5.4 Systematic uncertainties
5.4.2 Initial / final state radiation
5.4 Systematic uncertainties For the evaluation of some of the systematic uncertainties, only varied tt → WWbb templates are available. A dedicated study is conducted to evaluate the effect of neglecting the variation of the smaller contributions fromtt →WWqbandtt→WWqqusing, as an example, the ISR/FSR uncertainty.
Measurement ofRbandσtt¯
present in the selected events of the nominal Protost¯tsample in order to reduce the number of categories to a reasonable amount, 25 in total. But not all of these categories have a significant contribution in the selected events. Table5.5 presents only the categories with relative contributions exceeding 2% to at
Category %
tt →WWbb tt→WWqb tt →WWqq
bb 38.7 0.2 0.0
bbq 22.8 0.9 0.0
bbqq 11.1 0.8 0.2
bq 13.4 44.6 0.3
bqq 4.9 25.3 0.4
bqqq 2.0 12.5 0.5
qq 0.8 7.7 51.5
qqq 0.1 2.8 28.6
qqqq 0.1 1.0 13.9
bbb 2.3 0.1 0.0
cqqq 0.3 0.9 2.0
others 3.6 3.3 2.7
Table 5.5: Fractions of selected events in a given category for the threett¯templates, as estimated from the nominal Protossample. The “bbqq”, “bqqq”, “qqqq” and “cqqq” categories are inclusive with respect to the number of light jets. The “bbb” category is inclusive with respect to the number ofb-jets. All other categories presented in this table are exclusive.
least one of the three templates. As it can be seen from the table, at most four categories significantly contribute to a given template. Figure5.14shows distributions of the number ofb-tagged jets for the
“bb” category of thett →WWbbevents. The nominal Protosand variation distributions, as obtained from the corresponding ISR/FSR variation samples, are presented together.
The ISR/FSR variation is expected to have a symmetric effect in each bin of the nominal distribution of a given category. The deviation from this expectation (see figure5.14) are likely due to statistical fluctuations, the use of different generators and the use of full vs. fast simulation. In order to symmetrise the ISR/FSR distributions to the corresponding nominal distributions, a set of scale factors is determined for each category by equation5.4,
SFi= 2ynomi
ymini + ymaxi , (5.4)
where ynomi , ymaxi and ymini are respectively the nominal value and the higher and the lower values of the ISR/FSR distributions of the number ofb-tagged jets in thei-th bin, where all distributions are normalised to unity. The scale factors, SFi, are applied to the corresponding bins of the original ISR/FSR distributions (not normalised to unity). This leads to the symmetrisation of the ISR/FSR variation with respect to the nominal distribution.
The symmetrised ISR/FSR distributions (dashed red and blue histograms) of the “bb” category for thett →WWbbtemplate is shown in figure5.14, together with the original ISR/FSR (solid red and blue histograms) and the nominal (solid black histogram) distributions. All distributions are normalised to the integrated luminosity in data. The approximate 4-times difference in normalisation of the nominal and the ISR/FSR distributions is due to theRb = 0.5 parameter used in the generation of the nominal Protossample. The normalisation of the ISR/FSR distributions is not necessarily preserved after the
90
5.4 Systematic uncertainties
Number of b-tagged jets
0 1 2 ≥ 3
1 / events
0 0.2 0.4 0.6 0.8
bb category of WWbb template
Nominal IFSR up, original IFSR down, original
Number of b-tagged jets
0 1 2 ≥ 3
Events
0 500 1000 1500 2000
bb category of WWbb template
Nominal IFSR up, original IFSR down, original IFSR up, symmetrized IFSR down, symmetrized
Number of b-tagged jets
0 1 2 ≥ 3
Events
0 100 200 300 400 500
bb category of WWbb template
Nominal IFSR up, normalized IFSR down, normalized
Figure 5.14: Distributions of number ofb-tagged jets for the “bb” category of the selectedtt→ WWbbevents, for the nominal Protosand ISR/FSR variations for the original (left), the symmetrised distributions (middle) and after normalisation (right). The solid black histograms correspond to the nominal distribution. The solid red and blue histograms stand for the original distributions of the ISR/FSR “up” and “down” variation respectively.
The dashed red and blue histograms correspond to the modified distributions of the ISR/FSR “up” and “down”
variation respectively.
symmetrisation procedure. For example, the solid red and the dashed red histograms in figure 5.14 might not correspond to the same number of events.
In the next step, the symmetrised distributions are normalised to the number of events in the cor-responding nominal distributions. The normalisation factor is the ratio of the number of events in the nominal distribution over the averaged number of events in the symmetrised ISR/FSR “up” and “down”
distributions. As an example, the resulting normalised ISR/FSR distributions for the “bb” category of thett→WWbbtemplate are shown in figure5.14together with the nominal distribution. The normal-ised distributions of ISR/FSR “up” (“down”) variation of all categories are summed and then normalised to one in order to produce derivedtt → WWbbISR/FSR “up” (“down”) templates. They are shown together with the nominal and the original ISR/FSR templates in figure5.15.
Number of b-tagged jets
0 1 2 ≥ 3
1 / events
0 0.2 0.4 0.6 0.8 1
Nominal IFSR up, derived IFSR down, derived IFSR up, original IFSR down, original
WWbb template
Figure 5.15: The nominal tt → WWbbtemplate from Protos sample (solid black) and its original (solid red and blue) and derived (dashed red and blue) ISR/FSR “up” and “down” variation templates. All templates are normalised to unity.
The distribution of the number ofb-tagged jets only depends on the event category and not on the type
Measurement ofRbandσtt¯
of the template. This can be seen from the plots in figure5.16. The same categories but corresponding
a.
0 1 2 3 4
1 / events
0.2 0.4 0.6 0.8
bq in WWbb bq in WWbq
Number of b-tagged jets
0 1 2 ≥ 3
Ratio
0.5 1
b.
0 1 2 3 4
1 / events
0.2 0.4
0.6 bqq in WWbb
bqq in WWbq
Number of b-tagged jets
0 1 2 ≥ 3
Ratio
0.5 1
c.
0 1 2 3 4
1 / events
0.2 0.4 0.6 0.8
qq in WWbb qq in WWbq
Number of b-tagged jets
0 1 2 ≥ 3
Ratio
0.5 1
d.
0 1 2 3 4
1 / events
0.2 0.4 0.6 0.8
qq in WWbb qq in WWqq
Number of b-tagged jets
0 1 2 ≥ 3
Ratio
0.5 1
e.
0 1 2 3 4
1 / events
0.2 0.4 0.6 0.8
qq in WWbq qq in WWqq
Number of b-tagged jets
0 1 2 ≥ 3
Ratio
0.5 1
f.
0 1 2 3 4
1 / events
0.2 0.4
0.6 qqq in WWbq
qqq in WWqq
Number of b-tagged jets
0 1 2 ≥ 3
Ratio
0.5 1
Figure 5.16: Distributions of number ofb-tagged jets for the “bq” (a), “bqq” (b), “qq” (c-e) and “qqq” (f) categor-ies compared for pairs of relevant templatestt →WWbb tt → WWqbandtt→WWqq. The ratio bands show the statistical uncertainty and are centered at the ratio value.
to different templates are compared to each other. The categories, which contribute significantly in at least one of the three templates are chosen for comparison. All ratios are compatible with one within the statistical precision. Based on this finding, the ISR/FSR variations of the categories of thett →WWbb template can be used to derive variations for the categories of the tt → WWqb and tt → WWqq templates.
The procedure of symmetrisation described before is used to derive similar symmetrised distributions for the categories of the other templates. The set of scale factors for each category is calculated with equation5.4. Now, the normalised nominal distribution of a given category of thett→WWbbtemplate, from which theynomi values are obtained, is replaced with the template under study,tt → WWqb or tt→WWqq. Obviously, the corresponding distributions of the ISR/FSR variation, from which theymini andymaxi values are obtained, are taken from thett →WWbbtemplate. The scale factors, SFi, are used to derive symmetrised distributions (similar to the distributions shown in figure5.14) for all categories of thett → WWqbandtt → WWqq templates. In the next step, the distributions are normalised to the number of events in the corresponding nominal distributions, as was explained above. Finally, the normalised distributions of all categories are summed to create the “up” and “down” ISR/FSR variation of thett → WWqb and tt → WWqq templates. These derived templates are shown in figure 5.17 together with the corresponding nominal templates.
To estimate the effect of using the derived templates described above, three different sets of
pseudo-92
5.4 Systematic uncertainties
Number of b-tagged jets
0 1 2 ≥ 3
1 / events
0 0.2 0.4 0.6 0.8 1
WWbq template
Nominal IFSR up, derived IFSR down, derived
Number of b-tagged jets
0 1 2 ≥ 3
1 / events
0 0.2 0.4 0.6 0.8 1
WWqq template
Nominal IFSR up, derived IFSR down, derived
Figure 5.17: The tt → WWqb (left) andtt → WWqq(right) templates and their derived ISR/FSR “up” and
“down” variations.
experiments are performed. Results obtained using the original ISR/FSRtt → WWbbtemplate (tt → WWqbandtt → WWqqare taken from Protos), results obtained with only thett → WWbbtemplate derived, including the symmetrisation procedure (tt→WWqbandtt→WWqqare taken from Protos) and results where all derived templates tt → WWbb, tt → WWqb and tt → WWqq are used, are compared and the resulting ISR/FSR uncertainty are shown in table5.6.
Setting ∆Rb ∆σdilepton
Varied sample (tt→WWbbonly) 0.016 0.15 pb
Derivedtt→WWbbtemplate 0.018 0.16 pb
Derivedtt→WWbb,tt →WWqb,tt→WWqq 0.018 0.16 pb
Table 5.6: Uncertainty onRbandσdileptonin different scenarios concerning the variation of thett→WWbb,tt→ WWqbandtt→WWqqtemplates obtained from pseudo-experiments withRb=0.99830 andσdilepton=11.33 pb.
No significant discrepancy is observed when comparing the results forRbandσdilepton for the three different approaches.