10. Analysis strategy 101
10.4. Choice of observables
The 8 TeV analysis showed that the choice of observables used in the fit affects the total un-certainty of the measurement. The 13 TeV measurement utilises different observables in the individual analysis regions. If the observables from a single channel are uncorrelated, the defini-tion of the likelihood from Equadefini-tion (10.10) can be easily extended by multiplying the likelihoods for the considered distributions for each observable. The observables are split into observables that are sensitive to the decay width of the top quark, and observables that are insensitive to the decay width but are sensitive to the dominant systematic uncertainties affecting the mea-surement. The observables sensitive to the decay width act assignal regions in the “standard”
profile likelihood measurements and observables insensitive to the decay width act as control regions. The control regions are designed to control and, potentially, constrain the dominant systematic uncertainties.
The m`b distribution has been proven to be the optimal variable for the measurement of the top-quark decay width in the 8 TeV analysis and it is also used for the 13 TeV measurement in both analysis channels. To minimise the NLO effect in the decay vertex, them`b distributions are restricted tom`b<150 GeV.
10.4.1. Lepton+jets channel
The choice of the m`b distribution in the lepton+jets channel allows distributions from the hadronic hemisphere of thet¯tdecay to be used, as these distributions originate from a different top quark than them`band are expected to be uncorrelated. A natural choice for a variable that is insensitive to the decay width of the top quark but is sensitive to the dominant systematic uncertainties, especially JES and JER, is the reconstructed mass of the W boson from the hadronically decayingW,mW. Additionally, the ratio of the reconstructed invariant top-quark mass and the reconstructed invariant mass of theW boson,R3/2, can be used in the measurement of the decay width due to its large sensitivity to the decay width and small sensitivity to scales and their uncertainties. The ambiguity in the jet-to-parton assignment is resolved following techniques summarised in Section 8.1. The m`b and R3/2 distributions for various top-quark decay widths, obtained from the reweighting technique described in Section10.1.1, are displayed in Figure10.6. Two different choices of variables in the lepton+jets channel are compared:
1. m`bdistribution from combined electron+jets and muon+jets events as a variable sensitive to the top-quark decay width, and mW from the combined electron+jets and muon+jets events as a control variable.
2. m`bdistribution from combined electron+jets and muon+jets events, andR3/2 from muon +jets events as variables sensitive to the top-quark decay width. The mW distribution from electron+jets events is used as a control variable. Distributions of R3/2 and mW are expected to be correlated and thus they need to come from an orthogonal selection, which is ensured by the split by charged lepton flavours.
Variables originating from the decays of the semileptonically and fully hadronically decaying top quark are expected to be uncorrelated but due to the reconstruction technique described in Section 8.1, a correlation can appear. The correlations are ≤ 0.01 and the variables are thus treated as uncorrelated further in the analysis. The corresponding figures are presented in AppendixE.
10.4. Choice of observables
Figure 10.6.: Template distributions for various values of Γt for the m`b distribution in the electron+jets channel (top-left), muon+jets channel (top-right) and for the R3/2 distribution in muon+jets channel (bottom). The additional requirement on the BDT score > 0.7 is applied. The bottom part of each plot shows the ratio of events for alternative top-quark decay widths with respect to the SM expected value Γt= 1.32 GeV. The first bins contain underflow events. The last bin contains overflow events in the case of R3/2 distribution.
10. Analysis strategy
10.4.2. Dilepton channel
The choice of the variables in the dilepton channel is limited considering that any techniques that reconstruct the full kinematics of the decaying top quarks suffer from large systematic un-certainties, mainly from the uncertainty ofETmissenergy scale and resolution. Despite the limited possibilities, the m`b observables can be easily reconstructed without the need to reconstruct the kinematics of the neutrinos. Thus, m`b is chosen as the observable sensitive to the decay width of the top quark. For the same reasons as in the lepton+jets channel, only values of m`b<150 GeV are considered in the fit. Due to two top quarks decaying leptonically present in the dilepton channel, twom`b variables can be reconstructed in the dilepton channel per event.
Both of these variables are considered, and the corresponding histograms are filled twice per event, the ambiguity of the pairing of the charged leptons and the b-jets is resolved with the technique described in Section8.2. The distribution of them`b variable in the dilepton channel for various values of the Γt is displayed in Figure 10.7.
The variable insensitive to the top-quark decay width and sensitive to the dominant systematic uncertainties is mbb, the invariant mass of the twob-jet system. mbb does not originate from a decay of an unstable particle and is not expected to have a distinct peak in the distribution.
Nevertheless,mbb is the variable that is sensitive to the JES and JER and especially to the JES uncertainties related to the b-jets, which may differ from those for the light-flavour jets. The combination of the lepton+jets channel and the dilepton channel thus provides strong control of the jet related uncertainties viamW and mbb observables. Since m`b and mbb observables from the dilepton channel are expected to be correlated, an orthogonal event selection is used for the variables. Them`b distribution is built from events containing exactly one electron and one muon, theeµ channel, while the mbb distribution is built from combined events with the same flavour of the leptons,eeandµµ, which ensures the statistical independence of the distributions.
Them`bdistribution exploitseµevents as these events have a higher expected number of events, due to the branching ratio of the process and looser selection criteria but also due to the smaller expected background.
10.4.3. Lepton+jets optimisation
To identify the optimal selection on the BDT score (see Section8.1), different selection values of the BDT output have been tested in the early stages of the analysis to identify the selection that leads to the lowest total expected uncertainty on Γt in the lepton+jets channel. Selection with a BDT output>0.3, >0.5, >0.7 and >0.9 are compared in a fit with the dominant systematic uncertainties, that were available during the time of the test, to the distributions representing Γt = 1.32 GeV and mt = 172.5 GeV. Table 10.5 shows the total expected uncertainties for various BDT selection requirements for the configuration with the R3/2 variable. Although some of the systematic uncertainties were not available during the test, it can be seen that the BDT selection criterion does not affect the expected uncertainty significantly between BDT requirements of 0.3–0.7. The requirement of BDT score> 0.7 has lead to the lowest expected total uncertainty on the decay width, and is used further in the analysis.
As described in Section 10.4.1 two sets of observables are tested in the lepton+jets channel, with and one set without the R3/2 observable. The expected uncertainties for both choices are compared in the fit to expected distributions for Γt = 1.32 GeV andmt= 172.5 GeV. The fit includes all considered systematic uncertainties for the analysis. Table10.6shows the expected uncertainties for both configurations. Since the setup that includes the R3/2 observable yields only marginally better expected uncertainty, other criteria have to be considered. One of the
10.4. Choice of observables
20 40 60 80 100 120 140
5000 10000 15000 20000 25000
Events / 5.2 GeV
1.32 GeV 0.50 GeV 1.00 GeV 2.00 GeV 3.00 GeV 4.00 GeV 5.00 GeV 6.00 GeV µ
e 2 b-tags
≥ = 13 TeV, 140.5 fb-1
s Full Run 2
50 100 150
[GeV]
mlb
0.9 1 1.1
/1.32 GeVtΓ
Figure 10.7.: Template distributions for various values of Γt for the m`b distribution in the eµ channel. The bottom part of the plot shows the ratio of events for alternative top-quark decay widths with respect to the SM expected value Γt = 1.32 GeV.
The first bin contains underflow events.
BDT score Total exp. uncertainty
>0.3 ±0.367 GeV
>0.5 ±0.355 GeV
>0.7 ±0.349 GeV
>0.9 ±0.490 GeV
Table 10.5.: Total expected uncertainty estimated in the early stages of the analysis in the lepton+jets channel for various requirements on the BDT score. The fit setup with theR3/2 variable is used. Some of the systematic uncertainties were not available for the comparison.
10. Analysis strategy
main disadvantages of using the setup with R3/2 is the stability of the fit. Including the R3/2 observable in the fit increases the number of bins significantly, thus making the fit less stable.
This results in frequent non-convergence with even minor changes in the fit, e.g. decorrelating some of the NP between the distributions. Therefore a setup that only includes distributions of m`band mW from the combined electron+jet and the muon+jets channel is used further in the analysis.
Fit setup Total exp. uncertainty
WithR3/2 +0.64 GeV
−0.49 GeV WithoutR3/2 +0.68 GeV
−0.53 GeV
Table 10.6.: Expected total uncertainties on Γtfor two sets of observables. The first set contains m`b, R3/2 and mW, the second set contains m`b and mW. The uncertainties are obtained from the fit to the expected distribution representing Γt= 1.32 GeV and mt= 172.5 GeV. The BDT score is required to be>0.7.