Wing Sheung Chan

98 Signal and background modelling 2. The statistical errors in the measured F p . The error in each p T ( τ had - vis ) and p T ( τ track ) bin is propagated to the final fake factor, F r , in the same bin and combined with the error in the estimation of k p and R p r . The combined errors are introduced as constrained nuisance parameters in the binned maximum-likelihood fits (one independent parameter per bin). 3. The statistical errors in the k p and R p r estimations. As described above, these errors are propagated to F r and combined with the errors in F p . In theory, there are also uncertainties associated to the extrapolations from the FRs to the SR. However, given the excellent data-model agreement observed in the same-sign region test (described in the next section), which validated the extrapolation and showed that there are enough degrees of freedom for the maximum-likelihood fits to describe the fakes accurately, no extra uncertainties for the extrapolations are needed to be considered. 5.5.3. The FR closure test and same-sign region test To validate the modelling of fakes, a closure test is performed, where the FF method is used to predict kinematic distributions in the four FRs. By definition, the predicted p T ( τ had - vis ) and | η ( τ had - vis ) | distributions should always agree with what is observed from data, barring the differences due to binning. Furthermore, assuming that the jet → τ had - vis misidentification rate depends only on p T ( τ had - vis ) and | η ( τ had - vis ) | , which is to a large extent a good approximation, the predicted distributions in other variables can also be expected to agree with the observed distributions within statistical and systematic uncertainties. Closure is observed for all the input and output variables of the NN classifiers. As examples, the predicted and observed m vis ( `, τ ) distributions are shown in Figures 5.6, 5.7, 5.8 and 5.9. Another validation for the FF method is the same-sign region test. The entire procedure of deriving and applying the fake factors are repeated in the SS FRs and the VRSS, instead of the FRs and the SR. The VRSS is dominated by fakes but is kinematically similar to the SR, which allows us to independently validate the fakes modelling in the SR. The predicted and observed distributions of the combined NN output are shown in Figure 5.10. The excellent data-model agreement observed in the VRSS shows that the difference in the compositions of fakes in the FRs and the SR is indeed correctly accounted for by the FF method. It also validated the extrapolation of the process-specific fake factors from the FRs to the SR. 5.6. Summary To summarise this chapter, the methods used to model each of the signal and background processes are listed in Table 5.2.