Validation of the Fit Method

The fit method is validated using so-calledcalibration curvesandpull distributions. A calibration curve aims at testing the linearity of the underlying fit. For this purpose, 1,000 pseudo-experiments for decay width values in the range 0.5≤Γt ≤4.0 GeV using steps of∆Γt=0.5 GeV were conducted.

For the resulting eight calibration points and an additional one for the nominal value, the templates were fitted to pseudo-data distributions obtained from the procedure described in Sec. 7.4. The average measured top quark decay width can be obtained from the mean of the histograms in which the fit results of all pseudo-experiments were filled for the chosen nine input decay width values.

The mean values were then plotted as a function of the input values. A linear fit to these points is expected to have a slope of one and an offset of zero - measured as an intercept of the ordinate - as long as the estimator is unbiased. The resulting calibration curve for the final configuration chosen in the template fit is shown in Fig. 7.7.

Figure 7.7: Calibration curve for the final fit configuration. The visible deviations from the theoretical expectation in the intercept are well understood and explained in the text. The lower panel shows the difference between the obtained fitted mean values,Γt^fitted, and the expected input parameter values,Γt^input.

The linear fit reveals small deviations from the theoretical expectation. The slope is still consistent with one within two standard deviations around the measured slope of 0.995 (the given uncertainty of 0.003 corresponds to 1σ). However, the intercept has an offset and is significantly different from zero. This offset is caused by the slightly shifted mean values of samples with small decay width values because only positive values ofΓt can be fitted, leading to the edge of Γt values at 0 GeV.

This effect is thus related to the decay width samples withΓt >0 GeV and, hence, the resulting

7 . 5 VA L I D AT I O N O F T H E F I T M E T H O D

curve follows the expectation for this particular analysis setup. This is further discussed in the next paragraphs.

Pull distributions are employed as well to assess the stability and the modelling of the fit. The pull is calculated as the difference between the fitted valueΓt and the expected input oneΓ_t^inputdivided by the estimated uncertainty on the fit resultσ(Γt):

Pull= Γt−Γ_t^input σ(Γt) .

The nine calibration points used for the calibration curve together with three additional points around a decay width of 1.0 GeV were considered with the same number of PEs per input option as before. Based on the definition of the pull, the average fitted pull value is anticipated to be zero, the corresponding pull width to be one. The latter quantity refers again to the standard deviation of the obtained pull distribution. The results of the average fitted pulls and their uncertainties for the tested values ofΓt are shown in Fig. 7.8a. The mean values of the expected statistical uncertainties σ(Γt) as part of the pull calculation are contained in Fig. 7.8b for the original nine calibration points.

(a) (b)

Figure 7.8:Results of the (a) pull distributions and (b) mean expected statistical uncertainties for the final fit setting in the range0.5<Γt <4.0GeV. Small deviations from the expectation can be observed for small decay width values which is why additional pull values are added in the region aroundΓt =1.0GeV. The visible deviations from the theoretical expectation are well understood and explained in the text.

As exhibited by the linear fit for the calibration curve, discrepancies arise from the theoretical expectation for small values of the decay width starting aroundΓt®1.1 GeV. As a result, additional

calibration points were added in this range to better determine the region of possible deviations.

Larger input decay widths return mean values included in the 1σ-region around the expectation.

The pull mean value is still consistent with zero forΓt =1.0 GeV while the width values start to deviate in the rangeΓt®1.1 GeV. Since negative decay width values cannot be physically motivated, they are not allowed in the fit and lead to a sharp edge at 0 GeV which explains the deviations visible in the pull values. Besides, according to the definition of the pull, the decrease in the pull width starting slightly aboveΓt≈1.0 GeV during which the mean values are still close to zero hints at an overestimation of uncertainties caused by the physical decay width edge. In other words, the uncertainty assigned to fit results lying in this region emerge from a more conservative uncertainty estimation than required. Apart from that, the region of decay width values around 1.0 or 1.1 GeV is not reached by the fit to data in this measurement and not touched by the associated systematic uncertainties. Consequently, taking all these arguments into account, the entire fit can be regarded as stable and unbiased, and no correction for low values ofΓt needs to be applied.

Selected output distributions of the fitted decay width values for 1,000 PEs are visualised in Fig. 7.9.

Figure 7.9: Decay width distributions obtained from 1,000 pseudo-experiments with the ex-pected mean values marked in grey in the range0.5≤ Γt ≤4.0GeV. Due to the sharp edge at 0 GeV, the Gaussian-shaped distributions are narrower with more distinct peak regions for smaller values ofΓt.

The shapes of the histograms support the statements from the last paragraph. They change visibly for decay width values smaller than or aroundΓt=1 GeV. The distributions in thisΓtregion have a narrower shape when approaching the limit of 0 GeV, which affects the calibration curves and pull distributions as outlined above. These narrower curves thus possess significantly smaller expected