10. Analysis strategy 101
10.5. Fit validation
Several procedures have been employed to validate the fitting procedure. The tests for the linearity of the response to the decay width variations rely on the fits of the width templates, as described in Section10.3, to a combination of background templates and a tt¯signal template with a fixed Γt, the so calledAsimov fits8. The tests are carried out to check the mean value of the fitted Γt. The fitted value should correspond to the value of Γtused to build the given Asimov dataset. Deviations from this value would correspond to a bias in the procedure. Furthermore, the post-fit mean values of the NP probability density functions, so called pulls9, should be centred at zero as all the Asimov templates are built from the nominal distributions. Pulls not-centred at zero would point to a non-linearity of the response to the decay width, as some of the width effects would be compensated by the pulls of systematic uncertainties. The post-fit standard deviations of the NP probability density functions represent the posterior uncertainty of the given NP. Table10.7 shows the fitted value of Γtwith the corresponding uncertainties in two fit scenarios: a) when no systematic uncertainties are considered (fit with statistics only) and b) fit with all considered systematic uncertainties as summarised in Chapter 9. Results of the Asimov fit for botht¯tdecay channels are shown. Good agreement between the fitted values and the input values is obtained, with the non-closure≤0.02 GeV. It can also be seen that the expected total uncertainty depends on the mean value of the fit.
The pull distributions for the Asimov fits with input Γt = 1.32 GeV are displayed in Fig-ure10.8 for the lepton+jets channel and in Figure 10.9 for the dilepton channel. No pulls are observed in the Asimov fits, while some of the NPs are constrained in the fit. These include mainly uncertainties for the tt¯modelling and, to some extent, also for the JES and JER un-certainties. The constraints partially arise from the fact that some of the uncertainties are
8Asimov dataset is named after a short story Franchise, by Issac Asimov (Isaac Asimov, Franchise, in Isaac Asimov: The Complete Stories, Vol. 1, Broadway Books, 1990), where elections are held by selection of a single person to represent the entire population.
9Sometimes the wordpull is used to represent both the shift of the mean value and the standard deviation.
10.5. Fit validation conservative, especially t¯t modelling uncertainties, while the JES and JER uncertainties are constrained because the dedicated calibrations do not use the full Run 2 dataset of 140 fb−1, and for JER the calibrations are done only with the dedicated 2017 LHC conditions and they do not reflect LHC conditions of the whole Run 2.
Correlations of the NP, obtained from the minimisation technique described in Section10.3.2, are displayed in AppendixG. Figure 10.10 shows the likelihood scan of Γt and the NP ranking plot for the lepton+jets channel, and similarly for the dilepton channel in Figure 10.11. The nuisance parameter ranking plot is obtained from running the Asimov fit to systematically shifted distributions for both up and down variation with the pre- and post-fit constraints, the difference between the fitted values and the nominal values are shown, which gives a hint on the impact of the individual uncertainty sources on the measurement. It can be seen that the lepton+jets channel is dominated by the uncertainties in the normalisation of the multijet background as well as tt¯modelling uncertainties. Although, the multijet process has only small contribution it has a distinct effect on the Γt as the multijet process populates mostly regions with low value ofm`b. But the regions with low value ofm`b are also very sensitive to the decay width as is displayed in Figures 10.6 and 10.7. The dilepton channel is dominated by the tt¯ modelling uncertainties. The likelihood scan is obtained by setting the POI, Γt, to a fixed value and then running the minimisation procedure to obtain the best likelihood value, these values are displayed on the vertical axis10.
To further validate the fitting procedure the pseudo-experiment approach is used to cross-check the expected statistical uncertainties obtained from the fit. Pseudo-data are generated 3000 times and then fitted with the procedure outlined in Section 10.3.3 while excluding systematic uncertainties. The mean values of the individual fits are then filled into a histogram. The resulting histograms are then fitted with a Gaussian function. The mean value and standard deviation of the Gaussian fits are compared to the input decay widths and expected statistical uncertainties obtained from the fit. Possible deviations of the mean values from the input widths would imply a bias in the technique. Deviations of the width of the Gaussian from the expected statistical uncertainties would suggest an incorrect estimation of the uncertainties. The mean values of the fits agree with the input width within uncertainties, for the input Γt >0.5 GeV.
Deviations from the Gaussian shape are observed for Γt<0.5 GeV which is reflected by the mean values and standard deviations. The deviations originate from the sharp edge at Γt= 0.2 GeV which is the smallest considered width template. For Γt>0.5 GeV the widths of the Gaussian curves correspond to the expected statistical uncertainties obtained from the fit as summarised in Table10.7. This provides a strong proof that the chosen fitting strategy is valid. The individual distributions are presented in AppendixF.
10The values are shifted by the likelihood value for the most probable value of Γt.
10. Analysis strategy
Lepton+jets Dilepton
Input Γt Fit setting Mean value Uncert. Mean value Uncert.
[GeV] [GeV] [GeV] [GeV] [GeV]
0.5 Stat 0.51 +0.14
0.51 +0.17
−0.13 −0.16
0.7 Stat 0.70 +0.15
0.70 +0.19
−0.14 −0.17
1.0 Stat 1.00 +0.16
1.00 +0.20
−0.15 −0.19
1.32 Stat 1.32 +0.16
1.32 +0.21
−0.16 −0.20
1.5 Stat 1.50 +0.16
1.50 +0.21
−0.16 −0.21
2.0 Stat 2.00 +0.16
2.00 +0.22
−0.16 −0.22
2.5 Stat 2.50 +0.16
2.50 +0.22
−0.16 −0.22
0.5 Stat+Syst 0.52 +0.38
0.51 +0.39
−0.22 −0.28
0.7 Stat+Syst 0.70 +0.49
0.70 +0.44
−0.31 −0.35
1.0 Stat+Syst 1.00 +0.60
1.00 +0.47
−0.42 −0.42
1.32 Stat+Syst 1.32 +0.68
1.32 +0.48
−0.53 −0.46
1.5 Stat+Syst 1.50 +0.71
1.50 +0.49
−0.57 −0.47
2.0 Stat+Syst 2.00 +0.78
2.00 +0.51
−0.66 −0.48
2.5 Stat+Syst 2.50 +0.83
2.50 +0.53
−0.73 −0.50
Table 10.7.: Summary of the tests of the fitting procedure with the statistics only fit and a fit with the full systematics model for the fixed template for various top-quark decay widths. Mean value and the expected uncertainty for each fit are shown for the lepton+jets channel and for the dilepton channel. Good agreement between the input width and the fitted width is observed.
10.5. Fit validation
Figure 10.8.: The pull distributions for the NPs used in the fit for the lepton+jets channel. The black dots represent the fit mean value of the NP, the lines represent the post-fit NP uncertainties. The green and the yellow bands represent the pre-post-fit one and two standard deviations, respectively.
10. Analysis strategy
Figure 10.9.: The pull distributions for the NPs used in the fit for the dilepton channel. The black dots represent the fit mean value of the NP, the lines represent the post-fit NP uncertainties. The green and the yellow bands represent the pre-post-fit one and two standard deviations, respectively.
10.5. Fit validation
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 θ
)/∆ θ0
θ -( (TemplateLjets bin 4)
γ
(TemplateLjets bin 16) γ
JER_EffectiveNP_4 (TemplateLjets bin 15) γ
JET_Flavor_Response (TemplateLjets bin 19) γ
JET_EffectiveNP_Modelling1 JET_Pileup_OffsetNPV JER_EffectiveNP_2 (TemplateLjets bin 6) γ
MultijetShape_ljets (TemplateLjets bin 24) γ
aMCNLO_ljets PowHer7_ljets MultijetNorm_ljets
2
− −1 ∆0Γt 1 2
t: Γ Pre-fit impact on
θ +∆ = θ
θ θ = θ-∆θ
t: Post-fit impact on Γ
θ +∆ = θ
θ θ = θ-∆θ
Nuis. Param. Pull s = 13 TeV, 140 fb-1
0.5 1 1.5 2 2.5 3 3.5
[GeV]
Γt
0 2 4 6 8 10
ln L∆- 2 b-tags≥ l+jets -1
= 13 TeV, 140 fb s
Figure 10.10.: The ranking plot (top) and the likelihood scan (bottom) for the top-quark decay width in the lepton+jets channel. Only the 15 highest ranking NPs are shown.
The boxes represent the effect on Γt with the full boxes representing the post-fit values and empty boxes representing the pre-fit values. The points with the error bars represent the pulls of the NPs. For the NPs related to the MC statistical uncertainties, γ, the nominal value is represented by 1, while for other NPs it is represented by 0. The input top-quark decay width is Γt= 1.32 GeV.
10. Analysis strategy
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 θ
)/∆ θ0
θ -( JER_EffectiveNP_1
(TemplateDilep bin 3) γ
JER_EffectiveNP_7restTerm JET_EffectiveNP_Modelling1 St_Wt_model_dilepton (TemplateDilep bin 2) γ
(TemplateDilep bin 4) γ
FSR_dilepton JET_BJES_Response JER_EffectiveNP_2 pileup bTagSF_B_0 JER_EffectiveNP_5 ISR_dilepton PowHer7_dilepton
1.5
− −1 −0.5 ∆0Γt 0.5 1 1.5
t: Γ Pre-fit impact on
θ +∆ = θ
θ θ = θ-∆θ
t: Post-fit impact on Γ
θ +∆ = θ
θ θ = θ-∆θ
Nuis. Param. Pull s = 13 TeV, 140 fb-1
0.5 1 1.5 2 2.5 3 3.5
[GeV]
Γt
0 2 4 6 8 10
ln L∆- 2 b-tags≥ dilepton-1
= 13 TeV, 140 fb s
Figure 10.11.: The ranking plot (top) and the likelihood scan (bottom) for the top-quark decay width in the dilepton channel. Only the 15 highest ranking NPs are shown.
The boxes represent the effect on Γt with the full boxes representing the post-fit values and empty boxes representing the pre-fit values. The points with the error bars represent the pulls of the NPs. For the NPs related to the MC statistical uncertainties, γ, the nominal value is represented by 1, while for other NPs it is represented by 0. The input top-quark decay width is Γt= 1.32 GeV.
CHAPTER 11
Results
This chapter summarises the results obtained from the fit to data. Section 11.1 focuses on the results obtained from the 8 TeV measurement using the concatenated distributions of the m`b and ∆Rmin(jb, jl) observables in eight orthogonal regions. The 8 TeV result has been published in Reference [245]. The results of the 13 TeV measurement are presented in Section 11.2 and Section 11.3 for the lepton+jets and the dilepton channels, respectively. Finally, Section 11.4 summarises the results of the combined lepton+jets and dilepton channel profile likelihood tem-plate fit to the data at√
s= 13 TeV.
11.1. 8 TeV results
Concatenated distributions ofm`band ∆Rmin(jb, jl), defined in Section10.2, are split into eight orthogonal regions and are simultaneously fitted to the data following the binned likelihood template fit described in Section 10.2.3. Figure 11.2 shows the post-fit comparison of the con-catenated distributions. For reasons of visibility, the jet|η|regions are split into two figures.
Figure 11.1 displays the likelihood curve of the fit of the 55 width templates as discussed in Section 10.2.3. A quadratic fit to the likelihood points, which follows the parabolic shape, is shown. The likelihood values, given as twice the negative logarithm of the likelihood, −2L, are shifted so that the minimum of the curve corresponds to−2∆L= 0. The statistical uncertainty, which contains the contributions from the finite number of the data events and normalisation of the backgrounds, is inferred from the likelihood curve as the width of the curve at−2∆L= 1 around the minimum.
The measured decay width reads
Γt= 1.76±0.33(stat.)+0.79−0.68(syst.) GeV = 1.76+0.86−0.76 GeV, (11.1) assuming the top-quark mass mt = 172.5 GeV. The result is in good agreement with the SM prediction of Γt= 1.322 GeV corresponding to NNLO corrections [132].
The pre-fit and post-fit yields for signal and backgrounds are summarised in Table11.1. Both absolute and relative differences are shown, the relative difference is also displayed in terms of