Wing Sheung Chan

88 Signal and background modelling Generally, the event weight can be interpreted as w ( p ) = 1 + s · h = 1 + p · h z , (5.1) where s is the τ polarisation vector, h is the polarimetric vector constructed using the hadronic currents, p is the polarisation of the τ lepton (equals − 1 for left-handed and 1 for right-handed τ leptons) and h z is the component of h in the travelling direction of the τ lepton. Therefore, the weight w ( − 1) provided by TauSpinner can also be used to derive weights for simulating scenarios with any assumed τ polarisation states: w ( p ) = 1 − p ( w ( − 1) − 1) . (5.2) This means that by applying the weight 2 − w ( − 1) to the nominal signal sample, the scenario where only right-handed τ leptons ( p = 1 ) are produced can also be simulated. Since a nominal signal sample is generated with unpolarised τ leptons and can therefore be viewed as an ensemble of events with 50% left-handed and 50% right-handed τ leptons, it is able to represent scenarios with any assumed τ polarisation states with a sufficient sample size. For this reason, the neural network classifiers described in Section 4.2 are only trained with the nominal signal sample. Figure 5.1 shows the combined NN output distributions of the signal events in the SR before and after the τ polarisation reweighting. As can be seen, the assumption for the τ polarisation has a rather significant impact on combined NN output distributions in the 1P regions. 5.3. Corrections to the simulated Z -boson production Events with Z -boson decays ( Z → τ τ , Z → `` and the signal Z → `τ ) are modelled based on MC simulations. These simulations come with uncertainties in the theoretical predictions (theory uncertainties). The Z -boson decays themselves are purely electroweak processes and can be predicted with negligible uncertainties. Nonetheless, the cross section, σ ( Z ) , and transverse momentum, p T ( Z ) , of the simulated Z bosons can only be predicted with much larger theory uncertainties, as they also depend on the simulation of QCD activities. These theory uncertainties could have a significant negative impact on the sensitivity of the analysis. In order to avoid that, data-driven corrections are applied to the overall prefit normalisations and the simulated p T ( Z ) spectra of the Z → τ τ , Z → `` and signal MC samples. The prefit normalisations of all the Z -decay MC samples are first corrected to match the measured Z -boson production cross section σ ( Z ) = 1 . 981 nb [62] . Then for the Z → τ τ and signal samples, their actual normalisations are determined in a maximum-likelihood fit to data in the SR and CRZ τ τ (see Section 6.1) . This allows us to make use of the large amount of data, especially in CRZ τ τ , to strongly constrain σ ( Z ) , along with the systematic uncertainties in the simulation of real `τ had - vis final states. For the Z → ``