Definition of the Likelihood

The binned likelihood fit set up to determineΓtuses the signal templates defined in the last section for the t¯t contribution. Templates for the background processes, involving the production of single top quarks, are fixed. Those enter the fit with their normalisations and associated uncertainties included in Table 6.1. The influence of missing templates with alternative values ofΓtfor the single top contribution was evaluated separately and is delineated in this section. The implementation of the fit is realised with dedicated commands exploiting the RooFit[289]tool, a part of the ROOT framework[279].

The normalisation of the signal template is a free parameter of the fit while the backgrounds are allowed to vary within Gaussian constraints. The number of expected events as used in the fit equals the sum of all template normalisations and can thus be written as:

n_exp=n_signal+n_singletop+n_W+b¯b/c¯c+n_W+c+n_W+light+n_Z+jets+n_diboson+n_multijet. An equivalent expression per biniyields, after summarising the backgroundsBby the index j:

n_exp,i=n_signal,i+ XB

j=1

n_bkg,ji.

Accordingly, the number of data events per bin is denoted asn_data,i. Based on these numbers, the likelihood for an observableO is defined as follows:

L(O |Γt) =

Nbins

Y

i=1

Poisson(O |n_data,i,n_exp,i,Γt)·

B

Y

j=1

p 1

2πσbkg,j

exp

‚−(nbkg,j−nˆ_bkg,j)² 2σ²_bkg,j

Œ

. (7.2)

The coefficientN_binsspecifies the number of bins in the templates where a Poisson regression based on the bin entries is exploited in the fit. The expected number of events from a background source j, n_bkg,j, is deduced fromn_bkg,ji by summing over all binsi. The number of background events is allowed to vary in the fit but is constrained by the Gaussian terms of Eq. (7.2) where ˆn_bkg,jis the expected number of background events for the contribution jandσbkg,jis its uncertainty.

The uncertainties in the background contributions used as constraints or Gaussian priors in Eq. (7.2) are the expected uncertainties of the individual background normalisations. The corresponding numbers of the normalisation of theW+jets background components obtained from the data-driven calibration (as shown in Section 5.3) amount to 7% forW+b¯bandW+c¯c, 25% forW+c, and 5%

forW+light jets events[270]. The uncertainty on the multijet background is 30% and originating from the matrix method[272]. For both diboson and Z+jets events, a 4% theory uncertainty in the inclusive cross-section is applied combined with a 24% uncertainty per additional jet taken in quadrature, which serves to cover the extrapolation to higher jet multiplicities according to MC studies, and adds up to a 48% uncertainty for events having four jets. Finally, the single top

7 . 3 D E F I N I T I O N O F T H E L I K E L I H O O D

quark background contribution is assigned with an uncertainty of 17%. This number accounts for the variation of initial and final state radiation in the MC samples for t-channel processes and incorporates extra jets in single top quark events. The numbers are summarised in Table 7.2.

All background fit parametersn_j are common across the utilised b-tag bins, lepton channels and

|η|regions, which means that one parameter comprises all eight regions, except for the multijet background. For the latter, each of the eight analysis regions is associated with an individual fit parameter, i.e. the number of multijet events is varied independently in all analysis regions.

Fit parameter σbkg,j

n_signal

-n_singletop 0.17 n_W+b¯b/c¯c 0.07

n_W+c 0.25

n_W+light 0.05

n_Z+jets 0.48

n_diboson 0.48

n_multijet 0.30

Table 7.2:Signal and background parameters that enter the binned likelihood fit and the relative normalisation uncertaintiesσbkg,j as used in the Gaussian constraints imposed on the varying background contributions. Shape uncertainties are discussed in Sec. 8.3.

Contrary to the background sources, the number of signal events is left unconstrained in the fit.

The uncertainty on the signal normalisation is thus not used in the fit but only considered for the uncertainties presented in Table 6.1. It amounts to 6.43% and is directly taken from the theory calculation[64–68, 71] of the t¯t cross-section at a centre-of-mass energy of 8 TeV presented in Sec. 5.2.

Within the decay width values used for the template reweighting, the difference of the observable distributions in the ratios with respect to the nominal template is up to around 5% or even less in the individual bins, demonstrated in the ratio panels of Fig. 7.3 and Fig. 7.4. As less than 5% of all events in these templates are presumed to originate from single top quarks, the entire effect of the single top events on theΓt templates is covered by the MC statistical uncertainty in the single top background. Studies which rest on a change of the single top mass in the event reconstruction with KLFitter also revealed that those mass variations do not bias the result. Because of the presence of one “fake” top quark in single top events these are not reweighted.

The fit is performed for 55 templates, the 54 templates obtained from the reweighting algorithm and the nominal one. The combined likelihood considering both observables is maximised for all available templates ofΓt. It is defined as the product of two Poisson terms as written in Eq. (7.2), one for each observable, multiplied by the Gaussian constraints. The measured value ofΓtis derived from the minimum of a quadratic fit to the negative logarithm of the likelihood values from the fits to all templates. The statistical uncertainty of the measurement is extracted from the width of this

likelihood curve at−2∆ln(L) =1 around the minimum, which is half of the distance between the abscissa positions having a function value of one above the minimum. Thus, the likelihood values are shifted in a way that the minimum coincides with−2∆ln(L) =0. Such a likelihood curve for the fit to data is shown in Fig. 10.3.

The templates used in the fit constitute concatenated distributions comprising all eight mutually exclusive analysis regions. These distributions are fitted for the two observables using all fit param-eters, one for signal and one for each background contribution except for the multijet background, for which eight free parameters are used. The different parameters, the pre-fit numbers of events and the associated uncertainties are summarised in Table 7.3.

Parameter Norm. unc. Pre-fit

t¯t 6.43% 153138

Single top 17% 6731

W+b¯b/c¯c 7% 8381

W+c 25% 3363

W+light 5% 1629

Z+jets 48% 2521

Diboson 48% 522

Multijete, 1 b-tag,|η| ≤1 30% 228 Multijete, 1 b-tag,|η|>1 30% 2493 Multijete,≥2 b-tags,|η| ≤1 30% 41 Multijete,≥2 b-tags,|η|>1 30% 538 Multijetµ, 1 b-tag,|η| ≤1 30% 195 Multijetµ, 1 b-tag,|η|>1 30% 1873 Multijetµ,≥2b-tags,|η| ≤1 30% 46 Multijetµ,≥2b-tags,|η|>1 30% 399

Table 7.3:Pre-fit yields in the eight channel combination with the related normalisation uncer-tainties. The numbers are not rounded in order to directly display the values used in the fit which leads to differences with the numbers listed in Table 6.1. The normalisation uncertainty for the t¯tsignal is not used in the fit as the number oft¯tsignal events is left unconstrained. However, this percentage number is reflected in thet¯tuncertainties listed in Table 6.1.

From the technical point of view, the different components of Eq. (7.2) are defined as particular RooFit objects, mainlyRooHistPdfobjects, where “Pdf” stands for probability density function. To account for the Gaussian constraints on the background, specificRooGaussianobjects are available within the RooFit framework. Since the likelihood fit combines Poisson terms for both observables and fits their templates simultaneously, furtherRooSimultaneousobjects, which incorporate the RooFit objects for the observablesm_`band∆R_min(j_b,j_l), enter the implementation of the likelihood fit as well.

7 . 4 E VA L U AT I O N O F S Y S T E M AT I C A N D E X P E C T E D S TAT I S T I C A L U N C E RTA I N T I E S

7.4 Evaluation of Systematic and Expected Statistical