx
meas for kaons and electrons.
In the fit, the followingχ2 is minimized χ2=
P
TracksRπmeas−P
TracksDtheo
σstat
. (4.10)
The fit was tested using Monte Carlo track samples. Here, the particle species for each individual track is known, and the distribution of the residual and the relative size of the contribution of each particle species can be compared to the result obtained in the fit. This is illustrated in Figure 4.14. In the upper row, the momentum distribution of all tracks considered in the fit is displayed separately for all three particle species.
The dots represent the distribution used in the Monte Carlo simulation and the histogram is the prediction of the fit as calculated from the fit parametersCK,e, π. The shaded area represents the uncertainty as obtained in the fit. Below, the residual for each individual particle species as obtained from the fit is compared to the expectation. In general, a good agreement between the Monte Carlo and the results of the fit is observed.
The scaling factor Perr for the experimental uncertainty of the energy loss measurement were found to be one within the errors of the fit. The corrections described in the previous section are relevant for data tracks only. They are not present in theOpaldetector simulation, therefore no scaling ofσdE/dxmeas is necessary. The same is true forPscalandPmean, which were found to one and zero within statistical uncertainty, respectively.
This fit is now applied to the track sample in data events. The tracks in 1-prong and in multi-prongτ decay environment are fitted separately for each year of data taking. The results obtained for the parameters of the fit are given in Table 4.4. For the error on the energy loss measurementσdE/dxmeas a scaling factor of up to 0.9 was obtained in the multi-prong environment. This means a 10% reduction of the experimental uncertainty.
Furthermore, in this environment, a shift of the order of 0.05 was observed. In the 1-prong environment, as expected only small corrections are obtained from the fit.
The effect of all corrections can be seen in figure 4.15. It shows the dE/dxpull distribution under a pion-hypothesis for all like-sign tracks from 3-prong tau decays. The full dots represent the result obtained using tracks in data events with all corrections applied. The solid line represents the prediction from the fit as explained above. In addition, the distribution of residuals with the defaultOpalcorrection only is shown as open dots. Due to the effects explained above, a significant excess of events is observed in the range where the kaon tracks are expected.
Momentum / GeV
Number of Tracks
Kaon Tracks
Momentum / GeV
Number of Tracks
Pion Tracks
Monte Carlo Fit
Momentum / GeV
Number of Tracks
Electron Tracks
0 100 200 300 400 500
0 20 40 0
500 1000 1500
0 20 40 0
50 100 150 200
0 20 40
Residual
Number of Tracks
Kaon Tracks
Residual
Number of Tracks
Pion Tracks
Residual
Number of Tracks
Electron Tracks
0 5 10 15 20 25
−7.5 −5 −2.5 0 0
100 200 300 400 500
−5 0 5 0
20 40 60
0 2.5 5 7.5 10
Fig. 4.14: Test of the fit procedure using tracks from Monte Carlo simulation. In the upper row, the momentum distribution in the Monte Carlo (dots) is compared to the distribution as obtained from the fit (histogram).
The shaded area represents the uncertainty from the fit. Below, the distribution of residuals for the individual particle species as obtained from the fit (histogram) is compared to the prediction from the Monte Carlo simulation.
1-prong
Year Pmean Perr Pscal
91 0.0077± 0.0015 0.9810 ± 0.0073 0.9989 ± 0.0022
92 0.0078± 0.0049 1.0128 ± 0.0054 0.9995 ± 0.0016
93 0.0048± 0.0027 1.0013 ± 0.0063 0.9995 ± 0.0070
94 0.0012± 0.0013 1.0018 ± 0.0090 0.9971 ± 0.0057
95 0.1128± 0.0032 1.0100 ± 0.0056 0.9994 ± 0.0062
91 0.0533± 0.0091 0.9293 ± 0.0077 1.0074 ± 0.0015
92 0.0517± 0.0056 0.9365 ± 0.0065 1.0099 ± 0.0012
93 0.0499± 0.0046 0.9021 ± 0.0074 1.0064 ± 0.0006
94 0.0371± 0.0039 0.9509 ± 0.0097 1.0085 ± 0.0010
95 0.0355± 0.0083 0.9657 ± 0.0120 1.0081 ± 0.0022
Year Pmean Perr Pscal
multi-prong
Tab. 4.4: Result of the fit to the residuals. The mean of the parametersPmean,PerrandPscalare given separately for each year of data taking and for tracks in 1-prong and multi-prong environment separately.
Number of Tracks
K e
π / µ
OPAL (corrected) OPAL (default) FIT
( dE/dxmeas − dE/dxexp )/ σdE/dxexp 10
−1
1 10 10 2 10 3
−8 −6 −4 −2 0 2 4 6 8 10
0 2
−8 −6 −4 −2 0 2 4 6 8 10
Fig. 4.15: Pull distribution obtained under a pion hypothesis for all tracks in 3-prongτ decays with a minimum momentum of 3 GeV and a minimum number of 20 hits in the dE/dx measurement. Only the tracks with the same charge as the decayingτ are shown. The full points with error bars are data after all corrections and the histograms shows the expectation as explained in the text. The open points in the range between -8 to -2 show the same distribution but without the corrections mentioned in the text. The smaller plot at the bottom shows the ratio of the full data points to the histogram.