Transfer factors method

As already mentioned, the aim of the transfer factor method is to reduce as much as possible the uncertainties on the BG estimation exploiting the information of the data in the CRs. In order to estimate the BG contribution of a given process in the SRs the number of data events in a CR is corrected with MC-based factors (called transfer factors or simply TFs). As an example, in order to estimate theZ(→νν)+jets contribution to the SR we can clearly make use of the events in theZ(→µµ)+jets CR.

Since theE_T^missis calorimeter based,Z(→νν)+jets events in the SR andZ(→µµ)+jets events in the CR have a very similar behavior¹. The differences are the related to:

• the branching ratios of the Z decays,

• the muon identification,

• the energy deposited in the calorimeters by the muons and by the radiated photons (typically few GeV),

• the contribution of the photon propagator and the interference with the Z.

All these differences between SR and CR events are taken into account in the TFs.

The TF method provides a separate estimation of every Z/W+jets process (Z(→ νν),W(→eν) ,W(→µν) ,W(→τ ν),Z(→τ τ), andZ(→µµ)²), using one of the four CRs defined previously (W(→ µν) ,Z(→µµ), inclusive muon and inclusive electron).

For each process a correspondent CR is chosen, and the number of SR events N_SR^est. is estimated with the formula:

N_SR^est.= (N_CR^{data T OT}−N_BG^{M C})×[N_SR^{M C}

N_CR^{M C}] , (4.1)

where:

• N_CR^{data T OT} is the number of all data events selected in CR

• N_BG^{M C} is the number of top and di-bosons events in CR estimated from MC

• N_SR^{M C} is the number of SR events of the process under investigation estimated from MC

1This is the reason why theET^misshas been defined without any correction for the muons.

2TheZ(→ee) +jetsis negligible because of theET^missrequirement.

• N_CR^{M C} is the number of Z/W+jets events in CR estimated from MC

• The ratio N_SR^{M C}/N_CR^{M C} is called transfer factor (TF).

The TFs are in fact the ratio of simulated events for the process in the SR over the number of simulated events in the CR. In this way the TF includes in one factor all effects related to lepton acceptance and efficiency as well as the production cross section and branching ratios of the different processes. The CRs have a contamination of di-bosons and top production (order of 3%) that are subtracted using MC estimates, as shown in formula 4.1. This subtraction has an impact on the total BG uncertainty of the order of 1%(see below).

In the example of the Z(→ νν)+jets estimation from the Z(→ µµ)+jets CR, the following formula is used:

N_SR^est.(Z(→νν)) = (N_CR^{data T OT}(Z(→µµ)) -N_BG^{M C}(Z(→µµ))) ×[^{NSRM C(Z(→νν))}

N_CR^{M C}(Z(→µµ))] TheZ(→νν)+jets contribution can also be estimated from theW(→µν)+jets CR, and from the inclusive muon CR, relying on MC for the ratios betweenZ+jets andW+jets processes. Figure 4.9presents the comparison of the results for the Z(→νν)+jets BG as determined from the different CR. The comparison demonstrates the consistency across different CRs. As expected, the results from the inclusive muon CR are very close to those from the Z(→ µµ)+jets andW(→µν) +jets CR. The comparison also illustrates how a data-driven procedure based on theZ(→µµ)+jets sample alone would suffer from large statistical uncertainties.

With the TF method most of the systematic uncertainties on the BG estimation are largely reduced. For example the luminosity uncertainty does not affect the TF, since is exactly cancelled in the ratio. The jet energy scale (JES) uncertainty is largely reduced as well (see below) since a change on the energy scale would change the denominator and denominator in a very similar way, so that the TF is mostly unchanged. To minimize the effects on the TF it’s important that CRs have a final state as close as possible to the SR process that we want to estimate. This argument has driven the choice of the CRs to be used for each SR estimate.

The Z(→ νν)+jets, W(→ τ ν)+jets, W(→ µν) +jets, and Z(→ µµ)+jets back-grounds are first estimated separately from the W(→ µν) +jets or Z(→ µµ)+jets CRs. Then, for each process the predictions from the two CRs are combined for the

BG estimation / MC estimation ν

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Figure 4.9: Comparison of Z(→ νν)+jets estimations using different methods for the four SRs (SR1 and SR2 in the upper plots, SR3 and SR4 in the lower ones). The error bands represent statistical errors only.

final estimation. The combination is an error-weighted average that takes into ac-count the correlations of the statistical errors. Instead the processes W(→ eν) +jets, Z(→τ τ)+jets, andZ(→ee)+jets are estimated using the inclusive electron CR.

As a cross-check, the W(→ τ ν)+jets estimation was carried out using W(→ µν) , Z(→ µµ) and inclusive electron CRs and the results were consistent to each other within uncertainties¹. For the final results, the estimation based on the W(→ µν) and Z(→ µµ) control samples was adopted, because of the higher statistics in these CRs. Nevertheless the use of the inclusive electron sample could also be justified since the τ dominantly decays into a narrow jet leaving energy in the calorimeters, so that W(→τ ν)+jet events behave similarly toW(→eν) +jet events.

The TF method can also be applied bin-by-bin to the different distributions, leading to a data-driven corrected shape in the SR. Technically this means that formula 4.1 is used for the estimation of the number of events in each bin of the distribution. It’s worth to mention here that estimating the total number of events from formula 4.1or by integrating the bin-by bin corrected distributions, give the same result.

Figures 4.10 presents the steps for building the Z(→ νν)+jets E_T^miss distribution in SR1 from the W(→ µν) +jets CR. The same procedure repeated from the Z(→ µµ)+jets CR is shown in figure4.11.

1This comparison consider both the statistical uncertainty and systematic uncertainties that are presented in the following.

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Figure 4.10: Procedure to build the Z(→ νν)+jets ET^miss distribution for SR1 from W(→µν) +jets CR. Top left plot shows the distribution ofET^missfor the data (black) and the MC (red) in theW(→µν) +jets CR and theZ(→νν)+jets MC in the SR (blue). The Top right plot shows the transfer factor (TF) as described in the text. The bottom left plot shows the estimatedZ(→ νν) ET^miss distribution (open circle) compared to the MC only prediction (blue). The bottom right plot shows the ratio of the data-driven and the MC-only estimated SR distribution. Only statistical errors are shown.

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Figure 4.11: Procedure to build the Z(→ νν)+jets ET^miss distribution for SR1 from Z(→µµ)+jets CR. Top left plot shows the distribution ofE^missT for the data (black) and the MC (red) in theZ(→µµ)+jets CR and theZ(→νν)+jets MC in the SR (blue). The Top right plot shows the transfer factor (TF) as described in the text. The bottom left plot shows the estimatedZ(→νν)ET^miss distribution (open circle) compared to the MC only prediction (blue). The bottom right plot shows the ratio of the data-driven and the MC-only estimated SR distribution. Only statistical errors are shown.