Wing Sheung Chan

106 Statistical interpretation and results The fits are implemented through the use of the statistical analysis packages RooFit [124] , RooStats [125] and HistFitter [126] . 6.1.1. Likelihood function and fit parameters The parameter of interest (POI) in the analysis is the signal strength modifier µ s , which quantifies the size of the LFV Z -decay branching fraction B ( Z → `τ ) = µ s ×B prefit ( Z → `τ ) ( B prefit can be set to any arbitrary value without affecting the physical interpretation of the results). The likelihood function L that is being maximised in the fits can be expressed as: L ( µ s , b , θ | n ) = Y i ∈ bin Pois( n i | λ i ( µ s , b , θ )) × Y j ∈ syst C j ( θ 0 j , θ j ) . (6.1) The first product in the equation corresponds to the Poisson measurements of n = { n i } , the observed number of events in each bin in the SR and CRZ τ τ . The Poisson expectation values λ i are functions of the predicted yields b for the different background contributions, the nuisance parameters (NP) θ that parameterise the effects of systematic uncertainties in the modelling, and the POI µ s . The second product in the equation is the constraints on the NPs θ . The background predictions b depend on three unconstrained NPs, also known as normalisation factors (NF): µ Z , µ 1P fakes and µ 3P fakes . They determine the overall yields of Z → τ τ , fakes in 1P regions and fakes in 3P regions respectively. The values of the NFs are mainly constrained by the data in CRZ τ τ and the low-NN-output region of the SR, which are dominantly Z → τ τ events and fakes. In the background-plus-signal model, the signal yield is determined jointly by µ Z and µ s . By fitting the predicted Z → τ τ yield to data, the Z -boson production cross section σ ( Z ) , the acceptance A ( Z → τ τ → `τ had - vis ) , and the combined trigger, reconstruction, identification and isolation efficiency ε ( Z → τ τ → `τ had - vis ) for the Z → τ τ events are determined. The determined value of σ ( Z ) × A ( Z → τ τ → `τ had - vis ) × ε ( Z → τ τ → `τ had - vis ) is then reflected by the value of µ Z , which equals to the ratio of the total postfit Z → τ τ yield to the total prefit Z → τ τ yield. Given that the signal and Z → τ τ samples are both normalised to the same measured σ ( Z ) before the fit, and that the A × ε for Z → τ τ events can be expected to be similar to that for the signal events in the SR, the value of µ Z determined by fitting Z → τ τ events can also be interpreted as σ ( Z ) × A ( Z → `τ → `τ had - vis ) × ε ( Z → `τ → `τ had - vis ) for the signal events. Therefore, by normalising the signal sample with µ Z × µ s , the POI µ s is effectively decoupled from σ ( Z ) , A ( Z → `τ → `τ had - vis ) and ε ( Z → `τ → `τ had - vis ) , and represents unequivocally the ratio B ( Z → `τ ) / B prefit ( Z → `τ ) . The NPs θ are constrained by the probability density functions C j ( θ 0 j , θ j ) for each systematic uncertainty j . Depending on the nature of the uncertainty, the constraint C j is either a Poisson or a unit-variance Gaussian distribution with a central value θ 0 j around which θ j can be varied. The total statistical uncertainties in the MC samples in each bin are modelled by NPs with Poisson constraints and are considered to be independent of