7. Analysis Strategy 113
7.3. Binned Likelihood Fit
and b-quark spin analysers. The data is considered validated and the measurement is performed as described in the next sections.
7.3. Binned Likelihood Fit
The spin correlation measurement must deduce thet¯tsignal – as well as its spin proper-ties – and the background contribution. This is realized with a template fit, based on the principle of a binned likelihood fit. Templates are created for each signal and background composition. MC simulation is used for all templates except the fake lepton background, which is derived from data. The measured dataset is split into several channels, which are explained in the next section. By using the templates from the prediction for signal and background events as well as the measured data distribution, it is possible to define a likelihood
L=
C
Y
j=1 N
Y
i=1
e−(sij+bij)(sij +bij)nij
nij! , (7.1)
whereCis the number of channels,N is the number of bins per template, andnij,sij and bij are the number of events in the i’th bin and the j’th channel in the data, signal and background distribution, respectively. The signal distribution is a linear combination of the two available signal samples: t¯t pairs with spins correlated as predicted by the SM and t¯tpairs with uncorrelated spins. The fraction fSM of SM like spin correlation defines the mixing:
sij =εsignalj Ntt¯·(fSM·pSMij t¯t+ (1−fSM)·punc.ij t¯t) (7.2) where pSMij tt¯and punc.ij tt¯are the entries in bin iof the normalized template for the SM and the sample with uncorrelated spins, respectively, in channelj. The totalt¯t yield is given by the parameterNt¯t. It can also be reformulated as the expected tt¯yield Nexp.t¯t
scaled by a factorc,Nt¯t=c·Nexp. tt¯. The efficiencyεj denotes the fraction of the total t¯tyield that is reconstructed in channel j.
The background contribution for biniand channelj breaks down to:
bij =
3
X
k=1
Nk·εkj·pijk (7.3)
summed over the different background contributions k, each having its own efficiencyε:
W+jets, fake lepton background and remaining backgrounds. Nk represents the total number of events from the background type k. εkj is the relative contribution of the background events of typek in the channelj to the total number of background events Nk. pijk is the entry of biniof the normalized background template for the background type kin the channel j.
Technically, this fitting is implemented by transforming the two normalized signal templatespSMt¯t and punc. tt¯intopsumt¯t and pdifft¯t via
psumt¯t= 1 2
pSMt¯t+punc.t¯t
, (7.4)
pdifft¯t= 1 2
pSMt¯t−punc.t¯t
. (7.5)
The signal contribution is a linear combination ofpsumt¯t and pdiff.t¯t :
sij =εsignalj (Nsumtt¯·psumt¯t,ij+Ndifft¯t·pdifftt,ij¯ ) (7.6) with the parameter values for the total yieldNsumt¯tand the parameter for the scaling of the difference of SM and uncorrelated eventsNdifftt¯. The values ofpsumt¯t,ij andpdifft¯t,ij
represent the entries of biniof the templatespsumt¯t and pdifft¯tin the channel j.
As these are just linear transformations, the fitted parameter valuesNsumt¯tandNdifftt¯
can be easily translated into the parameters of interest, namely the cross section scale factor cand the spin correlation fractionfSM.
The linear transformation is introduced to add numerical stability to the fit. With-out the transformation the two signal parameters would be fully anti-correlated. The transformation resolves this issue.
In addition to the basic fit parameters, further parameters are added. One set of parameters accounts for the systematic uncertainties, taking them into account as ad-ditionalnuisance parameters (NPs) by fitting their effects on the template to the data and thus constraining their impact. This procedure is described in Section7.3.3.
After the choice of channels is explained in the following section, the treatment of fit parameters is described in Section 7.3.5.
7.3.1. Analysis Channels
Each channel is a subset of the whole data available and has distinct properties: signal to background ratio, reconstruction efficiency, impact of systematic uncertainties and statistical uncertainty. Some channels are pre-defined as the e+ jets and the µ+ jets channel are reconstructed in different, orthogonal data streams. Others are defined by the analysis strategy. In the combination of down-type quark and b-quark results, the different analysers are treated as separate channels. Section 7.6 is dedicated to the question if the treatment of the down-type quark and the b-quark as independent variables is justified.
Further splitting of thee+ jets and the µ+ jets data is possible and reasonable. The first splitting divides the data into a channel with exactly four jets and one with at least five jets. As shown in Section6.7, this creates a subsample with a higher reconstruction efficiency for both the down-type quark and the b-quark, namely fornjet = 4. Another motivation for this splitting is the jet multiplicity mismodelling of theMC@NLOgenerator.
It is possible to introduce an additional parameter to the fit correcting the efficiencies of thenjet = 4 with respect to thenjet >5 channel, allowing an in-situ correction of the mismodelling (see Section7.3.4).
7.3. Binned Likelihood Fit The number of b-tagged jets is also a criterion to split the data sample into subsets of higher and lower reconstruction efficiency. As the b-tag multiplicity is also not per-fectly modelled (see Section 7.2), the introduced nuisance parameters dedicated to the b-tagging uncertainties can correct this mismodelling in-situ.
All channels used in the analysis are defined and listed in Table 7.1. Results are obtained for the individual channels, combinations with the same analyser and a full combination.
Channel Analyser Lepton Flavour Jet Multiplicity B-Tags 1
down-type quark
electron
= 4 =1
2 >1
3 >4 = 1
4 >1
5
muon
= 4 = 1
6 >1
7 >4 = 1
8 >1
9
b-quark
electron
= 4 =1
10 >1
11 >4 = 1
12 >1
13
muon
= 4 = 1
14 >1
15 >4 = 1
16 >1
Table 7.1.: Analysis channels.
7.3.2. Usage of Priors
As the fitting frameworkBATis using the Bayesian approach, a-priori information about the parameters can be included in the fit. This is realized by the addition of priors to each parameter pi. These are multiplied to the likelihood. Different types of priors exist: Delta priors fix a parameter to one certain value. Constant priors have no effect at all as they are constant in the parameter space. Gaussian priors are normalized Gaussian functions that take an expected value as central value and an uncertainty on the expectation as width.2 This kind of prior is used to constrain the background yields according to their normalization uncertainties. The Gaussian priors used in this analysis are explained in Section 7.3.5.
2The expression ’width’ of Gaussian distributions refers to the standard deviationσ and may not be confused with the ’Full Width at Half Maximum’, FWHM.
7.3.3. Systematic Uncertainties as Nuisance Parameters
Systematic uncertainties affect the results by changing both the shape and the yield of the measured distributions. Next to the more traditional way of evaluating their effect via ensemble tests, the option of including them already in the fit can be a good alternative if certain requirements are fulfilled. If for an uncertainty both the ±1σ variations lead to a well-defined3 template the effects on each bin can be quantified and linearly interpolated. This allows to assign an additional fit parameter (nuisance parameter) βi to the uncertainty i. Within the fit, the effect is considered via modified efficiencies used in Equations7.2 and 7.3:
˜
ε=ε+X
unc.
βi∆εi. (7.7)
Here, ∆εi is the relative change of yield per bin caused by the systematic effect i.
Values ofβ =±1 correspond to the effects of ±1σ deviations. By including systematic effects as nuisance parameters they can hence improve the data/MC agreement caused by miscalibration covered by systematic uncertainties. Systematic uncertainties included as nuisance parameters propagate their uncertainties into the fit uncertainty.
A complete list of nuisance parameters used in this analysis is provided in Section8.2.
7.3.4. Jet Multiplicity Correction
The division into subsets of njets = 4 and njets > 4 allows for a correction of the mis-modelling of jet multiplicity by MC@NLO. Such a correction needs to be applied to the signal samples as the predicted signal yield in the njets >4 subset is too low. No such correction was applied to the background samples.
As the efficiencies for each type of signal are set and fixed before the fit, there are two possibilities for the fit to fill the gap between prediction and data yield in thenjets >4 channels:
• The backgrounds will fill the gap and will thus be overestimated.
• As the efficiencies are deduced from the MC including deviating jet multiplicities, they will be incorrect. By filling the gap in thenjets ≥5 channels, the signal will also increase in thenjets= 4 channels. The fit will end up in a compromise between an overestimation of thenjets = 4 channels and an underestimation of thenjets>4 channels.
Both possibilities are inconvenient. Thus, an additional nuisance parameter is intro-duced. It modifies the efficiency of thenjets >4 channels by±10 % per integer value of the parameter. The method was tested first by performing a combined fit of all down-type quark and allb-quark channels for which the effect of jet multiplicity mismodelling is the same but the ∆φ shapes are different. With a starting value of 0 and a constant prior, the correction parameters were fitted to be 0.88±0.16 (down-type quark) and
3’Well-defined’ in a sense that the variations are not simply random fluctuations.
7.4. Method Validation