8 Systematic Uncertainties
8.6 Impact of the Top Quark Mass
According to the explanations in Sec. 2.3, the top quark decay width depends significantly on the top quark mass. Moreover, the decay width is measured exploiting mass distributions sensitive to Γt. It is thus a relevant part of this analysis to quantify the influence of a variation of the top quark mass on the decay width.
For this purpose, alternative MC signal samples based on different masses mt were employed. The samples with underlying masses closest to the nominal value ofmt=172.5 GeV differ by 2.5 GeV from this reference although latest mass measurements evaluate the top quark mass with a much better precision[111, 123]. These two variations samples withmt=170 GeV andmt=175 GeV are based on a full simulation of the ATLAS detector withhdamp=∞. They are treated as systematic variation samples and compared to a full simulationt¯tsample withhdamp=∞andmt=172.5 GeV.
Pseudo-experiments were conducted as for other systematic uncertainties. The mean values of the distributions resulting from 1,000 pseudo-experiments were compared to infer the effect of the top quark mass on this measurement. The numbers including the standard deviationσof the resulting Gaussian-shaped distributions are listed in Table 8.4.
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mt MeanΓt
170.0 GeV 2.49±0.49 GeV 172.5 GeV 1.34±0.29 GeV 175.0 GeV 1.21±0.28 GeV
Table 8.4: Results of mean decay width valuesΓt obtained from 1,000 pseudo-experiments based on samples with different underlying top quark massesmt. The uncertainties correspond to the standard deviations from the Gaussian-shaped distributions that contain the resulting values from the pseudo-experiments.
The value for the alternative mass of 175 GeV is relatively close to the nominal expected decay width ofΓt=1.33 GeV but the smaller mass sample reveals a notable deviation from this expectation. In the following, attempts to parametrise this difference are presented before studies to understand this asymmetric response ofΓton the top quark mass variation are discussed.
The three measured values are depicted in Fig. 8.8, together with the simplest possible parametri-sation, i.e. linear splines between the points. In addition, decay width values obtained from theory for the three massesmt=170, 172.5, 175 GeV are included to illustrate discrepancies between the theoretical calculation and the measured numbers. These theory values are computed using a ratio with respect to the nominal value:
Γt(mt) with a reference value that equals 1.33 GeV. The given expression is equivalent to the leading order formula given in Eq. (2.2). NLO or NNLO terms possess only small differences in the given mass range so that these effects cancel out to a significant extent in the ratio. A linear fit between the resulting numbers of Γt(170 GeV) = 1.28 GeV, Γt(172.5 GeV) = 1.33 GeV and Γt(175 GeV) = 1.38 GeV is shown in Fig. 8.8. A reference line with a constant function value of 1.33 GeV is added as well. The measured values do not coincide with the expected behaviour from theory. Instead of an increase inΓt, a distinct decrease is observed when going to higher masses.
The mass uncertainty of 0.70 GeV quoted by the latest ATLAS combination[111]can be translated into an uncertainty of Γt based on the linear splines. The two clearly different slopes shown in Fig. 8.8 yield a highly asymmetric response of+0.33 GeV and -0.03 GeV.
More complex parametrisations like a quadratic fit using the three mass points were tested as well but were not able to improve the description of the relationship betweenΓt andmt.
In order to understand the asymmetric response to mt, further studies were conducted. Pseudo-experiments similar to the ones described above with the alternative mass samples were repeated, based on modified configurations of the template fit. The following options are compared: (a) only
Figure 8.8: Illustration of the asymmetric response of Γt tomt. Shown are mean values of the top quark decay width obtained from pseudo-experiments for different MC mass variation samples atmt=170, 172.5, 175GeV, connected by linear splines. The theoretical prediction for the mass dependence of the top quark decay width is drawn as well, which is close to a constant reference line atΓt=1.33GeV (dashed), correspondent to a mass ofmt=172.5GeV. The large difference for the mass value of 170 GeV is too large to be true and the reason for this fit response is explained in the text.
events with at least two b-tagged jets are considered in the fit, (b) the mass in the single top MC samples along with the t¯t samples is set to the alternative values of 170 GeV and 175 GeV, (c) only one observable, eitherm`bor∆Rmin(jb,jl), is used, (d) a cut on the mass observable as introduced in the last section is applied (ignore events withm`b≥150 GeV). The mean values obtained from pseudo-experiments for these four fit configurations performed for the three mass samples are given in Table 8.5. The theoretical predictions for the alternative mass values are shown as well.
In compliance with the expectation,m`bis more sensitive to the top quark mass than∆Rmin(jb,jl). The observed asymmetry is mainly caused bym`b and increases if only this variable is used in the fit. Particularly the peak region ofm`b intensifies this effect. The result is almost independent of the used b-tag regions or additional mass variations in the single top event samples. The values of pseudo-experiments relying on fits with only∆Rmin(jb,jl)reveal an almost symmetric response to mt and the mean values of Γt increase with the underlying masses, following the theoretical calculations, although the slope of this rise is more distinct.
These results verify once more that the fit configuration with two observables is mainly driven by the observable which possesses a larger dependence onmt andΓt while∆Rmin(jb,jl)has a clear stabilising effect on the final result and the uncertainty evaluation.
Further tests demonstrated that the observed asymmetry is enhanced by jets in the forward detector
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mt 170.0 GeV 172.5 GeV 175.0 GeV
Γt(default option)[GeV] 2.49±0.49 1.34±0.29 1.21±0.28 Γt(≥2b-tag bin)[GeV] 2.56±0.77 1.33±0.33 1.44±0.29 Γt(mass varied single-top)[GeV] 2.51±0.60 1.34±0.29 1.21±0.24 Γt(m`bonly)[GeV] 2.99±0.60 1.34±0.36 1.09±0.24 Γt(m`b<150 GeV only)[GeV] ∼6±0.78 1.34±0.34 0.77±0.28 Γt(∆Rmin(jb,jl)only)[GeV] 1.02±0.43 1.33±0.60 1.56±0.61
Γt(theory)[GeV] 1.28 1.33 1.38
Table 8.5:Mean decay width valuesΓtobtained using samples with different top quark masses mt for various fit configurations and options as described in the text. The uncertainties corre-spond to the standard deviations from the Gaussian-shaped distributions of theΓt values from the pseudo-experiments. The option labelled as “default” is consistent with the fit setup also used for the comparison in Table 8.4. The last row contains the theoretical prediction based on Eq. (8.1). All quoted values are given in units of GeV.
regions and decreases for fit configurations resting on jets in the central region with |η| ≤1. In this central pseudorapidity region a mean value of about 2.2 GeV compared to the 2.5 GeV for the default setup is obtained from pseudo-experiments while fits with events from the pseudorapidity region with|η|>1 for at least one of the considered jets yield a mean value around 2.8 GeV.
To refute the possibility that the observed asymmetric behaviour is induced by fluctuations in the histogram bins, related to the relatively low statistics in the alternative mass samples with respect to the nominal templates, the number of bins per analysis region was reduced from 20 to 10. However, pseudo-experiments revealed even an increase of the asymmetry, the mean values amount toΓt(170 GeV)≈3.1 GeV andΓt(175 GeV)≈1.0 GeV, respectively.
An interpolation between the nominal observable distributions and those belonging to the 170 GeV and 175 GeV mass variations at reconstruction level, a so-called template morphing, yields a smooth transition between the different templates and pseudo-experiments mean values without unexpected slopes or jumps. This morphing is designed to reproduce the change of the templates in steps of 0.5 GeV in the top quark mass by dividing differences between the nominal and the alternative mass samples per bin by a factor of five. Thus, interpolated or morphed observable distributions for the intermediate masses ofmt=170.5, 171, 171.5, 172, 173, 173.5, 174, 174.5 GeV could be derived. For templates near the nominal value of 172.5 GeV, mean values from pseudo-experiments give decay width values significantly closer to the nominal result, i.e. Γt(172 GeV)≈ 1.52 GeV andΓt(173 GeV)≈1.31 GeV. TheΓt mean values for the intermediate mass points from 500 pseudo-experiments in the mass region below 172.5 GeV, where the deviations with regard to the nominal sample are most pronounced, are given in Table 8.6.
To assess the impact of the top quark mass onΓt in the range between the nominal and the two alternative mass points, a reweighting using Breit-Wigner functions was tested. For such a study, the procedure defined in Sec. 7.2 for the derivation of decay width templates can be adopted to create
mt MeanΓt
170.0 GeV 2.49±0.02 GeV 170.5 GeV 2.30±0.03 GeV 171.0 GeV 1.93±0.02 GeV 171.5 GeV 1.71±0.02 GeV 172.0 GeV 1.52±0.02 GeV 172.5 GeV 1.33±0.02 GeV
Table 8.6: Results of mean fitted decay-width valuesΓt including options based on morphed templates for different underlying top quark masses. The values formt =170GeV andmt = 172.5GeV are obtained from dedicated mass samples, the intermediate values from interpolating or morphing the mass distributions corresponding tomt = 170 GeV andmt = 172.5GeV as explained in the text. The given uncertainties represent statistical uncertainties on the mean values. The response to the decay width betweenmt =172.5GeV andmt=175GeV is similar in the given range fromΓt=1.21GeV andΓt=1.33GeV.
templates based on alternative mass values. Hence, mt is varied in Eq. (7.1) to create additional mass samples in the range 170≤mt≤175 GeV. But a closure test performed at reconstruction level showed that differences between the rescaled distributions corresponding tomt =170 GeV and mt =175 GeV and the respective alternative mass samples are beyond the statistical uncertainties in the bins due to acceptance effects. Since, furthermore, dedicated samples for top quark masses closer to 172.5 GeV are not available for the current analyses, the above delineated template morphing is the best approach to access themt range close to the nominal value ofmt=172.5 GeV.
These masses comply with uncertainties quoted in latest measurements ofmt.
According to the above listed studies, neither the binning nor specific b-tag, jet pseudorapidity or m`b regions are solely responsible for the observed asymmetric response to the top quark mass although some of these parameters amplify the observed asymmetry. The differences between the mass samples are of a similar order in the eight exclusive analysis regions, there is no obvious shift or fluctuation in one certain region between the two alternative mass distributions explaining why the fits behave so asymmetricly. This is highlighted in Fig. 8.9 for plots in the eight regions with the nominal sample as reference. Different values ofΓt were tested and reflect the same behaviour.
To understand the asymmetric fit response, templates based onΓt =1.33 GeV obtained from the alternative mass sample with mt = 170 GeV were compared with templates from the nominal sample with mt = 172.5 GeV that correspond toΓt = 1.0 GeV andΓt = 2.5 GeV. The resulting plots are shown in Fig. 8.10 for the pseudorapidity region with|η|>1, also including plots for the alternative sample withmt=175 GeV. App. F comprises further plots also for the|η| ≤1 region.
The distributions contrasted in Fig. 8.10 verify that the templates derived atmt=172.5 GeV are not able to cover the m`bdistribution of the t¯t mass samples of 170 GeV and 175 GeV. Since the alternative mass distributions are beyond the covered range of the nominalmt =172.5 GeV tem-plates, the fit is not able to reproduce a symmetric response tomt. This stresses the large sensitivity
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Figure 8.9:Distributions of the observablem`bin the eight exclusive analysis regions according to the labels. Templates based onΓt =1.33 GeV are compared for the nominal sample with mt =172.5GeV and for the alternative top quark masses of 170 and 175 GeV. The lower panels illustrate the ratio of the presented histograms with respect to the nominal signal sample.
Figure 8.10:Distributions of the observablem`bin the four|η|>1analysis regions. Templates based onΓt = 1.33GeV for the alternative mass sample withmt =170GeV (upper half) and withmt=175GeV (lower half) are compared to templates for two different decay width values, namelyΓt=1.0GeV andΓt=2.5GeV, from the nominal sample withmt =172.5GeV. The lower panels illustrate the ratio of the presented histograms with respect to the alternativet¯tsamples atmt=170GeV andmt=175GeV.
8 . 6 I M PA C T O F T H E T O P Q U A R K M A S S
ofm`b to the mass. For the determination of other systematic uncertainties, as described in the previous sections of this chapter, the different decay width templates encompass the systematic shifts, and thus reliable estimates of the underlying systematic uncertainty can be made.
These conclusions are supported byχ2tests, as introduced in Sec. 7.2. Fits of nominal templates to pseudo-data distributions of the nominalt¯tsample atmt =172.5 GeV, which yield the expected mean value of around 1.33 GeV, are characterised by χ2 values of around 0.6, after dividing by the numbers of degrees of freedom (ndf). Fits to pseudo-data distributions of the two alternative mass samples haveχ2/ndf values of around 1.4 (formt=170 GeV) and of around 1.6 (formt= 175 GeV), reflecting that the goodness of the fit is significantly worse, following the expectations pertaining to Fig. 8.10.
For the fit configurations implemented and compared so far, the nominal 172.5 GeV templates were fitted to pseudo-data distributions of the 170 GeV and 175 GeV samples, as it is the recommended procedure for systematic uncertainties described in Sec. 8.1. In contrast to this default procedure, the decay width templates derived from the 170 GeV and 175 GeV mass samples can be fitted to pseudo-data distributions of the nominal 172.5 GeV sample. This leads to mean values of Γt=1.28 GeV for a configuration withmt =170 GeV templates fitted tomt=172.5 GeV pseudo-data histograms while a mean value ofΓt=1.38 GeV is measured for the other possible setup with mt =175 GeV templates fitted to mt =172.5 GeV pseudo-data distributions. In this particular case, the potential effect of the decay width onmt is far less distinct in comparison to the values in Table 8.5 and even symmetric around the expectation of 1.33 GeV related to the nominal top quark mass.
A similar approach relies on the idea of fittingmt=170 GeV templates directly tomt=175 GeV pseudo-data histograms and vice versa. The resulting mean values from pseudo-experiments are around 1.25 GeV and 1.6 GeV. Despite the difference of∆mt=5 GeV between the two compared samples, the resulting difference inΓt is smaller compared to the numbers in Table 8.5 where the difference between the samples is merely∆mt=2.5 GeV.
These further cross-checks confirm the above description that the fit is not able to properly account for the differences between the alternative mass samples because the different decay width tem-plates do not cover the mass differences. Hence, the resulting numbers are so different for the various cross-checks without a clear tendency for the preference of a certain template.
As a consequence, a precise estimate of the impact of the top quark mass on theΓt measurement requires alternative MC mass samples much closer to the nominal value of mt = 172.5 GeV so that the resulting observable distributions can be covered by the different decay width templates.
The most reliable approach available forp
s=8 TeV is the template morphing for which a shift in the decay width of up to 0.2 GeV can be deduced from a mass variation of±0.5 GeV around the nominal value.
The top quark mass dependence is not included as an uncertainty in many analyses but given as a parametrisation. According to the above described studies, a parametrisation is not possible for this measurement and the final result ofΓt is quoted for a top quark mass of 172.5 GeV.