Fake Rate Estimation using the Matrix Method

The estimation of the rate of falsely identified leptons or fake rate from the data is important, since the number of events in the current MC samples is not sufficient to obtain a good estimation. The number of selected events with falsely identified leptons is small and the large statistical uncertainties together with the large cross section of e.g. W+jets orQCD result in a bad prediction. Furthermore, the rate is subject to instrumental effects that can be different from the simulation and it can vary with time. The following data-driven method tries to estimate the number of events that contain one falsely identified lepton. It is assumed that the rate of two falsely identified leptons, which are mainly from QCD multijet events, is low. The events also have low E/T, so that they are rejected already by the E/T cut.

The rate of fake events is derived from the fake rate, f, for single leptons and the efficiency, ε, to select real leptons. The fake rate is measured in fake lepton dominated events, the so-called control region, with a looser lepton selection. The efficiency is measured in Z → ℓ⁺ℓ⁻ events with dominating real leptons and the default lepton selection.

Table 6.6.: Top-quark pair dilepton acceptances and top-quark pair branching ratios estimated from MC@NLO Monte Carlo simulation for the different sub-channels.

ee [%] µµ [%] eµ [%]

A 16.5±0.4 26.1±0.4 26.5±0.3 BR(t¯t→ll)/BR(all) 1.67±0.05 1.64±0.05 3.40±0.10

Table 6.7.: Statistical uncertainty on the cross section σ and significance for the different sub-channels from MC based signal and backgrounds. The uncertainty on the number of observed events is Poissonian and the uncertainty on the back-ground estimation is taken from the statistical uncertainty of the MC sample.

statistical uncertainty ∆σ/σ [%] ee µµ eµ

L= 10 pb⁻¹ 34.3 27.6 18.3

50 pb⁻¹ 15.9 12.7 8.4

100 pb⁻¹ 11.7 9.3 6.1

200 pb⁻¹ 7.8 6.2 4.4

significance S/√

S+B ee µµ eµ

L= 10 pb⁻¹ 2.9 3.6 5.4

50 pb⁻¹ 6.5 8.0 12.0

100 pb⁻¹ 9.1 11.4 17.0

200 pb⁻¹ 12.9 16.1 24.1

The loose lepton selection is chosen such that it is kinematically similar to the actual lepton selection, but some cuts are relaxed, so that the probability for a fake lepton is high to avoid large statistical uncertainties. For muons this is achieved by not requiring isolation. For electrons theisEM:mediumrequirement and the isolation cut is reversed. The tight selection is the same as the default object selection.

The definition of the efficiency ε and the fake rate f is ε= Ntight,real

Ntight,real+Nloose,real

, (6.4)

f = N_tight,fake Ntight,fake+Nloose,fake

. (6.5)

The efficiency is measured inZ →ℓ⁺ℓ⁻ events that require two oppositely signed, same flavoured leptons with an invariant mass within ±5 GeV of the Z-boson mass

Figure 6.9.: E/T and dilepton invariant mass space with the labelled areas used for the data-driven background estimation of the Z →ℓ⁺ℓ⁻ background [232].

and E/T < 15 GeV. The number of tight leptons and loose leptons are counted and the equation above is applied.

While the selection of real lepton events is very clear, the selection for event dominated by loose leptons is tested in two regions. In addition special care has to be taken of events with W-bosons. The two regions are also useful to estimate the systematic uncertainty of this method and thus are chosen to be orthogonal.

Both regions require one loose or one tight lepton, the first region requires also /

ET < 15 GeV while the other requires E/T > 15 GeV and ∆φ(lepton, /ET) < 1 rad.

The requirement of low E/_T rejects 98% of W-boson events while the angular cut only rejects between 75−80% of the events. The number of remaining events with real leptons is estimated from W-boson and single-lepton t¯t MC events employing the control region selection cuts and an additional cut on the transverse mass² of mT > 60 GeV. This estimation is subtracted from the number of tight leptons in the control region that is used in the calculation of the fake rate f.

The single lepton efficiency and fake rate is correlated to the event fake rate as follows: The sources for events with two tight (TT), one tight and one loose lepton (TL, ordered by pT) and one loose and one tight lepton (LT) are events with two real leptons (RR), one real and one fake lepton (RF) or one fake and one real lepton (FR). E.g. the number of events with two tight leptons (N_TT) is related to the number of real/fake lepton events (NRR, NRF, NFR) such as NTT =

2The transverse mass is defined from the transverse lepton and the E/T vector.

ε₁ε₂N_RR +ε₁f₂N_RF +f₁ε₂N_FR, where the subscripts denote the first and second highestpT lepton. All cases can be summarised into a vector equation:



 NTT

NTL

NLT



=





ε1ε2 ε1f2 f1ε2

ε1(1−ε2) ε1(1−f2) f1(1−ε2) (1−ε1)ε2 (1−ε1)f2 (1−f1)ε2







 NRR

NRF

NFR



 (6.6)

The number of fake events is obtained by inverting this relation and adding the number of events with one fake lepton:

NRF+NFR =NFake =

f2(ε2−1) ε2−f2

+f1(ε1−1) ε1−f1

(6.7) The efficiencies and fake rates can also be expressed as functions of kinematic variables, so that the event fake rate is also a function of these variables. For example in Figures 6.10 the fake rate f is shown for electrons and muons as a function of pT and η. The fake event estimate is given as a function of the jet multiplicity in Figures 6.11 for the different sub-channels. From the figures it can be seen that the assumption made earlier that the fake rate estimation should be independent of the method is not very well fulfilled. Only in a small phase space of the fake event estimation do the two control regions result in the same prediction. This difference could be an effect of the different fake sources that were not distinguished, e.g. heavy-flavour decay or real instrumental effects. Until the effects of these two sources are not separated, the difference in the two control regions can be taken as a systematic uncertainty of this method.