Many different sets of tests are performed to prove the stability of the presented measurement and its ability of constraining uncertainties beyond their initial ±1σ variations. Most of the studies rely on pseudo-experiments or the exchange of the input templates for certain systematic uncertainties, and some of the studies will be presented here. Since the different jet energy scale uncertainties can be constrained significantly below their initial ±1σ range, several tests are performed to test the sensitivity of the analysis to modeling assumptions made for the JES uncertainty. For the tests described in the following a slightly different setup of the analysis is used as the default reference, with a fit result very close to the final presented one: β0 = 1.12±0.10, which translates intoσt¯t = 185+18−16 pb, and consistent results for the other fit parameters4. The resulting combined uncertainty from jet energy scale should cover a scenario in which all transverse momenta are miscalibrated globally,
4no systematic uncertainties external to the fit are estimated here, since only the fitter behavior is to be tested
JES Component (%)
ALPGEN Model + 0.87 - 0.88
PERUGIA tuning + 0.53 - 0.72 b-Jet Energy Scale + 0.87 - 0.66 Calorimeter Response + 1.02 - 0.97
Noise + 1.18 - 1.02
Flavor Composition + 2.99 - 1.92
Close-By Jet + 0.75 - 0.59
Pile-Up + 0.53 - 0.59
η-Extrapolation + 1.22 - 0.59 Total JES Uncertainty (%) + 3.93 - 2.90
Table 6.8.: Observed uncertainties from the different sources of jet energy scale uncertainties on the combined fit. A nuisance parameterδi is associated to each of the sources separately.
i.e. if the central jet energy scale is shifted. Such a scenario is created artificially by shifting the transverse momenta of all jets found in a MC simulated event down by 1.5%. The full analysis is repeated with this setup, including all systematic uncertainties and the creation of the necessary templates. While the shapes of the likelihood discriminant D do not change much in this scenario, the number of predicted events and especially the division of the events in different jet multiplicity bins does. Performing the full fit in this setup yields β0 = 1.13±0.09 or σt¯t = 187 pb. This corresponds to a shift of the central value of 0.8%, and a small reduction of uncertainties, clearly covered by the JES related sources of uncertainties, which account for +3.9%/-2.9%. Furthermore, as can be seen in figure ??, only the nuisance parameters related to jets differ with respect to the nominal fit, and tend to be fitted to larger negative values on average. This is expected, since the transverse momenta are shifted to smaller values.
Three different studies are performed to test assumptions made about correlations as a function of pseudorapidity and transverse momentum of the different jets in the event in the calorimeter response uncertainty. This uncertainty is chosen, since studies suggest that the correlation of this uncertainty for jet pairs well separated in pseudorapidity or transverse momentum can be smaller than one [77]. First, the uncertainties are modified to introduce an additional dependence on the transverse momentum, by varying an additional factorx ×σ fromx = 0.6 tox = 1.5 linearly over the range [0,200] GeV. Compared to the default setup, the fit result of β0 = 1.13±0.10, translated into σt¯t = 186+18−16 is obtained. This corresponds to a difference of 0.77% in the measured cross section, while the contribution from the calorimeter response uncertainty is measured to be approximately +0.6%/-0.8%, i.e. in agreement with the observed shift. In the second test the calorimeter response uncertainty is modified depending on the pseudorapidity of the jet, following the recipe
x =
1.1 for |η| <0.5.
0.9 for 0.5≥ |η| <1.0.
1.2 for 1.0≥ |η| <1.5.
0.8 for 1.5≥ |η| <2.0.
1.3 for 2.0≥ |η| <2.5.
(6.3)
While the actual values are rather arbitrary, the larger shifts of uncertainties are implemented in the less well-understood forward region. With this setup a fit parameter of β0 = 1.13±0.10 is obtained, corresponding to σt¯t = 187+17−16 pb, which is also covered by the uncertainty associated to the calorimeter response term. The agreement between the different fit parameters and the default setup can be seen in figure ??, showing deviations between the results only for the different JES parameters, as expected. Overall, the agreement between the different setups is found to be good, proving the validity of the models.
b-tag WP1 b-tag WP2 b-tag WP3 b-tag WP4 Mistag WP1 Mistag WP2 Mistag WP3 Mistag WP4 Jet ID JER Wbb/cc 3 Jets Wbb/cc 4 Jets Wbb/cc 5 Jets Wc 3 Jets Wc 4 Jets Wc 5 Jets Pile-Up SFsµ e SFs Resolutionµ e Resolution e Scale miss TE JES Eta JES Calo JES Alpgen JES Noise JES Pile-Up JES Perugia JES b-Jet JES flavor
Nuisance Parameter
-4 -3 -2 -1 0 1 2 3 4 5
nominal shifted jet pT
T) calo JES (p
) calo JES (η
b-tag WP1 b-tag WP2 b-tag WP3 b-tag WP4 Mistag WP1 Mistag WP2 Mistag WP3 Mistag WP4 Jet ID JER Wbb/cc 3 Jets Wbb/cc 4 Jets Wbb/cc 5 Jets Wc 3 Jets Wc 4 Jets Wc 5 Jets Pile-Up SFsµ e SFs Resolutionµ e Resolution e Scale miss TE JES Eta JES Calo JES Alpgen JES Noise JES Pile-Up JES Perugia JES b-Jet JES flavor
Nuisance Parameter
-4 -3 -2 -1 0 1 2 3 4 5
Figure 6.29.: Observed fit parameters for a nominal fit with the default settings and three different scenarios: (1) jet energy scale of all jets shifted down by 1.5%, (2) a pT dependent calorimeter response of the JES uncertainty and (3) aηdependent calorimeter response of the JES uncertainty.
The influence of a decorrelation of the calorimeter response uncertainties for jets with different transverse momenta is evaluated in a third test. The default setup makes the assumption of a 100%
correlation of uncertainties between jets with different transverse momenta, and studies show that this is true for the majority of jet pairs, while a smaller correlation of 80% is observed for few cases.
For the presented test a conservative assumption is made, and the calorimeter response uncertainty is separated into three different uncorrelated terms and three nuisance parameters in the fit:
• 50% uncertainty, fully correlated over the full range of jet pT.
• linear increase of the uncertainty from 0% to 50% in the range 0< pT <200 GeV.
• linear decrease of the uncertainty from 50% to 0% in the range 0< pT <200 GeV5.
The fit yieldsβ0 = 1.12±0.07 and σt¯t = 185+17−16 pb , i.e. no difference in the measured cross section is observed. The uncertainties are slightly reduced due to a better ability of the fit to adjust to data, and the fit parameters not corresponding to the calorimeter response change insignificantly within
5In both cases jets withpT > 200 GeV are assigned the same shift in uncertainty as jets atpT = 200 GeV.
their uncertainties. This test proves that there is no effect of over constraining the uncertainty from the low pT range to the high pT range or vice versa, since these are handled separately here.
More tests were performed, but not explained in detail here for the sake of brevity6. No significant deviations of the fitter performance and the measured cross section not covered by the uncertainties are observed when including shape variations, for example sinusoidal or step-functions, modifying the behavior within the ±1σ ranges. A more realistic model of the flavor composition and the related uncertainties for the dominant W+jets background is used, which leads to a reduction of the fit uncertainties, but does not change the resulting top quark pair production cross section. Instead of creating the pseudo-data set from a Gaussian sampling of the initial values βi = 1.0 and δi = 0.0, systematic variations of one or more nuisance parameters can be used. In such a setup one can test the ability of the fit to discover such variations, i.e. the ability to extract the correct input value for the nuisance parameter. Furthermore, the resulting shift in the signal cross section should be covered by the uncertainty associated to the varied source of systematic uncertainty. This is found to be true for a selection of nuisance parameters varied, and the overall uncertainty of the measurement is found to cover even very pessimistic scenarios in which several systematic uncertainties are shifted to their respective ±1σ uncertainties simultaneously.
A series of two consecutive fits is performed, reducing the ±1σ templates of one uncertainty to the templates created using the fitted ±1σ uncertainties. In a second iteration of the fit, the fit parameters should not change with respect to the initial fit, and the uncertainties of this special parameter should not be reduced any further. Taking into account the convolution with the Gaussian constraint, these assumptions are verified, showing that the reduction of uncertainties is not an artificial effect of the fit. The stability of the fit result is tested by changing the number of bins for the likelihood discriminant D and by performing fits to the separate input variables themselves or likelihood discriminants constructed from subsets of the input variables. Within the statistical uncertainties all these fits agree with the main result.